diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzqegu" "b/data_all_eng_slimpj/shuffled/split2/finalzzqegu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzqegu" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nIn \\cite{P}, Perelman {\\it proved} a Li-Yau-Hamilton type (also\ncalled differential Harnack) inequality for the fundamental solution\nof the conjugate heat equation, in the presence of the Ricci flow.\nMore precisely, let $(M, g_{ij}(t))$ be a solution to Ricci flow:\n\\begin{equation}\\label{1}\n\\frac{\\partial}{\\partial t} g_{ij}=-2R_{ij}\n\\end{equation}\non $M\\times [0, T]$ and let $H(x,y, t)=\\frac{e^{-f}}{(4\\pi\n\\tau)^{\\frac{n}{2}}}$ (where $\\tau=T-t$) be the fundamental solution\nto the conjugate heat equation $u_\\tau-\\Delta u+Ru=0$. (More\nprecisely we should write the fundamental solution as $H(y,t; x,\nT)$, which satisfies $ \\left(-\\frac{\\partial}{\\partial t}+\\Delta_{y}+R(y,\nt)\\right)H=0$ for any $(x, t)$ with $t0$, $\\int_M\nv_H(y) h(y)\\, d\\mu_{\\tau_*}(y)\\le 0$, for any smooth function\n$h(y)\\ge 0$ with compact support. In order to achieve this, in\n\\cite{P} the heat equation $\\left(\\frac{\\partial}{\\partial t}-\\Delta\\right)h(y,\nt)=0$ with the `initial data' $h(y, T-\\tau_*)=h(y)$ (more precisely\n$t=T-\\tau_*$), the given compactly supported nonnegative function,\nis solved. Applying (\\ref{2}) to $u(y, \\tau)=H(x, y, \\tau)$, one can\neasily derive as in \\cite{P}, via integration by parts, that\n\\begin{equation}\\label{3}\n\\frac{d}{ d\\tau}\\int_M v_H h\\, d\\mu_\\tau=-2\\int_M\n\\tau|R_{ij}+\\nabla_i\\nabla_j f-\\frac{1}{2\\tau}g_{ij}|^2Hh\\,\nd\\mu_\\tau\\le 0.\n\\end{equation}\nThe Li-Yau type inequality $v_H\\le 0$ then follows from the above\nmonotonicity, provided the claim that \\begin{equation}\\label{4}\n\\lim_{\\tau \\to 0}\\int_M v_Hh\\, d\\mu_\\tau \\le 0. \\end{equation}\n\nThe main purpose of this note is to prove (\\ref{4}), hence provide a\ncomplete proof of the claim $v_H\\le 0$. This will be done in\nSection 3 after some preparations in Section 2. It was written in\n\\cite{P} that `{\\it it is easy to see}' that $\\lim_{\\tau\\to 0}\\int_M\nv_Hh\\, d\\mu_\\tau = 0$. It turns out that the proof found here need\nto use some gradient estimates for positive solutions, a quite\nprecise estimate on the `reduced distance' and the monotonicity\nformula (\\ref{3}). (We shall focus on the proof of (\\ref{4}) for the\ncase when $M$ is compact and leave the more technical details of\ngeneralizing it to the noncompact setting to the later refinements.)\nIndeed the claim that $\\lim_{\\tau\\to 0}\\int_M v_Hh\\, d\\mu_\\tau = 0$\nfollows from a blow-up argument of \\cite{P}, after we have\nestablished (\\ref{4}). Since our argument is a bit involved, this\nmay not be {\\it the proof}.\n\nIn Section 4 we derive several monotonicity formulae, which improve\nvarious Li-Yau-Hamilton inequalities for linear heat equation\n(systems) as well as for Ricci flow, including the original\nLi-Yau's inequality. In Section 5 we illustrate the localization of\nthem by applying a general scheme of \\cite{EKNT}.\n\n\\section{Estimates and results needed} We shall collect some known\nresults and derive some estimates needed for proving (\\ref{4}) in\nthis section. We need the asymptotic behavior of the fundamental\nsolution to the conjugate heat equation for small $\\tau$. Let\n$d_\\tau(x,y)$ be the distance function with respect to the metric\n$g(\\tau)$. Let $B_\\tau(x, r)$ ($Vol_\\tau$) be the ball of radius\n$r$ centered at $x$ (the volume) with respect to the metric\n$g(\\tau)$.\n\n\\begin{theorem} \\label{Theorem 1} Let $H(x,y, \\tau)$ be the fundamental solution to the ({\\it\nbackward in $t$}) conjugate heat equation. Then as $\\tau \\to 0$ we\nhave that\n\\begin{equation}\\label{5}\nH(x,y,\\tau)\\sim \\frac{\\exp\\left(-\\frac{d_0^2(x,y)}{4\\tau}\\right)}\n{\\left(4\\pi\\tau\\right)^{\\frac{n}{2}}}\\sum_{j=0}^{\\infty}\\tau^ju_j(x,y,\\tau).\n\\end{equation} By (\\ref{5}) we mean that exists $T>0$ and sequence $u_j\\in\nC^{\\infty}(M\\times M\\times [0, T])$ such that\n$$\nH(x,y,\\tau)-\\frac{\\exp \\left(-\\frac{d_0^2(x,y)}{4\\tau}\\right)}\n{\\left(4\\pi\\tau\\right)^{\\frac{n}{2}}}\\sum_{j=0}^{k}\\tau^ju_j(x,y,\\tau)=w_k(x,y,\n\\tau)\n$$\nwith\n$$\nw_k(x,y, \\tau)=O\\left(\\tau^{k+1-\\frac{n}{2}}\\right)\n$$\nas $\\tau \\to 0$, uniformly for all $x, \\, y\\in M$. The function\n$u_0(x, y, \\tau)$ can be chosen so that $u_0(x, x, 0)=1$.\n\\end{theorem}\n\nThis result was proved in details, for example in \\cite{GL}, when\nthere is no zero order term $R(y, \\tau) u(y, \\tau)$ in the equation\n$\\frac{\\partial}{\\partial \\tau}u-\\Delta u+ Ru=0$ and replacing $d_0(x,y)$ by\n$d_\\tau(x,y)$. However, one can check that the argument carries over\nto this case if one assumes that the metric $g(\\tau)$ is $C^\\infty$\nnear $\\tau=0$. One can consult \\cite{SY, CLN} for intrinsic\npresentations.\n\nLet\n$$\n\\mathcal {W}_h(g, H, \\tau)=\\int_M v_Hh\\, d\\mu_\\tau\n$$\nwhere $h$ is the previously described solution to the heat equation.\nIt is clear that for any $\\tau$ with $T\\ge \\tau>0$,\n$\\mathcal{W}_h(g, H, \\tau)$ is a well-defined quantity. A priori it\nmay blow up as $\\tau\\to 0$. It turns out that in our course of\nproving that $\\lim_{\\tau\\to 0} \\mathcal{W}_h(g, H,\\tau)\\le 0$ we\nneed to show first that exists $C>0$, which may depends on the\ngeometry of the Ricci flow solution $(M, g(\\tau))$ defined on\n$M\\times [0, T]$, but independent of $\\tau$ (as $\\tau\\to 0$) so that\n$\\mathcal{W}_h(g, H, \\tau)\\le C$ for all $T \\ge\\tau>0$. The\nfollowing lemma supplies the key estimates for this purpose.\n\n\\begin{lemma}\\label{Lemma 2} Let $(M, g(t))$ be a smooth solution to the Ricci\nflow on $M\\times[0, T]$. Assume that there exist $k_1\\ge 0$ and\n$k_2\\ge0$, such that the Ricci curvature $R_{ij}(g(\\tau))\\ge -k_1\ng_{ij}(\\tau)$ and $\\max(R(y, \\tau), |\\nabla R|^2(y, \\tau))\\le k_2$,\non $M\\times [0, t]$.\n\n(i) If $u\\le A$ is a positive solution to the conjugate heat\nequation on $M\\times [0, T]$, then there exists $C_1$ and $C_2$\ndepending on $k_1$, $k_2$ and $n$ such that for $0< \\tau\\le \\min(1,\nT)$,\n\\begin{equation}\\label{6}\n\\tau\\frac{|\\nabla u|^2}{u^2}\\le \\left(1+C_1\\tau\\right)\\left(\\log\n\\left(\\frac{A}{u}\\right)+C_2\\tau\\right)\\end{equation}\n\n(ii) If $u$ is a positive solution to the conjugate heat equation\non $M\\times[0, T]$, then there exists $B$, depending on $(M,\ng(\\tau))$ so that for $0\\le\\tau\\le \\min(T, 1)$,\n\\begin{equation}\\label{7}\n\\tau \\frac{|\\nabla u|^2}{u^2}\\le \\left(2+C_1\\tau\\right)\\left(\\log\n\\left(\\frac{B}{u\\tau^{\\frac{n}{2}}}\\int_M u\\,\nd\\mu_{\\tau}\\right)+C_2\\tau\\right). \\end{equation}\n\\end{lemma}\n\\begin{remark}\\label{Remark1} Here and thereafter we use the same $C_i$ ($B$) at the\ndifferent lines if they just differ only by a constant depending on\n$n$. Notice that $\\int_M u\\, d\\mu_{\\tau}$ is independent od $\\tau$\nand equals to $1$ if $u$ is the fundamental solution. The proof to\nthe lemma given below is a modification of some arguments in\n\\cite{H}.\n\\end{remark}\n\\begin{proof} Direct computation, under a unitary frame, gives\n\\begin{eqnarray*}\n \\left(\\frac{\\partial}{\\partial \\tau}-\\Delta \\right)\\left(\\frac{|\\nabla\nu|^2}{u}\\right) &=&\n-\\frac{2}{u}\\left|u_{ij}-\\frac{u_iu_j}{u}\\right|^2+\\frac{|\\nabla\nu|^2}{u}\nR\\\\\n&\\, &+\\frac{-4R_{ij}u_i u_j -2\\langle \\nabla (R u), \\nabla\nu\\rangle}{u}\\\\\n&\\le& (4+n)k_1\\frac{|\\nabla u|^2}{u}+2|\\nabla R||\\nabla u|\\\\\n&\\le& \\left[(4+n)k_1+1\\right]\\frac{|\\nabla u|^2}{u}+k_2u\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\left(\\frac{\\partial}{\\partial \\tau}-\\Delta\n\\right)\\left(u\\log\\left(\\frac{A}{u}\\right)\\right)&=&\\frac{|\\nabla\nu|^2}{u}+Ru-Ru\\log\\left(\\frac{A}{u}\\right)\\\\\n&\\ge &\\frac{|\\nabla u|^2}{u}-nk_1u-k_2u\\log\n\\left(\\frac{A}{u}\\right).\n\\end{eqnarray*}\nCombining the above two equations together we have that\n$$\n\\left(\\frac{\\partial}{\\partial \\tau}-\\Delta \\right)\\Phi\\le 0\n$$\nwhere $$\\Phi=\\varphi\\frac{|\\nabla\nu|^2}{u}-e^{k_2\\tau}u\\log\\left(\\frac{A}{u}\\right)-(k_2+nk_1e^{k_2})\\tau\nu$$ with $\\varphi=\\frac{\\tau}{1+\\left[(4+n)k_1+1\\right]\\tau}$, which\nsatisfies\n$$\n\\frac{d}{d\\tau} \\varphi +\\left[(4+n)k_1+1\\right]\\varphi <1.\n$$\nBy the maximum principle we have that\n$$\n\\varphi\\frac{|\\nabla u|^2}{u}\\le\ne^{k_2\\tau}u\\log\\left(\\frac{A}{u}\\right) +(k_2+nk_1e^{k_2})\\tau u.\n$$\nFrom this one can derive (\\ref{6}) easily.\n\nTo prove the second part, we claim that for $u$, a positive solution\nto the conjugate heat equation, there exists a $C$ depending on $(M,\ng(\\tau))$ such that \\begin{equation}\\label{claim} u(y, \\tau)\\le\n\\frac{C}{\\tau^{\\frac{n}{2}}}\\int_M u(z, \\tau)\\, d\\mu_\\tau(z).\n\\end{equation}\nThis is a mean-value type inequality, which can be proved via, for\nexample the Moser iteration. Here we follow \\cite{H}. We may assume\nthat $\\sup_{y\\in M, 0\\le \\tau\\le 1}\\tau^{\\frac{n}{2}}u(y,\\tau)$ is\nfinite. Otherwise we may replacing $\\tau$ by $\\tau_\\epsilon =\\tau-\\epsilon$ and\nlet $\\epsilon\\to 0$ after establishing the claim for $\\tau_\\epsilon$. Now let\n$(x_0, \\tau_0)\\in M\\times [0,1]$ be such a space-time point that\n$\\max \\tau^{\\frac{n}{2}}u(y, \\tau) =\\tau_0^{\\frac{n}{2}}u(y_0,\n\\tau_0)$. Then we have that\n$$\n\\sup_{M\\times [\\frac{\\tau_0}{2}, \\tau_0]}u(y, t)\\le\n\\left(\\frac{2}{\\tau_0}\\right)^{\\frac{n}{2}}\\tau_0^{\\frac{n}{2}}u(y_0,\n\\tau_0)=2^{\\frac{n}{2}}u(y_0, t_0).\n$$\nNoticing this upper bound, we apply (\\ref{6}) to $u$ on\n$M\\times[\\frac{\\tau_0}{2}, \\tau_0]$, and conclude that\n$$\n\\frac{\\tau_0}{2}\\left(\\frac{|\\nabla u|^2}{u^2}\\right)(y, \\tau_0)\\le\n(1+C_1\\tau_0)\\left(\\log\\left( \\frac{2^{\\frac{n}{2}}u(y_0,\n\\tau_0)}{u(y, \\tau_0)}\\right)+C_2\\tau_0\\right).\n$$\nLet $g=\\log\\left( \\frac{2^{\\frac{n}{2}}u(y_0, \\tau_0)}{u(y,\n\\tau_0)}\\right)+C_2\\tau_0$. The above can be written as\n$$\n|\\nabla \\sqrt{g}|\\le \\sqrt{\\frac{1+C_1\\tau_0}{2\\tau_0}}\n$$\nwhich implies that\n$$\n\\sup_{B_{\\tau_0}\\left(y_0,\n\\sqrt{\\frac{\\tau_0}{1+C_1\\tau_0}}\\right)}\\sqrt{g}(y, \\tau_0)\\le\n\\sqrt{g}(y_0, \\tau_0)+\\frac{1}{\\sqrt{2}}.\n$$\nRewriting the above in terms of $u$ we have that\n$$\nu(y, \\tau_0)\\ge 2^{\\frac{n}{2}}u(y_0,\n\\tau_0)e^{-\\left(\\frac{1}{2}+\\frac{2}{\\sqrt{2}}\\sqrt{\\frac{n}{2}\\log\n2+C_2}\\right)}=C_3 u(y_0, \\tau_0)\n$$\nfor all $y\\in B_{\\tau_0}\\left(y_0,\n\\sqrt{\\frac{\\tau_0}{1+C_1\\tau_0}}\\right)$. Here we have also used\n$\\tau_0\\le 1$. Noticing that\n$$\nVol_{\\tau_0}\\left( B_{\\tau_0}\\left(y_0,\n\\sqrt{\\frac{\\tau_0}{1+C_1\\tau_0}}\\right)\\right)\\ge C_4\n\\tau_0^{\\frac{n}{2}}\n$$\nfor some $C_4$ depending on the geometry of $(M, g(\\tau_0))$.\nTherefore we have that\n$$\n\\frac{C_5}{\\tau_0^{\\frac{n}{2}}}\\int_M u(y, \\tau_0)\\,\nd\\mu_{\\tau_0}(y)\\ge u(y_0, \\tau_0)\n$$\nfor some $C_5$ depending on $C_3$ and $C_4$. By the way we choose\n$(y_0, \\tau_0)$ we have that\n$$\n \\tau^{\\frac{n}{2}}u(y, \\tau)\\le \\tau_0^{\\frac{n}{2}}u(y_0,\n\\tau_0) \\le C_5\\int_M u(y, \\tau_0)\\, d\\mu_{\\tau_0}(y) = C_5\\int_M\nu(y, \\tau)\\, d\\mu_{\\tau}(y).\n$$\nThis proves the claim (\\ref{claim}). Now the estimate (\\ref{7})\nfollows from (\\ref{6}), applying to $u$ on $M\\times[\\frac{\\tau}{2},\n\\tau]$, and the just proved (\\ref{claim}), which ensures the needed\nupper bound for applying the estimate (\\ref{6}).\n\\end{proof}\n\nIf $u=\\frac{e^{-f}}{\\left(4\\pi\\tau\\right)^{\\frac{n}{2}}}$ is the\nfundamental solution to the conjugate heat equation we have that\n$\\int_M u\\, d\\mu_\\tau=1$. Therefore, by (\\ref{7}), we have that\n\\begin{equation}\\label{8}\n\\int_M \\tau|\\nabla f|^2 uh \\, d\\mu_\\tau \\le\n\\left(2+C_1\\tau\\right)\\int_M \\left(\\log B +f +C_2\\tau\\right)uh\\,\nd\\mu_\\tau. \\end{equation} On the other hand, integration by parts\ncan rewrite\n\\begin{eqnarray*}\n\\mathcal{W}_h(g, u, \\tau)&=&\\int_M \\tau|\\nabla f|^2uh\\,\nd\\mu_\\tau-2\\tau\\int_M \\langle \\nabla f, \\nabla h\\rangle u\\,\nd\\mu_\\tau \\\\\n&\\quad& +\\tau \\int_M R uh\\, d\\mu_\\tau +\\int_M (f-n)uh\\,\nd\\mu_\\tau\\\\&=&I+II+III+IV.\n\\end{eqnarray*}\nThe $I$ term can be estimated by (\\ref{8}), whose right hand side\ncontains only one `bad' term $\\int_M fuh\\, d\\mu_\\tau$ in the sense\nthat it could possibly blow up. The second term\n$$\nII=2\\tau\\int_M \\langle \\nabla u, \\nabla h\\rangle \\,\nd\\mu=-2\\tau\\int_M u\\Delta h \\, d\\mu_\\tau\n$$\n is clearly bounded as $\\tau \\to 0$. In fact $II\\to 0$ as $\\tau\\to\n0$. The same conclusion\n obviously holds for $III$. Summarizing above, we reduce the\nquestion of bounding from above the quantity $\\mathcal{W}_h(u, g,\n\\tau)$ to bounding one single term\n$$\nV=\\int_M fuh\\, d\\mu_\\tau\n$$\nfrom above (as $\\tau \\to 0$). We shall show later that $\n\\lim_{\\tau\\to 0} V\\le 0.$ To do this we need to use the `reduced\ndistance', introduced by Perelman in \\cite{P} for the Ricci flow\ngeometry.\n\nLet $x$ be a fixed point in $M$. Let $\\ell(y, \\tau)$ be the reduced\ndistance in \\cite{P}, with respect to $(x, 0)$ (more precisely\n$\\tau=0$). We collect the relevant properties of $\\ell(y, \\tau)$\nin the following lemma (Cf. \\cite{Ye, CLN}).\n\n\\begin{lemma}\\label{Lemma 3} Let $\\bar L(y, \\tau)=4\\tau \\ell(y, \\tau)$.\n\n(i) Assume that there exists a constant $k_1$ such that\n$R_{ij}(g(\\tau))\\ge -k_1g_{ij}(\\tau)$, $\\bar L(y, \\tau)$ is a local\nLipschitz function on $M\\times[0, T]$;\n\n(ii) Assume that there exist constant $k_1$ and $k_2$ so that\n$-k_1g_{ij}(\\tau)\\le R_{ij}(g(\\tau))\\le k_2g_{ij}(\\tau)$. Then\n\\begin{equation}\\label{9}\n\\bar L(y, \\tau)\\le e^{2k_2\\tau}d^2_0(x, y)+\\frac{4k_2 n}{3}\\tau^2\n\\end{equation} and\n\\begin{equation}\\label{10}\nd^2_0(x, y)\\le e^{2k_1\\tau}\\left(\\bar L(y, \\tau)+\\frac{4k_1\nn}{3}\\tau^2\\right); \\end{equation}\n\n(iii) \\begin{equation}\\label{11}\\left(\\frac{\\partial}{\\partial \\tau}-\\Delta\n+R\\right)\\left(\\frac{\\exp\\left(-\\frac{\\bar L(y,\n\\tau)}{4\\tau}\\right)}{(4\\pi\\tau)^{\\frac{n}{2}}}\\right)\\le 0.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} The first two claims follow from the definition by straight forward\nchecking. For (iii), it was proved in Section 7 of \\cite{P}. By now\nthere are various resources where the detailed proof can be found.\nSee for example \\cite{Ye} and \\cite{CLN}.\n\\end{proof}\n\nThe consequence of (\\ref{9}) and (\\ref{10}) is that\n$$\n\\lim_{\\tau \\to 0}\\frac{\\exp\\left(-\\frac{\\bar L(y,\n\\tau)}{4\\tau}\\right)}{(4\\pi\\tau)^{\\frac{n}{2}}}=\\delta_x(y),\n$$\nwhich together with (\\ref{11}) implies that $H$, the fundamental\nsolution to the conjugate heat equation, is bounded from below as\n$$H(x, y, \\tau)\\ge \\frac{\\exp\\left(-\\frac{\\bar L(y,\n\\tau)}{4\\tau}\\right)}{(4\\pi\\tau)^{\\frac{n}{2}}}.$$ Hence\n\\begin{equation}\\label{12}\nf(y, \\tau)\\le \\frac{\\bar{L}(y, \\tau)}{4\\tau}. \\end{equation} This\nwas proved in \\cite{P} out of the claim $v_H\\le 0$. Since we are in\nthe middle of proving $v_H\\le 0$, we have to show the above\nalternative of obtaining (\\ref{12}).\n\n\n \\section{ Synthesis }\n\nNow we assembly the results in the previous section to prove\n(\\ref{4}). As the first step we show that $\\mathcal{W}_h(g, H,\n\\tau)$ is bounded (thanks to the monotonicity (\\ref{3}), it is\nsufficient to bound it from above) as $\\tau \\to 0$, where $H(x, y,\n\\tau)$ is the fundamental solution to the conjugate heat equation\nwith $H(x, y,0)=\\delta_x(y)$. By the reduction done in the previous\nsection we only need to show that\n$$\nV=\\int_M fuh\\, d\\mu_\\tau\n$$\nis bounded from above as $\\tau \\to 0$. By (\\ref{12}) we have that\n\\begin{eqnarray*}\n &\\, & \\limsup_{\\tau \\to 0}\\int_M fHh\\, d\\mu_\\tau \\le \\limsup_{\\tau \\to\n0}\\int_M \\frac{\\bar L (y, \\tau)}{4\\tau} H(x, y, \\tau) h(y, \\tau)\\, d\\mu_\\tau(y)\\\\\n&\\quad &\\quad \\le \\limsup_{\\tau \\to 0}\\int_M \\frac{d_0^2(x,\ny)}{4\\tau}H(x, y, \\tau)\nh(y, \\tau)\\, d\\mu_\\tau(y)\\\\\n&\\quad &\\quad +\\lim_{\\tau \\to 0}\\int_M\n\\left(\\frac{e^{k_2\\tau}-1}{4\\tau}d_0^2(x,\ny)+\\frac{k_2n}{3}\\tau\\right)H(x, y, \\tau) h(y, \\tau)\\, d\\mu_\\tau(y).\n\\end{eqnarray*}\nHere we have used (\\ref{9}) in the last inequality. By Theorem\n\\ref{Theorem 1}, some elementary computations give that\n$$\n\\lim_{\\tau \\to 0}\\int_M \\frac{d_0^2(x, y)}{4\\tau}H(x, y, \\tau) h(y,\n\\tau)\\, d\\mu_\\tau(y)=\\frac{n}{2} h(x, 0).\n$$\nSince $\\frac{e^{k_2\\tau}-1}{4\\tau}d_0^2(x, y)+\\frac{k_2n}{3}\\tau$ is\na bounded continuous function even at $\\tau=0$, we have that\n$$\n\\lim_{\\tau \\to 0}\\int_M \\left(\\frac{e^{k_2\\tau}-1}{4\\tau}d_0^2(x,\ny)+\\frac{k_2n}{3}\\tau\\right)H(x, y, \\tau) h(y, \\tau)\\,\nd\\mu_\\tau(y)=0.\n$$\nThis completes our proof on finiteness of $\\limsup_{\\tau\\to 0}\n\\int_M f H h\\, d\\mu_\\tau.$ In fact we have proved that\n\\begin{equation}\\label{13} \\limsup_{\\tau\\to 0} \\int_M (f-\\frac{n}{2}) H h\\,\nd\\mu_\\tau\\le 0. \\end{equation} By the just proved finiteness of\n$\\mathcal{W}_h(g, H, \\tau)$ as $\\tau\\to 0$, and the (entropy)\nmonotonicity (\\ref{3}), we know that the limit $\\lim_{\\tau \\to 0}\n\\mathcal{W}_h(g, H, \\tau)$ exists. Let\n$$\n\\lim_{\\tau \\to 0} \\mathcal{W}_h(g, H, \\tau)=\\lim_{\\tau \\to 0} \\int_M\nv_H h\\, d\\mu_\\tau =\\alpha\n$$\nfor some finite $\\alpha$. Hence $\\lim_{\\tau \\to 0}\n\\left(\\mathcal{W}_h(g, H, \\tau)-\\mathcal{W}_h(g, H,\n\\frac{\\tau}{2})\\right)=0$. By (\\ref{3}) and the mean-value theorem\nwe can find $\\tau_k\\to 0$ such that\n$$\n\\lim_{\\tau_k \\to 0} \\tau_k^2\\int_M |R_{ij}+\\nabla_i\\nabla_j\nf-\\frac{1}{2\\tau_k}g_{ij}|^2Hh\\, d\\mu_{\\tau_k}=0.\n$$\nBy the Cauchy-Schwartz inequality and the H\\\"older inequality we\nhave that\n$$\n\\lim_{\\tau_k \\to 0} \\tau_k\\int_M \\left(R+\\Delta f\n-\\frac{n}{2\\tau_k}\\right)Hh\\, d\\mu_{\\tau_k}=0.\n$$\nThis implies that\n$$\n\\lim_{\\tau \\to 0}\\mathcal{W}_h(g, H, \\tau)=\\lim_{\\tau_k\\to 0}\\int_M\n\\left(\\tau_k(\\Delta f -|\\nabla f|^2)+f-\\frac{n}{2}\\right)H h\\,\nd\\mu_{\\tau_k}.\n$$\nAgain the integration by parts shows that\n\\begin{eqnarray*}\n\\int_M \\tau_k(\\Delta f -|\\nabla f|^2)Hh\\, d\\mu_{\\tau_k}&=&\\int_M\n\\tau_k \\langle \\nabla H, \\nabla h\\rangle\\,\nd\\mu_{\\tau_k}\\\\\n&=&-\\tau_k\\int_M H\\Delta h\\, d\\mu_{\\tau_k}\\to 0.\n\\end{eqnarray*}\nHence by (\\ref{13}) $$ \\lim_{\\tau \\to 0}\\mathcal{W}_h(g, H,\n\\tau)=\\lim_{\\tau_k\\to 0}\\int_M (f-\\frac{n}{2})Hh\\, d\\mu_{\\tau_k}\\le\n0.\n$$\nThis proves $\\alpha \\le 0$, namely (\\ref{4}).\n\nThe claim that $\\alpha=\\lim_{\\tau \\to 0}\\mathcal{W}_h(g, H, \\tau)=0$\ncan now be proved by the blow-up argument as in Section 4 of\n\\cite{P}. Assume that $\\alpha<0$. One can easily check that this\nwould imply that $\\lim_{\\tau \\to 0}\\mu(g, \\tau)<0$. Here $\\mu(g,\n\\tau)$ is the invariant defined in Section 4 of \\cite{P}. In fact,\nnoticing that $h(y, \\tau)>0$ for all $\\tau\\le \\tau_*$ (where the\n$\\tau_*$ is the one we fixed in the introduction). Therefore by\nmultiple $\\frac{1}{h(x, 0)}$ (more precisely $\\frac{1}{h(x, \\cdot)}$\nat $\\tau=0$) to the original $h(y, \\tau)$, we may assume that $h(x,\n0)=\\int_M H(x, y, \\tau)h(y, \\tau)\\, d\\mu_\\tau=1$. Let $\\tilde u(y,\n\\tau)=H(x,y,\\tau)h(y, \\tau)$ and $\\tilde f=-\\log \\tilde u\n-\\frac{n}{2}\\log(4\\pi)$. Now direct computation yields that\n$$\n\\mathcal{W}_h( g, H, \\tau)=\\mathcal{W}(g, \\tilde u, \\tau)+\\int_M\n\\left(\\tau\\left(\\frac{|\\nabla h|^2}{ h}\\right)-h\\log h\\right)H\\,\nd\\mu_\\tau.\n$$\nNoticing that the second integration goes to $0$ as $\\tau\\to 0$, we\ncan deduce that $\\mathcal{W}(g, \\tilde u, \\tau)<0$ for sufficient\nsmall $\\tau$ if $\\alpha <0$. This, together with the fact $\\int_M\n\\tilde u\\, d\\mu=1$, implies that $\\mu(g, \\tau)<0$ for sufficiently\nsmall $\\tau$.\n Now Perelman's blow-up argument in the Section 4 of \\cite{P} gives a contradiction with the\nsharp logarithmic Sobolev inequality on the Euclidean space\n\\cite{G}. (One can consult, for example \\cite{N1, STW}, for more\ndetails of this part.)\n\n\\begin{remark}\\label{Remark 2} The method of proof here follows a similar idea used\nin \\cite{N1}, where the asymptotic limit of the entropy as $\\tau\\to\n\\infty$ was computed. Note that we have to use properties of the\nreduced distance, introduced in Section 7 of \\cite{P}, in our proof,\nwhile the similar, but slightly easier, claim that $\\lim_{\\tau\\to\n0}\\mathcal{W}(g, H, \\tau)=0$ appears much earlier in Section 4 of\n\\cite{P}.\n\\end{remark}\n\n\nThe proof can be easily modified to give the asymptotic behavior of\nthe entropy defined in \\cite{N1} for the fundamental solution to the\nlinear heat equation, with respect to a fixed Riemannian metric.\nIndeed if we restrict to the class of complete Riemannian manifolds\nwith non-negative Ricci curvature we have the following estimates.\n\\begin{proposition}\\label{heat-est} For any $\\delta>0$, there exists\n$C(\\delta)$ such that\n\\begin{equation} \\label{grad-log3}\n\\frac{|\\nabla H|^2}{H}(x, y, \\tau)\\le 2\\frac{H(x, y,\n\\tau)}{\\tau}\\left(C(\\delta)+\\frac{d^2(x,y)}{(4-\\delta)\\tau}\\right)\n\\end{equation}\nand\n\\begin{equation}\\label{lapla-log2}\n\\Delta H (x, y, \\tau) +\\frac{|\\nabla H|^2}{H}(x, y, \\tau)\\le\n2\\frac{H(x, y,\n\\tau)}{\\tau}\\left(C(\\delta)+4\\frac{d^2(x,y)}{(4-\\delta)\\tau}\\right).\n\\end{equation}\n\\end{proposition}\nThe previous argument for the Ricci flow case can be transplanted to\nshow that\n$$\n\\tau(2\\Delta f -|\\nabla f|^2)+f-n\\le 0\n$$\nwhere $H(y, \\tau; x, 0)=\\frac{1}{(4\\pi \\tau)^{n\/2}}e^{-f}$ is the\nfundamental solution to the heat operator $\\frac{\\partial}{\\partial \\tau}-\\Delta$.\nThis gives a rigorous argument for the inequality (1.5) (Theorem\n1.2) of \\cite{N1}, for both the compact manifolds and complete\nmanifolds with non-negative Ricci (or Ricci curvature bounded from\nbelow). For the full detailed account please see \\cite{CLN}.\n\n\n\\section{Improving Li-Yau-Hamilton estimates via monotonicity\nformulae}\n\nThe proof of (\\ref{4}) indicates a close relation between the\nmonotonicity\n formulae and the differential inequalities of Li-Yau type. The hinge\nis simply Green's second identity. This was discussed very generally\nin \\cite{EKNT}. Moreover if we chose $h$ in the introduction to be\nthe fundamental solution to the time dependent heat equation\n($\\frac{\\partial}{\\partial t}-\\Delta$) centered at $(x_0, t_0)$ we can have a better\nupper bound on $v_H(x_0, t_0)$ in terms of the a weighted integral\nwhich is non-positive. In fact, this follows from the\nrepresentation formula for the solutions to the non-homogenous\nconjugate heat equation. More precisely, since $h(y, t; x_0, t_0)$\n is the fundamental\nsolution to the heat equation (to make it very clear, $v_H$ is\ndefined with respect to $H=H(y, t; x, T)$, the fundamental solution\nto the conjugate heat equation centered at $(x, T)$ with $T>t_0$),\nwe have that\n$$\n\\lim_{t\\to t_0}\\int_M h(y, t; x_0, t_0) v_H(y, t)\\, d\\mu_t(y)\n=v_H(x_0, t_0).\n$$\nOn the other hand from (\\ref{2}) we have that (by Green's second\nidentity)\n$$\n\\frac{d}{d t} \\int_M h v_H\\, d\\mu_t =2\\tau\\int_M\n|R_{ij}+f_{ij}-\\frac{1}{2\\tau}|^2H h\\, d\\mu_t.\n$$\nTherefore\n$$\n\\lim_{t\\to T}\\int_M hv_H\\, d\\mu_t - v_H(x_0,\nt_0)=\\int_{t_0}^T2\\tau\\int_M\n|R_{ij}+f_{ij}-\\frac{1}{2\\tau}g_{ij}|^2H h\\, d\\mu_t\\, dt.\n$$\nUsing the fact that $\\lim_{t\\to T}v_H=0$ we have that\n$$\nv_H(x_0, t_0)=-2\\int_{t_0}^T (T-t)\\int_M\n\\left|R_{ij}+f_{ij}-\\frac{1}{2(T-t)}\\right|^2Hh\\, d\\mu_t \\, dt \\le\n0,\n$$\nwhich sharpens the estimate $v_H\\le 0$ by providing a non-positive\nupper bound. Noticing also the duality $h(y, t; x_0, t_0)=H(x_0,\nt_0; y, t)$ for any $t>t_0$ we can express everything in terms of\nthe fundamental solution to the {\\it (backward) conjugate heat\nequation}.\n\n\nBelow we show a few new monotonicity formulae, which expand the list\nof examples shown in the introduction of \\cite{EKNT} on the\nmonotonicity formulae, and more importantly improve the earlier\nestablished Li-Yau-Hamilton estimates in a similar way as the above.\n\nFor the simplicity let us just consider the K\\\"ahler-Ricci flow case\neven though often the discussions are also valid for the Riemannian\n(Ricci flow) case, after replacing the assumption on the\nnonnegativity of the bisectional curvature by the nonnegativity of\nthe curvature operator whenever necessary.\n\nWe first let $(M, g_{\\alpha{\\bar{\\beta}}}(x, t))$ ($m=\\dim_\\Bbb C M$) be a solution to\nthe K\\\"ahler-Ricci flow:\n$$\n\\frac{\\partial}{\\partial t} g_{\\alpha{\\bar{\\beta}}}=-R_{\\alpha{\\bar{\\beta}}}.\n$$\nLet $\\Upsilon_{\\alpha{\\bar{\\beta}}}(x,t)$ be a Hermitian symmetric tensor defined on\n$M\\times [0,T]$, which is deformed by the complex\nLichnerowicz-Laplacian heat equation (or L-heat equation in short):\n$$\n\\left(\\frac{\\partial}{\\partial t}-\\Delta \\right)\\Upsilon_{\\gamma{\\bar{\\delta}}}= R_{\\beta\n\\bar{\\alpha}\\gamma{\\bar{\\delta}}}\\Upsilon_{\\alpha{\\bar{\\beta}}}-\n\\frac{1}{2}\\left(R_{\\gamma\\bar{p}}k_{p{\\bar{\\delta}}}+\nR_{p{\\bar{\\delta}}}\\Upsilon_{\\gamma\\bar{p}}\\right).\n$$\nLet $div(\\Upsilon)_\\alpha= g^{\\gamma{\\bar{\\delta}}}\\nabla_\\gamma \\Upsilon_{\\alpha{\\bar{\\delta}}}$ and\n$div(\\Upsilon)_{{\\bar{\\beta}}}= g^{\\gamma{\\bar{\\delta}}}\\nabla_{{\\bar{\\delta}}}\\Upsilon_{\\gamma {\\bar{\\beta}}}$.\nConsider the quantity\n\\begin{eqnarray*}\n Z& =&g^{\\alpha{\\bar{\\beta}}}g^{\\gamma{\\bar{\\delta}}}\\left[\\frac{1}{2} \\left\n(\\nabla_{{\\bar{\\beta}}}\\nabla_\\gamma +\\nabla_\\gamma \\nabla_{{\\bar{\\beta}}}\\right)\n\\Upsilon_{\\alpha{\\bar{\\delta}}}+R_{\\alpha{\\bar{\\delta}}}\\Upsilon_{\\gamma{\\bar{\\beta}}}\\right.\\\\\n&\\,& +\\left.\\left(\\nabla_\\gamma\\Upsilon_{\\alpha{\\bar{\\delta}}}V_{{\\bar{\\beta}}}+\n\\nabla_{{\\bar{\\beta}}}\\Upsilon_{\\alpha{\\bar{\\delta}}}V_{\\gamma}\\right)+\\Upsilon_{\\alpha{\\bar{\\delta}}}V_{{\\bar{\\beta}}}V_\\gamma\n\\right]\n +\\frac{K}{t}\\\\\n&=&\\frac12[g^{\\alpha{\\bar{\\beta}}}\\nabla_{\\bar{\\beta}} div(\\Upsilon)_\\alpha+g^{\\gamma{\\bar{\\delta}}}\\nabla_\\gamma div(\\Upsilon)_{\\bar{\\delta}}]\\\\\n&\\quad&+g^{\\alpha{\\bar{\\beta}}}g^{\\gamma{\\bar{\\delta}}}[R_{\\alpha{\\bar{\\delta}}}\\Upsilon_{\\gamma{\\bar{\\beta}}}+\\nabla_\\gamma\n\\Upsilon_{\\alpha{\\bar{\\delta}}}V_{{\\bar{\\beta}}}+\n\\nabla_{{\\bar{\\beta}}}\\Upsilon_{\\alpha{\\bar{\\delta}}}V_{\\gamma}+\\Upsilon_{\\alpha{\\bar{\\delta}}}V_{{\\bar{\\beta}}}V_\\gamma\n]+\\frac{K}{t}\n\\end{eqnarray*}\nwhere $K$ is the trace of $\\Upsilon_{\\alpha{\\bar{\\beta}}}$ with respect to\n$g_{\\alpha{\\bar{\\beta}}}(x,t)$.\n In \\cite{NT} the following result,\nwhich is the K\\\"ahler analogue of an earlier result in \\cite{CH},\nwas showed by the maximum principle.\n\n \\begin{theorem} \\label{Theorem 4} Let $\\Upsilon_{\\alpha{\\bar{\\beta}}}$ be a\nHermitian symmetric tensor satisfying the L-heat equation on\n$M\\times[0,T]$. Suppose $\\Upsilon_{\\alpha{\\bar{\\beta}}}(x,0)\\ge0$ (and satisfies some\ngrowth assumptions in the case $M$ is noncompact). Then $Z\\ge0$ on\n$M\\times(0,T]$ for any smooth vector field $V$ of type $(1,0)$.\n\\end{theorem}\n\nThe use of the maximum principle in the proof can be replaced by the\nintegration argument as in the proof of (\\ref{4}). For any $T\\ge\nt_0>0$, in order to prove that $Z\\ge 0$ at $t_0$ it suffices to show\nthat\n when $t=t_0$, $\\int_M t^2Z h\\, d\\mu_t\\ge 0$ for any\n compact-supported nonnegative function $h$. Now we solve the {\\it\n conjugate heat equation} $\\left(\\frac{\\partial }{\\partial \\tau} -\\Delta\n +R\\right)h(y,\\tau)=0$ with $\\tau=t_0-t$ and $h(y, \\tau=0)=h(y)$,\n the given compact-supported function at $t_0$. By the perturbation\n argument we may as well as assume that $\\Upsilon>0$. Then let\n $Z_m(y,t)=\\inf_{V} Z(y, t)$. It was shown in \\cite{NT} that\n $$ \\left(\\frac{\\partial}{\\partial t}-\\Delta \\right) Z_m =\nY_1+Y_2-2\\frac{Z_m}{t}\n$$\nwhere \\begin{eqnarray*}Y_1&=&\\Upsilon_{\\bar{p}q}\\left(\\Delta R_{p\\bar{q}}\n+R_{p\\bar{q}\\alpha{\\bar{\\beta}}}R_{\\bar{\\alpha}\\beta} + \\nabla_\\alpha\nR_{p\\bar{q}}V_{\\bar{\\alpha}}+\n\\nabla_{\\bar{\\alpha}}R_{p\\bar{q}}V_{\\alpha}\\right.\\\\\n&\\,& +\\left. R_{p\\bar{q}\\alpha{\\bar{\\beta}}}V_{\\bar{\\alpha}}V_\\beta +\n\\frac{R_{p\\bar{q}}}{t}\\right)\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\nY_2&= &\\Upsilon_{\\gamma\\bar{\\alpha}}\\left[ \\nabla_p V_{\\bar{\\gamma}}- R_{p\\bar{\\gamma}}\n-\\frac{1}{t}g_{p\\bar{\\gamma}}\\right]\\left[\\nabla_{\\bar{p}}V_\\alpha- R_{\\alpha\\bar{p}}\n-\\frac{1}{t} g_{\\bar{p}\\alpha}\\right]\\\\\n&\\, &+\n\\Upsilon_{\\gamma\\bar{\\alpha}}\\nabla_{\\bar{p}}V_{\\bar{\\gamma}}\\nabla_pV_{\\alpha}\\\\\n&\\ge& 0.\n\\end{eqnarray*}\nNotice that in the above expressions, at every point $(y,t)$ the $V$\nis the minimizing vector. This implies the monotonicity\n$$\n\\frac{d}{d t}\\int_M t^2Z_m h\\, d\\mu_t =t^2\\int_M\n\\left(Y_1+Y_2\\right)h\\, d\\mu_t\\ge 0.\n$$\nSince $\\lim_{t\\to 0}t^2Z_m =0$, which is certainly the case if\n$\\Upsilon$ is smooth at $t=0$ and can be assumed so in general by\nshifting $t$ with a $\\epsilon>0$, we have that $\\int_M t^2Z_m h\\, d\\mu\n|_{t=t_0}\\ge 0$. This proof via the integration by parts implies the\nfollowing monotonicity formula.\n\n\\begin{proposition}\\label{Proposition 5} Let $(M, g(t))$, $\\Upsilon$ and $Z$ be as in Theorem\n\\ref{Theorem 4}. For any space-time point $(x_0, t_0)$ with\n$00$. Let $\\tau=t_0-t$. Then\n\\begin{eqnarray}\\label{19}\n &\\,& \\frac{d}{d t}\\left(\\int_M t^2 Q (y, t)\n\\hat H(x_0, y, \\tau)\\, d\\mu(x)\\right)\\\\\n& \\ge &2t^2\\int_M \\left(\\left|\\nabla_i\\nabla_j \\log\nu+\\frac{1}{2t}g_{ij}\\right|^2+R_{ij}\\nabla_i \\log u \\nabla_j \\log\nu\\right)u \\hat H\\, d\\mu\\ge 0, \\nonumber\\end{eqnarray} where $\\hat\nH(x_0, y, \\tau)= \\frac{1}{(4\\pi\n\\tau)^{\\frac{n}{2}}}\\exp\\left(-\\frac{d^2(x_0, y)}{4\\tau}\\right)$\nwith $d(x_0, y)$ being the distance function between $x_0$ and $y$.\nIn particular, we have that\n\\begin{eqnarray}\\label{20}\n &\\,& \\left(u\\Delta \\log u+\\frac{n}{2t}u\\right)\n(x_0, t_0)\\\\\n&\\ge &\\frac{2}{t_0^2}\\int_0^{t_0}t^2\\int_M\n\\left(\\left|\\nabla_i\\nabla_j \\log\nu+\\frac{1}{2t}g_{ij}\\right|^2+R_{ij}\\nabla_i \\log u \\nabla_j \\log\nu\\right)u \\hat H.\\nonumber\n\\end{eqnarray}\n\\end{proposition}\n\nIt is clear that (\\ref{20}) improves the estimate of Li-Yau slightly\nby providing the lower estimate, from which one can see easily that\nthe equality (for Li-Yau's estimate) holding somewhere implies that\n$M=\\Bbb R^n$ (this was first observed in \\cite{N1}, with the help of an\nentropy formula). The expression in the right hand side of\n(\\ref{17}) also appears in the linear entropy formula of \\cite{N1}.\n\nOne can write down similar improving results for the Li-Yau type\nestimate proved in \\cite{N1}, which is a linear analogue of\nPerelman's estimate $v_H\\le 0$, and the one in \\cite{N2}, which is a\nlinear version of Theorem 4 above. For example, when $M$ is a\ncomplete Riemannian manifold with the nonnegative Ricci curvature,\nif $u=H(x,y,t)=\\frac{e^{-f}}{(4\\pi t)^{\\frac{n}{2}}}$, the\nfundamental solution to the heat equation centered at $x$ at $t=0$,\nletting $W=t(2\\Delta f -|\\nabla f|)+f-n$, we have that $W\\le 0$. If\n$\\hat H$ is the `pseudo backward heat kernel' defined as in\nProposition \\ref{Proposition 7} we have that\n\\begin{eqnarray*}\n &\\,& \\frac{d}{dt}\\int_M (-W)u \\hat H(x_0, y, \\tau)\\,\nd\\mu(y)\\\\\n&=&2t\\int_M\\left(\\left|\\nabla_i \\nabla_j\nf-\\frac{1}{2t}g_{ij}\\right|^2+R_{ij}\\nabla_i f\\nabla_j f\\right)u\n\\hat H(x_0, y, \\tau)\\, d\\mu(y)\\ge 0\\end{eqnarray*} and $$\n\\left(-Wu\\right)(x_0, t_0)\\ge\n2\\int_0^{t_0}t\\int_M\\left(\\left|\\nabla_i \\nabla_j\nf-\\frac{1}{2t}g_{ij}\\right|^2+R_{ij}\\nabla_i f\\nabla_j f\\right)u\n\\hat H.\n$$\nIf we assume further that $M$ is a complete K\\\"ahler manifold with\nnonnegative bisectional curvature and $u(y,t)$ is a strictly\nplurisubharmonic solution to the heat equation with $w=u_t$, then\n$$\n\\frac{d}{ dt} \\int_M t^2Z^w_m \\hat H(x_0, y, t) \\, d\\mu(y)\n=t^2\\int_M Y_5 \\hat H(x_0, y, t)\\,\\, d\\mu(y)\\ge 0,\n$$\nwhere\n$$\nZ^w_m(y, t)=\\inf_{V\\in T^{1, 0}M}\\left(w_t+\\nabla_\\alpha w\nV_{{\\bar{\\alpha}}}+\\nabla_{{\\bar{\\alpha}}} w V_\\alpha\n+u_{\\alpha{\\bar{\\beta}}}V_{{\\bar{\\alpha}}}V_{\\beta}+\\frac{w}{t}\\right)\n$$and\n\\begin{eqnarray*}\n Y_5 &=&u_{\\gamma\n\\bar{\\alpha}}\\left[\\nabla_pV_{\\bar{\\gamma}}-\\frac{1}{t}g_{p\\bar{\\gamma}}\\right]\\left[\\nabla_{\\bar{p}}V_{\\alpha}\n-\\frac{1}{t}g_{\\bar{p}\\alpha}\\right]\\\\\n&\\, &+ u_{\\gamma\n\\bar\\alpha}\\nabla_{\\bar{p}}V_{\\bar{\\gamma}}\\nabla_pV_{\\alpha} +R_{\\alpha{\\bar{\\beta}}\ns\\bar{t}}u_{\\bar{s}t}V_{\\beta}V_{\\bar{\\alpha}}\\\\&\\ge& 0\n\\end{eqnarray*}\nwith $V$ being the minimizing vector in the definition of $ Z^w_m$.\nIn particular,\n$$\n\\left(\\frac{\\partial^2}{\\partial (\\log t)^2} u(y, t) \\right)(x_0,\nt_0)\\ge\\int_0^{t_0}t^2\\int_M Y_5 \\hat H(x_0, y, t)\\,\\, d\\mu(y)\\, dt.\n$$\nThis sharpens the logarithmic-convexity of \\cite{N2}.\n\nFinally we should remark that in all the discussions above one can\nreplace the `pseudo backward heat kernel' $\\hat H(y,t; x_0,\nt_0)=\\frac{\\exp(-\\frac{r^2(x_0, y)}{4(t_0-t)})}{(4\\pi\n(t_0-t))^{\\frac{n}{2}}}$ (or $\\frac{\\exp(-\\ell(y, \\tau))}{(4\\pi\n\\tau)^{\\frac{n}{2}}}$, centered at $(x_0, t_0)$ in the case of Ricci\nflow), which we wrote before as $\\hat H(y, x_0, \\tau)$ by abusing\nthe notation, by the fundamental solution to the backward heat\nequation (even by constant $1$ in the case of compact manifolds).\nAlso it still remains interesting on how to make effective uses of\nthese improved estimates, besides the rigidity results out of the\ninequality being equality somewhere. There is also a small point\nthat should not be glossed over. When the manifold is complete\nnoncompact, one has to justify the validity of the Green's second\nidentity (for example in Proposition \\ref{Proposition 7} we need to\njustify that $\\int_M \\left(\\hat H\\Delta Q-Q\\Delta \\hat H\\right)\\, d\\mu =0$).\nThis can be done when $t_0$ is sufficiently small together with\nintegral estimates on the Li-Yau-Hamilton quantity (cf. \\cite{CLN}).\nThe local monotonicity formula that shall be discussed in the next\nsection provides another way to avoid possible technical\ncomplications caused by the non-compactness.\n\n\n\\section{ Local monotonicity formulae}\n\nIn \\cite{EKNT}, a very general scheme on localizing the monotonicity\nformulae is developed. It is for any family of metrics evolved by\nthe equation $\\frac{\\partial}{\\partial t} g_{ij}=-2\\kappa_{ij}$. The\nlocalization is through the so-called `heat ball'. More precisely\nfor a smooth positive space-time function $v$, which often is the\nfundamental solution to the {\\it backward conjugate heat equation}\nor the `pseudo backward heat kernel' $\\hat H(x_0, y,\n\\tau)=\\frac{e^{-\\frac{r^2(x_0, y)}{4\\tau}}}{(4\\pi\n\\tau)^{\\frac{n}{2}}}$ (or $\\frac{e^{-\\ell(y, \\tau)}}{(4\\pi\n\\tau)^{\\frac{n}{2}}}$ in the case of Ricci flow), with $\\tau=t_0-t$,\none defines the `heat ball' by $E_r=\\{(y, t)|\\, v\\ge r^{-n}; t<\nt_0\\}$. For all interesting cases we can check that $E_r$ is compact\nfor small $r$ (cf. \\cite{EKNT}). Let $\\psi_r=\\log v +n\\log r$. For\nany `Li-Yau-Hamilton' quantity ${\\mathcal Q}$ we define the local quantity:\n$$\nP(r):=\\int_{E_r}\\left(|\\nabla \\psi_r|^2+\\psi_r({\\text tr}_g\n\\kappa)\\right){\\mathcal Q}\\, d\\mu_t \\,dt.\n$$\nThe finiteness of the integral can be verified via the localization\nof Lemma \\ref{Lemma 2}, a local gradient estimate. The general form\nof the theorem, which is proved in Theorem 1 of \\cite{EKNT}, reads\nas the following.\n\n\\begin{theorem}\\label{Theorem 8} Let $I(r)=\\frac{P(r)}{r^n}$. Then\n\\begin{eqnarray}\\label{21}\nI(r_2)-I(r_1)&=&-\\int_{r_1}^{r_2}\n\\frac{n}{r^{n+1}}\\int_{E_r}\\left[\\left(\\left(\\frac{\\partial}{\\partial\nt}+\\Delta-{\\text tr}_g\n\\kappa\\right)v\\right)\\frac{{\\mathcal Q}}{v}\\right.\\\\\n&\\, &\\left.+\\psi_r\\left(\\frac{\\partial}{\\partial t}-\\Delta \\right){\\mathcal Q}\\right] \\,\nd\\mu_t \\, dt\\, dr. \\nonumber\\end{eqnarray}\n\\end{theorem}\nIt gives the monotonicity of $I(r)$ in the cases that ${\\mathcal Q} \\ge 0$,\nwhich is ensured by the Li-Yau-Hamilton estimates in the case we\nshall consider, and both $\\left(\\frac{\\partial}{\\partial t}+\\Delta-{\\text tr}_g\n\\kappa\\right)v $ and $ \\left(\\frac{\\partial}{\\partial t}-\\Delta \\right){\\mathcal Q}$ are\nnonnegative. The nonnegativity of $\\left(\\frac{\\partial}{\\partial t}+\\Delta-{\\text\ntr}_g \\kappa\\right)v $ comes for free if we chose $v$ to be the\n`pseudo backward heat kernel'. The nonnegativity of $\n\\left(\\frac{\\partial}{\\partial t}-\\Delta \\right){\\mathcal Q}$ follows from the computation,\nwhich we may call as in \\cite{N3} the {\\it pre-Li-Yau-Hamilton\nequation}, during the proof of the corresponding Li-Yau-Hamilton\nestimate. Below we illustrate examples corresponding to the\nmonotonicity formulae derived in the previous section. These new\nones expand the list of examples given in Section 4 of \\cite{EKNT}.\n\n\n\n For the case of Ricci\/K\\\"ahler-Ricci flow, for\na fixed $(x_0, t_0)$, let $v=\\frac{e^{-\\ell(y, \\tau)}}{(4\\pi\n\\tau)^{\\frac{n}{2}}}$, the `pseudo backward heat kernel', where\n$\\ell$ is the reduced distance centered at $(x_0, t_0)$.\n\n \\begin{example}\\label{Example 1} Let $Z_m$, $Y_1$ and $Y_2$ be as\nin Proposition \\ref{Proposition 5}. Let ${\\mathcal Q}=t^2Z_m$. Then\n$$\n\\frac{d}{d r}I(r)\\le\n-\\frac{n}{r^{n+1}}\\int_{E_r}\\left[t^2\\psi_r\\left(Y_1+Y_2\\right)\\right]\\,\nd\\mu_t \\, dt\\le 0\n$$\nand\n$$\n{\\mathcal Q}(x_0, t_0)\\ge\nI(\\bar{r})+\\int_0^{\\bar{r}}\\frac{n}{r^{n+1}}\\int_{E_r}\\left[t^2\\psi_r\\left(Y_1+Y_2\\right)\\right]\\,\nd\\mu_t \\, dt\\, dr.\n$$\n\\end{example}\n\n\\begin{example}\\label{Example 2} Let $u$, ${\\mathcal Q}=t^2Q$, $\\Upsilon_{\\alpha{\\bar{\\beta}}}$, $Y_3$ and\n$Y_4$ be as in Proposition \\ref{Proposition 6}. Then\n$$\n\\frac{d}{d r}I(r)\\le\n-\\frac{n}{r^{n+1}}\\int_{E_r}t^2\\psi_r\\left(\\frac{1}{u}|\\Upsilon_{\\alpha{\\bar{\\beta}}}|^2+u\\left|\\nabla_\\alpha\\nabla_\\beta\n\\log u\\right|^2+Y_3+Y_4\\right)\\le 0\n$$and\n$$ {\\mathcal Q}(x_0, t_0)\\ge I(\\bar{r})+\\int_0^{\\bar{r}}\\frac{n}{r^{n+1}}\\int_{E_r}t^2\\psi_r\\left(\\frac{1}{u}|\\Upsilon_{\\alpha{\\bar{\\beta}}}|^2+u\\left|\\nabla_\\alpha\\nabla_\\beta\n\\log u\\right|^2+Y_3+Y_4\\right). $$\n\\end{example}\n\nFor the fixed metric case, we may choose either $v=H(x_0, y, \\tau)$,\nthe {\\it backward heat kernel} or $v=\\hat H(x_0, y,\n\\tau)=\\frac{e^{-\\frac{d^2(x_0, y)}{4\\tau}}}{(4\\pi\n\\tau)^{\\frac{n}{2}}}$, the `pseudo backward heat kernel'.\n\n\\begin{example}\\label{Example 3} Let $u$ and $Q$ be as in Proposition \\ref{Proposition 7}. Let\n${\\mathcal Q}=t^2 Q$ and $f=\\log u$. Then\n$$\n\\frac{d}{d r}I(r)\\le -\\frac{2n}{r^{n+1}}\\int_{E_r}t^2u\\psi_r\n\\left(\\left|\\nabla_i\\nabla_j\nf+\\frac{1}{2t}g_{ij}\\right|^2+R_{ij}\\nabla_i f \\nabla_j f\\right)\\,\nd\\mu\\, dt\\le 0\n$$\nand $$ {\\mathcal Q}(x_0, t_0)\\ge I(\\bar{r})+\\int_0^{\\bar{r}}\n\\frac{2n}{r^{n+1}}\\int_{E_r}t^2u\\psi_r\\left(\\left|\\nabla_i\\nabla_j\nf+\\frac{1}{2t}g_{ij}\\right|^2+R_{ij}\\nabla_i f \\nabla_j f\\right) .$$\n\\end{example}\n\n\n\\begin{example}\\label{Example 4} Let $u=\\frac{e^{-f}}{(4\\pi t)^{\\frac{n}{2}}}$ be\nthe fundamental solution to the (regular) heat equation. Let\n$W=t(2\\Delta f-|\\nabla f|^2)+f-n$ and ${\\mathcal Q}=-uW$. Then\n$$\n\\frac{d}{d r}I(r)\\le\n-\\frac{2n}{r^{n+1}}\\int_{E_r}tu\\psi_r\\left(\\left|\\nabla_i \\nabla_j\nf-\\frac{1}{2t}g_{ij}\\right|^2+R_{ij}\\nabla_i f\\nabla_j f\\right) \\,\nd\\mu\\, dt\\le 0\n$$\nand $$\n {\\mathcal Q}(x_0, t_0)\\ge I(\\bar{r})+\\int_0^{\\bar{r}}\\frac{2n}{r^{n+1}}\\int_{E_r}tu\\psi_r\\left(\\left|\\nabla_i \\nabla_j\nf-\\frac{1}{2t}g_{ij}\\right|^2+R_{ij}\\nabla_i f\\nabla_j f\\right).\n$$Note that this provides another localization of entropy other than\nthe one in \\cite{N3} (see also \\cite{CLN}).\n\\end{example}\n\n\\begin{example}\n\\label{Example 5} Let $M$ be a complete K\\\"ahler manifold with\nnonnegative bisectional curvature. Let $u$, $Z^w_m$ and $Y_5$ be as\nin the last case considered in Section 4. Let ${\\mathcal Q}=t^2Z^w_m$. Then\n$$\n\\frac{d}{d r}I(r)\\le -\\frac{n}{r^{n+1}}\\int_{E_r} t^2 Y_5 \\psi_r \\,\nd\\mu \\, dt\n$$ and\n$$\n\\left(\\frac{\\partial^2}{\\partial (\\log t)^2} u(x, t) \\right)(x_0, t_0)\\ge\nI(\\bar{r})+\\int_0^{\\bar{r}}\\frac{n}{r^{n+1}}\\int_{E_r} t^2 Y_5\n\\psi_r \\, d\\mu \\, dt\\, dr.\n$$\n\\end{example}\n\n\n\\medskip\n\n{\\it Acknowledgement}. We would like to thank Ben Chow and Peng Lu\nfor continuously pressing us on a understandable proof of (\\ref{4}).\nWe started to seriously work on it after the visit to Klaus Ecker\nin August and a stimulating discussion with him. We would like to\nthank him for that, as well as Dan Knopf and Peter Topping for\ndiscussions on a related issue.\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Introduction}\n\n\nFerroelectricity driven by magnetic ordering in so-called type-II multiferroics\nexhibits a high potential for technological applications. Switching ferroelectric polarization by a magnetic field or magnetization by an electric field\noffers unprecedented applications in modern energy-effective electronic data storage technology.\\cite{Auciello1998,Spaldin2010}\nHowever, the link of magnetic order and ferroelectricity in type-II multiferroics still remains an intriguing question.\\cite{Mostovoy2006,Cheong2007,Khomskii2009,Tokura2010}\nTo elucidate this issue, lately much attention has been focused on the magnetic and magnetoelectric (ME) properties of quasi-one-dimensional (1D) antiferromagnetic (afm) quantum chain systems, which exhibit incommensurate cycloidal magnetic ordering.\\cite{Brink2008} Such systems lose inversion symmetry and appear to be suitable candidates for multiferroicity.\nIncommensurate spin-spiral magnetic ordering occurs in magnetic systems consisting of 1D chains when\nthe intrachain nearest-neighbor (nn) and next-nearest-neighbor (nnn) spin exchange interactions ($J_{\\rm {nn}}$ and $J_{\\rm {nnn}}$, respectively) are spin-frustrated, as found for compounds with CuX$_2$ ribbon chains made up of CuX$_4$ plaquettes, where X is a suitable anion, e.g. oxygen or a halide. Current examples include LiCuVO$_4$, NaCu$_2$O$_2$, CuCl$_2$.\\cite{Gibson2004, Enderle2005, Capogna2005, Kimura2008, Banks2009, Capogna2010, Mourigal2011} It is typical that the Cu-X-Cu superexchange $J_{\\rm {nn}}$ is ferromagnetic (fm), the Cu-X$\\ldots$X-Cu super-superexchange $J_{\\rm {nnn}}$ is afm and larger in magnitude.\\cite{Banks2009,Koo2011}\nA cycloidal spin-spiral along a 1D chain induces a macroscopic electric polarization,\n$\\vec{P}$ $\\propto$ $\\vec{e}_{ij} \\times (\\vec{S}_i\\times \\vec{S}_j)$,\nwhere $e_{ij}$ is the vector linking the moments residing on adjacent spins $\\vec{S_{i}}$ and $\\vec{S_{j}}$.\\cite{Katsura2005, Sergienko2006, Xiang2007}\n\nIn an ongoing effort to identify new quantum spin chain systems which potentially exhibit spiral magnetic order and ferroelectric polarization, we recently focused our attention on compounds crystallizing with ribbon chains, mainly those belonging to the CrVO$_4$ structure-type. The aforementioned structure-type features MO$_2$ ribbon chains where M is a magnetic 3$d$ transition metal. Such compounds were recently shown to exhibit exotic magnetic ground-states.\\cite{Attfield1985,Glaum1996,Law2010,Law2011} Here, we report on the magnetic and ME properties of another member of this structure-type, CuCrO$_4$.\nOur density functional calculations indicate $J_{\\rm {nn}}$ to be about twice as strong as $J_{\\rm {nnn}}$ putting CuCrO$_4$ in the vicinity of the Majumdar-Ghosh point for which the ground state can by exactly solved.\\cite{Majumdar1969} This feature makes CuCrO$_4$ uniquely exceptional since all of the CuX$_2$ ribbon chain systems investigated so far exhibit fm $J_{\\rm {nn}}$ and afm $J_{\\rm {nnn}}$ spin exchange, where $J_{\\rm {nnn}}$ is considerably larger in magnitude than $J_{\\rm {nn}}$.\\cite{Enderle2005, Enderle2010, Banks2009, Koo2011} We demonstrate that CuCrO$_4$ exhibits long-range afm ordering below $\\sim$~8.2~K, which is accompanied by a ME anomaly due to possible spin-spiral ordering in the CuO$_2$ ribbon chains.\n\n\n\\section{Crystal Structure}\\label{Structure}\n\nCuCrO$_4$ crystallizes in the CrVO$_4$ structure-type\\cite{Brandt1943,Seferiadis1986} (SG: \\textit{Cmcm}, No. 63) with Cu$^{2+}$ (\\emph{d}$^9$, $S$~=1\/2) and Cr$^{6+}$ (\\emph{d}$^0$) ions. In the crystal structure of CuCrO$_4$, the axially-elongated CuO$_6$ octahedra share edges to form chains running along the $c$-axis (Fig. \\ref{Fig1}(a)). These chains are interconnected by CrO$_4$ tetrahedra such that each CrO$_4$ tetrahedron is linked to three CuO$_4$ chains by corner-sharing (Fig. \\ref{Fig1}(b)). The x$^2$-y$^2$ magnetic orbital of each CuO$_6$ octahedron is contained in the CuO$_4$ plaquette with four short Cu-O bonds.\\cite{Whangbo2003} Thus, as far as the magnetic properties are concerned, CuCrO$_4$ consists of corrugated CuO$_2$ ribbon chains running along the $c$-axis (Fig. \\ref{Fig1}(a)). At room temperature the Cu$\\ldots$Cu distance is 2.945(2)~\\AA\\ and the Cu-O-Cu $\\angle$ is 98.1(1)$^{\\rm o}$.\n\n\n\\begin{figure}\n \n \\includegraphics[width=7.5cm]{Fig1.eps}\\\\\n \\caption{(Color online)(a): The crystal structure of CuCrO$_4$. The (blue) octahedra are the CuO$_6$ units while the (green) tetrahedra are the CrO$_4$ units. The interchain spin exchange pathways $J_{1}$ and $J_{2}$ are also indicated. (b): A section of the CuO$_2$ ribbon chain highlighting the edge sharing CuO$_4$ plaquettes, with the nn $J_{\\rm {nn}}$ and nnn $J_{\\rm {nnn}}$ spin exchange pathways labeled.}\\label{Fig1}\n\\end{figure}\n\n\n\n\n\\section{Spin Exchange Interactions}\\label{SecTheory}\n\nTo examine the magnetic properties of CuCrO$_4$, we consider the four spin exchange paths defined in Fig. \\ref{Fig1}; the two intra-chain exchanges $J_{\\rm {nn}}$ and $J_{\\rm {nnn}}$ as well as the inter-chain exchanges $J_{1}$ and $J_{2}$. To determine the values of $J_{\\rm {nn}}$, $J_{\\rm {nnn}}$, $J_{1}$ and $J_{2}$, we examine the relative energies of the five ordered spin states depicted in Fig. \\ref{Fig2} in terms of the Heisenberg spin Hamiltonian,\n\n\\begin{equation}\\label{eq1}\nH = -\\sum J_{ij} \\vec{{S_i}}\\vec{{S_j}},\n\\end{equation}\n\nwhere $J_{\\rm ij}$ is the exchange parameter (i.e., $J_{\\rm {nn}}$, $J_{\\rm {nnn}}$, $J_{1}$ and $J_{2}$) for the interaction between the spin sites $i$ and $j$. Then, by applying the energy expressions obtained for spin dimers with \\textit{N} unpaired spins per spin site (in the present case, \\textit{N} = 1),\\cite{Dai} the total spin exchange energies of the five ordered spin states, per four formula units (FUs), are given as summarized in Fig. \\ref{Fig2}. We determine the relative energies of the five ordered spin states of CuCrO$_4$ on the basis of density functional calculations with the Vienna \\textit{ab initio} simulation package, employing the projected augmented-wave method, \\cite{Kresse1993,Kresse1996a,Kresse1996b} the generalized gradient approximation (GGA) for the exchange and correlation functional,\\cite{Perdew1996} with the plane-wave cut-off energy set to 400 eV, and a set of 64 \\textbf{k}-points for the irreducible Brillouin zone. To account for the strong ele\n ctron correlation associated with the Cu 3$d$ state, we performed GGA plus on-site repulsion (GGA+$U$) calculations with $U_{\\rm eff}$ = 4 and 6 eV for Cu.\\cite{Dudarev1998} The relative energies of the five ordered spin states obtained from our GGA+$U$ calculations are summarized in Fig. \\ref{Fig2}. Then, by mapping these relative energies onto the corresponding relative energies from the total spin exchange energies,\\cite{Whangbo2003,Koo2008a,Koo2008b,Kang2009,Koo2010} we obtain the values of the spin exchange parameters, $J_{\\rm {nn}}$, $J_{\\rm {nnn}}$, $J_1$, and $J_2$ as summarized in Table \\ref{Table1}.\n\n\\begin{figure}[htp]\n\\includegraphics[width=8cm ]{Fig2.eps}\n\\caption{Five ordered spin states used to extract the values of $J_{\\rm {nn}}$, $J_{\\rm {nnn}}$, $J_1$, and $J_2$, where the Cu$^{2+}$ sites with different spins are denoted by filled and empty circles. For each ordered spin state, the expression for the total spin exchange energy per 4 FUs is given, and the two numbers in square bracket (from left to right) are the relative energies, in meV per 4 FUs, obtained from the GGA+$U$ calculations with $U_{\\rm eff}$ = 4 and 6 eV, respectively.}\n\\label{Fig2}\n\\end{figure}\n\n\n\\begin{table}[tbh]\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n $J_i$ & \\multicolumn{2}{c}{$U_{\\rm eff}$ = 4 eV} & \\multicolumn{2}{c}{$U_{\\rm eff}$ = 6 eV} & experiment\\\\\n\\hline\n$J_{\\rm {nn}}$ & -199.7(1.0)& -63.8 & -115.9(1.0)& -55.4 & -54 \\\\ \\\\\n$J_{\\rm {nnn}}$ & -85.8(0.43)& -27 & -56.5(0.49) &-27 &-27 \\\\ \\\\\n$J_1$ & -8.6(0.04)& -2.7 & -6.00(0.05) & -2.9 &- \\\\ \\\\\n$J_2$ & +31.1(0.16)& +9.8 & +22.3(0.19)& +10.7& +12 \\\\\n\\\\\n$\\theta _{\\rm CW}$& & -43.2&&-38.8&-56\/-60\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\caption[]{Spin exchange parameters $J_{\\rm {nn}}$, $J_{\\rm {nnn}}$ $J_1$ and $J_2$ (in K) of CuCrO$_4$ obtained from GGA+$U$ calculations with $U_{\\rm eff}$ = 4 and 6 eV. The left column for each $U_{\\rm eff}$ contains the theoretical results, while the values in the right column are the scaled theoretical results such that $J_{\\rm nnn}$ equals the experimental finding, -27~K. The rightmost column summarizes the experimentally found spin exchange values. The final row show the Curie-Weiss temperatures of the scaled GGA+$U$ spin exchange parameters, calculated using the mean field expression; $\\theta_{\\rm CW}$~=~$\\frac{1}{3}\\sum_{\\substack{i}}z_iJ_{\\rm{i}}S(S+1)$, where $z_i$ is the number of neighbor with which a single atom interacts with the spin exchange $J_i$, and the experimentally observed values (see below).}\n\\label{Table1}\n\\end{table}\n\n\nThe intra-chain spin exchanges $J_{\\rm {nn}}$ and $J_{\\rm {nnn}}$ are both afm and constitute the two dominant spin exchanges in CuCrO$_4$. The inter-chain parameter $J_2$, connecting Cu atoms related by a translation along $a$, is fm and, depending on the onsite repulsion parameter $U_{\\rm eff}$, its magnitude amounts to 15 to 20\\% of the intra-chain spin exchange $J_{\\rm {nn}}$. $J_1$, which couples adjacent spin moments which are related by a translation along [110], is afm and comparatively small. Therefore, to a first approximation, CuCrO$_4$ can be described as a quasi 1D Heisenberg magnet with nn and nnn spin exchange interactions, both being afm. Since these 1D chains are connected by weak inter-chain exchanges ($J_{1}$ and $J_{2}$), long range ordering will eventually take place at low temperatures.\n\n\n\n\\subsection{Experimental}\n\nA polycrystalline sample of CuCrO$_4$ was prepared by separately dissolving equimolar amounts of anhydrous Copper(II)acetate and Chromium(VI)oxide in distilled water, similar to the recipe given by Arsene \\emph{et al.}\\cite{Arsene1978}. The two solutions were mixed and boiled to dryness. The resulting powder was heat treated in air at a temperature of 150$^{\\rm {\\circ}}$C for 2 days. The phase purity of the sample was checked by x-ray powder diffraction measurements using a STOE STADI-P diffractometer with monochromated Mo-$K$${\\alpha_1}$ radiation. The powder pattern was analyzed using the Rietveld profile refinement method employed within the Fullprof Suite.\\cite{Fullprof} No other reflections besides those of CuCrO$_4$ were observed.\n\nPowder reflectance spectra of CuCrO$_4$ were collected at room temperature using a modified CARY 17 spectrophotometer, equipped with an integrating sphere. The spectrometer was operated in the single-beam mode using BaSO$_4$ as reflectance (white) standard.\nCuCrO$_4$ powder was mixed with BaSO$_4$ in a volumetric ratio CuCrO$_4$:BaSO$_4$ $\\sim$1 : 5.\n\n\nTemperature dependent electron paramagnetic resonance (EPR) spectra of a $\\sim$~5~mg polycrystalline sample, contained within an EPR low-background suprasil$^\\copyright$ quartz tube, were collected using $\\sim$~9.5~GHz microwave radiation (Bruker ER040XK microwave X-band spectrometer, Bruker BE25 magnet equipped with a BH15 field controller calibrated against Diphenylpicrylhydrazyl (DPPH)).\n\nThe molar magnetic susceptibilities, $\\chi_{\\rm mol}$, of a polycrystalline sample weighting $\\sim$~84~mg were measured with various fields between 2~K and 350~K using a SQUID magnetometer (MPMS, Quantum Design). The raw magnetization data were corrected for the magnetization of the sample container.\n\nThe specific heats, $C_p$, of a powder sample weighting $\\sim$~2.4~mg were determined as a function of the temperature and magnetic field with a relaxation-type calorimeter (PPMS, Quantum Design) for the temperature range 0.4~K to 50~K and magnetic fields up to 9~T.\n\nThe relative dielectric constant, $\\epsilon_{\\rm r}$, was measured at a constant frequency and excitation voltage, 1000~Hz and 15~V, respectively, with an Andeen-Hagerling 2700A capacitance bridge on a compacted powder (thickness: $\\sim$~0.8~mm, $\\varnothing$: 3.6~mm).\n\n\n\n\n\n\n\\subsection{Results and Discussion}\n\nFigure \\ref{Fig3} shows the measured and simulated x-ray powder diffraction patterns of the sample of CuCrO$_4$ used for all subsequent characterization. The refined atomic parameters and the lattice parameters are summarized in Table \\ref{Table2} and were found to be in good agreement with the previously published single crystal results.\\cite{Seferiadis1986}\n\n\n\\begin{figure}[htp]\n\\includegraphics[width=8cm ]{Fig3.eps}\n\\caption{(Color online) (o): Measured x-ray diffraction pattern of CuCrO$_4$ (wavelength 0.709 \\AA\\, Mo-$K$${\\alpha_1}$ radiation). Solid (red) line: Fitted pattern ($R_p$~=~3.42~\\%, reduced $\\chi^2$~=~1.15) using the parameters given in Table \\ref{Table2}. Solid (blue) line (offset): Difference between measured and calculated patterns.\nThe positions of the Bragg reflections used to calculate the pattern are\nmarked by the (green) vertical bars in the lower part of the\nfigure.}\n\\label{Fig3}\n\\end{figure}\n\n\\squeezetable\n\\begingroup\n\\begin{table}\n\\squeezetable\n\\centering\n\\begin{tabular}{ c c c c c c }\n\\hline\\hline\natom&Wyckoff site&x&y&z&$B_{\\rm{iso}}$ (\\AA $^2$)\\\\\n\\hline\nCu&4a&0&0&0&0.09( 8)\\\\ \\\\\nCr&4c&0&0.3700(3)&0.25&0.90( 8)\\\\ \\\\\nO1&8f&0&0.2652(5)&0.0320(9)&0.80(12)\\\\ \\\\\nO2&8g&0.2326(7)&-0.0198(6)&0.25&0.80(12)\\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{Atomic positional parameters of CuCrO$_4$ (SG: \\textit{Cmcm}) as obtained from a profile refinement of the x-ray powder diffraction pattern, collected at room temperature. The lattice parameters amount to $a$~=~5.4388(5)~\\AA, $b$~=~8.9723(8)~\\AA\\ and $c$~=~5.8904(6)~\\AA.}\\label{Table2}\n\\end{table}\n\\endgroup\n\n\n\nFigure \\ref{Opt} displays the optical spectrum of CuCrO$_4$ which is consistent with the deep brownish-red color of the CuCrO$_4$ powder. The spectrum is dominated by a strong absorption band centered at 21500~cm$^{-1}$ (466 nm) which we attribute to an O$^{2-}$~$\\rightarrow$~Cr$^{6+}$ charge transfer transition, in agreement with observations for other hexavalent chromates.\\cite{Johnson1970,Lever1984}\nIn the near infrared regime (NIR) the spectrum exhibits a maximum at\n$\\tilde\\nu_3$~=~13000 cm$^{-1}$ with a tail extending down to $\\sim$~7000~cm$^{-1}$. Two subsequent faint shoulders are seen within the slope at\n$\\tilde\\nu_2$~=~11000~cm$^{-1}$ and $\\tilde\\nu_1 \\sim$~8000~cm$^{-1}$.\n\n\\begin{figure}[!h]\n\\includegraphics[width=8cm ]{Fig4.eps}\n\\caption{(Color online) Powder reflectance spectrum of CuCrO$_4$. Black vertical bars mark the ligand-field transition energies, for the CuO$_6$ distorted octahedron, obtained from AOM calculations. We show the Kubelka-Munk relation, (1-$R_f$)$^2$\/(2 $R_f$), where $R_f$ = $I$(CuCrO$_4$)\/$I$(BaSO$_4$)\nand $I$(CuCrO$_4$) and $I$(BaSO$_4$) are the reflected light intensities of the sample and the BaSO$_4$ standard, respectively.\\cite{Kortum1969}}\n\\label{Opt}\n\\end{figure}\n\n\n\nUsing ligand-field considerations (see below) the observed absorption bands ($\\tilde\\nu_1$, $\\tilde\\nu_2$ and $\\tilde\\nu_3$) can be assigned to Cu$^{2+}$ $d$~-~$d$ transitions,\n$^2B_{1g}$~$\\rightarrow$~$^2A_{1g}$ ($z^2$~$\\rightarrow$~$x^2$~-~$y^2$),\n$^2B_{1g}$~$\\rightarrow$~$^2B_{2g}$ ($xy$~$\\rightarrow$~$x^2$~-~$y^2$), and\n$^2B_{1g}$~$\\rightarrow$~$^2E_{g}$ ($xz, yz$~$\\rightarrow$~$x^2$~-~$y^2$), respectively.\\cite{Jorgensen1963,Richardson1993,Larsen1974,Figgis2000}\n\nFrom $\\tilde\\nu_1$, $\\tilde\\nu_2$, and $\\tilde\\nu_3$ the crystal field splitting, 10$Dq$ for CuO$_6$, can be calculated using the relation,\n\n\\begin{equation*}\n 10Dq=\\tilde\\nu_3-(\\frac{\\tilde\\nu_3-\\tilde\\nu_2}{3})-(\\frac{\\tilde\\nu_1}{2}),\n\\end{equation*}\n\nwhich yields a value of\n10$Dq$~$\\sim$~8300~cm$^{-1}$. This value is similar to crystal field splitting values previously reported e.g. for Cu$^{2+}$ aquo-complexes.\\cite{Figgis2000}\n\nUV\/vis spectra for CuCrO$_4$ have been reported before by Baran and an assignment of the observed transitions has been been proposed.\\cite{Baran1994}\nBased on calculations within the framework of the angular overlap model (AOM)\\cite{Jorgensen1963,Richardson1993,Larsen1974,Figgis2000} we argue that this assignment has to be revised.\n\n\nWithin the AOM model the pairwise interactions of the ligands with the $d$-orbitals are encoded into the parameters, $e_{\\sigma}$, $e_{\\pi,x}$ and $e_{\\pi,y}$ which take care of interaction along and perpendicular to the Cu~-~O$_i$ ($i$~=~1, \\ldots, 6) bond, respectively. The energies of the individual $d$-orbitals are obtained by summation over all pairwise interactions.\nThe variation of the AOM parameters $e_{\\sigma i}$ with the Cu~-~O$_i$ distance has been taken care of by,\n\n\\begin{equation*}\n e_{\\sigma i} \\propto 1\/r_i^n.\n\\end{equation*}\n\nAn exponent of $n\\approx$~5 is derived from electrostatic and covalent theoretical bonding considerations.\\cite{Kortum1969,Smith1969,Bermejo1983}\nMeasurements of the pressure dependence of $10Dq$ pointed to a similar exponent\n5~$\\leq$~$n$~$\\leq$~6.\\cite{Drickamer1973} For the sake of simplicity we have chosen $e_{\\pi,x}$~=~$e_{\\pi,y}$~=~1\/4$e_{\\sigma}$. AOM calculations have been performed using the program CAMMAG.\\cite{Gerloch1983,Cruse1980} Table \\ref{AOMTable} summarizes the parameters which have been used for these calculations. The resulting transition energies marked by vertical bars in Fig. \\ref{Opt} are in good agreement with the centers of the experimentally observed absorption features.\n\n\\begin{table}[!h]\n\\begin{tabular}{ccc}\\hline\\hline\n & O$_{\\rm eq}$ &O$_{\\rm ax}$ \\\\\n\\hline\n $d$ (Cu-O) (\\AA) & 1.965 (4$\\times$) & 2.400 (2$\\times$)\\\\ \\\\\n $e_{\\sigma}$ (cm$^{-1}$) & 5600 & 2061 \\\\ \\\\\n $e_{\\pi,x}$ (cm$^{-1}$) & 1400 & 515 \\\\ \\\\\n $e_{\\pi,y}$ (cm$^{-1}$) & 1400 & 515 \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{Parameter used in the AOM calculations. The equatorial plane forms a rectangle with the equatorial O$_{\\rm eq}$~-~Cu~-~O$_{\\rm eq}$ bonds enclosing an $\\angle$ of 81.92$^{\\rm o}$ and 98.08$^{\\rm o}$, respectively. The Racah parameters amounted to $B$~=~992~cm$^{-1}$, $C$~=~3770~cm$^{-1}$ yielding a ratio $C$\/$B$~=~3.8, as given for the free Cu$^{2+}$ ion.\\cite{Figgis2000} As for the aquo-complex the nephelauxetic ratio $\\beta$ was chosen to be 0.80, and the spin-orbit coupling parameter $\\zeta$~=~664~cm$^{-1}$ was reduced by 20\\% as compared to the free ion value.\\cite{Figgis2000,McClure1959}}\\label{AOMTable}\n\\end{table}\n\nIn addition to the energy of the excited electronic states of the isolated CuO$_6$ unit, its magnetic properties are also obtained from the AOM calculations. The parametrization leads to an average $g_{\\rm av}$ = 2.18 and a strongly anisotropic $g$-tensor with $g_x$~=~2.07, $g_y$~=~2.07, and $g_z$~=~2.39 along the principle axes. The $z$-direction of the $g$-tensor lies along the Cu~-~O$_{\\rm ax}$ bond direction.\n\n\nThe results of the specific heat measurements for magnetic fields of 0~T and 9~T are displayed in Fig \\ref{Fig4}. The 0~T data reveal a rather broad, smeared, $\\lambda$-type anomaly centered at 8.2(5)~K marking the onset of long-range magnetic ordering. Within experimental error the data measured in a magnetic field of 9~T are identical to those obtained at 0~T.\nThe plot of $C_p\/T$ versus $T$ given in the low right inset of Fig. \\ref{Fig4} enables the estimation of the entropy contained within the anomaly, which equates to $\\sim$~0.6~J\/molK or $\\sim$~10~$\\%$ of the expected entropy of a $S$ = 1\/2 system, $R$\\,ln(2), where $R$ is the molar gas constant. 90\\% of the entropy has already been removed by short-range afm ordering above $T_{\\rm N}$.\n\n\n\nAt low temperatures, the heat capacity comprises of a phonon and magnon contribution.\nThe temperature dependence of the phonon contributions to the heat capacity can be described by a Debye-$T^3$ power law. The magnon heat capacity at low temperatures varies with a power law depending on the spin wave dispersion relation and the dimensionality of the lattice.\nFor a three-dimensional (3D) magnetic lattice, one obtains a $T^3$ power law for afm magnons, and a $T^{3\/2}$ power law for fm magnons.\\cite{deJongh}\nThe $C_p\/T^{3\/2}$ versus $T^{3\/2}$ plot shown in the upper left inset of Fig. \\ref{Fig4} demonstrates that at low temperatures the heat capacity conforms well to a $T^{3\/2}$ power law, with the coefficient of the fm magnon contribution given by the non-zero intercept with the ordinate, $\\gamma$, according to,\n\n\\begin{equation}\\label{eqcp}\nC_p\/T^{3\/2} = \\beta T^{3\/2} + \\gamma,\n\\end{equation}\n\nwhere $\\beta$ is related to the Debye temperature, $\\theta_{\\rm D}(0)$ at zero temperature via,\n\n\\begin{equation}\\label{eqDeb}\n\\beta = M R \\frac{12\\,\\pi^4\\,}{5} (\\frac{1}{\\theta_{\\rm D}})^3,\n\\end{equation}\n\nwith $M$ = 6 being the number of atoms per formula unit of CuCrO$_4$. While $\\gamma$ can be expressed as,\\cite{Martin1967}\n\n\n\\begin{equation}\\label{eqgamma}\n\\gamma = A (\\frac{k_{\\rm B}}{J_{\\perp}S})^{3\/2}.\n\\end{equation}\n\nHere we have assumed that the Cu ribbon chains are coupled to neighboring chains by $J_{\\perp}$, which we associate with the fm inter-chain spin exchange constant $J_2$ (see Table \\ref{Table2}).\n\nBy using eq. (\\ref{eqDeb}) we ascertain $\\theta_D(T \\rightarrow 0)$ to be,\n\n\\begin{equation*}\\label{eqDeb1}\n\\theta(T \\rightarrow 0) = 138(3)\\,\\,{\\rm K},\n\\end{equation*}\n\n\nand from the intercept and eq. (\\ref{eqcp}) we obtain $\\gamma$ as\n\n\\begin{equation*}\\label{eqDeb2}\n\\gamma = 1.03(2) \\times 10^{-2}\\,\\,\\, {\\rm J\/molK^{5\/2}}.\n\\end{equation*}\n\nBy using $J_{\\perp} \\sim J_2 \\sim$ 12 K (see below) and $S$=1\/2, the pre-factor $A$ in eq. (\\ref{eqgamma}) amounts to\n\n\\begin{equation*}\\label{eqDeb3}\nA \\approx 0.15\\,\\,\\, {\\rm J\/molK}.\n\\end{equation*}\n\n\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{Fig5.eps}\\\\\n\\caption{(Color online) (Black) o and (red) $\\bigtriangleup$: Heat capacity of CuCrO$_4$ at 0~T and 9~T, respectively. The latter data have been shifted by +0.5 J\/molK. Upper left inset: $C_p\/T^{3\/2}$ plotted versus $T^{3\/2}$ to highlight the low-temperature $T^{3\/2}$ power law. The (red) solid line is a fit of the data to eq. (\\ref{eqcp}) with parameters given in the text.\nLower right inset: $C_p\/T$ depicted against $T$ in the low-temperature regime.}\\label{Fig4}\n\\end{figure}\n\n\nFigure \\ref{EPR} summarizes the results of our EPR measurements.\nNear 3.4~kOe a single rather broad (peak-to-peak linewidth $\\Delta H_{\\rm pp} \\approx$ 0.8 - 1 kOe) symmetric resonance line was observed. It can be well fitted to the derivative of a single Lorentzian absorption line with a small contribution $|\\alpha |\\leq$ 0.04 of dispersion according to\n\n\\begin{equation}\\label{EPRpow}\n\\frac{dP_{\\rm abs}}{dH} \\propto \\frac{d}{dH}\\frac{\\Delta H + \\alpha(H - H_{\\rm res})}{(H - H_{\\rm res})^2 + \\Delta H^2} + \\frac{\\Delta H + \\alpha(H+H_{\\rm res})}{(H + H_{\\rm res})^2 + \\Delta H^2}.\n\\end{equation}\n\nAs the linewidth (half-width at half-maximum (HWHM)), $\\Delta H$, is of the same order of magnitude as the resonance field, $H_{\\rm res}$ (see Fig. \\ref{EPR}(a)), in eq. (\\ref{EPRpow}) we took into account both circular components of the exciting\nlinearly polarized microwave field and therefore also included the\nresonance at negative magnetic fields centered at $-H_{\\rm res}$.\n\n\n\n\n\nThe resonance field of the room temperature powder spectrum corresponds to a \\emph{g}-factor of 2.117(2). Upon cooling a slight increase of the \\emph{g}-factor with saturation to a value of $\\sim$~2.125 below 150~K was observed (Fig. \\ref{EPR}(c)).\nSuch a value is somewhat lower than the expected average value $g_{\\rm av}$ ascertained from the AOM calculations.\nThe resonance line is too broad to resolve the anisotropic \\emph{g}-factors which range between $\\sim$~2.39 and $\\sim$~2.07 (see above).\n\n\n\nThe integrated intensity of the EPR resonance, $I(T)$ which is proportional to the spin-susceptibility, increases with decreasing temperature down to $\\sim$~15~K where a hump occurs. Above $\\sim$~150~K, $I(T)$ follows a Curie-Weiss type temperature-dependence,\n\n\n\\begin{equation}\\label{eqIntInt}\nI (T)\\propto \\frac{1}{T - \\theta_{\\rm EPR}},\n\\end{equation}\n\nwith\n\\begin{equation*}\\label{eqIntTheta}\n \\theta_{\\rm EPR} \\approx -60(5)\\,\\,{\\rm K}.\n\\end{equation*}\n\nThe negative $T$-axis intercept indicates predominant afm spin exchange interactions. Deviations from the Curie-Weiss type temperature-dependence are ascribed to short-range afm correlations, which start to develop below $\\sim$~150~K, similar to the behavior of the dc magnetic susceptibility (see below). The decrease of the integrated intensity below $\\sim$~15~K signals the onset of long-range ordering.\n\nThe magnitude and temperature-dependence of the EPR linewidth, $\\Delta H$, are similar to\nthose observed for the inorganic spin-Peierls system CuGeO$_3$ or the frustrated afm 1D system LiCuVO$_4$.\\cite{Yamada1996,Krug2002}\nThe linewidth exhibits a concave temperature dependence with a linear increase at low temperatures and for $T \\rightarrow \\infty$ one extrapolates a saturation value of $\\sim$~1.4~kOe.\n\nIf we assume that the temperature-dependant broadening\nof the EPR resonance line is due to anisotropic or antisymmetric components in the exchange Hamiltonian, the constant high-temperature value can be estimated from the Kubo-Tomita limit as,\\cite{Kubo1954}\n\\begin{equation}\\label{eqKT}\n\\Delta H (T \\rightarrow \\infty) \\approx \\frac{1}{g \\mu_{\\rm B}} \\frac{\\delta^2}{J},\n\\end{equation}\n\nwhere $\\delta$ indicates the deviations from the symmetric Heisenberg spin exchange and \\textit{J} is the afm symmetric intrachain exchange. If for CuCrO$_4$ we associate\n$J$ with the nn spin exchange, $\\sim$~60~K (see below), we can estimate a $\\delta$ of $\\sim$~3~K, i.e. 5\\% of the symmetric exchange.\n\nThe linear slope of the linewidth at low temperatures can be explained using the formulism put forth by Oshikawa and Affleck\\cite{Oshikawa1999,Oshikawa2002} predicting\n\n\\begin{equation}\\label{eqKT2}\n\\Delta H (T) \\propto \\frac{\\delta^2}{J^2} T.\n\\end{equation}\n\nWe find a linear slope, indicative of 1D afm system, of $\\sim$~2.5~Oe\/K, similar to that observed for CuGeO$_3$ ($\\sim$~4.5~Oe\/K).\\cite{Yamada1996}\n\n\n\n\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{Fig6.eps}\\\\\n\\caption{(Color online) Results of the EPR measurements on a polycrystalline sample of CuCrO$_4$.\n(a) (o) Inverse of the integrated intensity. The (red) solid line is a fit of eq. (\\ref{eqIntInt}) to the high temperature data ($T \\geq$~150~K).\n(b) (o) The fitted half-width-at-half-maximum (HWHM) versus temperature.\n(c) (o) $g$-factor versus temperature.\n(d) (o) EPR spectrum of CuCrO$_4$ measured at RT with $\\sim$~9.45~GHz versus applied magnetic field. The (red) solid line represents the fitted derivative of a Lorentzian absorption line (eq. \\ref{EPRpow}) to the measured spectrum.}\\label{EPR}\n\\end{figure}\n\n\n\n\n\nThe magnetic susceptibility of a polycrystalline sample of CuCrO$_4$ was measured in magnetic fields of 1, 3, 5 and 7 Tesla. Above $\\sim$~20~K the susceptibilities are independent of the magnetic field indicating negligible ferromagnetic impurities. The susceptibilities, $\\chi_{\\rm mol}(T)$, above $\\sim$~150~K follow the modified Curie-Weiss law,\n\n\\begin{equation}\\label{eq2}\n\\chi_{\\rm mol}(T) = \\frac{C}{T - \\Theta} + \\chi_{\\rm dia} + \\chi_{\\rm VV}.\n\\end{equation}\n\n\n\n$C$ is the Curie constant pertaining to the spin susceptibility of the Cu$^{2+}$ entities, \\mbox{$C$ = $N_{\\rm A}$$g^2$\\,$\\mu_{\\rm B}^2$\\,$S(S+1)$\/3$k_{\\rm B}$}. $\\chi_{\\rm dia}$ refers to the diamagnetic susceptibilities of the electrons in the closed shells, that can be estimated from the increments given by Selwood, which equates to -62$\\times$10$^{-6}$ cm$^3$\/mol.\\cite{Selwood1956}\n\n\n\n\nAt high temperatures, $T \\geq$~150~K, we fitted the molar susceptibility to the aforementioned modified Curie-Weiss law (eq. \\ref{eq2}).\nWe found best agreement with the following parameters:\n\n\\begin{equation*}\ng = 2.17(2)\\,\\,\\,\\,\\,\\, \\rm{and} \\,\\,\\,\\,\\,\\, \\theta = -56(1) K\n\\end{equation*}\n\nand\n\n\\begin{equation*}\n\\chi_{\\rm dia} + \\chi_{\\rm VV}\\approx +20\\times 10^{-6}\\,\\,{\\rm cm}^3\/{\\rm mol}.\n\\end{equation*}\n\nThis puts the Van Vleck contribution to $\\approx$~+80$\\times$ 10$^{-6}$cm$^3$\/mol which is in reasonable agreement with what has been found for other Cu$^{2+}$ compounds (see Ref. \\onlinecite{Banks2009} and refs. therein). The fitted $g$-factor is in good agreement with optical spectroscopy and the Curie-Weiss temperature is negative and in accordance with $\\theta_{\\rm EPR}$.\n\n\n\n\n\n\nBelow 150~K there are deviations from the Curie-Weiss law attributed to increasing afm short range correlations. The susceptibility passes through a broad shoulder with a subsequent kink at $\\sim$~8~K whereupon it becomes field dependent, with a tendency to diverge for small fields. With increasing fields the divergence is suppressed and the kink becomes more apparent. By 7 T a pronounced rounded hump with a maximum at 14.2(2) K and a subsequent dip at 8.0(5) K become clearly visible.\n\n\n\nIn general, GGA+$U$ calculations overestimate the spin exchange constants typically by a factor up to 4, in our case 2. \\cite{Koo2008a,Koo2008b,Xiang2007b} By taking this into account and by using a mean field approach one calculates, from the spin exchange parameters summarized in Table \\ref{Table2}, a (negative) Curie-Weiss temperatures ranging between -38~K to -45~K, consistent with the experimental observations.\n\n\n\nOur GGA+$U$ calculations indicate that CuCrO$_4$ can be described by a Heisenberg 1D chain with afm nn and afm nnn spin exchanges, with significantly weak inter-chain interactions ($J_2$\/$J_{\\rm {nn}}$~$<$~0.19).\nTherefore, we modeled the magnetic susceptibility of CuCrO$_4$ against exact diagonalization results for the susceptibility $\\chi_{\\rm chain}(g,\\alpha,J_{\\rm{nnn}})$ of a single chain provided by Heidrich-Meissner \\textit{et al.},\n\\cite{Heidrich2006,Heidrichweb} with\n\n\\begin{equation}\\label{eqalpha}\n\\alpha = J_{\\rm {nn}}\/J_{\\rm {nnn}}.\n\\end{equation}\n\n\nInterchain spin exchange is treated within a mean-field approach according to,\n\\cite{Carlin1986}\n\n\n\n\\begin{equation}\\label{eqMF}\n\\chi_{\\rm mol}(T) = \\frac{\\chi_{\\rm chain}(T)}{1 - \\lambda\\,\\chi_{\\rm chain}(T)} + \\chi_0.\n\\end{equation}\n\n\n\n\n\nBy using the already known values, $\\chi_{\\rm{0}}$ = $\\chi_{\\rm{dia}}$ + $\\chi_{\\rm{VV}}$ = +20$\\times$10$^{-6}$\\,\\,cm$^3$\/mol as found from the fit of the high temperature magnetic susceptibility and a $g$-factor of 2.13 obtained from the EPR measurements, the simulated results can be compared to experimental data. The mean-field parameter, $\\lambda$, in eq. (\\ref{eqMF}) can be ascribed to the inter-chain spin exchange interactions according to\\cite{Carlin1986}\n\n\\begin{equation}\\label{eqlambda}\n\\lambda = (z_1\\,J_1+z_2\\,J_2)\/N_{\\rm A}g^2\\mu_{\\rm B}^2,\n\\end{equation}\n\nwherein, $z_1$~=~4 and $z_2$~=~2 count the number of spin moments with which a chain spin interacts through the inter-chain spin exchange interactions, $J_1$ and $J_2$, respectively.\nGuided by the GGA+$U$ results, the ratio $\\alpha$ is positive and in the regime of 1.5 to 2.5. Within this range for $\\alpha$ we find best agreement of our experimental data with the model calculations for,\n\n\n\\begin{equation*}\\label{eqMFres2}\n\\alpha \\approx 2, \\,\\,\\,\\,\\,\\, {\\rm{implying}} \\,\\,\\,\\,\\,\\, J_{\\rm {nnn}} = -27(2)~{\\rm{K}},\n\\end{equation*}\n\n\n\n\n\nand a positive $\\lambda$, which amounts to\n\n\n\\begin{equation*}\\label{eqMFres4}\n\\lambda = 7(1)\\, {\\rm mol}\/{\\rm cm^3}.\n\\end{equation*}\n\nFigure \\ref{Fig6} shows a comparison of the measured data and the mean-field corrected exact diagonalization results.\n\n\n$\\lambda >$ 0 indicates that the dominant inter-chain spin exchange is fm, consistent with our density functional calculations. The DFT calculations indicate $J_1$~$\\approx$~-1\/4$\\times J_2$, irrespective of $U_{\\rm eff}$. From eq. (\\ref{eqlambda}) using $\\lambda$~=~7(1)~mol\/cm$^3$ we derive a value for $J_2$ which amounts to\n\n \\begin{equation*}\\label{eqMFres5}\nJ_2 = 12(2) {\\rm K}.\n\\end{equation*}\n\nThis value is in good agreement with the scaled DFT result, see Table \\ref{Table1}.\n\n\n\n\n\n\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{Fig7.eps}\\\\\n\\caption{(Color online) (main panel) (o) Temperature dependence of the molar magnetic susceptibility, $\\chi$$_{\\rm m}$, taken at 7~T. Colored solid lines represent the exact diagonalization results by Heidrich-Meissner \\textit{et al.} for various ratios of $J_{\\rm nn}$\/$J_{\\rm nnn}$, 1.5, 1.75, 2 (red solid line), 2.25 and 2.5, from top to bottom, respectively. See text for more details. The dashed line is the magnetic susceptibility of a $S$=1\/2 Heisenberg chain with afm uniform nn spin exchange of -27~K.\\cite{Johnston2000}\n(a) red symbols: heating data, blue symbols: cooling data. $\\chi_{\\rm mol}$ versus temperature for various magnetic field. (b) (o) Reciprocal molar susceptibility versus temperature with a fit ((red) solid line) to a modified Curie-Weiss law (eq. (\\ref{eq2})).}\\label{Fig6}\n\\end{figure}\n\nThe inter-chain spin exchange can also be estimated from the N\\'{e}el temperature, $T_{\\rm N}$, which, according to the heat capacity data, amounts to (see above);\n\n\\begin{equation*}\\label{eqTN}\nT_{\\rm N} \\approx 8.2(5) {\\rm K}.\n\\end{equation*}\n\n\nYasuda \\textit{et al.} calculated the N\\'{e}el temperature of a quasi 1D Heisenberg antiferromagnet on a cubic lattice with the isotropic inter-chain coupling $J_{\\perp}$, inducing 3D long-range magnetic ordering at a N\\'{e}el temperature, $T_{\\rm N}$;\\cite{Yasuda2005}\n\n\\begin{equation}\\label{eq8Yasuda}\nT_{\\rm N}\/|J_{\\perp}| = 0.932\\,\\sqrt{ln(A) + \\frac{1}{2}ln\\,ln(A)},\n\\end{equation}\n\nwhere $A$= 2.6\\,$J_{\\rm{\\parallel}}$\/$T_{\\rm N}$ and $J_{\\rm{\\parallel}}$ is the intrachain spin exchange constant. If we assume $J_{\\rm{\\parallel}}$ to be our $J_{\\rm {nn}}~\\sim$~-60 K we find the inter-chain coupling to be\n\n \\begin{equation*}\\label{eqJperp}\n|J_{\\perp}| \\approx 5 {\\rm K},\n\\end{equation*}\n\nconsistent with the value obtained, from $\\lambda$. The differences may arise, since our real system has two different inter-chain coupling constants, $J_1$ and $J_2$, as indicated by our density functional calculations. Additionally, CuCrO$_4$ has a nnn spin exchange $J_{\\rm {nnn}}$, which is not included in Yasuda's model.\n\n\nFigure \\ref{epsilon} displays the temperature and magnetic field dependence of the relative dielectric constant, $\\epsilon_{\\rm r}$, of a compacted polycrystalline sample of CuCrO$_4$.\n\n\n\\begin{figure}\n\\includegraphics[width=9cm]{Fig8.eps}\\\\\n\\caption{\n(Color online) (a) Colored symbols represent the relative dielectric constants, $\\epsilon_{\\rm r}$, versus temperature for different applied magnetic fields, as given in the legend. (b) The zero field relative dielectric constant is shown by the solid black line within a greater temperature range. (c) (o) The relative dielectric constant versus applied magnetic field at a temperature of 5.2(1)~K.\n}\\label{epsilon}\n\\end{figure}\n\n\n\nAt room temperature, a value of $\\sim$~48 was found for $\\epsilon$$_{\\rm r}$. With decreasing temperature, $\\epsilon_{\\rm r}$ is seen to decrease in a smooth fashion, until it passes through a shallow double maximum between 35 and 15 K, possibly indicating some magnetostriction induced by short range magnetic ordering processes above $T_{\\rm N}$ (see Fig. \\ref{epsilon} inset (b)). At 10~K a value of $\\epsilon_{\\rm}~\\sim$~4.35 was measured.\nLong-range magnetic ordering leads to a sizeable ME effect as evidenced in the $\\epsilon_{\\rm r}$, however, with a rather broad anomaly extending over the whole temperature range down to 3~K. Indication for a sharp spike near $T_{\\rm N}$, as is frequently found in multiferroic systems, has not been seen. Similar broad anomalies, originating at $T_{\\rm N}$, have been in seen in CuCl$_2$ and CuBr$_2$.\\cite{Seki2010,Kremer2010}\nIn zero field a steep increase of $\\epsilon_{\\rm r}$ is seen to occur below $\\sim$~8.5~K with a broad slightly asymmetric hump centered at $\\sim$~5.35~K.\nIn zero field the increase of $\\epsilon_{\\rm r}$ from the paramagnetic phase to the maximum of the hump amounts to $\\sim$~6\\%. Applying a magnetic field decreases the ME anomaly and moves the maximum to higher temperatures. The onset of the ME anomaly is not seen to move, in accordance with the aforementioned $C_p$ measurements (see Fig. \\ref{epsilon}, inset (c)).\nThe decrease of $\\epsilon_{\\rm r}$ with a magnetic field starts above $ \\sim$~1~T and tends to saturation at sufficiently high fields.\n\n\nIn summary,\nCuCrO$_4$ represents a new 1D quantum antiferromagnet with a remarkable pronounced ME anomaly below the N\\'{e}el temperature of 8.2 K. Our density functional calculations indicate that, to a first approximation, the spin lattice of CuCrO$_4$ is a 1D Heisenberg chain with the unique situation that both, nn and nnn, spin exchanges are afm.\n$J_{\\rm {nn}}$\/$J_{\\rm {nnn}}$ is found to be close to 2, which places CuCrO$_4$ in the vicinity of the Majumdar-Ghosh point. The presence of sizeable ferromagnetic inter-chain spin exchange interaction leads to long-range magnetic ordering. The occurrence of the rather large ME anomaly below the N\\'{e}el temperature is taken as evidence for non-collinear, possibly helicoidal, spin ordering in the 1D chains. CuCrO$_4$ therefore represents a new interesting example for an unusual type-II multiferroicity system.\nNeutron scattering investigations are scheduled to clarify the exact nature of the magnetic ground state of CuCrO$_4$.\n\n\n\\begin{acknowledgments}\nThe Authors would like to thank S. H\\\"ohn, E. Br$\\ddot{\\rm u}$cher and G. Siegle for experimental assistance and T. Dahmen for the sample preparation. Work at NCSU by the Office of Basic Energy Sciences, Division of Materials Sciences, U. S. Department of Energy, under Grant DE-FG02-86ER45259, and also by the computing resources of the NERSC center and the HPC center of NCSU.\n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe recent observation in \\cite{Belkin&al} of double phase slips \nin superconducting nanowires and the interpretation of these there \nas paired quantum phase slips have brought to the fore the question of to \nwhat extent a nanowire can\nbe modeled as a lumped superconducting element, i.e., described by a single phase\nvariable, similarly to a Josephson junction (JJ). In \\cite{Belkin&al}, \nsuch a description has been \nfound to provide good fits to the experimental data. Moreover, in that work, as well as\nin some of the earlier experiments \\cite{Li&al,Aref&al}, some of the behavior\ncharacteristic\nof a JJ has been identified in nanowires also for the classical, thermally activated \nvariety of phase slips. This concerns specifically the \nscaling law followed by the activation barrier at currents near the critical.\n\n\nRecall that, for a JJ with critical current $I_c$, under biasing current \n$I_b$, the activation barrier scales as\n\\begin{equation}\n\\Delta E = \\mbox{const} \\times (1 - \\bm{i}_b)^{3\/2} \n\\label{3\/2}\n\\end{equation}\nas $\\bm{i}_b = I_b\/I_c \\to 1$ \\cite{Fulton&Dunkleberger}. \nThis scaling is different from the one obtained \nfor a nanowire (an extended one-dimensional superconductor) in the framework of the\nGinzburg-Landau (GL) theory. In that case, the barrier is\ndetermined by the\nenergy of the Langer-Ambegaokar (LA) \\cite{LA} saddle-point and scales as\n$(1 - \\bm{i}_b)^{5\/4}$ \\cite{Tinkham&al}. Somewhat surprisingly,\nas a description of the experimental data\non nanowires, the power-3\/2 scaling characteristic of a JJ typically works quite \nwell \\cite{Belkin&al,Li&al,Aref&al}, although there is one type of wires \n(the crystalline wires\nof Ref.~\\cite{Aref&al}) where the power-5\/4 scaling has been found to work better.\n\nOf course, a priori there is no reason to expect that the GL theory will work at\ntemperatures where (\\ref{3\/2}) is typically\nobserved (which are well below the critical $T_c$),\nso one may take the abovementioned experimental results simply as an indication \nthat one should use a different model of nanowire.\\footnote{\nIn this respect, it may be significant that the crystalline wires of Ref.\n\\cite{Aref&al}---the one case where the power-5\/4 is found to work better---have \nlower values of $T_c$ than their amorphous counterparts.} \nOne alternative\nis to consider the wire as a discrete set of nodes connected by\nsuperconducting links, with a phase variable defined on each node.\nThe distance between the nodes plays the role of the ultraviolet cutoff and\ncorresponds to the ``size of a Cooper pair.'' Phase slips whose cores are\nof that size are not seen in the standard GL theory (which treats the pairs as\npointlike) and thus represent an alternative channel of supercurrent decay.\n\nA discrete model of superconductor has been used, \nfor instance, in a study \\cite{Matveev&al}\nof the current-phase relation (CPR) of superconducting nanorings.\nVarious point of similarity\nwith the results of the GL theory have been noted.\nThe CPR, however, is primarily a low-current effect \n(unless, that is, one starts looking at higher bands in\nthe CPR band structure). In contrast, here\nour focus is on the novel features that\na discrete model (similar but not identical to that in \\cite{Matveev&al})\ncan bring in at currents close to the critical (depairing) current $I_c$.\n\nAt first glance, it seems easy to explain (\\ref{3\/2}) in a discrete model. \nFor example, let the supercurrent through each link be\na sinusoidal function of\nthe phase difference\n$\\theta_{j+1} - \\theta_j$ (where $j$ labels the nodes). The metastable \nground state at a biasing current $I_b < I_c$ corresponds \nto all $\\theta_{j+1} - \\theta_j = \\vartheta_{gs}$,\nwhere $\\vartheta_{gs}$ is the smaller root of\n\\begin{equation}\n\\sin \\vartheta_{gs} = \\bm{i}_b\n\\label{sine_eq}\n\\end{equation}\non the interval $[0,\\pi]$. \nConsider now the configuration in which \nthe phase difference on one of the links is instead the larger root of \n(\\ref{sine_eq}), namely, $\\pi - \\vartheta_{gs}$. \nIt seems reasonable to assume that this configuration is \nthe ``critical droplet''---the saddle point \nwhose energy determines the height of the activation \nbarrier. Since, compared to the ground state, the phase difference changes on\nonly one link, that height is exactly the same as for a single junction and, in\nparticular, scales according to (\\ref{3\/2}).\n\nA potential difficulty with this explanation becomes apparent if we observe that\nthe activation process envisioned above requires a change in the total phase\ndifference between the ends of the wire: if there is a total of $N$ links, this \nphase difference equals $N \\vartheta_{gs}$ for the ground state and \n\\[\n\\pi - \\vartheta_{gs} + (N - 1) \\vartheta_{gs} > N \\vartheta_{gs} \n\\]\nfor the purported critical droplet. Consider a very long wire or a short wire connected to\nbulk superconducting leads. In either case, one expects that the phases at the ends\nwill not be able to react\ninstantaneously to whatever changes occur in the middle; there will be a delay. \nIn the limit of a large delay, the correct boundary conditions\nare that the phases at the ends do not change at all during the\nactivation process and can only change during the subsequent real-time evolution.\n(This is similar to how, in a first-order phase transition, \na bubble of the new phase nucleates locally and then expands to fill the entire \nsample.) \n\nFor a long wire, one may suspect that the change in the saddle-point energy brought\nabout by the boundary conditions is slight. To argue that more rigorously, one may allude\nto the ``theorem of small increments'' \\cite{LL:Stat}\n(which relates the changes of different thermodynamic\npotentials under small perturbations), as has been done by McCumber\n\\cite{McCumber} for the case of the LA saddle point in the GL theory.\\footnote{Incidentally, \nfor currents near the critical, this argument\n(based on the premise that a localized nucleation event can be considered as a small \nperturbation)\nappears more straightforward in the discrete model than in the original GL case. \nThis is because in the former case the spatial size \nof the saddle point remains fixed in the limit $\\bm{i}_b \\to 1$, while in the latter \nit grows as $(1- \\bm{i}_b)^{-1\/4}$.} On the other hand, for a short wire (connected to \nbulk superconducting leads), there is no\nreason to expect that the role of boundary conditions will be small. On general grounds,\none may expect that the activation barrier in this case will be higher than for a long \nwire, reflecting the tendency of superconductivity to be more robust in the presence of\nthe leads. The question, however, is whether this tendency will lead only to a larger value\nof the constant in (\\ref{3\/2}), or it is capable of modifying the scaling exponent itself. \n\nIn the present paper, we would like to answer this question. We would also like to \nunderstand how the transition from thermal activation to tunneling, as the main mechanism\nof phase slips, occurs as the temperature is lowered. \nWe describe the discrete model used in this paper in more detail in Sec.~\\ref{sec:model}. \nWe show (in Sec.~\\ref{sec:static}) that, in this model,\nthe scaling law (\\ref{3\/2}) holds at least as long as the number $N$ of the links satisfies\n$N > 4$,\ndespite the presence of boundary conditions\nthat suppress fluctuations at the ends. (Smaller values of $N$ constitute special cases, \nwhich we have not studied in detail.)\n\nWe then consider (in Secs.~\\ref{sec:cross} and \\ref{sec:comp}) the \ncrossover to tunneling. In the semiclassical approximation, \nthe rate of tunneling is determined\nby classical solutions (instantons) that depend nontrivially on the Euclidean time\n$\\tau$ and are periodic in $\\tau$ with period $\\beta = \\hbar \/ T$, where $T$ is the\ntemperature. We expect the semiclassical approximation to apply when \nthe instanton action is \nlarge. Following \\cite{periodic}, we consider solutions (periodic instantons) \nthat have two turning points---states where all canonical momenta vanish \nsimultaneously---one at $\\tau = 0$ and the other at $\\tau = \\beta\/2$. These states can\nbe interpreted as the initial and final states of tunneling. We search for periodic\ninstantons numerically and find that they \nexist at any temperature below a certain crossover temperature $T_q$. The latter \nscales near the critical current according to\n\\[\nT_q = \\mbox{const} \\times (1 - \\bm{i}_b)^{-1\/4} \\, .\n\\] \nAt any $T < T_q$, the action of the periodic instanton is smaller than the activation\nexponent $\\Delta E \/ T$, which makes tunneling the main mechanism of phase slips at \nthese temperatures.\nAt $T=0$, the instanton action in the discrete model scales at $\\bm{i}_b \\to 1$ as\n\\[\nS_{inst}(T=0) = \\mbox{const} \\times (1 - \\bm{i}_b)^{5\/4} \\, .\n\\]\nNote that this scaling is different from the scaling law (\\ref{3\/2}) for the activation\nbarrier. The difference may be attributed to the ``critical slowing down'' of the \nEuclidean dynamics at currents near the critical.\n\n\n\\section{The discrete model} \\label{sec:model}\nWe consider a finite chain of nodes, labeled by $j=0,\\dots, N$, having\ncoordinates $x_j$ along a line in the physical space\nand ordered according to $x_j < x_{j+1}$. \nOn each node there is a phase variable, $\\theta_j$,\ninterpreted as the phase of the\nsuperconducting order parameter at the corresponding point in the wire. We assume that\nthe nodes are equally spaced:\n\\[\nx_{j+1} - x_j = \\Delta x\n\\]\nfor all $j = 0, \\dots, N-1$.\n\nAs we will see, solutions corresponding to phase slips in the present model involve\nchanges of $\\theta_j$ over a variety of spatial scales, including significant changes\non the scale $\\Delta x$ (i.e., from one node to the next). Clearly, then,\nin application to nanowires, results obtained with the help of this model\ncan be at best semiquantitative. Our main interest here is \nnot so much in precise quantitative detail (although we will present estimates for\nvarious quantities as we go) as in the scaling laws for rate\nexponents at currents close to the critical. One may hope those to be\nto some degree universal.\n\nTo estimate $\\Delta x$ (and so also the number of nodes needed to describe\nexperimentally relevant wire lengths), we \ninterpret the ratio $(\\theta_{j+1} - \\theta_j)\/\\Delta x$ as the gradient of the\nphase of the order parameter, i.e. (up to a factor of $\\hbar$), \nthe momentum of a Cooper pair in the link\n$(j,j+1)$. The value of this ratio corresponding to the maximum (critical)\ncurrent will be the critical momentum. To find it in the present model, we \nneed an expression for the\nsupercurrent $I_{j,j+1}$ in the link as a function\nof $\\theta_{j+1} - \\theta_j$. In the main text, we use the simplest $2\\pi$-periodic\nexpression, the sinusoidal\n\\begin{equation}\nI_{j,j+1} = I_c \\sin(\\theta_{j+1} - \\theta_j) \\, ,\n\\label{cur}\n\\end{equation}\nbut this choice is more or less arbitrary. As shown in the Appendix,\nmany results, including the scaling law (\\ref{3\/2}), generalize to \na much broader class of $2\\pi$-periodic functions. \n\nThe choice (\\ref{cur}), together with our later choice of the kinetic term \nfor $\\theta_j$, \nmakes our system equivalent to a particular model of a chain of Josephson \njunctions, the ``self-charging'' model of Ref.~\\cite{Bradley&Doniach}, where\nthe dynamics of this model has been considered in the limit of large $N$ and zero\ncurrent. Note that, \nto describe a uniform wire, we have taken the critical current \n$I_c > 0$ to be the same for all the links. \n\nThe current (\\ref{cur}) reaches maximum when the phase difference equals $\\pi \/ 2$,\nso the critical momentum, as defined above, is\n$p_c = \\pi \\hbar \/ (2 \\Delta x)$. If the wire were in the clean limit, we could \nestimate $\\Delta x$ by comparing this to the expression from the microscopic theory,\n$p_c = \\Delta_0 \/ v_F$,\nwhere $\\Delta_0$ is the gap at $T=0$, and $v_F$ is the Fermi velocity. This\nmicroscopic formula relates $p_c$ to the distance that an electron in a clean sample\nwould travel \nduring the time $\\hbar \/ \\Delta_0$ (up to a numerical factor of order one,\nthat distance coincides with Pippard's coherence length $\\xi_0$).\nAs nanowires are more \nappropriately described by the dirty limit, we replace that distance with the one that\nan electron, now in the presence of disorder, will diffuse over the same timescale:\n\\[\n\\xi_D = \\left( \\frac{\\hbar D}{\\Delta_0} \\right)^{1\/2} \\, ,\n\\]\nwhere $D$ is the diffusion coefficient. We then obtain\n\\[\n\\Delta x = \\frac{\\pi \\xi_D}{2} \\, .\n\\]\nTaking for estimates $D = 1.2 \\times 10^{-4}$ m$^2$\/s, as appropriate for amorphous\nMoGe \\cite{Bezryadin:book}, and $\\Delta_0 = 0.76$ meV (corresponding to $T_c = 5$ K), \nwe find $\\xi_D = 10$ nm and $\\Delta x = 16$ nm. Thus,\na 150 nm long wire corresponds to $N \\sim 10$.\n\nNote that the physical meaning of the distance $\\xi_D$ is that of the ``size of a Cooper\npair.'' As such, $\\xi_D$ is physically distinct from the GL coherence length $\\xi_{GL}$\nand, indeed, will not be even seen in the standard GL theory (which treats the pairs as\npoint-like). In this respect, a phase slip here, which takes place on the scale $\\Delta x$,\nand the one mediated by the LA saddle point \\cite{LA} of the GL theory \nrepresent two different channels of\nsupercurrent decay. In this paper, we consider only temperatures significantly below\nthe critical and will not ask how the effective GL description at $T$ close to\n$T_c$ arises in the discrete model.\n\nNext, we formulate the equations of motion for the phases $\\theta_j$. We take the dynamics to\nbe entirely Lagrangian at the interior nodes \nbut to have a dissipative component at the ends. The Lagrangian is\n\\begin{equation}\nL = \\frac{1}{2} \\sum_{j = 0}^N C_j (\\partial_t \\theta_j)^2 - U \\, ,\n\\label{L}\n\\end{equation}\nwhere the potential energy is\n\\begin{equation}\nU = - I_c \\sum_{j=0}^{N-1} \\cos (\\theta_{j+1} - \\theta_j) - I_b (\\theta_N - \\theta_0) \\, .\n\\label{pot_ene}\n\\end{equation}\nHere and in what follows, we use the system of units in which $\\hbar$\nand the charge of a pair are\nset equal to 1: $\\hbar = 1$, $2 e = 1$. Powers of $2e$ can be restored in the final\nanswers via the replacements \n\\begin{equation}\nI_c \\to I_c \/ 2e \\, , \\hspace{3em} C_j \\to C_j \/ (2e)^2 \\, .\n\\label{phys_units}\n\\end{equation}\nThe kinetic term in (\\ref{L}) corresponds to all nodes having finite capacitances, $C_j$,\nto nearby conductors; these capacitances are assumed to the much larger than those\nbetween the nodes. In this respect, the present model is different from that used in\n\\cite{Matveev&al}. \n\nIn numerical computations, we will, for definiteness, assume\nthat all the $C_j$ are equal, except possibly for the two nodes at the ends. \nThe static solutions, described in the next section, \nare insensitive to this assumption. The numerical method used to obtain \nthe non-static solutions could be used also if the capacitances were distinct.\n\nIn (\\ref{pot_ene}), the cosine term corresponds to our choice of the sinusoidal\nexpression (\\ref{cur}) for\nthe current. The second, non-periodic term reflects the fact that we are considering the \nwire at a fixed biasing current, $I_b$. \nFormally, this term can be seen as a result of the Legendre \ntransform with respect to $\\phi = \\theta_N - \\theta_0$. Physically, it represents the work done by an external\nbattery to replenish the current back to the bias value. Note that the value of this term changes \n(i.e., the work done is nonzero) only when there is a change in $\\phi$, the total phase accumulation\nalong the wire.\n\nThe equation of motion at the interior points is derived from the Lagrangian and reads\n\\begin{equation}\nC_j \\ddot{\\theta}_j = I_c \\sin (\\theta_{j+1} - \\theta_j) - I_c \\sin (\\theta_j - \\theta_{j-1}) \\, ,\n\\label{eqm}\n\\end{equation}\n$j=1,\\dots,N-1$. At the endpoints, $j = 0$ and $N$, we include\ndissipative dynamics intended primarily\nto model suppression of quantum fluctuations by the bulk superconducting leads.\nIt also provides a mechanism for relaxation of the supercurrent back to the bias\nvalue after thermal activation or tunneling.\nWe describe it by two small impedances, $R_0$ and $R_N$, \nshunting the ends of the wire to the ground. In classical theory, their effect\nis represented by dissipative terms added to the equations of motion, as follows:\n\\begin{eqnarray}\nC_0 \\ddot{\\theta}_0 & = & I_c \\sin(\\theta_1 - \\theta_0) - I_b - R_0^{-1} \\dot{\\theta}_0 \\, \n\\label{eqm_0} \\\\\nC_N \\ddot{\\theta}_N & = & - I_c \\sin(\\theta_N - \\theta_{N-1}) + I_b - R_N^{-1} \\dot{\\theta}_N \\, .\n\\label{eqm_N}\n\\end{eqnarray}\nThe effect of the impedances {\\em during} tunneling is represented by a \nCaldeira-Leggett term \\cite{Caldeira&Leggett} in the Euclidean action. We do not\nwrite this term explicitly here but merely assume that \n$R_0$ and $R_N$ are small enough for\nit to suppress variations of the phase at the endpoints down to negligible\nvalues.\n\n\\section{Static solutions} \\label{sec:static}\nFor time-independent (static) solutions, the equations of motion become\n\\begin{equation}\n\\sin (\\theta_{j+1} - \\theta_j) - \\sin (\\theta_j - \\theta_{j-1}) = 0\n\\label{eqm_static}\n\\end{equation}\nat the interior points, and\n\\begin{eqnarray}\n\\sin (\\theta_1 - \\theta_0) - \\bm{i}_b & = & 0 \\, , \\label{sin_bc1} \\\\\n- \\sin (\\theta_N - \\theta_{N-1}) + \\bm{i}_b & = & 0 \\label{sin_bc2} \n\\end{eqnarray}\nat the ends. Here\n\\[\n\\bm{i}_b = I_b \/ I_c \\, ,\n\\]\nwhich without loss of generality can be assumed non-negative. Thus, it is \nin the range $0\\leq \\bm{i}_b \\leq 1$.\n\nIn this section, we consider three types of solutions to these equations. The first type\nis the ground states, one for each value of\n\\begin{equation}\n\\vartheta_{gs} = \\arcsin \\bm{i}_b \\, .\n\\label{vgs}\n\\end{equation}\nAs we will see, these ground states are stable against small fluctuations\nfor all $0\\leq \\vartheta_{gs} < \\pi\/2$.\nOn the other hand, for any $\\bm{i}_b > 0$ the potential (\\ref{pot_ene}) is unbounded from\nbelow, so these states are not absolutely stable but only metastable, i.e., \nsubject to decay via large fluctuations, either\nthermal or quantum. \n\nThe other two types of solutions considered in this section correspond to certain\nintermediate states in the decay of the (metastable) ground states. We view such a \ndecay, whether it is classical or quantum, as a two-stage process, where at the first stage\nthe system reaches an intermediate state, which sits either at \nthe top of the potential barrier (in the classical case) or on the other side of it (in\nthe quantum case). Crucially, we assume that the boundary values of $\\theta_j$ in this intermediate \nstate are the same as they were in the ground state. \nThis is consistent with our earlier discussion of how\nthe correct boundary conditions during the activation process must reflect \nthe role of the superconducting leads. \nThe current in such an intermediate state is different from the biasing current. At the second \nstage, the system slowly \nadjusts the boundary values of the phase to return the current back to the bias value.\nNote that, since any resistive effect of a phase slip requires a change\nin $\\phi = \\theta_N - \\theta_0$, all such effects are relegated to the second stage.\n\nIn the two-stage picture, the configurations corresponding to the intermediate states need\nto solve only the interior equations (\\ref{eqm_static}) and not the boundary equations\n(\\ref{sin_bc1})--(\\ref{sin_bc2}). Instead, the boundary values of the phase in these states\nare determined\nby the boundary conditions\n\\begin{eqnarray}\n\\theta_0 & = & 0 \\, , \\label{theta_bc1} \\\\\n\\theta_N & = & \\vartheta_{gs} N \\label{theta_bc2}\n\\end{eqnarray}\n(the phase $\\theta_0$ can be arbitrarily set to zero, because the static equations involve\nonly the phase differences).\n\nInstead of the phase variable $\\theta_j$, we will often use the ``reduced'' variable $\\tilde{\\theta}_j$, \nwith a linear growth subtracted away as follows:\n\\begin{equation}\n\\tilde{\\theta}_j = \\theta_j - \\vartheta_{gs} j \\, .\n\\label{tth}\n\\end{equation}\nFor this, the boundary conditions (\\ref{theta_bc1})--(\\ref{theta_bc2}) become simply \n\\begin{equation}\n\\tilde{\\theta}_0 = \\tilde{\\theta}_N = 0 \\, .\n\\label{bc_tth}\n\\end{equation}\nAn example of all three types of solutions (a ground state and the two\nstates mediating its decay) is shown in Fig.~\\ref{fig:gs}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.25in]{gs.eps}\n\\end{center} \n\\caption{{\\small Different configurations of the reduced phase (\\ref{tth}) \nthat solve the static equations of motion (\\ref{eqm_static}) at the interior points. \nThe ground state is $\\tilde{\\theta}_j \\equiv 0$; it solves also the static equations\n(\\ref{sin_bc1})--(\\ref{sin_bc2}) at the ends. Short dashes represent the saddle point\n(critical droplet) that sits at the top of the potential barrier responsible for metastability of \nthe ground state. The solid polyline is a state on the other side of the barrier; it is\nreferred to as a state with a $2\\pi$ jump. Thermal decay corresponds to a transition\nthrough a vicinity of the critical droplet, and tunneling {\\em at low currents}\nto transition from the ground state to the state with a $2\\pi$ jump. \n(For tunneling at a high current, the final state needs to be found by the method described\nin Sec.~\\ref{sec:comp}.)\nThe state with a $2\\pi$ jump does not solve the static equations at the ends, and so\nwill be subject to forces there. Those will induce its evolution, over a longer timescale,\ntowards the state shown by the long dashes (the initial direction of the evolution is\nshown by the arrows).}\n} \n\\label{fig:gs} \n\\end{figure}\n\nFor the analysis of linear stability of these various solutions, we will need \nthe Hessian---the matrix of second derivatives of the potential energy (\\ref{pot_ene}).\nIt is a symmetric tridiagonal matrix, which we prefer to write in units of $I_c$:\n\\begin{equation}\nM_{jj'} = I_c^{-1} \\frac{\\partial^2 U}{\\partial \\theta_j \\partial \\theta_{j'}} \\, .\n\\label{sec_var}\n\\end{equation}\nConsistent with the boundary conditions (\\ref{theta_bc1})--(\\ref{theta_bc2}), we restrict\nvariations of $\\theta_j$ to be supported at the interior points only; thus, in (\\ref{sec_var}), \n$j,j' = 1, \\dots, N-1$. Then, the diagonal elements of the Hessian are\n\\begin{equation}\nM_{jj} = \\cos (\\theta_{j+1} - \\theta_j) + \\cos (\\theta_j - \\theta_{j-1}) \\, ,\n\\label{diag}\n\\end{equation}\nand the off-diagonal ones are\n\\begin{equation}\nM_{j,j+1} = M_{j+1,j} = - \\cos(\\theta_{j+1} - \\theta_j) \\, .\n\\label{off-diag}\n\\end{equation}\nWe now consider the different types of solutions in turn.\n\n\\subsection{Ground states}\n\\label{subsec:ground}\n\nThe simplest solution to (\\ref{eqm_static}) is the one where all the phase differences are\nequal, i.e., $\\theta_j$ grows linearly with $j$ with some slope $\\vartheta_{gs}$:\n\\begin{equation}\n\\theta_j = \\vartheta_{gs} j \\, .\n\\label{gs}\n\\end{equation}\nThis solves also the boundary equations (\\ref{sin_bc1})--(\\ref{sin_bc2}) provided the slope\nsatisfies\n\\begin{equation}\n\\sin \\vartheta_{gs} = \\bm{i}_b \\, .\n\\label{sin_vgs}\n\\end{equation}\nAs already done previously, for $\\bm{i}_b < 1$,\nwe take $\\vartheta_{gs}$ to be the smaller root of (\\ref{sin_vgs}) \non $[0,\\pi]$. Eq.~(\\ref{gs}) would remain a solution if we were to replace $\\vartheta_{gs}$ with\nthe larger root, $\\pi - \\vartheta_{gs}$. As we will see, however, that latter solution is linearly\nunstable. \n\nWe refer to (\\ref{gs}) as the ground state (corresponding to a given $\\bm{i}_b$).\nAs noted earlier, the ground state is at best metastable (i.e., cannot\nbe absolutely stable) for any $\\vartheta_{gs} > 0$.\nIn terms of the reduced phase (\\ref{tth}), the ground state becomes simply \n\\[\n\\tilde{\\theta}_j \\equiv 0 \\, .\n\\]\nThus, in general, \nwe can think of the reduced phase as measuring fluctuations relative to the ground\nstate.\n\nTurning to analysis of linear stability, we note that all the cosines \nin the expressions (\\ref{diag})--(\\ref{off-diag})\nare now equal to $\\cos\\vartheta_{gs}$. The eigenvectors of the Hessian are then readily found: they\nare\n\\[\n\\alpha_j^{(p)} = \\sin (p j) \\, ,\n\\]\nwhere $p$ is a positive integer multiple of $\\pi \/ N$.\nThe corresponding eigenvalues are\n\\begin{equation}\n\\lambda_p = 2 (1 - \\cos p) \\cos\\vartheta_{gs} \\, .\n\\label{lambda_gs}\n\\end{equation} \nWe see that the ground-state is linearly stable \nfor $0 \\leq \\vartheta_{gs} < \\pi \/ 2$. The solution, obtained by replacing $\\vartheta_{gs}$ with the\nlarger root of (\\ref{sin_vgs}), i.e., $\\pi -\\vartheta_{gs}$, is linearly unstable. \nFor the critical $\\vartheta_{gs} = \\pi\/2$, the Hessian vanishes, and stability \nis determined by the leading non-linear term. That is cubic, and so the critical\nsolution is unstable.\n\n\\subsection{$2\\pi$ jumps}\n\\label{subsec:jumps}\nThe equality of the sines in (\\ref{eqm_static}) does not require equality of the arguments, \nand indeed (\\ref{eqm_static}) has\nsolutions for which the phase differences across the links \nare not all equal. The simplest case is when\nthey are all equal except for one, which differs from the rest by $2\\pi$. For the phase\nitself, we then have\ntwo segments of linear growth joined together by a jump of approximately $2\\pi$\n(cf. Fig.~\\ref{fig:gs}).\nIn terms of the reduced phase (\\ref{tth}), the solution satisfying the boundary \nconditions (\\ref{bc_tth}) is \n\\begin{eqnarray}\n\\tilde{\\theta}_j & = & -\\frac{2\\pi j}{N} \\, , \\hspace{5em} j = 0,\\dots,k \\, , \\nonumber \\\\\n\\tilde{\\theta}_j & = & -\\frac{2\\pi j}{N} + 2\\pi \\, , \\hspace{3em} j = k + 1,\\dots, N \\, . \\label{jump}\n\\end{eqnarray}\nThe jump occurs on the link from $j = k$ to $j = k + 1$, where $k$ can be any integer from\n0 to $N-1$. \nWe refer to it as a ``$2\\pi$ jump'' even though the actual difference\n\\[\n\\tilde{\\theta}_{k+1} - \\tilde{\\theta}_k = 2 \\pi (1 - 1\/N)\n\\]\nis somewhat smaller than $2\\pi$.\n\nAnalysis of linear stability is completely parallel to that for the ground state, with\nstability now determined by the sign of \n$\\cos(\\vartheta_{gs} - 2\\pi \/ N)$. For \n\\begin{equation}\nN > 4 \\, , \n\\label{N>4}\n\\end{equation}\nthe solution is linearly stable for all $0\\leq \\vartheta_{gs} < \\pi\/2$.\nSmaller values of $N$ constitute special cases. For $N =2$, \nthe solution is unstable for any of the above $\\vartheta_{gs}$. For $N = 3$, it is\nunstable for $\\vartheta_{gs} < \\pi \/ 6$, while for $N = 4$ it is stable as long as $\\vartheta_{gs} \\neq 0$ \nwith the\nHessian vanishing at $\\vartheta_{gs} = 0$. In what follows, we do not pursue detailed analysis \nof these special cases but simply assume that the condition (\\ref{N>4}) is satisfied.\n\nWhat is the relevance of these solutions to decay of supercurrent? Our numerical \nresults (a sample of which will be presented later) indicate that at temperatures\n$T\\to 0$ and {\\em low currents}, \nthe state with a $2\\pi$ jump is the final state of the tunneling process, by which\nthe system, originally in a vicinity of the \nground state, escapes through the potential barrier. The evolution after tunneling\n(the second stage in our two-stage description) \nproceeds in real time, according to the equations of motion\n(\\ref{eqm}) and (\\ref{eqm_0})--(\\ref{eqm_N}) with $\\tilde{\\theta}_j$ = (\\ref{jump}) and\n$\\partial_t \\tilde{\\theta}_j \\equiv 0$ as the initial conditions. \nNote that (\\ref{jump}) solves\nthe static equations at the interior points but not at the ends. Thus, the initial\ndirection of the evolution is determined by the net forces in the\nboundary equations\n(\\ref{eqm_0})--(\\ref{eqm_N}). The force at $j=0$ is equal to $I- I_b$, where\n\\[\nI = I_c \\sin (\\vartheta_{gs} - 2\\pi \/ N )\n\\]\nis the current on the solution, and $I_b = I_c \\sin \\vartheta_{gs}$ is the biasing current;\nthe force at $j=N$ is $I_b - I$.\nFor $N > 4$ and \n$0\\leq \\vartheta_{gs} < \\pi \/2$, the difference $I - I_b$ is negative. This means \nthat the initial direction of the real-time \nevolution is as indicated by arrows in Fig.~\\ref{fig:gs},\nthat is the system evolves\ntowards the state shown in Fig.~\\ref{fig:gs} by the long dashes. \n\nThe state shown by the long dashes has $\\tilde{\\theta}_j = c$ (a constant) for $j \\leq k$ and \n$\\tilde{\\theta}_j = c + 2\\pi$ for $j \\geq k +1$. \nBecause each $\\tilde{\\theta}_j$ is defined modulo $2\\pi$, this state differs from the ground\nstate $\\tilde{\\theta}_j \\equiv 0$ essentially by a constant shift of the phase. \nThe transition to it from\n$\\tilde{\\theta}_j \\equiv 0$, however, is observable as\nit generates a voltage pulse in the external circuit and\ninvolves work done by an external battery. This transition constitutes a phase slip.\nThe work done by the battery is given by the negative of the second, non-periodic\nterm in the potential energy (\\ref{pot_ene}). For instance, for the case\nshown in Fig.~\\ref{fig:gs}, it equals $2\\pi I_b$.\n\nWe wish to reiterate that this special role of (\\ref{jump}) as the final state of\ntunneling is characteristic of the low-current regime. At large currents, the final\nstates need to be found by the method described in Sec.~\\ref{sec:comp}.\n\n\\subsection{Saddle points (critical droplets)}\nOne more way to solve the sine equation (\\ref{eqm_static}) \nis to let all $\\theta_{j+1} - \\theta_j$ \nexcept one be equal, as in the case of a $2\\pi$ jump,\nbut let that one be $\\pi$ minus any other.\nThat is, if \n\\begin{equation}\n\\theta_{j+1} - \\theta_j = \\vartheta_{gs} + \\gamma\n\\label{lin}\n\\end{equation}\nfor $j \\neq k$ (where $\\gamma$ is an as yet undetermined constant), then\n\\begin{equation}\n\\theta_{k+1} - \\theta_k = \\pi - (\\vartheta_{gs} + \\gamma) \\, .\n\\label{pi_minus}\n\\end{equation}\nIn terms of the reduced phase (\\ref{tth}), the solution corresponding to (\\ref{lin}) and\nsatisfying the boundary conditions (\\ref{theta_bc1})--(\\ref{theta_bc2}) is\n\\begin{eqnarray}\n\\tilde{\\theta}_j & = & \\gamma j \\, , \\hspace{5em} j = 0,\\dots,k \\, , \\nonumber \\\\\n\\tilde{\\theta}_j & = & \\gamma (j - N) \\, , \\hspace{2em} j = k + 1,\\dots, N \\, . \\label{drop}\n\\end{eqnarray}\nEq.~(\\ref{pi_minus}) then becomes an equation for the slope $\\gamma$.\nIt has a solution,\n\\begin{equation}\n\\gamma = - \\frac{\\pi - 2 \\vartheta_{gs}}{N - 2} \n\\label{gamma}\n\\end{equation}\nfor any $N > 2$. Note that $\\gamma < 0$ for any $0\\leq \\vartheta_{gs} < \\pi\/2$ and is zero for the \ncritical $\\vartheta_{gs} = \\pi\/2$. In the latter case, the solution coincides\nwith the ground state.\n\nEven though the solution exists for any $N > 2$, we restrict our attention to cases\nwhen \n\\begin{equation}\n\\cos(\\gamma + \\vartheta_{gs}) = \\cos \\frac{N \\vartheta_{gs} - \\pi}{N - 2} > 0 \\, ,\n\\label{cos}\n\\end{equation}\nbecause then the solution has a special interpretation, discussed below. \nThis inequality is always satisfied\nfor $\\vartheta_{gs} < \\pi\/2$ and $N > 4$, a condition we have already imposed.\n\nWe will refer to the link form $j=k$ to $k+1$ as the ``core'' of the solution, even \nthough, as we will see shortly, the energy is by no means concentrated at the core:\nthe difference in the slope from the ground state is essential and leads to \na contribution\ndistributed over the entire length of the wire.\n\nThe solution is shown in Fig.~\\ref{fig:gs} by the short dashes. The way it appears\nthere implies\nthat the slope $|\\gamma|$ is smaller than $2\\pi \/ N$, the slope of the solution with\na $2\\pi$ jump. One can verify that the condition for that is precisely the same as\nthe condition of linear stability of the latter solution, derived previously. \n\nGiven that, as\nfar as the slopes go, the present solution lies between the ground state and the state\nwith a $2\\pi$ jump, one may expect that it is a saddle point that sits at the top of\nthe potential barrier separating the two states. We will now see that it is in fact\na {\\em critical droplet}---a saddle point that has exactly one negative mode and \nmediates thermally activated decay of the ground state. \n\nA negative mode is an eigenvector of the Hessian corresponding to a negative eigenvalue.\nThe cosines in eqs.~(\\ref{diag})--(\\ref{off-diag}) now are\n\\[\n\\cos(\\theta_{j+1} - \\theta_j) = \\left\\{ \\begin{array}{cc} \n\\cos(\\vartheta_{gs} + \\gamma) \\, , & j\\neq k \\, , \\\\\n-\\cos(\\vartheta_{gs} + \\gamma) \\, , & j = k \\, .\n\\end{array} \\right.\n\\]\nThus, the eigenvalue problem for the Hessian is a discrete version of one-dimensional \nquantum mechanics with a localized potential.\nA negative mode corresponds to a bound state. \nThe corresponding eigenvalue is of the form\n\\begin{equation}\n\\lambda_- = - \\Lambda \\cos(\\vartheta_{gs} + \\gamma) \\, ,\n\\label{lam_neg}\n\\end{equation}\nwhere $\\Lambda > 0$ depends only on $N$ and $k$. \n\nFor given $N$ and $k$, it is straightforward to find eigenvalues of the Hessian by \nnumerical diagonalization. We have done that for a few small values of $N$, to convince\nourselves that there is a unique negative mode in those cases. For $N \\gg 1$,\nit is possible to prove existence and uniqueness of a negative mode\nwithout resorting to numerics.\n\nAs a sample of these\nresults, consider the case when $N$ is odd and $k = \\frac{1}{2} (N - 1)$, meaning that the\ncore of the solution is directly in the middle of the wire. \nWe find, for instance, \n$\\Lambda = 1.303$ for $N = 5$ and $\\Lambda = 1.330$ for $N = 7$. The latter value is\nalready close to the asymptotic\n\\begin{equation}\n\\Lambda = \\frac{4}{3} \n\\label{Lam}\n\\end{equation}\nwhich corresponds to $N\\to \\infty$ and can be found analytically. \nThe eigenvector corresponding to the negative mode is\n\\begin{equation}\n\\alpha^{(-)}_j = \\left\\{ \\begin{array}{cc} \nA \\sinh (p j) \\, , & j\\leq \\frac{1}{2} (N -1) \\, , \\\\\n- A \\sinh [p (N - j)] \\, , & j\\geq \\frac{1}{2} (N + 1) \\, . \\end{array} \\right.\n\\label{neg_mode}\n\\end{equation}\nwhere $A$ is a normalization coefficient, and $p$ is related to the eigenvalue by\n\\[\n\\Lambda = 2 \\cosh p - 2 \\, .\n\\]\nFor $N \\to \\infty$, $p = \\ln 3$.\nMoving the core away from the middle causes $\\Lambda$ to decrease, but the negative \nmode persists for all $k$, even $k = 0$. In the latter case, the asymptotic value \nof at $N \\to \\infty$ is $p = \\ln 2$, that is $\\Lambda = \\frac{1}{2}$. \n\nNote that, unlike the droplet itself,\nwhich has linear ``tails'' extending to the ends\nof the wire (cf. Fig.~\\ref{fig:gs}), the negative mode is tightly localized: \nfor instance, $p = \\ln 3$ (in the $N\\to\\infty$ case) means that the \nmagnitude of (\\ref{neg_mode}) decreases by a factor of 3 per link\nas one moves away from the core. As a consequence, the asymptotic result \n(\\ref{Lam}), obtained for a droplet in the middle of a long\nwire, has exponential accuracy in $N$ and will hold well even for droplets\naway from the middle, except for very short wires or droplets very near the ends.\n\nWhen the droplet is strictly in the middle, \nthe negative mode is antisymmetric about\nthe core. This will hold approximately also away from the middle, except when\nthe droplet is close to one of the ends.\nThis antisymmetry has a simple interpretation: as clear from Fig.~\\ref{fig:gs},\nadding an antisymmetric \n$\\alpha^{(-)}_j$ with a small coefficient (i.e., moving along the negative mode)\nwill deform the droplet either towards the\nground state or towards the state with a $2\\pi$ jump,\nprecisely as expected \nof motion across the top of the potential barrier separating the two states.\n\n\\subsection{Activation barrier}\n\\label{subsec:act}\n\nThe energy of the critical droplet is obtained by substituting (\\ref{lin}) and\n(\\ref{pi_minus}) into (\\ref{pot_ene}). \nIn units of $I_c$, it equals\n\\begin{equation}\n{\\cal E}_{drop} \\equiv E_{drop} \/ I_c =\n- (N - 2) \\cos(\\gamma + \\vartheta_{gs}) - \\bm{i}_b (\\theta_N - \\theta_0) \\, .\n\\label{Edrop}\n\\end{equation}\nThis should be compared to the energy of the ground state,\n\\begin{equation}\n{\\cal E}_{gs} \\equiv E_{gs} \/ I_c = - N \\cos \\vartheta_{gs} - \\bm{i}_b (\\theta_N - \\theta_0) \\, .\n\\label{Egs}\n\\end{equation}\nThe difference between the two is the activation barrier for thermally activated phase \nslips (TAPS). Using (\\ref{cos}), we obtain\n\\begin{equation}\n{\\cal E}_{drop} - {\\cal E}_{gs} = N \\cos \\vartheta_{gs} - (N - 2) \\cos \\frac{N \\vartheta_{gs} - \\pi}{N - 2} \\, .\n\\label{Eact}\n\\end{equation}\nA curious property of (\\ref{Eact}) is that it is independent of $k$, the\nlocation of the droplet core. In a continuum model, that would imply existence of\na zero mode---a zero eigenvalue of the Hessian---associated with the translational \nsymmetry. In the\npresent case, there is no such mode, as the symmetry with respect to infinitesimal\ntranslations is broken by the lattice.\n\nAnother convenient expression for the activation energy is obtained by using,\ninstead of $\\vartheta_{gs}$, the parameter $\\epsilon$ defined by\n\\begin{equation}\n\\vartheta_{gs} = \\frac{\\pi}{2} - \\epsilon \\, .\n\\label{eps}\n\\end{equation}\nThen,\n\\begin{equation}\n{\\cal E}_{drop} - {\\cal E}_{gs} = N \\sin\\epsilon - (N - 2) \\sin \\frac{N \\epsilon}{N-2} \\, .\n\\label{Eact_eps}\n\\end{equation}\n\nEq.~(\\ref{Eact}) has two interesting limits. The first is $N \\gg 1$ with $\\vartheta_{gs}$ fixed.\nIn this case, \n\\begin{equation}\n{\\cal E}_{drop} - {\\cal E}_{gs} = 2 \\cos \\vartheta_{gs} - (\\pi - 2 \\vartheta_{gs}) \\sin \\vartheta_{gs} + O(1\/N) \\, .\n\\label{E_lim1}\n\\end{equation}\nWe see that the activation barrier remains finite in the limit of large length.\nThat does not\nmean, however, that the energy difference (\\ref{E_lim1}) is accumulated locally, around \nthe core of the droplet: the reduction (by the amount $|\\gamma|$) of the slope of \nthe droplet's ``tails'' relative to the ground state, cf. (\\ref{lin}), is essential\nand leads to a contribution distributed over the entire length of the wire.\n\nExpressing $\\vartheta_{gs}$ through the biasing current via (\\ref{sin_vgs}), we find that,\nin the limit $N\\to \\infty$, the activation barrier (\\ref{E_lim1}) coincides exactly\nwith that of a Josephson junction whose potential energy, as a function of the\nphase difference, is\n\\begin{equation}\nU(\\phi) = - I_c \\cos\\phi - I_b \\phi \\, .\n\\label{UJJ}\n\\end{equation}\nAs we show in the Appendix, this agreement is not limited to sinusoidal currents\nbut extends to a much broader class of current-phase relations.\nIt may not be obvious a priori, given especially that, in the wire, the activation process\ndoes not involve any changes of the phase difference between the ends, so in the computation\nof the activation barrier\nthe last terms in (\\ref{Edrop})\nand (\\ref{Egs}) simply cancel each other. In contrast, in the junction (where the ground state\nis $\\phi_{gs}= \\arcsin \\bm{i}_b$, and the critical\ndroplet is $\\phi_{drop} = \\pi - \\phi_{gs}$),\nthe contribution of the last term in (\\ref{UJJ}) is\nessential. Nevertheless, in the $N\\to \\infty$ limit, the agreement between the two cases\nis not entirely\nunexpected: it can be seen as an instance of the ``theorem of small\nincrements'' \\cite{LL:Stat}, as already noted by McCumber \\cite{McCumber} in the context of the\ncontinuous GL theory. \n\nThe second interesting limit of (\\ref{Eact}) is $\\vartheta_{gs} \\to \\pi \/ 2$ with $N$ fixed,\nwhich corresponds to currents close to the critical. In this case, it is convenient to\nuse the form (\\ref{Eact_eps}) of the activation energy: \nthe parameter $\\epsilon$ is now small, and we can expand (\\ref{Eact_eps}) in it. We obtain\n\\begin{equation}\n{\\cal E}_{drop} - {\\cal E}_{gs} = \\frac{2 N (N-1)\\epsilon^3}{3(N-2)^2} + O(\\epsilon^5) \\, .\n\\end{equation}\nIn terms of the biasing current,\n\\begin{equation}\n\\epsilon = \\sqrt{2} (1 - \\bm{i}_b)^{1\/2} + O[(1 - \\bm{i}_b)^{3\/2}] \\, .\n\\label{eps_scaling}\n\\end{equation}\nWe see that the activation barrier is higher for smaller $N$, as might be expected, but\nthe scaling at $\\bm{i}_b \\to 1$ is always the same \n$(1 -\\bm{i}_b)^{3\/2}$. This is the first of the results we have highlighted in the Introduction.\n\nFinally, we note that, for $N > 4$, \nthere is also a double droplet, a state where eq.~(\\ref{pi_minus}) \n(with a different value of $\\gamma$) is used on two links. Its energy,\nin the same notation as in (\\ref{Eact_eps}), is\n\\begin{equation}\n{\\cal E}_{double} - {\\cal E}_{gs} = N \\sin\\epsilon - (N - 4) \\sin \\frac{N \\epsilon}{N-4} \n\\end{equation}\nand is always larger than the energy of the single droplet.\n\n\\section{Euclidean solutions and the crossover temperature}\n\\label{sec:cross}\nThere can be no thermal activation at $T=0$, only quantum tunneling, so as the temperature\nis lowered past some point one mechanism of phase slips must give precedence to the other.\nIn the semiclassical approximation,\ntunneling is described by classical solutions (instantons)\nthat depend nontrivially on the Euclidean \ntime $\\tau = it$ and are periodic in $\\tau$ with period $\\beta = 1\/T$.\nFollowing \\cite{periodic}, we consider ``periodic instantons''---solutions with two turning \npoints (those are configurations where all the canonical momenta vanish simultaneously),\none at $\\tau = 0$, and the other at half the period, $\\tau = \\frac{1}{2} \\beta$.\nIn the present\ncase, the turning-point conditions are\n\\begin{equation}\n\\partial_\\tau \\tilde{\\theta}_j(0) = \\partial_\\tau \\tilde{\\theta}_j(\\beta \/2) = 0 \\, .\n\\label{bc_tau}\n\\end{equation}\nAs discussed in \\cite{periodic}, a periodic instanton \nis expected to saturate the microcanonical tunneling rate $\\Gamma_{micro}$,\ni.e., to give the most\nprobable tunneling path at a fixed energy $E$; the period of the instanton in that case\nis determined by the energy. The same instanton will then saturate also the canonical\n(fixed temperature) \ntunneling rate \n\\begin{equation}\n\\Gamma_{can}(\\beta) \\sim \\int dE e^{-\\beta E} \\Gamma_{micro}(E) \\, ,\n\\label{can}\n\\end{equation}\nprovided the integrand here is peaked about the corresponding energy.\nAs we discuss in more detail below, such is indeed the case in our present\nmodel.\n\nThe turning-point conditions (\\ref{bc_tau}),\ntogether with the spatial boundary conditions (\\ref{bc_tth}), define a boundary problem\nfor the rectangle $0 < j < N$, \n$0\\leq \\tau \\leq \\frac{1}{2} \\beta$.\nThe solution for $\\frac{1}{2} \\beta < \\tau \\leq \\beta$ is obtained via\n\\[\n\\tilde{\\theta}_j(\\tau) = \\tilde{\\theta}_j(\\beta - \\tau) \\, .\n\\]\nThe turning points $\\tilde{\\theta}_j(0)$ and $\\tilde{\\theta}_j(\\beta\/2)$ correspond, respectively, to the\ninitial and final states of tunneling, that is, the states in which the \nsystem enters and leaves the classically forbidden region of the configuration space.\nThe subsequent real-time evolution (the second stage in our two-stage description) \nproceeds with $\\tilde{\\theta}_j(\\beta\/2)$ as the initial state.\nThus, $\\tilde{\\theta}_j(\\beta\/2)$ must be real. Since, for $0 < j < N$, \nthe Euclidean equations of motion contain no complex coefficients, the entire $\\tilde{\\theta}_j(\\tau)$\nmust then be real, so we concentrate on real-valued solutions in what follows.\n\nAs shown in \\cite{periodic}, the main, exponential factor in the microcanonical \ntunneling rate is\ndetermined by the instanton's ``abbreviated'' (Maupertuis) action per period, $\\widetilde{S}(E)$,\nas follows:\n\\begin{equation}\n\\Gamma_{micro}(E) \\sim e^{-\\widetilde{S}(E)} \\, .\n\\label{micro}\n\\end{equation}\nSo, the conditions for the integrand in (\\ref{can}) to have a maximum at $E$ are\n\\begin{eqnarray}\n-d\\widetilde{S} \/ dE & = & \\beta \\, , \\label{dtS} \\\\\nd^2 \\widetilde{S} \/ dE^2 & > & 0 \\, . \\label{d2tS}\n\\end{eqnarray}\nIf these are satisfied, \n\\begin{equation}\n\\Gamma_{can}(\\beta) \\sim e^{-S_{inst}(\\beta)} \\, ,\n\\label{can_exp}\n\\end{equation}\nwhere\n\\begin{equation}\nS_{inst}(\\beta) = \\beta E + \\widetilde{S}(E) \\, .\n\\label{Sinst}\n\\end{equation}\nNote that the normalization of (\\ref{can}) assumes that the ground state energy is zero; \nthus, $E$ in (\\ref{Sinst}) is the energy of either turning point (by the\nEuclidean energy conservation, their energies are equal) relative to the ground\nstate. Also, it is understood that $E$ is expressed through $\\beta$ by means of (\\ref{dtS}).\nOne result of that is the relation \\cite{periodic}\n\\begin{equation}\ndS_{inst}\/d\\beta = E \\, .\n\\label{dSdb}\n\\end{equation}\n\nBy a standard theorem of mechanics \\cite{LL:Mech}\n(translated to the Euclidean time), the right-hand side of (\\ref{dtS}) is the period of the \nsolution, so (\\ref{dtS}) tells us that the period is the same as $\\beta$. This is not\nunexpected but does go to show formally \nthat (\\ref{Sinst}) is the full (non-``abbreviated'') action \nof the instanton, relative to that of the ground state.\nNote that the exponential factor (\\ref{can_exp}) accounts both for the suppression of \nthe rate due to the tunneling per se (as represented by $\\widetilde{S}$) and for that caused by \nthe need to populate an initial state of energy $E$.\n\nThe inequality (\\ref{d2tS}) is less trivial, in the sense that it may or may not be\nsatisfied in a given system for a given range of energies.\nAt small $E$, if the zero-energy instanton is known, one can construct an approximate \nperiodic instanton\nby alternating instantons and anti-instantons in the $\\tau$ direction \n\\cite{periodic} and check the condition (\\ref{d2tS}) for it. \nFor larger $E$, however, one typically has to resort\nto numerical studies. For the latter, the condition (\\ref{d2tS}) \ncan be somewhat more conveniently rewritten as\n\\begin{equation}\n\\frac{d^2 S_{inst}}{d\\beta^2} < 0 \\, .\n\\label{d2S}\n\\end{equation}\n(By virtue of the relation between $E$ and $\\beta$, the left-hand side here is \nthe negative of \n$(d^2 \\widetilde{S} \/ d E^2)^{-1}$.)\nCases of varying complexity in the behavior of $S_{inst}(\\beta)$ have \nbeen described in the literature (as briefly surveyed below), \nand we now discuss the specifics of the present case.\n\nIn general,\nthe critical droplet undergoes a bifurcation at some $\\beta = \\beta_{bif}$, where it develops, \nin addition to its single $\\tau$-independent negative mode, another, $\\tau$-dependent one.\nThe key distinction between different cases is in where that new negative mode \nleads at $\\beta$ just above $\\beta_{bif}$: it can lead\nto a limit cycle (a periodic instanton) in a vicinity of the critical droplet or, alternatively,\nto an entirely different region of the configuration space. This distinction is analogous\nto the one between the supercritical and subcritical forms of the usual Hopf bifurcation.\nFor the case at hand, we find\n(numerically) that the former case (a nearby limit cycle)\nis realized. In addition, we find that a real-valued periodic instanton exists only\nfor $\\beta > \\beta_{bif}$, and for all these $\\beta$ \nthe condition (\\ref{d2S}) is satisfied. \nIn these respects, the present system is similar to the \n2-dimensional Abelian Higgs model; the periodic instanton for that case was found \nnumerically in \n\\cite{Matveev}. The behavior is different from that in the various versions of the \n2-dimensional $O(3)$ sigma model or in the 4-dimensional SU(2) Yang-Mills-Higgs theory; \nnumerical solutions for those cases were found, respectively, \nin \\cite{Habib&al,Kuznetsov&Tinyakov} and \n\\cite{Frost&Yaffe:1998,Frost&Yaffe:1999,Bonini&al}.\n\nAt the bifurcation point, the condition (\\ref{dSdb}) (recall that $E$ in it is measured from \nthe ground state) implies \n\\begin{equation}\n \\frac{dS_{inst}}{d\\beta} (\\beta_{bif}) = E_{drop} - E_{gs} \\, .\n\\label{slope}\n\\end{equation}\nThe right-hand side here is the activation barrier of the preceding section.\nIn other words, at this point, the $S_{inst}(\\beta)$ curve touches the straight line\n\\begin{equation}\nS_{drop}(\\beta) = (E_{drop} - E_{gs}) \\beta \\, ,\n\\label{Sdrop}\n\\end{equation}\ncorresponding to the thermal activation exponent. \nThe condition (\\ref{d2S}) being satisfied for\nall $\\beta > \\beta_{bif}$ then implies that $S_{inst}$ deviates down\nfrom (\\ref{Sdrop}), so that, at least in the leading semiclassical approximation \n(where only the exponential factors count), tunneling has a larger rate than thermal \nactivation. Thus, in the present case, $\\beta_{bif}$ is the same as $\\beta_q$, the point\nwhere one mechanism of phase slips overtakes the other.\nNote that the crossover temperature $T_q = 1\/ \\beta_q$\ncan be measured experimentally \n\\cite{Li&al,Aref&al,Belkin&al}.\n\nComputation of $\\beta_{bif}$ and so, in the present case, also of $T_q$ is standard. For \nthe Lagrangian (\\ref{L}), the $\\tau$-dependent normal modes are\neigenfunctions of the operator\n\\begin{equation}\n\\hat{N}_{ij} = - C_i \\delta_{ij} \\partial_\\tau^2 + I_c M_{ij} \\, ,\n\\label{Nij}\n\\end{equation}\nwhere $i,j = 1,\\dots,N-1$, and $M_{ij}$ is the Hessian (\\ref{sec_var}). As already mentioned,\nin this paper, we set all the capacitances $C_j$ at the interior\npoints equal, $C_j = C$. Then, the eigenfunctions of (\\ref{Nij}) are of the form\n\\[\n\\psi_j^{(n)}(\\tau) = \\alpha_j \\cos(2\\pi T n \\tau) \\, ,\n\\]\nwhere $\\alpha_j$ is an eigenvector of the Hessian, and $n \\geq 0$ is an integer. The choice\nof the cosine, rather than sine, here respects the boundary conditions (\\ref{bc_tau}). \nIn this way, for each eigenvalue $\\lambda$ of the Hessian, the $\\tau$-dependent problem\ngenerates an infinite sequence of eigenvalues:\n\\begin{equation}\n\\lambda \\to \\lambda + \\left( \\frac{2\\pi T n}{\\omega_c} \\right)^2 \\, ,\n\\label{time-dep}\n\\end{equation}\nwhere\n\\begin{equation}\n\\omega_c = (I_c \/ C)^{1\/2} \\to (2e I_c \/ C)^{1\/2} \\, .\n\\label{omega_c}\n\\end{equation}\nIn the last relation,\nthe arrow signifies transition to the physical units via (\\ref{phys_units}).\nFor $\\lambda > 0$, (\\ref{time-dep}) can never produce a negative eigenvalue.\nFor $\\lambda = \\lambda_- < 0$, there is always the original one, corresponding to\n$n = 0$. Another one appears at\n\\begin{equation}\nT < T_q = \\frac{\\omega_c |\\lambda_-|^{1\/2}}{2\\pi} \\, ,\n\\label{T_q}\n\\end{equation}\nwhich determines the crossover temperature. \n\nRecall that $\\lambda_-$ depends not\nonly on $N$ and the biasing current $\\bm{i}_b$, \nwhich are the parameters of the system, but also\non $k$, the location of the droplet along the wire. The meaning of $T_q$ is that of\nthe highest temperature for which tunneling is the main mechanism of phase slips, so\nin (\\ref{T_q}) we choose the largest $|\\lambda_-|$ we can get at given $N$ and $\\bm{i}_b$. \nAs we have seen in the preceding section, this corresponds\nto $k$ in the middle of the wire. With this choice, $\\lambda_-$ is given by \n(\\ref{lam_neg}) where $\\Lambda$ now depends only on $N$. \n\n\nBecause $T_q$ can be measured \nexperimentally, (\\ref{T_q}) can be used to estimate\n$\\omega_c$. Let us use for this estimate\nthe asymptotic large-$N$ value $\\Lambda = 4\/3$,\nwhich, as we have seen, becomes a good approximation already at modest $N$.\nSetting $N= 11$ and the biasing current \nto 90\\% of the critical current\nresults in $\\omega_c = 7.5~T_q$. Since $I_c$ is also measurable,\nwe can use (\\ref{omega_c}) to convert this estimate of $\\omega_c$ into an estimate of $C$.\nFor $I_c = 10~\\mu\\mbox{A}$ and $T_q =0.9~\\mbox{K}$ (corresponding to the values in the\nexperiment of \\cite{Belkin&al}), we obtain\n$C = 3.9 \\times 10^{-14}~\\mbox{F}$. Note that this is the capacitance of \na single segment of length $\\Delta x$, not of the entire wire. We attribute this\nrelatively large value of $C$ to the wire being in close proximity to large \nconductors (e.g., the center conductor strips \\cite{Belkin&al}).\n\nNext, consider the scaling of various quantities near the critical current.\nThe Hessian matrix of the critical droplet is proportional to $\\cos(\\gamma + \\vartheta_{gs})$.\nIn view of (\\ref{cos}), this means that \nall its eigenvalues, including $\\lambda_-$, scale at $\\bm{i}_b \\to 1$ as the first\npower of $\\epsilon= \\pi\/2-\\vartheta_{gs}$, i.e., as $(1- \\bm{i}_b)^{1\/2}$. \nThen, according to (\\ref{T_q}), $T_q$ scales\nas $(1 -\\bm{i}_b)^{1\/4}$. \nThis implies that the system stays classical longer (i.e., until a lower temperature)\nas the current gets closer to the critical. The power-$1\/4$ dependence, though, \nis quite weak,\nand it is not clear if this effect can be observed experimentally.\n\nThe Hessian of the ground state is proportional to $\\cos\\vartheta_{gs} = \\sin\\epsilon$ and so also \nscales as $\\epsilon$. Thus, at $T \\ll T_q$ the characteristic frequencies of Euclidean motion \nnear the ground state and near the top of the barrier are both of order \n$\\omega_0 \\sim \\omega_c \\sqrt{\\epsilon}$.\nThis suggests that, at these $T$, we can estimate the instanton action $S_{inst}$\nby taking the product of the characteristic barrier height,\n$I_c (E_{sph}- E_{gs})$, and the common timescale $2\\pi \/ \\omega_0$. The result is\n\\begin{equation}\nS_{inst} \\sim 2\\pi (I_c C)^{1\/2} (1 - \\bm{i}_b)^{5\/4} \\, .\n\\label{Sinst_est}\n\\end{equation}\nWe will see that the power-$5\/4$ scaling law for $S_{inst}$ is well borne out numerically.\nNote that, due to the ``critical slowing down'' of the Euclidean dynamics\nat $\\bm{i}_b \\to 1$, this scaling law is different from the one for the barrier height\nitself. This is the second of the results highlighted in the\nintroduction.\n\n\\section{Computation of the tunneling exponent}\n\\label{sec:comp}\nWe now turn to a systematic study of periodic instantons in our system.\nThe Euclidean equations of motion at the interior points, with all $C_j$ set\nequal to $C$, read\n\\begin{equation}\n\\partial_\\tau^2 \\tilde{\\theta}_j + \\omega_c^2 \\left[ \\sin(\\tilde{\\theta}_{j+1} - \\tilde{\\theta}_j + \\vartheta_{gs}) \n- \\sin(\\tilde{\\theta}_j - \\tilde{\\theta}_{j-1} + \\vartheta_{gs}) \\right] = 0 \\, ,\n\\label{eqm_eucl}\n\\end{equation}\nwhere $\\omega_c$ is the frequency (\\ref{omega_c}). Recall that these\nare to be solved on the rectangle $0 < j < N$, $0\\leq \\tau \\leq \\frac{1}{2} \\beta$\nwith the boundary conditions (\\ref{bc_tth}) and (\\ref{bc_tau}). Note that\nthis boundary problem depends only on the following dimensionless parameters:\n$N$, $\\vartheta_{gs}$, and $\\omega_c \\beta$.\n\nThe Euclidean action corresponding to this boundary problem is \n\\begin{equation}\nS_E = \\int_0^\\beta \\!\\! d\\tau \\left[ \\frac{C}{2} \n\\sum_{j=1}^{N-1} (\\partial_\\tau \\tilde{\\theta}_j)^2 \n- I_c \\sum_{j=0}^{N-1} \\cos(\\tilde{\\theta}_{j+1} - \\tilde{\\theta}_j + \\vartheta_{gs}) \\right] . \n\\label{SE}\n\\end{equation}\nThe difference between (\\ref{SE}) computed for the instanton and that for the\nground state $\\tilde{\\theta}_j \\equiv 0$ gives the instanton action $S_{inst}$ of the preceding\nsection:\n\\begin{equation}\nS_{inst} = S_E + \\beta N I_c \\cos\\vartheta_{gs} \\, .\n\\label{diff}\n\\end{equation}\nUnder the conditions stated there, $S_{inst}$ determines the exponential factor in the\ntunneling rate at temperature $T = 1\/\\beta$, cf. eq.~(\\ref{can_exp}).\n\nIn the preceding, we concentrated on the dependence of $S_{inst}$ on the inverse \ntemperature $\\beta$. More generally, after extracting an overall factor, we\ncan write is as a function of the three dimensionless parameters\nmentioned earlier and, in addition, of the instanton's spatial location; the latter\nis labeled,\nas in the case of the critical droplet,\nby an integer $k$:\n\\begin{equation}\nS_{inst} = (I_c C)^{1\/2} \\sigma_{inst}(N, \\vartheta_{gs}, \\omega_c \\beta; k) \\, .\n\\label{resc_action}\n\\end{equation}\nThe function $\\sigma_{inst}$ will be referred to as the {\\em reduced action}.\nIn what follows, we present results for it for the case when the \ninstanton is in the middle of the wire; for odd\n$N$, this corresponds to $k = \\frac{1}{2} (N - 1)$. We have also found solutions, albeit with\nlarger actions, with cores away from the middle.\n\nNote that the activation exponent (\\ref{Sdrop}) can be written, similarly to\n(\\ref{resc_action}), as\n\\[\nS_{drop} = (I_c C)^{1\/2} \\sigma_{drop} (N, \\vartheta_{gs}, \\omega_c \\beta) \\, ,\n\\]\nwhere\n\\begin{equation}\n\\sigma_{drop} (N, \\vartheta_{gs}, \\omega_c \\beta) = ({\\cal E}_{drop} - {\\cal E}_{gs}) \\omega_c \\beta \n\\label{resc_drop}\n\\end{equation}\nand,\nas in Sec.~\\ref{sec:static}, ${\\cal E}$ denotes an energy measured in units of\n$I_c$.\nThus, a comparison of $S_{inst}$ to $S_{drop}$ amounts to a comparison of\n$\\sigma_{inst}$ to $\\sigma_{drop}$.\n\nFor numerical work, we discretize eq.~(\\ref{eqm_eucl}) on a uniform grid in the\n$\\tau$ direction and solve the resulting difference equation by the Newton-Raphson \nmethod. This is the same general approach as used, for instance, \nin \\cite{Frost&Yaffe:1999,Bonini&al}\nto find periodic instantons in the Yang-Mills-Higgs theory.\n\n\\subsection{Tunneling at low currents}\n\\label{subsec:low}\nWe take up\nthe case of high biasing currents in the next subsection. Here, we briefly discuss the case\n\\begin{equation} \n\\vartheta_{gs} \\leq \\pi \/ N \\, ,\n\\label{low_cur}\n\\end{equation}\nwhich we refer to as low currents. \n\nThe condition (\\ref{low_cur}) is a result of \ncomparing the energy (\\ref{Egs}) of the ground state to the energy of the state with\na $2\\pi$ jump, eq.~(\\ref{jump}). When (\\ref{low_cur})\nis satisfied,\n\\begin{equation}\nE_{jump} - E_{gs} = N [ \\cos\\vartheta_{gs} - \\cos(\\vartheta_{gs} - 2\\pi \/ N) ] > 0 \\, .\n\\label{up}\n\\end{equation}\nThe linear stability analysis in subsec.~\\ref{subsec:jumps} and the \ndiscussion there of the real-time evolution of the state with a $2\\pi$ jump suggest\nthat, for $N > 4$,\nthis state is the lowest-energy state in which the system can emerge after tunneling\nthrough the potential barrier. We have not rigorously proven\nthis assertion, but it is consistent with the numerical results, and in this\nsubsection we will\nproceed on the premise that it is correct. Then, (\\ref{up}) implies that, at low \ncurrents, a phase slip requires tunneling ``up'' in energy or, more precisely, that the\ninitial state of tunneling cannot be the ground state but must be thermally activated.\nThis means that (at low currents) the instanton action $S_{inst}$ retains a nontrivial dependence\non $\\beta$ for arbitrarily large $\\beta$ (i.e., low temperatures), namely, that in this limit\n\\begin{equation}\nS_{inst}(\\beta) \\approx (E_{jump} - E_{gs}) \\beta \\, .\n\\label{Sjump}\n\\end{equation}\nThe slope here, given\nby (\\ref{up}), is not as large as the full barrier height, eq.~(\\ref{Eact}), because\nthe system does not need to activate to the top of the barrier but only to an initial\nstate with enough energy to tunnel to the state with a $2\\pi$ jump. This activated\nbehavior can be traced back to the boundary conditions (\\ref{bc_tth}) the phase\nmust satisfy during tunneling and so is ultimately\nan effect of the bulk superconducting leads. As such,\nit was identified in a different (continuum) model of the wire \nin \\cite{Khlebnikov:CG}.\n\nAn example of numerically found low-current periodic instanton \nis shown in Fig.~\\ref{fig:small}. The front of the plot ($\\tau = 0$)\ncorresponds to the initial state\nof tunneling, and the back ($\\tau = \\frac{1}{2} \\beta$) to the final state. The difference\nbetween the initial state and the ground state $\\tilde{\\theta}_j \\equiv 0$ represents the\nthermal excitation required by the argument above. The final state coincides,\nas far as we can tell,\nwith the state with a $2\\pi$ jump.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.25in]{small.eps}\n\\end{center} \n\\caption{\\small A low-current periodic instanton. The plot shows\nthe reduced phase $\\tilde{\\theta}_j(\\tau)$, which is equal to\nthe difference between the original phase\nvariable $\\theta_j(\\tau)$ and the ground state $\\theta_j = \\vartheta_{gs} j$. \nThe values of the parameters are\n$N = 11$, $\\vartheta_{gs} = 0.2$, $\\omega_c \\beta = 20$, and $k = 5$. Recall that\n$\\sin\\vartheta_{gs}$ is the biasing current in units of the critical. Only the\ninterior points $j = 1,\\dots,10$ and half of the period $[0,\\frac{1}{2} \\beta]$ in the $\\tau$\ndirection are shown; $\\tau$ is in units of $\\omega_c^{-1}$. \n} \n\\label{fig:small} \n\\end{figure}\n\n\n\\subsection{Tunneling at arbitrary currents}\n\\label{subsec:arb}\n\n\nA sample result for $\\tilde{\\theta}_j(\\tau)$ for a comparatively large current, is shown\nin Fig.~\\ref{fig:large}. The key distinction between this instanton and the one for\nsmall current, Fig.~\\ref{fig:small}, is that now the system tunnels practically from\nthe ground state. That is so, even though the temperature for this plot\nis only about a factor of 3 smaller\nthan the crossover $T_q$. In the final state, the phase still has\na characteristic jump at the tunneling location, but the magnitude of the jump \nis reduced compared to the low-current case.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.25in]{large.eps}\n\\end{center} \n\\caption{\\small A high-current periodic instanton. The reduced phase $\\tilde{\\theta}_j(\\tau)$\n for the instanton with the same\n $N$, $\\omega_c \\beta$, and $k$ as in Fig.~\\ref{fig:small},\n but with a larger $\\vartheta_{gs} = 1$.\n} \n\\label{fig:large} \n\\end{figure}\n\n\nIn Fig.~\\ref{fig:action}, we plot the reduced action $\\sigma_{inst}$, \nas defined by (\\ref{resc_action}), as a function of the half-period for several \nvalues of $\\vartheta_{gs}$. These plots\nillustrate the crossover from the\nhigh-temperature (small $\\beta$) \nregime, where the main mechanism of phase slips is thermal activation,\nto the low-temperature (large $\\beta$) regime, where the main mechanism is tunneling.\nWe see that, after the instanton first appears at $\\beta= \\beta_q$, its action \nfor all $\\beta > \\beta_q$\nlies below the \nstraight line representing thermal activation---the\nbehavior announced in Sec.~\\ref{sec:cross}. The crossover \ntemperatures for all the curves shown are fairly close to one another \nand correspond to $\\frac{1}{2} \\omega_c \\beta_q \\approx 3$. For lower values of the current,\nthe crossover leads to a transition from the high-temperature activated behavior to\none with a smaller slope, as anticipated in subsec.~\\ref{subsec:low} \n(and found for a continuum model in \\cite{Khlebnikov:CG}). \nFor larger currents, the action quickly saturates \nat the zero-temperature value.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.25in]{action.eps}\n\\end{center} \n\\caption{\\small \nSolid lines: the reduced instanton actions $\\sigma_{inst}$\nfor $N = 11$, $k =5$ (instanton in the middle of the wire), and several values of\n$\\vartheta_{gs}$. These are\nplotted as functions of the half-period $\\frac{1}{2} \\beta$, where $\\beta$ is units of \n$\\omega_c^{-1}$. The values of $\\vartheta_{gs}$ increase in \nincrements of 0.1 from top to bottom.\nDashed lines: the straight lines (\\ref{resc_drop}) representing the over-barrier activation\nexponents for the same $\\vartheta_{gs}$.\nThe three smaller values\nof $\\vartheta_{gs}$ correspond to the low-current regime (cf. subsec.~\\ref{subsec:low}), where\nthe instanton action displays the activated (linear) behavior (\\ref{Sjump}) at large $\\beta$\n(though with a slope reduced compared to the full barrier height). \n} \n\\label{fig:action} \n\\end{figure}\n\nNext, consider dependence of the results on $N$, the length of the wire. \nThe low-current condition (\\ref{low_cur}) is obviously\nsensitive to $N$. The results at higher currents, however, are not particularly so.\nFor instance, for $\\vartheta_{gs} \\geq 1$, doubling the length from $N=11$ to $N=21$ makes the\nactions smaller by only a few percent.\n\n\nFinally, let us discuss the case of currents close to the critical, $\\vartheta_{gs} \\to \\pi\/2$.\nIn this regime, we are mostly interested in the scaling law obeyed by the\nasymptotic value of $\\sigma_{inst}$ at $\\beta \\to \\infty$,\nas a function of $\\epsilon = \\pi \/2 - \\vartheta_{gs}$. We find that \nthe power-$5\/4$ scaling anticipated in (\\ref{Sinst_est}) is well borne out. \nNumerically, for $N =11$,\n\\begin{equation}\n\\sigma_{inst}(\\beta\\to \\infty) = 5.5 ~\\epsilon^{5\/2} \n\\label{sigma}\n\\end{equation}\nat $\\epsilon \\to 0$; for $N=21$, the coefficient changes from 5.5 to 5.2.\nSubstituting (\\ref{eps_scaling}) for $\\epsilon$, we find that (\\ref{sigma}) corresponds to \n\\begin{equation}\nS_{inst}(T\\to 0) \\approx 13 \\left( \\frac{\\hbar I_c C}{8 e^3} \\right)^{1\/2} (1 - \\bm{i}_b)^{5\/4} \\, ,\n\\label{T=0}\n\\end{equation}\nwhere we have also used (\\ref{phys_units}) to convert $(I_c C)^{1\/2}$ to the physical units.\nFor $I_c = 10~\\mu{\\mbox A}$ and $C = 3.9 \\times 10^{-14}~\\mbox{F}$\n(the estimate obtained at\nthe end of Sec.~\\ref{sec:static}), eq.~(\\ref{T=0}) gives\n$S_{inst} \\approx 460 (1 - \\bm{i}_b)^{5\/4}$. This estimate suggests that, even for this\nlarge\nvalue of $C$, raising the current to within\n10\\% of the critical will make the rate large enough for quantum phase slips to become \nobservable.\n\n\n\n\\section{Discussion}\n\\label{sec:conc}\nIn this paper, we have looked at both classical and quantum \nmechanisms of decay of supercurrent in superconducting nanowires. These mechanisms \ncorrespond, respectively, to over-barrier activation and tunneling. For the former,\nour main conclusion is that the power-3\/2 \nscaling law (\\ref{3\/2}) for the activation barrier, often observed experimentally,\nis readily reproduced in a discrete model of the wire. That is so even though we\nassume\nthat the values of the phase at the ends remain unchanged during the\nactivation process (a boundary condition attributed to the presence of bulk \nsuperconducting leads) and keep $N$, the length of the wire, finite and possibly small \nwhen sending the current to the critical.\n\nNext, we have found that, in this discrete model, \nthe crossover to the quantum regime occurs in a continuous \nmanner similar to a supercritical Hopf bifurcation. We have found numerically\nthe Euclidean solutions (periodic instantons) that describe tunneling for temperatures\nranging from $T$ just below the crossover $T_q$ to $T$ close to zero. We have also observed\nthat the slowing down of the Euclidean dynamics near the critical current, leads\nto the power-5\/4 scaling law for the tunneling exponent at $T \\ll T_q$.\n\nPhysically, the discrete model represents the idea that the spatial size of a phase slip\nis determined by the size of a Cooper pair (Pippard's coherence\nlength in the clean limit or its counterpart, discussed in Sec.~\\ref{sec:model}, in the \ndirty limit). Such a phase slip will appear point-like in any local theory that deals\nwith the order \nparameter alone, for instance, in the standard GL theory. In other words, the \nactivation path described here is physically distinct from that mediated by the LA \nsaddle point \\cite{LA} of the GL model (and it is, then, perhaps not surprising that \nthe activation barrier follows a different scaling law).\n\nFinally, let us return to the question asked in the beginning of this paper, namely, \nwhether it is possible to represent the results obtained so far\nas consequences of the\ndynamics of a single phase variable. We note at once that this variable cannot be \nthe phase\ndifference between the ends of the wire, because we assume that the phases at the ends\ndo not change at all during either activation or tunneling. \nA variable that does look suitable is the jump,\n$\\vartheta_{core} = \\theta_{k+1} - \\theta_k$, of the phase across the core of a phase\nslip. This variable is ``emergent,'' in the sense that it refers specifically to\nsolutions describing phase slips: critical droplets and periodic instantons.\nIn Sec.~\\ref{sec:comp},\nwe have seen that, at large $N$ and low currents, $\\vartheta_{core}$ changes\nduring tunneling\nby nearly the full $2\\pi$. On the other hand, it changes by only \na small amount at currents near the critical. These properties \nare consistent with the intuition \nabout how the right variable should behave. We expect that a phenomenological theory\nof $\\vartheta_{core}$, based on a suitable effective potential,\nwill be able to\nreproduce the various scaling laws discussed in the present paper.\n\n\nThe author thanks A. Bezryadin for comments on the manuscript.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn what follows let a {\\em residual} set be a dense $G_{\\delta}$ set and we call a property {\\em generic} if it is satisfied on at least a residual set of the underlying space.\nThe roots of studying generic properties in dynamical systems can be derived from the article by Oxtoby and Ulam from 1941 \\cite{OxUl} in which they showed that for a finite-dimensional compact manifold with a non-atomic measure which is positive on open sets, the set of ergodic measure-preserving homeomorphisms is generic in the strong topology.\nSubsequently, Halmos in 1944 \\cite{Ha44},\\cite{Ha44.1} introduced approximation techniques to a purely metric situation: the study of interval maps which are invertible almost everywhere and preserve the Lebesgue measure and showed that the generic invertible map is weakly mixing, i.e., has continuous spectrum.\nThen, Rohlin in 1948 \\cite{Ro48} showed that the set of (strongly) mixing measure preserving invertible maps is of the first category.\nTwo decades later, Katok and Stepin in 1967 \\cite{KaSt} introduced the notation of a speed of approximations. One of the notable applications of their method is the genericity of ergodicity and weak mixing for certain classes of interval exchange transformations.\nOne of the most outstanding result using approximation theory is the Kerckhoff, Masur, Smillie result on the existence of polygons for which the billiard flow is ergodic \\cite{KeMaSm}, as well as its quantitative version by Vorobets \\cite{Vo}.\nMany more details on the history of approximation theory can be found in the surveys\n\\cite{BeKwMe}, \\cite{ChPr}, \\cite{Tr}.\n\nIn what follows we denote $I:=[0,1]$, $\\mathbb{S}^1$ the unit circle and $\\lambda$ the Lebesgue measure on an underlying manifold. Our present study focuses on topological properties of generic non-invertible maps on the interval resp.\\ circle preserving the Lebesgue measure $C_{\\lambda}(I)$, resp.\\ $C_{\\lambda}(\\mathbb{S}^1)$. For the rest of the paper we equip the two spaces with the uniform metric, which makes the spaces complete. The study of generic properties on $C_{\\lambda}(I)$ was initiated in \\cite{B} and continued recently in \\cite{BT}. It is well known that every such map has a dense set of periodic points (see for example \\cite{BT}). Furthermore, except for the two exceptional maps $\\mathrm{id}$ and $1-\\mathrm{id}$, every such map has positive metric entropy. Recently, basic topological and measure-theoretical properties of generic maps from $C_\\lambda(I)$ were studied in \\cite{BT}. We say that an interval map $f$ is \\emph{locally eventually onto (leo)} if for every open interval $J\\subset I$ there exists a non-negative integer $n$ so that $f^n(J)=I$. This property is also sometimes referred in the literature as \\emph{topological exactness}.\nThe $C_{\\lambda}(I)$-generic function\n\\begin{enumerate}[(a)]\n\t\\item is weakly mixing with respect to $\\lambda$ \\cite[Th. 15]{BT},\n\t\\item is leo \\cite[Th. 9]{BT},\n\t\\item satisfies the periodic specification property \\cite[Cor. 10]{BT},\n\t\\item has a knot point at $\\lambda$ almost every point \\cite{B},\n\t\\item maps a set of Lebesgue measure zero onto $[0,1]$ \\cite[Cor. 22]{BT},\n\t\\item has infinite topological entropy \\cite[Prop. 26]{BT},\n\t\\item has Hausdorff dimension = lower Box dimension = 1 $<$ upper Box dimension = 2 \\cite{SW95}.\n\\end{enumerate}\n It was furthermore shown that the set of mixing maps in $C_{\\lambda}(I)$ is dense \\cite[Cor. 14]{BT} and in analogy to Rohlin's result \\cite{Ro48} that this set is of the first category \\cite[Th. 20]{BT}. \n \n In this paper we delve deeper in the study of properties of generic Lebesgue measure preserving maps on manifolds of dimension $1$.\n Our choice of $C_{\\lambda}(I)$ for further investigation is that they are one-dimensional versions of volume-preserving maps, or more broadly, conservative dynamical systems. On the other hand, they represent variety of possible one-dimensional dynamics as highlighted in the following.\n\n\\begin{remark*}Let $f$ be an interval map. The following conditions are equivalent.\n\\begin{itemize}\n \\item[(i)] $f$ has a dense set of periodic points, i.e., $\\overline{\\mathrm{Per}(f)}=I$.\n \\item[(ii)] $f$ preserves a nonatomic probability measure $\\mu$ with $\\mathrm{supp}~\\mu=I$. \n \\item[(iii)] There exists a homeomorphism $h$ of $I$ such that $h\\circ f\\circ h^{-1}\\in C_{\\lambda}(I)$.\n\\end{itemize}\n\\end{remark*}\nTo see the above equivalence it is enough to combine a few facts from the literature. The starting point is \\cite{BM}, where the dynamics of interval maps with dense set of periodic points had been described; while this article is purely topological it easily implies that such maps must have non-atomic invariant measures with full support.\nThe Poincar\\'e Recurrence Theorem and the fact that in dynamical system given by an interval map the closures of recurrent points and periodic points coincide \\cite{CoHe80} provides connection between maps preserving \na probability measure with full support and dense set of periodic points. Finally, for $\\mu$ a non-atomic probability measure with full support the map $h\\colon~I\\to I$ defined as $h(x)=\\mu([0,x])$ is a homeomorphism of $I$; moreover, if $f$ preserves $\\mu$ then $h\\circ f\\circ h^{-1}\\in C_{\\lambda}(I)$ (see the proof of Theorem~\\ref{t-PP} for more detail on this construction). Therefore, the topological properties that are proven in \\cite{BT} and later in this paper are generic also for interval maps preserving measure $\\mu$.\n\nA basic tool to understand the dynamics of interval maps is to understand the structure, dimension and Lebesgue measure of the set of its periodic points.\nFor what follows let $f\\in C_{\\lambda}(I)$. Since generic maps from $C_{\\lambda}(I)$ are weakly mixing with respect to $\\lambda$ it holds that the Lebesgue measure on the periodic points is $0$. However, it is still natural to ask: \n\n\\begin{question}\n\t\\textit{What is the cardinality, structure and dimension of periodic points for generic maps in $C_{\\lambda}(I)$?}\n\\end{question}\n\nAkin et.~al.~proved in \\cite[Theorems 9.1 and 9.2(a)]{AHK} that the set of periodic points of generic homeomorphisms of $\\mathbb{S}^1$ is a Cantor set.\nIn an unpublished sketch, Guih\\'eneuf showed that the set of periodic points of a generic\nvolume preserving homeomorphism $f$\nof a manifold of dimension at least two (or more generally preserving a good\nmeasure in the sense of Oxtoby and Ulam \\cite{OxUl})\n is a dense set of measure zero and for any $\\ell \\ge 1$ the set of fixed points of $f^{\\ell}$ is either empty or a perfect set \\cite{PAG}.\nOn the other hand, Carvalho et.~al.~have shown that the upper box dimension of the set of periodic points is full for generic homeomorphisms on compact manifolds of dimension at least one \\cite{PV}\\footnote{this statement only appears in the published version of \\cite{PV}.}.\n\nIn the above context we provide the general answer about the cardinality and structure of periodic points of period $k$ for $f$ (denoted by $\\mathrm{Per}(f,k)$), of fixed points of $f^k$ (denoted by $\\mathrm{Fix}(f,k)$) and of the union of all periodic points of $f$ (denoted by $\\mathrm{Per}(f)$) and its respective lower box, upper box and Hausdorff dimensions. Namely, we prove:\n\n\\begin{thm}\\label{t8}\n\tFor a generic map $f \\in C_{\\lambda}(I)$, for each $k \\geq 1$:\n\t\\begin{enumerate}\n\t\t\\item \\label{pp1} the set $\\mathrm{Fix}(f,k)$ is a Cantor set, \n\t\t\\item \\label{pp2} the set $\\mathrm{Per}(f,k)$ is a Cantor set, \n\t\t\\item \\label{pp3} the set $\\mathrm{Fix}(f,k)$ has Hausdorff dimension and lower box dimension zero. In particular, $\\mathrm{Per}(f,k)$ has Hausdorff dimension and lower box dimension zero.\n\t\t\\item \\label{pp4} the set $\\mathrm{Per}(f,k)$ has upper box dimension one. Therefore, $\\mathrm{Fix}(f,k)$ has upper box dimension one as well.\n\t\t\\item \\label{pp5} the Hausdorff dimension of $\\mathrm{Per}(f)$ is zero.\n\t\\end{enumerate}\n\\end{thm}\n\nThe proof of the above theorem works also for the generic continuous maps which by our knowledge is not known in the literature yet.\nFurthermore, we can also address the setting of $C_{\\lambda}(\\mathbb{S}^1)$, however, due to the presence of rotations, we need to treat degree $1$ maps separately (for the related statement of the degree one case we refer the reader to Theorem~\\ref{thm:C_p}).\n\nRelated to the study above, there is an interesting question about the possible Lebesgue measure on the set of periodic points for maps from $C_{\\lambda}(I)$. \n\n\\begin{question}\\label{q:B}\n\t\\textit{Does there exist a transitive (or even leo) map in $C_{\\lambda}(I)$ with positive Lebesgue measure on the set of periodic points?}\n\\end{question}\n\nAs mentioned already above, generic maps from $C_{\\lambda}(I)$ will have Lebesgue measure $0$ since $\\lambda$ is weakly mixing. Therefore, the previous question asks about the complement of generic maps from $C_{\\lambda}(I)$ and requires on the first glance contradicting properties. The discrepancy between topological and measure theoretical aspect of dynamical systems again comes to display and we obtain the following result. We answer Question~\\ref{q:B} and even prove a stronger statement.\n\n\n\\begin{thm}\\label{t-PP}\n\tThe set of leo maps in $C_{\\lambda}(I)$\n\twhose periodic points have full Lebesgue measure and whose periodic points of period $k$ have positive measure for each $k \\ge 1$\n\tis dense in $C_{\\lambda}(I)$.\n\\end{thm}\n\nAnother motivation for the study in this paper was the following natural question. \n\n\\begin{question}\\label{q3}\n\t\\textit{Is shadowing property generic in $C_{\\lambda}(I)$?}\n\\end{question}\n\nShadowing is a classical notion in topological dynamics and it serves as a tool to determine whether any hypothetical orbit is actually close to some real orbit of a topological dynamical system; this is of great importance in systems with sensitive dependence on initial conditions, where small errors may potentially result in a large divergence of orbits.\nPilyugin and Plamenevskaya introduced in \\cite{Pi} a nice technique to prove that shadowing is generic for homeomorphisms on any smooth compact manifold without a boundary. This led to several subsequent results that shadowing is generic in topology of uniform convergence, also in dimension one (see \\cite{Med, KoMaOpKu} for recent results of this type). On the other hand, there are many cases known, when shadowing is not present in an open set in $C^1$ topology (see survey paper by Pilyugin \\cite{PilSur} and \\cite{PilBook,PilBook2} for the general overview on the recent progress related with shadowing).\n\nFor continuous maps on manifolds of dimension one, Mizera proved that shadowing is indeed a generic property \\cite{Mizera}. In the context of volume preserving homeomorphisms on manifolds of dimension at least two (with or without boundary), the question above was solved recently in the affirmative by Guih\\'eneuf and Lefeuvre \\cite{GuLe18}. \n\nOur last main theorem provides the affirmative answer on Question~\\ref{q3}.\n\n\\begin{thm}\\label{t-pshadow}\n\tShadowing and periodic shadowing are generic properties for maps from $C_{\\lambda}(I)$.\n\\end{thm}\n\nLet us briefly describe the structure of the paper. In Preliminaries we give general definitions that we will need in the rest of the paper. In particular, our main tool throughout the most of the paper will be controlled use of approximation techniques which we introduce in the end of Section~\\ref{sec:Preliminaries}. In Section~\\ref{sec:PP} we turn our attention to the study of periodic points and prove Theorem~\\ref{t8}.\nThe proof relies on a precise control of perturbations introduced in Section~\\ref{sec:Preliminaries} which turns out to be particularly delicate.\n With some additional work we consequently obtain Theorem~\\ref{t-PP}. We conclude the section with the study of periodic points for maps from $C_{\\lambda}(\\mathbb{S}^1)$ in Subsection~\\ref{subsec:PPcirc}. \nIn Section~\\ref{sec:shadowing} we provide a proof of Theorem~\\ref{t-pshadow}. Similarly as in \\cite{KoMaOpKu} we use covering relations, however the main obstacle is the preservation of Lebesgue measure which makes obtaining such coverings a more challenging task. We conclude the paper with Subsection~\\ref{subsec:s-limit} where we address a notion stronger than shadowing called the s-limit shadowing (see Definition~\\ref{def:shadowing}) in the contexts of $C_{\\lambda}(I)$. We prove that s-limit shadowing is dense in the respective environments. The approach resembles the one taken in \\cite{MazOpr}, however due to our more restrictive setting our proof requires better control of perturbations. This result, in particular, implies that limit shadowing is dense in the respective environments as well.\n\n\\section{Preliminaries}\\label{sec:Preliminaries}\nLet $\\mathbb{N}:=\\{1,2,3,\\ldots\\}$ and $\\mathbb{N}_0:=\\mathbb{N}\\cup\\{0\\}$.\nLet $\\lambda$ denote the Lebesgue measure on the unit interval $I:=[0,1]$. We denote by $C_{\\lambda}(I)\\subset C(I)$ the family of all continuous Lebesgue measure preserving functions of $I$ being a proper subset of the family of all continuous interval maps equipped with the \\emph{uniform metric} $\\rho$:\n$$\\rho (f,g) := \\sup_{x \\in I} |f(x) - g(x)|.$$\nA \\emph{critical point} of $f$ is a point $x\\in I$ such that there exists no neighborhood of $x$ on which $f$ is strictly monotone. Denote by $\\mathrm{Crit}(f)$ the set of all critical points of $f$. A point $x$ is called \\emph{periodic of period $N$}, if there exists $N\\in\\mathbb{N}$ so that $f^N(x)=x$ and we take the least such $N$. Let us denote by $\\Xi(f)$ the set of points from $I$ for which no neighborhood has a constant slope under $f$. Obviously, $\\mathrm{Crit}(f)\\subset \\Xi(f)$. Let $\\mathrm{PA}(I)\\subset C(I)$ denote the set of \\emph{piecewise affine} \nfunctions; i.e., functions that are affine on every interval of monotonicity and have finitely many points in the set $\\Xi(f)$.\n Let $\\mathrm{PA}_{\\lambda}(I)\\subset C_{\\lambda}(I)$ denote the set of piecewise affine functions that preserve Lebesgue measure and $\\mathrm{PA}_{\\lambda(\\textrm{leo})}(I)\\subset\\mathrm{PA}_{\\lambda}(I)$ such functions that are additionally \\emph{locally eventually onto} (i.e., the image under sufficiently large iterations of nonempty open sub-intervals cover $I$). For some $\\xi>0$ and $h\\in C_{\\lambda}(I)$ define \n $$B(h,\\xi):=\\{f\\in C_{\\lambda}(I): \\rho(f,h)<\\xi\\}.$$\n By $d(x,y)$ we denote the Euclidean distance on $I$ between $x,y\\in I$.\n For a set $U\\in I$ and $\\xi>0$ we will also often use, by the abuse of notation, \n $$B(U,\\xi):=\\{x\\in I: d(u,x)<\\xi \\text{ for some } u\\in U\\}.$$\n \n\n\n\n\\subsection{Window perturbations in Lebesgue preserving setting} In this subsection we briefly discuss the setting of Lebesgue measure preserving interval maps and introduce the techniques that we will apply in the rest of the paper.\nThe proof of the following proposition is standard and we leave it for the reader.\n\n\\begin{proposition}\\label{prop:1}$(C_{\\lambda}(I),\\rho)$ is a complete metric space. \\end{proposition}\n\n\\begin{definition} We say that continuous maps $f,g:[a,b]\\subset I\\to I$ are \\emph{$\\lambda$-equivalent} if for each Borel set $A \\in \\mathcal{B}$,\n$$\\lambda(f^{-1}(A))=\\lambda(g^{-1}(A)).\n$$\nFor $f\\in C_{\\lambda}(I)$ and $[a,b]\\subset I$ we denote by $C(f;[a,b])$ the set of all continuous maps $\\lambda$-equivalent to $f| [a,b]$. We define\n$$C_*(f;[a,b]):=\\{h\\in C(f;[a,b]):h(a)=f(a),~h(b)=f(b)\\}.$$\n\\end{definition}\n\nThe following definition is illustrated by Figure~\\ref{fig:perturbations}.\n \n\\begin{definition}\\label{eq:1} Let $f$ be from $C_{\\lambda}(I)$ and $[a,b]\\subset I$. For any fixed $m\\in\\mathbb{N}$, let us define the map $h=h\\langle f;[a,b],m\\rangle\\colon~[a,b]\\to I$ by ($j\\in\\{0,\\dots,m-1\\}$):\n\\begin{equation*}\nh(a + x) := \\begin{cases}\nf\\left (a+m \\Big (x-\\frac{j(b-a)}{m}\\Big) \\right )\\text{ if } x\\in \\left [\\frac{j(b-a)}{m},\\frac{(j+1)(b-a)}{m} \\right ],~j\\text{ even}, \\\\\nf\\left (a+m \\Big (\\frac{(j+1)(b-a)}{m}-x \\Big ) \\right )\\text{ if } x\\in \\left [\\frac{j(b-a)}{m},\\frac{(j+1)(b-a)}{m} \\right ],~j\\text{ odd}.\n\\end{cases}\n\\end{equation*}\nThen $h\\langle f;[a,b],m\\rangle\\in C(f;[a,b])$ for each $m$ and $h\\langle f;[a,b],m\\rangle\\in C_*(f;[a,b])$ for each $m$ odd.\n\\end{definition}\n \n\\begin{figure}[!ht]\n\t\\centering\n\t\\begin{tikzpicture}[scale=4]\n\t\\draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0);\n\t\\draw[thick] (0,1)--(1\/2,0)--(1,1);\n\t\\node at (1\/2,1\/2) {$f$};\n\t\\node at (7\/16,-0.1) {$a$};\n\t\\node at (5\/8,-0.1) {$b$};\n\t\\draw[dashed] (7\/16,0)--(7\/16,1\/4)--(5\/8,1\/4)--(5\/8,0);\n\t\\node[circle,fill, inner sep=1] at (7\/16,1\/8){};\n\t\\node[circle,fill, inner sep=1] at (5\/8,1\/4){};\n\t\\end{tikzpicture}\n\t\\hspace{1cm}\n\t\\begin{tikzpicture}[scale=4]\n\t\\draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0);\n\t\\draw[thick] (0,1)--(7\/16,1\/8)--(14\/32+1\/48,0)--(1\/2,1\/4)--(1\/2+1\/24,0)--(9\/16,1\/8)--(9\/16+1\/48,0)--(5\/8,1\/4)--(1,1);\n\t\\draw[dashed] (1\/2,0)--(1\/2,1\/4);\n\t\\draw[dashed] (9\/16,1\/4)--(9\/16,0);\n\t\\node at (1\/2,1\/2) {$h$};\n\t\\node at (7\/16,-0.1) {$a$};\n\t\\node at (5\/8,-0.1) {$b$};\n\t\\draw[dashed] (7\/16,0)--(7\/16,1\/4)--(5\/8,1\/4)--(5\/8,0);\n\t\\node[circle,fill, inner sep=1] at (1\/2,1\/4){};\n\t\\node[circle,fill, inner sep=1] at (9\/16,1\/8){};\n\t\\node[circle,fill, inner sep=1] at (7\/16,1\/8){};\n\t\\node[circle,fill, inner sep=1] at (5\/8,1\/4){};\n\t\\end{tikzpicture}\n\t\\caption{For $f\\in C_{\\lambda}(I)$ shown on the left, on the right\nwe show the the regular $3$-fold window perturbation of $f$ by $h=h\\langle f;[a,b],3\\rangle\\in C_*(f;[a,b])$.}\\label{fig:perturbations}\n\\end{figure}\n\nFor more details on the perturbations from the previous definition we refer the reader to \\cite{BT}.\n\n\n\\begin{definition}\\label{def:perturb}\nFor a fixed $h\\in C_*(f;[a,b])$, the map $g=g\\langle f,h\\rangle\\in C_{\\lambda}(I)$ defined by\n\\begin{equation*}\ng(x) := \\begin{cases}\nf(x)\\text{ if } x\\notin [a,b],\\\\\nh(x)\\text{ if } x\\in [a,b]\n\\end{cases}\n\\end{equation*}\nwill be called the \\emph{window perturbation} of $f$ (by $h$ on $[a,b]$). In particular, if $h=h\\langle f;[a,b],m\\rangle$, $m$ odd, (resp. $h$ is piecewise\naffine), we will speak of\n\\emph{regular $m$-fold (resp. piecewise affine) window perturbation} $g$ of $f$ (on $[a,b]$).\n\\end{definition}\n\n\n\\section{Cardinality and dimension of periodic points for generic Lebesgue measure preserving interval and circle maps}\\label{sec:PP}\nSince generic maps from $C_{\\lambda}(I)$ are weakly mixing (Theorem 15 from \\cite{BT}) it follows that the Lebesgue measure of the periodic points of generic maps from $C_{\\lambda}(I)$ is $0$. The main result of this section is Theorem \\ref{t8} which describes the structure, cardinality and dimensions of this set.\n\nLet \n$$\\mathrm{Fix}(f,k) := \\{x: f^k(x) = x\\}$$\n$$\\mathrm{Per}(f,k) := \\{x : f^k(x) = x \\text{ and } f^i(x) \\ne x \\text{ for all } 1 \\le i < k\n\\}$$\n$$k(x) := k \\text{ for } x \\in \\mathrm{Per}(f,k)$$\nand\n$$\\mathrm{Per}(f) := \\bigcup_{k \\ge 1} \\mathrm{Per}(f,k) = \\bigcup_{k \\ge 1} \\mathrm{Fix}(f,k).$$\n\n\n\n\\begin{definition}\\label{transverse}\nA periodic point $p \\in \\mathrm{Per}(f,k) $ is called \\emph{transverse} if there exist three adjacent intervals\n$A = (a_1,a_2) ,B = [a_2,c_1] ,C = (c_1,c_2)$, with $p \\in B$, $B$ possibly reduced to a point,\nsuch that (1) $f^k(x) = x$ for all $x \\in B$ and either (2.a) $f^k(x) > x$ for all $x \\in A$ and $f^k(x) < x$ for all $x \\in C$ or (2.b) $f^k(x) x$ for all $x \\in C$.\n\\end{definition}\n\nTo prove Theorem~\\ref{t8} we will use the following lemma.\n\n\\begin{lemma} For each $k \\ge 1$ \nthere is a dense set $\\{g_i\\}_{i\\geq 1}$ of maps in $C_{\\lambda}(I)$ such that\n$g_i \\in \\mathrm{PA}_{\\lambda}(I)$, $\\mathrm{Per}(g_i,k) \\not = \\emptyset$, and for each $i$ all points in $\\mathrm{Fix}(g_i,k)$ are transverse.\n\\end{lemma}\n\n\\begin{proof}\nThe set $\\mathrm{PA}_{\\lambda}(I)$ is dense in $C_{\\lambda}(I)$ (\\cite{B}, see also Proposition 8 in \\cite{BT}). \nEach $f \\in \\mathrm{PA}_{\\lambda}(I)$ (in fact each $f \\in C_{\\lambda}(I)$) has a fixed point, so using a 3-fold window perturbation around the fixed point we can approximate $f$ arbitrarily\nwell by a map $f_1 \\in\\mathrm{PA}_{\\lambda}(I)$ with $\\mathrm{Per}(f_1,k) \\not = \\emptyset$.\n\nFix $f \\in\\mathrm{PA}_{\\lambda}(I)$ with $\\mathrm{Per}(f,k) \\not = \\emptyset$.\nWe claim that by an arbitrarily small perturbation of $f$\n we can construct a map $g \\in\\mathrm{PA}_{\\lambda}(I)$ such that\n \\begin{equation}\n \\mathrm{Per}(g,k) \\not = \\emptyset \\text{ and\n all points in } \\mathrm{Fix}(g,k) \\text{ are transverse.}\\label{e1}\n \\end{equation}\n \n We do this in several steps. The first step is to perturb $f$ to $g$ in such a way that the points $0$ and $1$ are not in $\\mathrm{Fix}(g,k)$;\n we will treat only the point $0$, the arguments for the point $1$ are analogous.\nIf $0$ is a fixed point we can make an arbitrarily small window perturbation as in Figure\n\\ref{fig:Per1} so that this is no longer the case.\n\n\\begin{figure}[!ht]\n\t\\begin{tikzpicture}[scale=2]\n\t\\draw (0,1)--(0,0)--(1\/2,0);\n\t\\draw[dashed] (0,1)--(1\/2,1)--(1\/2,0);\n\t\\draw[dashed] (0,0)--(1\/2,1\/2);\n\t\\draw[thick] (0,0)--(1\/2,1);\n\t\\draw[thick,red] (0,1)--(0.25,0)--(1\/2,1);\n\t\\node at (-0.1,-0.1) {$\\scriptstyle 0$};\n\t\\node at (1\/2,-0.1) {$\\scriptstyle a$};\n\t\\node at (0.25,-0.12) {{\\color{red}$\\scriptstyle \\frac{a}{2}$}};\n\t\\node[circle,fill,red, inner sep=1] at (0.25,0){};\n\t\\node[circle,fill, inner sep=1] at (0,0){};\n\t\\node at (-0.1,1) {$\\scriptstyle b$};\n\t\\end{tikzpicture}\n\t\\caption{Small perturbations of a map $f\\in \\mathrm{PA}_{\\lambda}(I)$ near 0. }\\label{fig:Per1}\n\\end{figure}\n\nNow consider the case when $f^{j}(0) =0$, where $j > 1$ is\nthe period of the point $0$, and $j |k$. We assume that $a$ is so small that $f^i([0,a]) \\cap [0,a] = \\emptyset$ for $i=1, 2, \\dots, j-1$\nand choose $a$ so that $f^j(a) \\ne 0$. Let $g$ be the map resulting from a regular 2-fold window perturbation of $f$ on the interval $[0,a]$\n(see Figure \\ref{fig:Per2}). Thus $g(0) = f(a)$ and $g^{j -1} = f^{j -1} $ on the interval $[f(0), f(a)]$, and so \n$g^j(0) = g^{j-1} \\circ f(a) = f^j(a) \\ne 0$.\n\n\n\n\n\\begin{figure}[!ht]\n\t\\begin{tikzpicture}[scale=2]\n\t\\draw (0,1)--(0,-0.2)--(1\/2,-0.2);\n\t\\draw[dashed] (0,1)--(1\/2,1)--(1\/2,-0.2);\n\t\\draw[thick] (0,0)--(1\/2,1);\n\t\\draw[thick,red] (0,1)--(0.25,0)--(1\/2,1);\n\t\\node at (-0.2,0) {$\\scriptstyle f(0)$};\n\t\\node at (-0.2,1) {$\\scriptstyle g(0)$};\n\t\\node at (0,-0.3) {$\\scriptstyle 0$};\n\t\\node at (-0.15,-0.2) {$\\scriptstyle 0$};\n\t\\node at (1\/2,-0.3) {$\\scriptstyle a$};\n\t\\node at (0.25,-0.34) {{\\color{red}$\\frac{a}{2}$}};\n\t\\node[circle,fill,red, inner sep=1] at (0.25,-0.2){};\n\t\\node[circle,fill, inner sep=1] at (0,0){};\n\t\\end{tikzpicture}\n\t\\hspace{0.1cm}\n\t\\begin{tikzpicture}[scale=4]\n\t\\draw[dashed] (.1, 1\/5) --(3\/5,1\/5) -- (3\/5,1) -- (.1,1);\n\t\\draw (.1, 1\/5) -- (.1,1) ;\n\t\\draw [thick] (1\/5,1\/5)--(2\/5,1) -- (3\/5,1\/2);\n\n\t\\node at (.2, 0.12) {$\\scriptstyle f(0)$};\n\t\\node[circle,fill, inner sep=1] at (.2,0.2){};\n\t\\node[red] at (.6, 0.12) {$\\scriptstyle g(0)=f(a)$};\n\t\\node[circle,fill, red, inner sep=1] at (0.6,0.2){};\n\t\\node at (0.1,0.12) {$\\scriptstyle 0$};\n\t\\end{tikzpicture}\n\n\t\\caption{Left: small perturbations of a map $f\\in \\mathrm{PA}_{\\lambda}(I)$ near 0; Right: $g^{j -1} =f^{j-1}$ on $[f(0),f(a)]$.}\\label{fig:Per2}\n\\end{figure}\n\n\nThus we can choose a dense set of \n $f \\in\\mathrm{PA}_{\\lambda}(I)$ with $\\mathrm{Per}(f,k) \\not = \\emptyset$,\nand $\\mathrm{Fix}(f,k) \\cap \\{0,1\\} = \\emptyset$. Fix such a map $f$.\nWe claim that by an arbitrarily small perturbation of $f$\n we can construct a $g \\in\\mathrm{PA}_{\\lambda}(I)$ with $\\mathrm{Per}(g,k) \\not = \\emptyset$ and $\\mathrm{Fix}(g,k) \\cap \\{0,1\\} = \\emptyset$ such that for every \n $c \\in \\mathrm{Crit}(g)$ we have $g^i(c) \\not \\in \\mathrm{Crit}(g)$ for all $1\\leq i \\le k$.\n\nSuppose \nthat for some $c_1,c_2 \\in \\mathrm{Crit}(f)$ we have an $\\ell\\geq 1$ such that $f^{\\ell}(c_1) = c_2$\nand $f^i(c_1) \\not \\in \\mathrm{Crit}(f)$ for $1 \\le i \\le \\ell -1$. \nWe call this orbit a \\textit{critical connection of length $\\ell$}. \nChoose $c_1,c_2$ with the minimal such $\\ell$, if there are several choices fix one of them.\nWe will perturb $f$ to a map $g$ for which this critical connection is destroyed, so $g$ has\none less critical connection of length $\\ell$.\nSince there are finitely many critical connections of a given length, a finite number of such\nperturbations will remove all of them, and a countable sequence of perturbations will finish the proof\nof the claim.\n\nIf $c_1 \\ne c_2$ it suffices to use a small window perturbation around $c_2$ as in Figure\n\\ref{fig:Per};\nif the window perturbation is disjoint from the orbit segment $f^i(c_1)$ for $i \\in \\{0,1,\\dots, \\ell -1\\}$\nthen for the resulting map $g$ we have $g^{\\ell}(c_1) = c_2 < \\bar{c}_2$ and $g^i(c_1) \\not \\in\\mathrm{Crit}(g) = \\{\\bar{c}_2\\} \\cup \\mathrm{Crit}(f) \\setminus \\{c_2\\}$ for $i=1,2,\\dots,\\ell - 1$, \nthus we have destroyed the critical connection.\n Let $Q=\\{f^i(c): 0 \\le i< \\ell \\text{ and } c\\in \\mathrm{Crit}(f)\\} \\setminus \\{c_2\\}$. \nNotice that $g(\\bar{c}_2) =f(c_2)$, thus taking the neighborhood for the perturbations sufficiently small to be disjoint from $Q$\nguaranties that we did not create a new critical connections of length $\\ell$ or shorter.\n\n\n\\begin{figure}[!ht]\n\t\\begin{tikzpicture}[scale=4]\n\t\\draw[dashed] (1\/10,1\/5)--(1\/10,3\/5)--(3\/10,3\/5)--(3\/10,1\/5)--(1\/10,1\/5);\n\t\\draw [thick] (1\/10,3\/5)--(2\/10,1\/5)--(3\/10,3\/5);\n\t\\draw [thick,red] (1\/10,3\/5)--(5\/20,1\/5)--(3\/10,3\/5);\n\t\\node[circle,fill, inner sep=1] at (0.2,0.2){};\n\t\\node[circle,fill,red, inner sep=1] at (0.25,0.2){};\n\t\\node at (0.2,0.12) {$c_2$};\n\t\\node at (0.28,0.12) {{\\color{red}$\\bar{c}_2$}};\n\t\\end{tikzpicture}\n\t\\caption{Small perturbations of a map $f\\in \\mathrm{PA}_{\\lambda}(I)$.}\\label{fig:Per}\n\\end{figure}\n\n\n Now consider the case $c_1 = c_2$.\nSuppose that $c_1$ is a local minimum of $f$; the other cases are\nsimilar. If $\\ell = 1$ then we again move the peak using the window perturbation as in Figure \\ref{fig:Per} to destroy\nthe connection. If $\\ell > 1$ then by assumption the map $f^{\\ell-1}$ in a neighborhood $U = (a,b)$ of the point $f(c_1)$ is strictly monotone.\nUsing a window perturbation around $c_1$ as in Figure~\\ref{fig:Pert},\nyields a map $g \\in C_{\\lambda}(I)$ with $\\mathrm{Per}(g,k) \\not = \\emptyset$\nsuch that\n$$\\mathrm{Crit}(g) = \\{\\bar{c}_1\\} \\cup \\mathrm{Crit}(f) \\setminus \\{c_1\\}.$$ \n\\begin{figure}[!ht]\n\t\\begin{tikzpicture}[scale=4]\n\t\\draw[dashed] (1\/10,1\/5)--(1\/10,3\/5)--(3\/10,3\/5)--(3\/10,1\/5)--(1\/10,1\/5);\n\t\\draw [thick] (1\/10,3\/5)--(2\/10,1\/5)--(3\/10,3\/5);\n\t\\draw [thick,red] (1\/10,3\/5)--(5\/20,1\/5)--(3\/10,3\/5);\n\t\\node[circle,fill, inner sep=1] at (0.2,0.2){};\n\t\\node[circle,fill,red, inner sep=1] at (0.25,0.2){};\n\t\\node at (0.2,0.12) {$\\scriptstyle c_1$};\n\t\\node at (0.28,0.12) {{\\color{red}$\\scriptstyle \\bar{c}_1$}};\n\t\\node at (0.28,-0.05) {\\phantom{1}};\n\t\\end{tikzpicture}\n\t%\n\t\\hspace{0.1cm}\n\t%\n\t\\begin{tikzpicture}[scale=4]\n\t\\draw[dashed] (1\/5, 1\/5) -- (3\/5,1\/5) -- (3\/5,1) -- (1\/5,1) -- (1\/5,1\/5);\n\t\\draw [thick] (1\/5,1\/5)--(3\/5,1);\n\t\\node at (1\/5, 0.12) {$\\scriptstyle a$};\n\t\\node at (.4, 0.12) {$\\scriptstyle f(c_1)$};\n\t\\node at (.4, 0.05) {$\\scriptstyle =g(\\bar{c}_1)$};\n\t\\node[circle,fill, inner sep=1] at (0.4,0.2){};\n\t\\node at (3\/5, 0.12) {$\\scriptstyle b$};\n\t\\end{tikzpicture}\n\t\\caption{Left: small perturbations of a map $f\\in \\mathrm{PA}_{\\lambda}(I)$;\n\tRight: $g^{\\ell-1} =f^{\\ell-1}$.}\\label{fig:Pert}\n\\end{figure}\nThe critical set $\\mathrm{Crit}(f)$ is finite since $f$ is piecewise affine.\nIf this perturbation is small enough to be disjoint from the set $\\mathcal{U} := \\cup_{i=1}^{\\ell -1} f^i(U)$\nthen the\nresulting map $g| \\mathcal{U} = f| \\mathcal{U}$, and so $g^{\\ell -1}| U = f^{\\ell-1}| U$. Furthermore, if the perturbation is small enough\nso that $g(\\bar{c}_1) \\in U$ then\n $$g^{\\ell-1}\\circ g (\\bar{c}_1) = f^{\\ell - 1} \\circ g (\\bar{c}_1) \n= f^{\\ell -1} \\circ f(c_1) = c_1.$$\n Moreover, we can choose the perturbation so small that these two points are arbitrarily close,\ni.e.,\n $g^{\\ell} (\\bar{c}_1) \\in ( c_1 - \\varepsilon, c_1)$, for any fixed $\\varepsilon > 0$.\n \n\nIf $\\varepsilon$ is small enough then there are no critical points of $f$ in the interval\n $(c_1 - \\varepsilon,c_1)$. Furthermore, since $\\bar{c}_1 > c_1$, then if the perturbation\n and $\\varepsilon$ are small enough it holds that $\\bar{c}_1$ is also not in this interval.\nThis procedure possibly creates new critical connections but of length at least $\\ell+1$; but inductive application of both cases will eventually get rid of the critical connections of length at most $k$.\n\n\n If we choose the interval of perturbation small enough then no new critical connections of length at most $k$ can be created, thus the proof of the claim is finished. \n\\end{proof}\n\nNow we have prepared all the tools to give the proof of Theorem~\\ref{t8}.\nIn what follows, by $\\underline{\\mathrm{dim}}_{Box}$, $\\overline{\\mathrm{dim}}_{Box}$ and $\\mathrm{dim}_H$ we denote the lower box dimension, the upper box dimension and the Hausdorff dimension of the underlying sets respectively.\n\n\\begin{proof}[Proof of Theorem~\\ref{t8}]\nFirst note that \\ref{pp5}) follows from \\ref{pp3}) since \n $$\\mathrm{dim}_H(\\mathrm{Per}(f)) \\le \\sup_{k \\ge 1} \\mathrm{dim}_H(\\mathrm{Fix}(f,k)) \\le \\sup_{k \\ge 1} \\underline{\\mathrm{dim}}_{Box}(\\mathrm{Fix}(f,k)) = 0.$$\n\nFor the proofs of \\ref{pp1}), \\ref{pp2}), \\ref{pp3}) and \\ref{pp4}) we fix $k \\in \\mathbb{N}$.\n\n \n Thus we can choose a countable set \\{$g_i\\}_{i\\geq 1} \\subset \\mathrm{PA}_{\\lambda}(I)$, such that no $g_i$ has slope $\\pm 1 $ on any interval, which is dense in $C_{\\lambda}(I)$ with\neach $g_i$ satisfying \\eqref{e1}. The advantage of such $g_i$ is that for each point in $\\mathrm{Fix}(g_i,k)$, there is at least one corresponding periodic\npoint in $\\mathrm{Fix}(g,k)$ if the perturbed map $g$ if is sufficiently close to $g_i$.\n\nConsider the shortest length\n$$\\gamma_i:=\\min\\{|c-c'|: c,c'\\in \\mathrm{Crit}(g_i) \\cup \\{0,1\\} \\text{ and } c\\neq c'\\}$$\nof the intervals of monotonicity of $g_i$, note that $\\gamma_i>0$ since $g_i\\in \\mathrm{PA}_{\\lambda}(I)$. \n\nSince $g_i \\in \\mathrm{PA}_{\\lambda}(I)$ do not have slope $\\pm 1$ the set $\\mathrm{Fix}(g_i,k)$ is finite, suppose it consists of $\\ell_i$ disjoint orbits and the set $\\mathrm{Per}(g_i,k)$ consists of $\\bar \\ell_i \\le \\ell_i$ distinct orbits.\nIn particular \n\\begin{equation}\\label{e-count}\n\\ell_i \\le \\# \\mathrm{Fix}(g_i,k) \\le k \\ell_i \\text{ and } \\mathrm{Per}(g_i,k) = k \\bar \\ell_i.\n\\end{equation}\nChoosing one point from each of the orbits in $\\mathrm{Fix}(g_i,k)$ defines the set\n $\\{x_{l,i}: 1 \\le l \\le \\ell_i \\} \\subset \\mathrm{Fix}(g_i,k)$.\n\n \nBy the definition of $g_i$, the minimal distance \n$$\\eta_i:=\\min\\{|g_i^{m}(x_{l,i})-c|: 0 \\le m \\le k(x_{l,i})-1, \\ \\ 1 \\le l \\le \\ell_i , \\ c \\in \\mathrm{Crit}(g_i) \\cup \\{0,1\\} \\}$$ of the periodic orbits to the \nset $\\mathrm{Crit}(g_i) \\cup \\{0,1\\}$ is strictly positive.\n \n\n\n If $k=\\ell_i =1$ let $\\beta_i := 1$, otherwise we consider the minimal distance\n $$\\beta_i := \\frac{1}{2} \\min\\{|x - x'|: x \\ne x' \\in \n \\mathrm{Fix}(g_i,k)\\}.$$\n\n\n\nLet $\\tau_i$ be a positive real number such that the slope of every $|(g_i^k)'(x)| < \\tau_i$ for every point $x$\nwhere $g_i^k$ is differentiable.\n\nThe construction in the proof depends on integers $n_i \\ge 1$ which will be defined in the proof, for most of the estimates it suffices to have\n$n_i =1$, but for the upper box dimension estimates we will need $n_i$ growing sufficiently quickly.\nWe define a new map $h_ i\\in \\mathrm{PA}_{\\lambda}(I)$ by applying a regular $2n_i + 1$-fold window perturbation of $g_i$ of diameter $ a_i \\le \\frac{1}{2\\tau_i} \\min(\\frac{1}{i k \\ell_i},\\eta_i,\\gamma_i,\\beta_i, (k\\ell_i)^{-i})$\naround each of the points $x_{l,i}$ keeping the map $g_i$ unchanged elsewhere, in particular it\nis unchanged around the other points in $\\mathrm{Fix}(g_i,k)$.\nThe perturbations are disjoint from one another (perturbation around $x_{l,i}$ and $x_{l',i}$) by the definition of $a_i$. \n The bound on $a_i$ guarantees that these maps satisfy the following properties:\n\\begin{enumerate}[i)]\n\\item The collection $\\{h_i\\}_{i\\geq 1}$ is dense in $C_{\\lambda}(I)$ (since the total perturbation size is bounded by $k \\ell_i a_i \\to 0$);\n\\item \\label{p2} Suppose $x_{l,i} \\in \\mathrm{Fix}(g_i,k)$. \n\\begin{enumerate}[a)]\n\\item\n The map $h_i^{k(x_{l,i})}$ has exactly $2n_i + 1$ fixed points\nin the interval $I_{l,i} := [x_{l,i} -a_i, x_{l,i} + a_i]$,\n\\item \\label{p2''} The map $h_i^k$ has\n $(2n_i+1)^{k\/k(x_{l,i})}$ fixed points in this interval.\n \\item \\label{p2'} The full branches of $h_i^{k\/k(x_{l,i})}$ have length $a_i\/(2n_i+1)^{k\/k(x_{l,i})}$, thus\neach subinterval of $I_{l,i}$ of length $2a_i\/(2n_i+1)^{k\/k(x_{l,i})}$ contains at least one full branch and at most parts of three branches, and thus at least one fixed point and at most 3 fixed point of $h_i^k$.\n \\end{enumerate}\n \\item \\label{p3} The total number $N_{l,i}$ of fixed points of $h_i^k$ arising from the orbit of $x_{l,i}$ satisfies\n$$ N_{l,i} = (2n_i + 1)^{k\/k(x_{l,i})} k(x_{l,i}).$$\nSumming over the points $x_{l,i}$ and using $1 \\le k(x_{l,i}) \\le k$ yields\n$$\\max( (2n_i + 1) \\ell_i , (2n_i + 1)^k)\n \\le \\# \\mathrm{Fix}(h_i,k) = \\sum_{l=1}^{\\ell_i} N_{l,i} \\le (2n_i + 1)^{k} k \\ell_i.$$\n\\item If $x_{l,i} \\in \\mathrm{Per}(g_i,k)$ (i.e., $k(x_{l,i}) = k$) then the $N_{l,i} = (2n_i + 1)k$ points are\nnot only in $\\mathrm{Fix}(h_i,k)$ but also in $\\mathrm{Per}(h_i,k)$;\nthus $\\# \\mathrm{Per}(h_i,k) \\ge (2n_i + 1)k \\bar \\ell_i$;\n\\item\\label{p5} Any interval of length $a_i\/(2n_i+1)^k$ covers at most two points of $\\mathrm{Fix}(h_i,k)$\n(since $h_i$ restricted to an interval of length $a_i\/(2n_i+1)$ has at most one critical\npoint).\n\\end{enumerate}\n\nConsider $\\delta_i > 0$ and\n $$G:= \\bigcap_{j \\in\\mathbb{N}} \\bigcup_{i \\ge j} B(h_i ,\\delta_i).$$\n The set $G$ is a dense $G_\\delta$ set.\n \n (\\ref{pp1}) We claim that if $\\delta_i > 0$ goes to zero sufficiently quickly then $\\mathrm{Fix}(f,k)$ is a Cantor set for each $f \\in G$.\n The set $\\mathrm{Fix}(h_i,k)$ is finite, choose $\\zeta_i$ so small that the balls of radius $\\zeta_i$ around distinct points of $\\mathrm{Fix}(h_i,k)$ are \ndisjoint and such that $\\zeta_i \\to 0$.\nWe can choose $\\delta_i$ so small that if $f \\in B(h_i,\\delta_i)$ then $\\mathrm{Fix}(f,k) \\subset B(\\mathrm{Fix}(h_i,k),\\zeta_i)$; in particular the set\n$\\mathrm{Fix}(f,k)$ can not contain an interval whose length is longer than $2\\zeta_i$.\n Fix $f \\in G$, thus $f \\in B(h_{i_j},\\delta_{i_j})$ for some subsequence $i_j$.\nSince $\\zeta_{i _j}\\to 0$ $\\mathrm{Fix}(f,k)$ can not contain an interval.\n \nBy its definition the set $\\mathrm{Fix}(f,k)$ is closed. \nConsider the open cover of $\\mathrm{Fix}(h_i,k)$ by pairwise disjoint intervals of length \n$a_{i_j}$, by \\eqref{p2} these intervals contain $(2n_i+1)^{k\/k(x_{l,i})}$ fixed points for some $k(x_{l,i})$.\n Fix these covering intervals and choose $\\delta_i$ \nsufficiently small so that all fixed points of $f^k$ of any $f \\in B(h_{i},\\delta_{i})$ are contained \nin the covering intervals and so that there are at least $ \\mathrm{Fix}(h_i,k) \\ge \\#(2n_i + 1) \\ell_i$ such points (\\ref{p3}). \nThus\nthere are no isolated points in $\\mathrm{Fix}(f,k)$ since for any periodic point of period $k$ we can find another point from $\\mathrm{Fix}(f,k)$ arbitrary close.\nThis completes the proof of \\eqref{pp1}.\n\nTo prove (\\ref{pp2}) we need to make some adjustments to the above argument.\nWe again fix $f \\in G$, thus $f \\in B(h_{i_j},\\delta_{i_j})$ for some subsequence $i_j$.\nFor the same reasons as before the set $\\mathrm{Per}(f,k)$ is closed and\ncan not contain any intervals. \nWe consider the same open cover as above and impose the same restrictions on\n $\\delta_i$ as above.\n\nWe claim that if $x_{l,j} \\in \\mathrm{Fix}(g_i,k)$ then $\\# \\mathrm{Per}(h_i,k) \\cap I_{l,i} > 1$.\nIf $x_{l,i} \\in \\mathrm{Per}(g_i,k)$ has period $k$ then the $(2n_i+1)$ fixed points of $h_i$ in the interval $I_{l,i}$ all have period $k$ and so the claim holds in this case.\nSuppose now $x_{l,i} \\in \\mathrm{Fix}(g_i,k)$ has period $k(x_{l,i})$ (a strict divisor of $k$), then the corresponding $(2n_i +1)$ points in $\\mathrm{Fix}(h_i,k)\n\\cap I_{l,i}$ have period $k(x_{l,i})$. If we consider the map $h_i^{k(x_{l,i})}$ restricted to $I_{l,i}$ then it is a $(2n_i+1)$-fold tent map, thus it has periodic orbits of all periods. In particular, \nperiodic points of $h_i^{k(x_{l,i})}$ with period $k\/k(x_{l,i})$\nbelongs to the set $\\mathrm{Per}(h_i,k)$. The number \n of such periodic points is strictly larger than $1$, and the claim follows.\n\nWe additionally require that $\\delta_i$ is\nsufficiently small so that for any $f \\in B(h_{i},\\delta_{i})$\nnot only are \nall the fixed points of $f^k$ contained \nin the covering intervals but also that $f$ restricted to the covering interval around $x_{l,i_j}$ has\nat least 2 periodic points of period $k$.\n\nThus\nthere are no isolated points in $\\mathrm{Per}(f,k)$ since for any periodic point of period $k$ we can find another point from $\\mathrm{Per}(f,k)$ arbitrary close.\nThis completes the proof of \\eqref{pp2}.\n\n\n \n(\\ref{pp3}) Since $\\mathrm{Per}(f,k) \\subset \\mathrm{Fix}(f,k)$ it suffices to prove the statement for $\\mathrm{Fix}(f,k)$.\nRemember that the number $k \\ge 1$ and the sequence $\\ell_i$ are fixed.\nWe claim that if $\\delta_i > 0$ goes to zero sufficiently quickly then\nthe lower box dimension of the $\\mathrm{Fix}(f,k)$ is zero for any $f \\in G$.\n\nTo prove the claim fix $f \\in G$, thus $f \\in B(h_{i_j},\\delta_{i_j})$ for some subsequence $i_j$.\nConsider the open cover of $\\mathrm{Fix}(h_{i_j},k)$ by intervals of length \n$a_{i_j}$ guaranteed by \\eqref{p2}. Fix these covering intervals and choose $\\delta_{i_j}$ sufficiently small so that all points of $\\mathrm{Fix}(f,k)$ of any $f \\in B(h_{i_j},\\delta_{i_j})$ are contained in the covering intervals.\n\nLet $N(\\varepsilon)$ denote the number of intervals of length $\\varepsilon > 0$ needed to cover $\\mathrm{Fix}(h_{i_j},k)$. By the choice of $\\delta_{i_j}$, these intervals of length $a_{i_j}$\nalso cover $\\mathrm{Fix}(f,k)$.\nEquation \\eqref{e-count} combined with \\eqref{p2} implies that $\\ell_{i_j} \\le N(a_{i_j}) \\le k \\ell_{i_j}$.\nCombining this with the fact that $a_{i_j} \\le (k \\ell_{i_j})^{-i_j}$ yields\n$$\\frac{\\log (N(a_{i_j})) }{\\log(1\/a_{i_j})} \\le \\frac{ \\log(k \\ell_{i_j})}{\\log(1\/a_{i_j})} \\le \\frac{1}{i_j}$$\nand thus the lower box dimension of $\\mathrm{Fix}(f,k)$\ndefined as \n$$\\liminf_{\\varepsilon \\to 0} \\frac{\\log (N(\\varepsilon)) }{\\log(1\/\\varepsilon)}$$\nis 0.\n\n(\\ref{pp4}) We begin by calculating the upper box dimension of $\\mathrm{Fix}(f,k)$. Here we will need to choose the sequence $n_i$ growing sufficiently quickly.\n Instead of covering $\\mathrm{Fix}(h_i,k)$ by intervals of length $a_i$ we cover it by intervals of length $b_i := 2 a_i\/(2n_i+1)^k$. \nBy \\eqref{p2'} each such interval covers at most three points of $\\mathrm{Fix}(h_i,k)$.\nThus we need at least $(\\# \\mathrm{Fix}(h_i,k))\/3$ such intervals to cover $\\mathrm{Fix}(h_i,k)$; so by \\eqref{p3} we need\n at least $(2n_i+1)^{k}\/3$ such intervals to cover $\\mathrm{Fix}(h_i,k)$.\nFix such a covering and choose $\\delta_i$ sufficiently small so that all periodic points of period $k$ of any $f \\in B(h_{i_j},\\delta_{i_j})$ are contained in the covering intervals.\n\n\nThus \n\\begin{equation}\\label{ubd}\n\\frac{\\log (N(b_{i_j})) }{\\log(1\/b_{i_j})} \\ge \\frac{ \\log((2n_i+1)^k\/3)}{\\log(1\/b_{i_j})} \n= \\frac{ \\log((2n_i+1)^k) - \\log(3)}{\\log((2n_i+1)^k) - \\log(2a_i)}.\n\\end{equation}\n\nThe sequence $a_i$ has been fixed above, \nif $n_i$ grows sufficiently quickly the last term in \\eqref{ubd} approaches one. \nWe can not cover $\\mathrm{Fix}(h_i,k) $ by fewer intervals, and thus we can not cover $\\mathrm{Fix}(f,k)$ by fewer interval, thus\n it follows that the upper box dimension of $\\mathrm{Fix}(f,k)$\ndefined as \n$$\\limsup_{\\varepsilon \\to 0} \\frac{\\log (N(\\varepsilon)) }{\\log(1\/\\varepsilon)}$$\nis 1.\n\n{(5)} We modify the above proof to calculate the upper box dimension of $\\mathrm{Per}(f,k)$ for $k \\ge 2$.\nInstead of $\\mathrm{Fix}(h_i,k)$ we consider $\\mathrm{Per}(h_i,k)$. As before, an interval of length $b_i$ covers at most three points\nof this set, thus we need at least $\\#(\\mathrm{Per}(h_i,k))\/3$ such intervals to cover it. Let $x_{l,i}$ be a fixed point and $I_{l,i}$ the\nassociated interval, then\n\\begin{eqnarray*}\n\\#(\\mathrm{Per}(h_i,k)) &\\ge& \\#(\\mathrm{Fix}(h_i,k) \\cap I_{l,i}) - \\sum_{\\ell|k, 1 \\le \\ell < k} \\#(\\mathrm{Fix}(h_i,\\ell) \\cap I_{l,i})\\\\\n& = & (2n_i +1)^k - \\sum_{\\ell|k, 1 \\le \\ell < k} (2n_i +1)^\\ell\\\\\n& \\ge & (2n_i +1)^k - \\sum_{\\ell=1}^{\\lfloor k\/2 \\rfloor} (2n_i +1)^\\ell\\\\\n& = & (2n_i+1)^k - \\frac{(2n_i+1)^{1+ \\lfloor k\/2 \\rfloor} - 1}{2n_i}\\\\\n& = & (2n_i+1)^k \\cdot \\left [\n1 - \\frac{1}{2n_i} \\left ( \\frac{1}{ (2n_i+1)^{k - 1 - \\lfloor k\/2 \\rfloor }} + \\frac{ 1}{(2n_i+1)^k} \\right )\n \\right ].\n \\end{eqnarray*} \nIf we additionally suppose that $n_i \\ge 2$ then for any $k \\ge 2$ we have\n\\begin{eqnarray*}\n1 - \\frac{1}{2n_i} \\left ( \\frac{1}{ (2n_i+1)^{k - 1 - \\lfloor k\/2 \\rfloor }} + \\frac{ 1}{(2n_i+1)^k} \\right )\n& \\ge & 1 - \\frac{1}{4} \\left ( \\frac{1}{ (2n_i+1)^{k - 1 - \\lfloor k\/2 \\rfloor }} + \\frac{ 1}{(2n_i+1)^k} \\right ) \\ge \\frac{3}{4}.\n \\end{eqnarray*} \nThus the estimate \\eqref{ubd} becomes\n\\begin{eqnarray*}\n\\frac{\\log (N(b_{i_j})) }{\\log(1\/b_{i_j})} &\\ge& \\frac{ \\log(\\frac{3}{4} (2n_i+1)^k \/3) }{\\log(1\/b_{i_j})} \\\\\n&=& \\frac{ \\log( (2n_i+1)^k) - \\log(4) }{\\log((2n_i+1)^k) - \\log(2a_i)}\n\\end{eqnarray*}\nand the rest of the proof follows in a similar manner.\n\\end{proof}\n\n\\begin{remark}\nThe proof of Theorem~\\ref{t8} can easily be adapted to show that the generic map in $C(I)$ has the same properties, this does not seem to be known in our setting.\nRelated results have been proven for homeomorphisms on manifolds of dimension at least two in \\cite{PAG} (unpublished sketch) and \\cite{PV}.\n\\end{remark}\n\n\nWhile positive Lebesgue measure of periodic points can not be realized for ergodic maps, it turns out it can be visible in many leo Lebesgue measure preserving maps. To this end let us first introduce some needed definitions.\n\n\nLet $M_f(I)$ be the space of invariant\nBorel probability measures on $I$ equipped with the {\\it Prohorov metric} $D$ defined by\n$$\nD(\\mu, \\nu)=\\inf\\left\\{\\varepsilon\\colon\n\\begin{array}{l l}\n& \\mu(A) \\le \\nu(B(A,\\varepsilon))+\\varepsilon \\text{ and }\n \\nu(A) \\le \\mu(B(A,\\varepsilon))+\\varepsilon \\\\\n& \\text{ for any Borel subset } A \\subset I\n\\end{array}\n\\right\\}\n$$\nfor $\\mu, \\nu \\in M_f(I)$. The following (asymmetric) formula\n$$D(\\mu, \\nu)=\\inf\\{\\varepsilon\\colon\n\\mu(A)\\leq \\nu(B(A,\\varepsilon))+\\varepsilon \\text{ for all Borel subsets } A\\subset I\\}$$\nis equivalent to original definition, which means we need to check only one of the inequalities.\nIt is also well known, that the topology induced by $D$ coincides with the {\\it weak$^*$-topology} for measures, in particular $(M_f(I), D)$ is a compact metric space (for more details on Prohorov metric and weak*-topology the reader is referred to \\cite{Huber}).\n\n\n\\begin{lem}\\label{lem:nonatomic}\nAssume that $\\mathrm{Per}(f,k)$ is a Cantor set. Fix $x\\in \\mathrm{Per}(f,k)$ and $\\varepsilon>0$. Let $\\mu_x$ be the unique $f$-invariant Borel probability measure supported on the orbit of $x \\in \\mathrm{Per}(f,k)$. Then\nthere is a non-atomic measure $\\nu$ supported on $\\mathrm{Per}(f,k)$ such that $D(\\mu_x, \\nu)<\\varepsilon$.\n\\end{lem}\n\\begin{proof}\nThere exists a Cantor set $C\\subset \\mathrm{Per}(f,k)$ such that $x\\in C$ and sets $f^i(C)$ are pairwise disjoint with $\\diam(f^i(C))<\\varepsilon$ for $i=0,\\ldots,k-1$.\nLet $\\hat{\\nu}$ be any non-atomic probability measure on $C$ and put \n$\\nu=\\frac{1}{k}\\sum_{i=0}^{k-1}\\hat{\\nu}\\circ f^i$. Clearly $\\nu$ is $f$-invariant. \nNote that $\\nu(B(f^i(x),\\varepsilon))\\geq \\nu(f^i(C))=1\/k$, which yields that \n$D(\\mu_x, \\nu)<\\varepsilon$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{t-PP}]\nUsing Theorem \\ref{t8} and the results of \\cite{BT} we can choose\na map $f\\in C_{\\lambda}(I)$ that is leo, ergodic and $\\mathrm{Per}(f,k)$ is a Cantor set\nfor each $k$. By result of Blokh, every mixing interval map has the periodic specification property \\cite{Blokh} (see also \\cite{BT}, Corollary 10). By a well known result of Sigmund \\cite{Sig1,Sig2}, so called $\\mathrm{CO}$-measures, i.e., ergodic measures supported on periodic orbits, are dense in the space of invariant probability measures for maps with periodic specification property. In our context it means that Lebesgue measure can be approximated arbitrarily well by a $\\mathrm{CO}$-measure supported on a periodic orbit.\nAs a consequence, Lemma~\\ref{lem:nonatomic} implies that there exists a sequence $\\mu_k$ of non-atomic measures supported on a subset of $\\mathrm{Per}(f)$\nsuch that $\\lim_{k\\to \\infty} D(\\mu_k,\\lambda)=0$.\n\nLet us fix any $\\varepsilon>0$ and without loss of generality assume that $D(\\mu_k,\\lambda)<\\varepsilon$ for every $k$.\n\nConsider the measure \n$$\\nu := \\sum_{k=1}^{\\infty} \\frac{1}{2^k}\\cdot \\mu_k.$$\nBy definition $\\nu$ is an $f$-invariant Borel probability measure, so $f$ preserves both measures $\\lambda$ and $\\nu$.\nAs a combination of non-atomic measures, $\\nu$ is non-atomic,\nand since $\\lim_{k\\to \\infty} D(\\mu_k,\\lambda)=0$, $\\nu$ has full support, i.e., $\\mathrm{supp}~\\nu=I$.\n \n We define a map $h\\colon~I\\to I$ by $h(x):=\\nu([0,x])$, since $\\nu$ has full support and is non-atomic, the map $h$ is a homeomorphism. \nNote that by the definition of the metric $D$ we have\n$$\n\\nu([0,x])\\leq \\lambda([0,x+\\varepsilon])+\\varepsilon=x+2\\varepsilon\n$$\nand\n$$\nx-\\varepsilon=\\lambda([0,x-\\varepsilon])\\leq \\nu([0,x])+\\varepsilon\n$$\nhence $|x-h(x)|<2\\varepsilon$.\n \nFor each Borel set $A$ in $I$ we can equivalently write\n\\begin{equation}\\label{e:1}\n\\lambda(h(A))=\\nu(A)\\text{ or } \\lambda(A)=\\nu(h^{-1}(A)).\n\\end{equation}\nWe claim that $g :=h\\circ f\\circ h^{-1}\\in C_{\\lambda}(I)$. Using (\\ref{e:1}) for any Borel set $A$ in $I$ we have\n\\begin{equation*}\n \\lambda(A)=\\nu(h^{-1}(A))=\\nu(f^{-1}(h^{-1}(A)))=\n \\lambda(h(f^{-1}(h^{-1}(A))))=\\lambda(g^{-1}(A)).\n\\end{equation*}\nMoreover, the maps $g$ and $f$ are topologically conjugated, so the map $g$ is also leo and $h(\\mathrm{Per}(f))= \\mathrm{Per}(g)$. But by (\\ref{e:1}) again\n$$\\lambda(\\mathrm{Per}(g))=\\lambda(h(\\mathrm{Per}(f)))=\\nu(\\mathrm{Per}(f))= \\sum_{k=1}^{\\infty} \\frac{1}{2^k}\\cdot \\mu_k(\\mathrm{Per}(f)) = 1.\n$$\n\nIn the above construction, we may take $\\varepsilon$ arbitrarily small, therefore $g$ can be arbitrarily small perturbation of $f$.\n\nNow assume that we are given a leo mao $f\\in C_\\lambda(I)$ for which $\\lambda(\\mathrm{Per}(f))=1$.\nConsider the measure \n$$\\eta := \\sum_{k=1}^{\\infty} \\frac{1}{2^k}\\cdot \\eta_k,$$\nwhere each $\\eta_k$ is obtained by application of Lemma~\\ref{lem:nonatomic} to a point $x\\in \\mathrm{Per}(f,k)$.\nThen $\\eta$ is no-natomic and $\\eta(\\mathrm{Per}(f,k))>0$ for every $k$.\n\nWe repeat the above proof (construction of map $g$) using measures $\\nu_{\\varepsilon_i} := \n \\varepsilon_i \\cdot \\eta + (1 -\\varepsilon_i) \\cdot \\lambda$\nwhere the sequence $0 < \\varepsilon_i < 1$ decreases to $0$.\nThe resulting maps $\\{g_{\\varepsilon_i}\\}$ satisfy $\\rho(g_{\\varepsilon_i}, f) \\to 0$, completing the proof.\n\\end{proof}\n\n\n\\subsection{Periodic points for generic circle maps}\\label{subsec:PPcirc}\n \nLet $C_{\\lambda,d}(\\mathbb{S}^1)$ denote the set of degree $d$ maps in $C_{\\lambda}(\\mathbb{S}^1)$. The proof of Theorem \\ref{t8} immediately shows:\n\n\\begin{thm}\nTheorem \\ref{t8}\nholds for generic maps in $C_{\\lambda,d}(\\mathbb{S}^1)$ for each $d \\in \\mathbb{Z} \\setminus \\{1\\}$.\n\\end{thm}\n\n\nFor $C_{\\lambda,1}(\\mathbb{S}^1)$ the situation is more complicated, consider the open set\n$$C_p :=\\{f \\in C_{\\lambda,1}(\\mathbb{S}^1): f \\text{ has a transverse periodic point of period } p\\}.$$\nIn this setting the proof of Theorem \\ref{t8} yields the following result, \n\n\\begin{thm}\\label{thm:C_p}\nFor any $f$ in a dense $G_\\delta$ subset of $\\overline{C}_p$ we have that for each\n$k \\in \\mathbb{N}$\n\\begin{enumerate}\n\\item the set $\\mathrm{Fix}(f,kp)$ is a Cantor set;\n\\item there exists a Cantor set $P_{kp} \\subset \\mathrm{Per}(f,kp)$;\n\\item the sets $P_{kp} \\subset \\mathrm{Per}(f,kp) \\subset \\mathrm{Fix}(f,kp)$ have Hausdorff dimension and lower box dimension zero, while the upper box dimension of these sets is one, and\n\\item the Hausdorff dimension of $\\mathrm{Per}(f)$ is zero.\n\\end{enumerate}\n\\end{thm}\n\n\n\\begin{remark} As in the interval case, the\n proof of the previous two results can easily be adapted to show that the generic degree $d$ map in $C(\\mathbb{S}^1)$ has the same properties, again this does not seem to be known in our setting.\n\\end{remark}\n\nTo interpret this result we investigate the set $C_{\\infty} := C_{\\lambda,1}(\\mathbb{S}^1) \\setminus \\cup_{p \\ge 1} \\overline{C_p}$. As we already saw in the proof of Theorem \\ref{t8}, a periodic point can\nbe transformed to a transverse periodic point by an arbitrarily small perturbation of the map, thus\nthe set $C_{\\infty}$ consists of maps without periodic points. Using the same argument we see that $\\cup_{p \\ge 1} \\overline{C_p}$ contains an open dense set. Therefore, $C_{\\infty}$ is nowhere dense in $C_{\\lambda,1}(\\mathbb{S}^1)$. \n\n\n\\begin{proposition}\nThe set $C_{\\infty}$ consists of irrational circle rotations.\n\\end{proposition}\n\n\\begin{proof}\nClearly $C_{\\infty}$ contains all irrational circle rotations.\n\n\nWe claim that any $f \\in C_{\\infty}$ must be invertible. \nFor each point $z$ denote by $J_z$ the largest interval containing $z$ such that $f^n(z)\\not\\in J_z$\nfor all $n>0$. \nSuppose that $f(x)=f(y)$ for some $x \\ne y$.\nThen by \\cite[Theorem~1]{AK} we obtain that $J_x=J_y$, in particular both are nondegenerate intervals.\nBy the same result intervals $f^n(J_x)$ are pairwise disjoint for all $n\\geq 0$.\nThe Poincar\\'e recurrence theorem states that almost every point is recurrent, which is a contradiction since interior of $J_x$\nconsists of non-recurrent points. The proof is completed.\n\\end{proof}\n\n\n\\section{Shadowing is generic for Lebesgue measure preserving interval and circle maps}\\label{sec:shadowing}\n\nFirst we recall the definition of shadowing and its related extensions that we will work with in the rest of the paper. For $\\delta> 0$, a sequence $(x_n)_{n\\in \\mathbb{N}_0}\\subset I$ s called a \\emph{$\\delta$-pseudo orbit} of $f\\in C(I)$ if $d(f(x_n), x_{n+1})< \\delta$ for every $n\\in \\mathbb{N}_0$. A \\emph{periodic $\\delta$-pseudo orbit} is a $\\delta$-pseudo orbit for which there exists $N\\in\\mathbb{N}_0$ such that $x_{n+N}=x_n$, for all $n\\in \\mathbb{N}_0$. We say that the sequence $(x_n)_{n\\in\\mathbb{N}_0}$ is an \\emph{asymptotic pseudo orbit} if $\\lim_{n\\to\\infty} d(f(x_n),x_{n+1})=0$.\nIf a sequence $(x_n)_{n\\in\\mathbb{N}_0}$ is a $\\delta$-pseudo orbit and an asymptotic pseudo orbit then we simply say that it is an asymptotic $\\delta$-pseudo orbit.\n\n\\begin{defn}\\label{def:shadowing}\nWe say that a map $f\\in C(I)$ has the:\n\\begin{itemize}\n \\item \\emph{shadowing property} if for every $\\varepsilon > 0$ there exists $\\delta >0$ satisfying the following condition: given a $\\delta$-pseudo orbit $\\mathbf{y}:=(y_n)_{n\\in \\mathbb{N}_0}$ we can find a corresponding point $x\\in I$ which $\\varepsilon$-traces $\\mathbf{y}$, i.e.,\n$$d(f^n(x), y_n)< \\varepsilon \\text{ for every } n\\in \\mathbb{N}_0.$$\n\\item\n\\emph{periodic shadowing property} if for every $\\varepsilon>0$ there exists $\\delta>0$ satisfying the following condition: given a periodic $\\delta$-pseudo orbit $\\mathbf{y}:=(y_n)_{n\\in\\mathbb{N}_0}$ we can find a corresponding periodic point $x \\in I$, which $\\varepsilon$-traces $\\mathbf{y}$.\n\\item \\emph{limit shadowing} if for every sequence $(x_n)_{n\\in \\mathbb{N}_0}\\subset I$ so that $$d(f(x_n),x_{n+1})\\to 0 \\text{ when } n\\to \\infty$$\nthere exists $p\\in I$ such that\n$$d(f^n(p),x_n)\\to 0 \\text{ as } n\\to \\infty.$$\n\\item \\emph{s-limit shadowing} if for every $\\varepsilon>0$ there exists $\\delta>0$ so that\n\\begin{enumerate}\n\\item for every $\\delta$-pseudo orbit $\\mathbf{y}:=(y_n)_{n\\in \\mathbb{N}_0}$ we can find a corresponding point $x\\in I$ which $\\varepsilon$-traces $\\mathbf{y}$,\n\\item for every asymptotic $\\delta$-pseudo orbit $\\mathbf{y}:=(y_n)_{n\\in \\mathbb{N}_0}$ of $f$, there is $x\\in I$ which $\\varepsilon$-traces $\\mathbf{y}$ and\n$$ \\lim_{n\\to \\infty}d(y_n,f^n(x)) = 0.$$\n\\end{enumerate}\n\\end{itemize}\n\\end{defn}\n\n The notions of shadowing and periodic shadowing are classical but let us comment less classical notions of limit and s-limit shadowing. While limit shadowing seems completely different than shadowing, it was proved in \\cite{Limit} that transitive maps with limit shadowing also have shadowing property. \n In general it can happen that for an asymptotic pseudo orbit which is also a $\\delta$-pseudo orbit, the point which $\\varepsilon$-traces it and the point which traces it in the limit are different \\cite{Barwel}. This shows that possessing a common point for such a tracing is a stronger property than the shadowing and limit shadowing properties together and this property introduced in \\cite{Sakai} is called the s-limit shadowing.\n\n\\begin{obs}\\label{rem:implies}\nS-limit shadowing implies both classical and limit shadowing.\n\\end{obs}\n\n\\subsection{Proof of genericity of shadowing} The main step in the proof of genericity of \n the shadowing property in the context of maps from $C_{\\lambda}(I)$\n is the following lemma.\n\n\\begin{lem}\\label{lem:shadowingdense}\nFor every $\\varepsilon>0$ and every map $f\\in C_\\lambda(I)$\n there are\n$\\delta<\\frac{\\varepsilon}{2}$ and $F\\in C_{\\lambda}(I)$ such that:\n\\begin{enumerate}\n\\item $F$ is piecewise affine and $\\rho(f,F)<\\frac{\\varepsilon}{2}$,\n\\item if $g\\in C_\\lambda(I)$ and $\\rho(F,g)<\\delta$ then every $\\delta$-pseudo orbit $\\mathbf{x}:=\\set{x_i}_{i=0}^\\infty$ for $g$ is $\\varepsilon$-traced by a point $z\\in I$.\nFurthermore, if $\\mathbf{x}$ is a periodic sequence, then $z$ can be chosen to be a periodic point.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n\\noindent \\textbf{Step 1. Partition.}\nFirst, let \t$0<\\gamma<\\varepsilon\/2$ be such that, if $|a-b|<\\gamma$ then $|f(a)-f(b)|<\\varepsilon\/2$.\n\n Let us assume that $f$ is piecewise affine with the absolute value of the slope at least $4$ on every piece of monotonicity. Indeed, we can assume that $f$ is piecewise affine due to Proposition 8 from \\cite{BT}. Furthermore, we can also assume that the absolute value of the slope of $f$ is at least $4$ on every piece of monotonicity by using regular window perturbations from Definition~\\ref{def:perturb} and thus we can approximate arbitrarily well any piecewise affine map from $C_\\lambda(I)$ by a piecewise affine map from $C_\\lambda(I)$ having absolute value of the slope at least $4$ on every piece of monotonicity. We set $\\gamma$ to be smaller than the length of the shortest piece of monotonicity of $f$.\n Since $f$ preserves the Lebesgue measure it must have non-zero slope on every interval of monotonicity.\nThus we can assume we have a partition $0=a_00$ such that for each $j=0,\\ldots, n$ we have\n$$\n\\{i : f([a_j,a_{j+1}])\\cap (a_i,a_{i+1})\\neq \\emptyset\\}= \\{i : B(f([a_j,a_{j+1}]),3\\delta)\\cap (a_i,a_{i+1})\\neq \\emptyset\\}.\n$$\nWe may also assume that $\\delta$ is sufficiently small, so that if $f([a_j,a_{j+1}])\\cap [a_i,a_{i+1}]\\neq \\emptyset$ then\n\\begin{equation}\\label{eq:cover1}\nf([a_j,a_{j+1}])\\supset [a_i, a_i+2\\delta] \\quad \\text{ or }\\quad f([a_j,a_{j+1}])\\supset [a_{i+1}-2\\delta,a_{i+1}].\n\\end{equation}\nNow, repeating the construction behind Proposition~8 of \\cite{BT} we construct a map $F$ by replacing each $f| [a_i,a_{i+1}]$ by its regular $m$-fold window perturbation (see Definition~\\ref{def:perturb} and Figure~\\ref{fig:perturbations}), with odd $m$\nand large enough to satisfy $1\/m<\\delta$. This way $F$ is still piecewise affine and its minimal slope is larger than the maximal slope of $f$ and such that\n\\begin{equation}\\label{eq:cover2}\nF([a_i,a_i+\\delta])=F([a_{i+1}-\\delta,a_{i+1}])=F([a_i,a_{i+1}])=f([a_i,a_{i+1}]).\n\\end{equation}\nSince $C_{\\lambda}(I)$ is invariant under window perturbations we conclude $F\\in C_\\lambda(I)$.\n\n\\noindent \\textbf{Step 3. $\\varepsilon$-shadowing.} For some $x\\in I$ in what follows denote $\\mathrm{dist}(x,J):=\\inf\\{d(x,y): y\\in J\\subset I\\}$. Also, for an interval $J\\subset I$ let $\\diam(J):=\\sup\\{d(x,y): x,y\\in J\\}$.\nTake any $g\\in C_\\lambda(I)$ such that $\\rho(F,g)<\\delta$ and let $\\mathbf{x}:=\\set{x_i}_{i=0}^\\infty$ be a $\\delta$-pseudo orbit for $g$. We claim that there is a sequence of intervals $J_i$ such that\n\\begin{enumerate}\n\\item $\\diam J_i \\leq \\gamma$ and if $i>0$ then $J_{i}\\subset g(J_{i-1})$,\\label{con:s1}\n\\item $\\mathrm{dist}(x_{i},J_i)< \\gamma$,\\label{con:s2}\n\\item for every $i$ there is $p$ such that $F(J_i)=F([a_p,a_{p+1}])$ and $x_i\\in [a_p,a_{p+1}]$.\\label{con:s3}\n\\end{enumerate}\nTake $p\\geq 0$ such that $[a_p,a_{p+1}]\\ni x_0$ and put $J_0=[a_p,a_{p+1}]$. Then conditions \\eqref{con:s1}--\\eqref{con:s3} are satisfied for $i=0$.\n\nNext assume that for $i=0,\\ldots,m$ there are intervals $J_i$ such that conditions \\eqref{con:s1}--\\eqref{con:s3} are satisfied. We will show how to construct $J_{m+1}$.\nDenote $F(J_m)=:[a,b]$. By \\eqref{con:s3} and the definition of $F$, namely \\eqref{eq:cover1} and \\eqref{eq:cover2}, there are nonnegative\nintegers $\\hat{i}, \\hat{j}$, $\\hat{j}-\\hat{i}\\geq 2$ such that\n$$\n[a_{\\hat{i}+1}-2\\delta, a_{\\hat{j}-1}+2\\delta]\\subset [a,b]\\subset [a_{\\hat{i}}, a_{\\hat{j}}].\n$$\nFurthermore, if $a_{\\hat i}\\neq 0$ then $a>a_{\\hat{i}}+2\\delta$ and if $b_{\\hat j}<1$ then $b 0$ and $\\varepsilon_n\\to 0$ as $n\\to \\infty$. Let us also fix a dense collection of maps $\\{f_k\\}_{k\\in \\mathbb{N}}\\subset C_{\\lambda}(I)$. Define the set\n$$\nA_n:=\\{f\\in C_{\\lambda}(I): \\exists \\delta>0 \\text{ so that every } \\delta\\text{-pseudo orbit is } \\varepsilon_n\\text{-traced}\\}.\n$$\nLet us fix $k,n\\in \\mathbb{N}$. By Lemma~\\ref{lem:shadowingdense} it holds that for every $f\\in C_{\\lambda}(I)$ and for all integers $s>1\/\\varepsilon_n$ there exist $F_{k,s}\\in C_{\\lambda}(I)$ and $\\xi_{k,s}>0$ so that $\\rho(F_{k,s},f_{k})<1\/s$ and $B(F_{k,s},\\xi_{k,s})\\subset A_n$. Define\n$$\nQ_n:=\\bigcup_{s>\\frac{1}{\\varepsilon_n}}\\bigcup_{k=1}^{\\infty} B(F_{k,s},\\xi_{k,s})\\subset A_n.\n$$\nObserve that since $f_k$ is in the closure of $Q_n$ for all $k\\in\\mathbb{N}$ it follows that $Q_n$ is dense in $C_{\\lambda}(I)$. Also $B(F_{k,s},\\xi_{k,s})$ is an open set and thus $Q_n$ is open in $C_{\\lambda}(I)$ as well.\nNow, taking the intersection of the collection $\\{Q_n\\}_{n\\in\\mathbb{N}}$ we thus get a dense $G_\\delta$ set $Q\\subset C_{\\lambda}(I)$. Clearly, if $f\\in Q$ then for every $\\varepsilon>0$ there is $\\delta>0$ so that every $\\delta$-pseudo orbit is $\\varepsilon$-traced by some trajectory of $f$ and if $\\delta$-pseudo orbit is periodic then such trajectory of $f$ can be required to be periodic as well.\n\\end{proof}\n\n\n\\subsection{S-limit shadowing for Lebesgue measure preserving interval and circle maps}\\label{subsec:s-limit}\n\nIn this subsection we address the level of occurrence of the strongest of the above presented notions related with shadowing.\n\nLet us put $\\mathrm{LS}_{\\lambda}(I):=\\{f\\in C_{\\lambda}(I)\\colon~f\\text{ has the s-limit shadowing property}\\}$.\n\n\n\\begin{proposition}\\label{p:1}\n\tThe set $\\mathrm{LS}_{\\lambda}(I)$ is dense in $C_{\\lambda}(I)$.\n\\end{proposition}\n\n\n\\begin{proof}Choose $\\varepsilon>0$. Let $g_0\\in \\mathrm{PA}_{\\lambda(\\mathrm{leo})}(I)$. We will show how to perturb $g_0$ to obtain a map $g\\in C_{\\lambda}(I)$ close to $g_0$ - it will be specified later - which has the limit shadowing property. We will proceed analogously as in the proof of Lemma~\\ref{lem:shadowingdense}. In that proof for a given $\\varepsilon>0$ a perturbation of $f$ defining $F$ assumes a special finite partition ${\\mathcal P}$ and related positive parameters $\\gamma,\\delta,m$. We will call the whole procedure (such a map $F$) $(\\varepsilon,{\\mathcal P},\\gamma,\\delta,m)$-perturbation of $f$.\n\t\n\tFix a decreasing sequence $(\\varepsilon_n)_{n\\ge 1}$ of positive numbers such that\n\t\n\t\\begin{equation}\\label{e:18}\\varepsilon_1<\\varepsilon~\\text{ and }~\\varepsilon_n\\to 0,~n\\to\\infty.\n\t\\end{equation}\n\t\n\t\\noindent {\\bf Step~1.} We put $f=g_0$ and consider\n\t$$F=g_1\\text{ as } (\\varepsilon_1,{\\mathcal P}_1,\\gamma_1,\\delta_1,m_1)-\\text{perturbation of}~f.\n\t$$\n\tWe assume that $g_0| P^1_i$ is monotone for each $P^1_i\\in{\\mathcal P}_1$. From Lemma~\\ref{lem:shadowingdense} (1) it follows that $\\rho(g_0,g_1)<\\varepsilon_1\/2$ and Lemma~\\ref{lem:shadowingdense} (2) implies that for each $g\\in B(g_1,\\delta_1)$ (hence also for $g_1$ itself) every $\\delta_1$-pseudo orbit is $\\varepsilon_1$-traced. In addition we can require $\\delta_1<\\varepsilon\/2$.\n\t\n\t\n\t\\noindent {\\bf Step~2.} We put $f=g_1$ and consider\n\t$$F=g_2\\text{ as } (\\varepsilon_2,{\\mathcal P}_2,\\gamma_2,\\delta_2,m_2)-\\text{perturbation of}~f.\n\t$$\n\tWe assume that $g_1| P^2_i$ is monotone for each $P^2_i\\in{\\mathcal P}_2$. Moreover, we choose ${\\mathcal P}_2$ to be a refinement of ${\\mathcal P}_1$, i.e., each element of ${\\mathcal P}_1$ is a union of some elements of ${\\mathcal P}_2$. We consider $\\gamma_2$ and $\\delta_2$ so small that $$B(g_1,\\delta_1)\\supset B(g_2,\\delta_2);$$\n\tLemma~\\ref{lem:shadowingdense} implies that for each $g\\in B(g_2,\\delta_2)$ (hence also for $g_2$ itself) every $\\delta_2$-pseudo orbit is $\\varepsilon_2$-traced.\n\t\n\t\n\t\\noindent {\\bf Step~n.} We put $f=g_{n-1}$ and consider\n\t$$F=g_n\\text{ as }(\\varepsilon_n,{\\mathcal P}_n,\\gamma_n,\\delta_n,m_n\\})-\\text{perturbation of}~f.\n\t$$\n\tWe assume that $g_{n-1}| P^n_i$ is monotone for each $P^n_i\\in{\\mathcal P}_n$ and choose ${\\mathcal P}_n$ to be a refinement of ${\\mathcal P}_{n-1}$. We consider $\\gamma_n$ and $\\delta_n$ so small that\n\t\\begin{equation}\\label{e:11}B(g_{n-1},\\delta_{n-1})\\supset B(g_n,\\delta_n);\\end{equation}\n\tLemma~\\ref{lem:shadowingdense}(2) implies that for each $g\\in B(g_n,\\delta_n)$ (hence also for $g_n$ itself) every $\\delta_n$-pseudo orbit is $\\varepsilon_n$-traced.\n\t\n\t The proof of Lemma~\\ref{lem:shadowingdense} shows that for a fixed map $g\\in B(g_n,\\delta_n)$, for every $\\delta_n$-pseudo orbit $(x_i)_{i\\ge 0}$, if $x_i\\in [a^n_{q(i)},a^n_{q(i)+1}]\\in{\\mathcal P}_n$ for each $i\\ge 0$, there exists a sequence of intervals \\begin{equation}\\label{e:9}J^n_i\\in \\{[a^n_{q(i)},a^n_{q(i)}+\\delta_n],[a^n_{q(i)+1}-\\delta_n,a^n_{q(i)+1}]\\}\\end{equation}\n\tsuch that \\begin{equation}\\label{e:17}g(J^n_{i-1})\\supset J^n_i\\end{equation}\n\tand a point $z\\in\\bigcap_{i=0}^{\\infty}g^{-i}(J^n_i)$ satisfies\n\t\\begin{equation}\\label{e:8}\n\t\\vert g^i(z)-x_i\\vert<\\varepsilon_n\n\t\\end{equation}\n\tfor each $i\\ge 0$.\n\t\n\tBy our construction, the convergence of the sequence $(g_n)_{n\\geq 0}$ is uniform in $C_{\\lambda}(I)$ hence $\\lim_{n\\to \\infty}g_n=G\\in C_{\\lambda}(I)$. Moreover, since by (\\ref{e:11}),\n\t$$G\\in \\bigcap_{n}B(g_n,\\delta_n),$$ and by the previous the map $G$ has the shadowing property, i.e., for every $\\varepsilon>0$ there is $\\delta>0$ such that every $\\delta$-pseudo orbit is $\\varepsilon$-traced.\n\t\n\tLet us show that the map $G$ has the s-limit shadowing property. Due to the definition of s-limit shadowing let us assume that a sequence $(x_i)_{i\\ge 0}$ is satisfying\n \\begin{equation*}\\label{e:12}\n\t\t\\vert G(x_i)-x_{i+1}\\vert\\to 0,~i\\to\\infty.\n\t\t\\end{equation*}\n\tObviously there is an increasing sequence $(\\ell(n))_{n\\ge 1}$ of nonnegative integers (w.l.o.g. we assume that $\\ell(1)=0$, i.e., $(x_i)_{i\\ge 0}$ is an asymptotic $\\delta_1$-pseudo orbit) such that\n\t\\begin{equation*}\\label{e:10}\n\t\t\\vert G(x_i)-x_{i+1}\\vert<\\delta_n,~i\\ge \\ell(n),\n\t\t\\end{equation*}\n\ti.e., each sequence $(x_i)_{i\\ge \\ell(n)}$ is a $\\delta_n$-pseudo orbit. Now we repeatedly use the procedure describe after the equation (\\ref{e:11}) and containing the equations (\\ref{e:9})-(\\ref{e:8}). By that procedure, for each $n\\in\\mathbb{N}$ we can find sequences $(J^n_i)_{i\\ge \\ell(n)}$ (to simplify our notation on the $n$th level we index $J^n_i$ from $\\ell(n)$) such that for each\n\t\\begin{equation}\\label{e:13}\n\t\tz\\in\\bigcap_{i=\\ell(n)}^{\\infty}G^{-i}(J^n_i)\\ \\ \\ G^{\\ell(n)}(z)\\ \\ \\ \\varepsilon_n-\\text{traces}~(x_i)_{i\\ge \\ell(n)}\\text{ for }G.\n\t\\end{equation}\n\tBut by \\eqref{eq:cover2} and \\eqref{def:FLR} of Step 3 in the proof of Lemma \\ref{lem:shadowingdense}, we have $G(J^n_i)=g_n(J^n_i)$ for each $n$ and $i$ and the sequence $({\\mathcal P}_n)_{n\\ge 1}$ is nested, so by \\eqref{def:FLR} of Step 3 in the proof of Lemma \\ref{lem:shadowingdense}, \\eqref{e:9} and \\eqref{e:17} for each $n$ we get\n\t\t\\begin{equation}\\label{e:14}\n\t\tG(J^n_{\\ell(n+1)-1})\\supset J^{n+1}_{\\ell(n+1)}.\n\t\t\\end{equation}\n\t\tIf we define a new sequence $(K_i)_{i\\ge 0}$ of subintervals of $I$ by\n\t\\begin{equation*}\\label{e:15}\n\tK_i=J^n_i,~\\ell(n)\\le i\\le \\ell(n+1)-1,\n\t\\end{equation*}\n\tthen by (\\ref{e:14}) the intersection\n\t\\begin{equation*}\\label{e:16}\n\t\tK=\\bigcap_{i=0}^{\\infty}G^{-i}(K_i)\n\t\t\\end{equation*}\n\tis nonempty. It follows from (\\ref{e:13}) and (\\ref{e:18}) that for each $z\\in K$, $\\vert G^i(z)-x_i\\vert\\to 0$, $i\\to\\infty$.\n\tIf asymptotic pseudo orbit was $\\delta$-pseudo orbit at start, then the choice of intervals $J^1_i$ in the first step ensures $\\varepsilon$-tracing.\n\t\n\tIn order to finish the proof let us recall that we have chosen $\\varepsilon_1<\\varepsilon$ and $\\delta_1<\\varepsilon\/2$ hence\n\t$$\n\t\\rho(g_0,G)<\\rho(g_0,g_1)+\\rho(g_1,G)<\\varepsilon\/2+\\varepsilon\/2=\\varepsilon.\n\t$$\n\tSince the set $\\mathrm{PA}_{\\lambda(\\mathrm{leo})}(I)$ is dense in $C_{\\lambda}(I)$, the conclusion of our theorem follows.\n\\end{proof}\n\n\\section*{Acknowledgements}J. Bobok was supported by the European Regional Development Fund, project No.~CZ 02.1.01\/0.0\/0.0\/16\\_019\/0000778.\nJ. \\v Cin\\v c was supported by the FWF Schr\u00f6dinger Fellowship stand-alone project J 4276-N35.\nP. Oprocha was supported by National Science Centre, Poland (NCN), grant no. 2019\/35\/B\/ST1\/02239. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{Intro}\n\nRecently, mixed ultracold quantum gases including Bose-Bose, Fermi-Fermi, Bose-Fermi, and Bose-impurity mixtures\nhave attracted a great deal of interest due to their fascinating properties.\nPrecision measurements and novel phase transitions are among a few prominent examples provided by such mixtures.\n\nExperimentally, binary states can be realized by using different hyperfine levels ${}^{87}$Rb \\cite{Mya, Hall, Mad,Cab, Sem}, different isotopes of the same species\n${}^{87}$Rb-${}^{85}$Rb \\cite{Pap}, ${}^{168}$Yb-${}^{174}$Yb \\cite{Sug}, \ndifferent atomic species ${}^{87}$Rb-${}^{41}$K \\cite{Mog}, ${}^{87}$Rb-${}^{133}$Cs \\cite{McC, Ler}, ${}^{87}$Rb-${}^{84}$Sr and ${}^{87}$Rb-${}^{88}$Sr \\cite{Pasq}, \n${}^{87}$Rb-${}^{39}$K \\cite{Wack}, and ${}^{87}$Rb-${}^{23}$Na \\cite{Wang}, and with different statistics ${}^{6}$Li-${}^{7}$Li \\cite{Igor}.\nThese achievements allow one to study collective modes \\cite{Mad, Igor}, phase separation between the constituents \\cite{Pap, McC, Wack, Wang, Tojo, Nick}, \nthe observation of heteronuclear Effimov resonances \\cite {Bar}, and the production of polar molecules \\cite{Mol}.\n\nTheoretical investigations of degenerate binary Bose mixtures have mainly addressed the determination \nof the ground state and the density profiles of trapped systems \\cite {Tin, Pu, Esry}, the stability, and the phase separation \\cite{ Trip, Esry, Tim, Esry1, Rib, Jez, Svid}.\nThe dynamics of the center-of-mass oscillation (dipole modes) of two-component Bose-Einstein condensates (BECs) was studied analytically and numerically\nby Sinatra et {\\it al}. \\cite{Sinatra}, whereas, the excitations of quadrupole and scissors modes have been explored by Kasamatsu et {\\it al}. \\cite{Kasa}. \nFurthermore, the properties of homogeneous double condensate systems were analyzed in \\cite {Larsen, Bass, YNep, Sor, Tom} using the Bogoliubov theory.\n\n\n\n\nAt finite temperature, uniform binary Bose gases have been worked out using the Bogoliubov approach \\cite{CFetter}, Hartree-Fock theory \\cite{Scha} \nand a large-$N$ approximation \\cite{Chien}. The phase separation, the dynamics, and the thermalization mechanisms of trapped binary mixtures \nat finite temperatures have been also examined utilizing the local-density approximation \\cite {Shi}, HFB-Popov theory \\cite {Arko}, \nand the Zaremba-Nikuni-Griffin (ZNG) model \\cite {Edm, Lee, Lee1}.\nVery recently, effects of quantum and thermal fluctuations in a two-component Bose gas with Raman induced spin-orbit coupling \nhave been analyzed using the HFB-Popov theory \\cite {Hui1}. \n\n\n\n\nAlthough the above theories received great success in describing the behavior of two-component BECs, \nmuch remains to be investigated regarding effects of quantum and thermal fluctuations on \nthe phase separation and collective excitations of such mixtures.\nThe present work deals with the static and the dynamic properties of homogeneous and inhomogeneous Bose-Bose mixtures at finite temperature \nusing the TDHFB theory \\cite {Boudj, Boudj1, Boudj2, Boudj3, Boudj4, Boudj5, Boudj6, Boudj7, Boudj8, Boudj9, Boudj10, Boudj11}.\nOur scheme provides an excellent starting point to study the dynamics of Bose systems \nand has been successfully tested against experiments in a wide variety of problems namely, collective modes, vortices, solitons and Bose polarons. \n\nIn this paper we show that the TDHFB theory offers a rigorous and self-consistent framework to analyze the full dynamics of the two condensates, thermal clouds \nand pair anomalous correlations, including coupling between the two thermal clouds and anomalous components.\nIn addition, the TDHFB equations allow us to examine the role of anomalous fluctuations in the phenomenon of phase separation in trapped dual Bose condensates.\nThe anomalous density has a crucial contribution in the stability, excitations, superfluidity, and solitons in a single component BEC \n\\cite {Boudj1, Boudj2, Griffin, Burnet, Yuk, Burnet,Giorg, Boudj10, Boudj11,Bulg}. \nBased on experimentally relevant parameters, we demonstrate that a large anomalous density may lead to a transition from miscible to immiscible regime. \nWe find also that the relative motion of the centers of mass of the BECs and thermal clouds is strongly damped when the anomalous density is present \nat both zero and finite temperatures. \n\nIn the spirit of the generalized random-phase approximation (RPA), linearized TDHFB equations are derived in order to investigate \nthe collective excitations in a homogeneous mixture at finite temperature. The developed theory can be referred to as the TDHFB-RPA. \nNeglecting the intraspecies interactions and keeping only terms of second order in coupling constants, the TDHFB-RPA reduces to the \nfinite temperature second-order Beliaev theory \\cite{Griffin}.\nThe ultraviolet divergence of the anomalous averages is properly regularized obtaining useful analytical expression.\nEffects of quantum and thermal fluctuation corrections in the excitations and the thermodynamics are deeply analyzed.\n\n\nThe rest of the paper is organized as follows. In Sec.\\ref {Model}, we outline the general features of the TDHFB equations derived for \nbinary Bose condensates. \nWe discuss also the main hindrances encountered in our model and present the resolution of these problems.\nThe finite temperature stability condition of the mixture is accurately identified. \nSection \\ref{TBBM} deals with harmonically trapped Bose-Bose mixtures and is divided into two subsections related to several subjects.\nSection \\ref{DP} is devoted to solving our equations numerically in a three-dimensional (3D) case and \nanalyzing the profiles of the condensed, noncondensed, and anomalous densities in terms of temperatures for miscible and immiscible mixtures.\nWe will look at in particular how the anomalous fluctuations enhance the degree of the overlap between both the condensates and thermal clouds.\nIt is found that the phase separation between the condensates is suppressed as the temperature is increased in good agreement with the HFB-Popov approximation \\cite{Arko}.\nIn Sec.\\ref{TE} we analyze the dynamics of two trapped BECs in the presence of\nthe thermal cloud and the pair anomalous correlation at both zero and finite temperatures.\nWe relate our findings to those of previous experimental and theoretical treatments.\nIn Sec.\\ref{HM} we solve our TDHFB equations to second order in the interaction coupling constants for uniform mixture at finite temperature using the generalized RPA.\nWe show that the TDHFB-RPA method constitutes a finite-temperature extension of the Beliaev approximation discussed in a single component Bose condensed gas \nwith contact interaction \\cite{Griffin, Beleav} and dipole-dipole interactions \\cite{Boudj2015}.\nMeaningful analytical expressions are obtained for the excitations spectrum, the condensed depletion, the anomalous density, the equation of state (EoS) and the ground state energy.\nFinally, we conclude in Sec.\\ref{concl}.\n\n\n\\section {TDHFB Theory} \\label{Model}\n\nWe consider weakly interacting two-component BEC with the atomic mass $m_j$ confined in external traps $V_j ({\\bf r})$.\nThe many-body Hamiltonian describing such mixtures reads \n\\begin{align} \\label{eq4}\n\\hat H &= \\sum_{j=1}^2\\int d{\\bf r} \\, \\hat\\psi_j^\\dagger ({\\bf r}) \\left[h_j^{sp} +\\frac{g_j}{2} \\hat \\psi_j^\\dagger ({\\bf r})\\hat \\psi_j ({\\bf r})\\right]\\hat\\psi_j ({\\bf r}) \\\\ \\nonumber\n&+g_{12}\\int d{\\bf r} \\, \\hat\\psi_2^\\dagger ({\\bf r}) \\hat \\psi_2({\\bf r})\\hat \\psi_1^\\dagger ({\\bf r}) \\hat\\psi_1({\\bf r}),\n\\end{align}\nwhere $\\hat\\psi_j^\\dagger$ and $\\hat\\psi_j$ are the boson destruction and creation field operators, respectively, satisfying the usual canonical commutation rules \n$[\\hat\\psi_j({\\bf r}), \\hat\\psi_j^\\dagger (\\bf r')]=\\delta ({\\bf r}-{\\bf r'})$.\nThe single particle Hamiltonian is defined by $h_j^{sp}=-(\\displaystyle\\hbar^2\/\\displaystyle 2m_j) \\Delta + V_j$.\nThe coefficients $g_j=(4\\pi \\hbar^2\/m_j) a_j$ and $g_{12}=g_{21}= 2\\pi \\hbar^2 (m_1^{-1}+m_2^{-1}) a_{12}$ with \n$a_j$ and $a_{12}$ being the intraspecies and the interspecies scattering lengths, respectively. \n\nAt finite temperature, we usually perform our analysis in the mean-field framework relying on the TDHFB equations.\nFor Bose mixtures, the TDHFB equations are given by \\cite {Boudj5, Boudj6}\n\\begin{equation} \\label {TDH1}\ni\\hbar \\frac{d \\Phi_j}{d t} =\\frac{d{\\cal E}}{d \\Phi_j},\n\\end{equation}\n\\begin{equation} \\label {TDH2}\ni\\hbar \\frac{d \\rho_j}{d t} =-2\\left[\\rho_j, \\frac{d{\\cal E}}{d\\rho_j} \\right],\n\\end{equation}\nwhere ${\\cal E}=\\langle \\hat H\\rangle$ is the energy of the system.\nIn Eqs.(\\ref{TDH2}), $\\rho_j ({\\bf r},t)$ is the single particle density matrix of a thermal component defined as\n$$\n\\rho_j=\\begin{pmatrix} \n\\langle \\hat{\\bar{\\psi}}^\\dagger\\hat{\\bar{\\psi}}\\rangle & -\\langle\\hat{\\bar{\\psi}}\\hat{\\bar{\\psi}}\\rangle\\\\\n\\langle\\hat{\\bar{\\psi}}^\\dagger\\hat{\\bar{\\psi}}^\\dagger\\rangle& -\\langle\\hat{\\bar{\\psi}}\\hat{\\bar{\\psi}}^\\dagger\\rangle\n\\end{pmatrix}_j,\n$$\nwhere $\\hat{\\bar \\psi}_j({\\bf r})=\\hat\\psi_j({\\bf r})- \\Phi_j({\\bf r})$ is the noncondensed part of the field operator with $\\Phi_j({\\bf r})=\\langle\\hat\\psi_j({\\bf r})\\rangle$ \nbeing the condensate wave-function. \nEquations (\\ref{TDH1}) and (\\ref{TDH2}) are obtained using the Balian-V\\'en\\'eroni variational principle \\cite{BV} \nthat optimizes a generating functional related to the observables of interest.\nThe single component BEC version of Eqs.(\\ref{TDH1}) and (\\ref{TDH2}) was derived in \\cite{Ben}.\n\nAn important feature of the TDHFB formalism is that it allows unitary evolution of $\\rho_j$.\nThen it follows that\n\\begin{align} \\label{Invar}\n\\rho_j (\\rho_j +1)= ( I_j-1)\/4,\n\\end{align} \nwhere $I$ is often known as the Heisenberg invariant \\cite{Cic,Ben, Boudj}. It represents the variance of the number of noncondensed particles. \nFor pure state and at zero temperature, $I=1$.\n\n\nThe total energy can be easily computed yielding: \n\\begin{align} \\label{egy}\n{\\cal E} &= \\sum_{j=1}^2 \\bigg[ \\int d{\\bf r} \\, \\left( \\Phi_j^* h_j^{sp} \\Phi_j + \\hat{\\bar \\psi}_j^\\dagger h_j^{sp} \\hat{\\bar \\psi}_j \\right) \\\\\n&+ \\frac{g_j}{2} \\int d{\\bf r} \\bigg( n_{cj}^2+ 4\\tilde n_j n_{cj} +2\\tilde n_j^2 +|\\tilde m_j|^2 \\nonumber\\\\\n&+ \\tilde m_j^*\\Phi_j^2 + \\tilde m_j {\\Phi_j^*}^2 \\bigg) \\bigg] \\nonumber \\\\\n&+ g_{12}\\int d{\\bf r}\\, (n_{c1}+\\tilde n_1) (n_{c2}+\\tilde{n}_2), \\nonumber\n\\end{align}\nwhere $n_{cj}=|\\Phi_j|^2$ is the condensed density, $\\tilde n_j=\\langle\\hat{\\bar {\\psi}}_j ^\\dagger\\hat{\\bar {\\psi}}_j\\rangle$ is the noncondensed density, \nand $\\tilde m_j= \\langle\\hat {\\bar {\\psi}}_j\\hat{\\bar {\\psi}}_j\\rangle$ is the anomalous density.\n\nUpon introducing the expression (\\ref{egy}) into Eqs.(\\ref {TDH1}) and (\\ref {TDH2}), one obtains the explicit TDHFB equations for the two-component BECs\n\\begin{subequations}\\label {T:DH}\n\\begin{align} \ni\\hbar \\dot{\\Phi}_j & = \\left[ h_j^{sp}+g_j (n_{cj}+2\\tilde n_j) + g_{12} n_{3-j} \\right]\\Phi_j \\label{T:DH1} \\\\\n&+ g_j\\tilde m_j \\Phi_j^{*} , \\nonumber \\\\ \ni\\hbar \\dot{\\tilde n}_j &= g_j\\left(\\tilde m_j^{*}\\Phi_j^2-\\tilde m_j {\\Phi_j^{*}}^2\\right) , \\label{T:DH2} \\\\ \ni\\hbar \\dot{\\tilde m}_j &= 4\\left[ h_j^{sp}+2g_j n_j+\\frac{ g_j }{4} \\left (2\\tilde n_j +1\\right)+g_{12} n_{3-j} \\right] \\tilde m_j \\nonumber \\\\\n&+g_j (2\\tilde n_j +1)\\Phi_j^2 \\label{T:DH3},\n\\end{align}\n\\end{subequations}\nwhere $n_j=n_{cj}+\\tilde n_j $ is the total density.\nSetting $g_{12} =0$, one recovers the usual TDHFB equations \\cite {Boudj, Boudj1, Boudj2, Boudj3, Boudj4, Boudj7, Boudj10, Boudj11} \ndescribing a degenerate Bose gas at finite temperature.\nIn a highly imbalanced mixture where $g_1=0$ or $g_2=0$, Eqs.(\\ref{T:DH}) coincide with our TDHFB equations \nrecently employed in Bose-polaron systems \\cite {Boudj5, Boudj6, Boudj8, Boudj9}. \nFor $\\tilde n_j=\\tilde m_j=0$, they reduce to the coupled Gross-Pitaevskii (GP) equations for binary condensates at zero temperature.\nIn the case of a Fermi-Fermi mixture, Eq.(\\ref{T:DH1}) has no analog, while Eqs.(\\ref{T:DH2}) and (\\ref{T:DH3}) stand for the Hartree-Fock and the gap equations, respectively.\nIn the semiclassical limit, the TDHFB is equivalent to the collisionless Boltzmann equation for the particle distribution function \\cite{Giorg}.\n\nIndeed, the TDHFB theory, as the standard HFB approximation, runs into trouble. \nThe first problem is the destruction of the gaplessness of the TDHFB theory due to the inclusion of the anomalous density \nsignaling that the theory satisfies neither the Hugenholtz-Pines theorem \\cite{HP} nor the Nepomnyashchy identity \\cite {NP}. \nSecondly, the anomalous pair average which in general leads to a double counting of the interaction effects is ultraviolet divergent \\cite{DStoof}. \nPhysically this comes from the contact interaction potential, which treats collisions of different momenta with the same probability.\nTo reinstate the gaplessness of the spectrum, one should renormalize the intraspecies coupling constants $g_j$ \nfollowing the procedure outlined in Refs \\cite{Burnet, Boudj, Boudj4, Boudj6} for a single BEC. This gives\n\\begin{align} \\label{Ren}\n\\bar g_j= g_j(1+\\tilde m_j\/\\Phi_j^2).\n\\end{align} \nDespite the dilute nature of the system, the spatially dependent effective interaction $\\bar g_j $ may modify the static and the dynamics of the mixture.\nFurthermore, $\\bar g_j$ have substantial implications for the stability condition. \nIt is worth noticing that this technique renders the TDHFB equations (\\ref{T:DH}) gapless but leaves the anomalous density divergent as we shall see in Sec.\\ref{HM}.\\\\\nGiven Eq.(\\ref{Ren}), the renormalized TDHFB equations read\n\\begin{subequations}\\label {RT:DH}\n\\begin{align} \ni\\hbar \\dot{\\Phi}_j & = \\left[ h_j^{sp}+ \\bar g_j n_{cj}+2g_j \\tilde n_j + g_{12} n_{3-j} \\right]\\Phi_j, \\label {RT:DH1} \\\\\ni\\hbar \\dot{\\tilde m}_j &= 4\\left[ h_j^{sp}+2g_j n_j+G_j \\left (2\\tilde n_j +1\\right)+g_{12} n_{3-j} \\right] \\tilde m_j \\label {RT:DH2},\n\\end{align}\n\\end{subequations}\nwhere $G_j$ is related to $\\bar g_j$ via $G_j=g_j \\bar g_j \/4(\\bar g_j-g _j)$.\nEquations (\\ref{RT:DH}) are appealing since they permit us to study the behavior of the thermal cloud and the pair anomalous density of Bose-Bose atomic mixtures at any temperature.\nIt is easy to check that they satisfy the energy and number conserving laws.\n\nEquilibrium states can be readily determined via the transformations:\n$\\Phi_j ({\\bf r},t)= \\Phi_j ({\\bf r}) \\exp(-i \\mu_j t\/\\hbar)$ and $\\tilde m_j ({\\bf r},t)= \\tilde m_j ({\\bf r}) \\exp (-i \\mu_j t\/\\hbar)$, \nwhere $\\mu_j$ are chemical potentials related with each components. Here $\\mu_j$ must be calculated self-consistently employing the normalization condition \n$N_j=\\int n_j d {\\bf r}$, where $N_j=N_{cj}+\\tilde N_j$ is the single condensate total number of particles with \n$N_{cj}=\\int n_{cj} d {\\bf r}$ and $\\tilde N_j=\\int \\tilde n_j d {\\bf r}$ being respectively, the condensed and noncondensed number of particles in each component.\n\nA useful relation between the normal and anomalous densities can be given via Eq.(\\ref {Invar}) \n\\begin{equation} \\label{Inv1}\nI_j= (2\\tilde n_j+1)^2- 4|\\tilde m_j |^2.\n\\end{equation}\nThis equation clearly shows that when $I \\rightarrow $1 or equivalently $T\\rightarrow 0$, the absolute value of the anomalous density is larger than the noncondensed density.\nIn the quasiparticle space, one has $\\tilde n_j=\\sum_k \\left[v_{kj}^2+(u_{kj}^2+v_{kj}^2)N_{kj}\\right]$ and $\\tilde m_j=-\\sum_k \\left[u_{kj} v_{kj} (2N_{kj}+1)\\right]$,\nwhere $N_{kj}=[\\exp(\\varepsilon_{kj}\/T)-1]^{-1}$ are occupation numbers for the excitations and \n$ u_{kj},v_{kj}=(\\sqrt{\\varepsilon_{kj}\/E_k}\\pm\\sqrt{E_{kj}\/\\varepsilon_{kj}})\/2$ are the Bogoliubov functions with $E_{kj}$ being the energy of the free particle \nand $\\varepsilon_k$ is the excitations energy. \nCombining the expressions of $\\tilde m_j$ and $\\tilde n_j$ and using the fact that $2N (x)+1= \\coth (x\/2)$, we obtain\n$I_{kj} =\\text {coth} ^2\\left(\\varepsilon_{kj}\/2T \\right)$.\nFor a noninteracting Bose gas where the anomalous density vanishes, $I_{kj} =\\text {coth} ^2\\left(E_{kj}\/2T\\right)$ \\cite{Boudj10}.\nFor an ideal trapped case, the Heisenberg invariant keeps the same form as Eq.(\\ref{Inv1}) \nwith only setting $ \\varepsilon_{kj} \\rightarrow \\varepsilon_j ({\\bf p, r})=p_j^2\/2m +V_j({\\bf r})$,\nwhich can be calculated within the semiclassical approximation. \nEquation (\\ref{Inv1}) allows us to determine in a very convenient manner the critical temperatures of the mixture.\n\n\n\n\n\\section {Trapped Bose-Bose mixture} \\label{TBBM}\n\n\\subsection {Density profiles} \\label{DP}\n\nAs a starting point, it is useful to establish the stability condition.\nWorking in the Thomas-Fermi (TF) approximation which consists in neglecting the kinetic terms in Eqs.(\\ref {RT:DH1}) and valid for large number of particles. \nThe resulting equations for the condensed density distributions $n_{c1}$ and $n_{c2}$ are given by\n\\begin{align}\nn_{c1}&= \\frac{\\Delta}{\\bar g_1 (\\Delta -1)} \\bigg [ \\mu_1-V_1-2g_1\\tilde n_1-g_{12} \\tilde n_2 \\label{TF1} \\\\\n&- \\frac{g_{12}}{\\bar g_2} \\left(\\mu_2-V_2-2g_2 \\tilde n_2-g_{12} \\tilde n_1 \\right) \\bigg], \\nonumber \\\\\nn_{c2}&= \\frac{\\Delta}{\\bar g_2 (\\Delta -1)} \\bigg [\\mu_2-V_2-2g_2\\tilde n_2-g_{12} \\tilde n_1 \\label{TF2} \\\\\n&-\\frac{g_{12}}{\\bar g_1} \\left(\\mu_1-V_1-2g_1 \\tilde n_1-g_{12} \\tilde n_2 \\right) \\bigg], \\nonumber \n\\end{align}\nwhere $\\Delta=\\bar g_1 \\bar g_2\/ g_{12}^2$ is often known as the miscibility parameter.\nIn our case the mixture can be miscible if $\\Delta>1$ or immiscible when $\\Delta<1$. \nThe transition between the two regimes was previously observed in Bose-Bose mixtures in different spin states \\cite{Tojo, Nick}, \nBose-Bose mixtures of two Rb isotopes \\cite{Pap}, and in heteronuclear Bose-Fermi mixtures \\cite{Osp, Zac}.\nIf $g_{12}=0$ and one component vanishes (say $n_1=0$) in a certain space region, Eqs.(\\ref{TF1}) and (\\ref{TF2}) simplify to the one-component TF equation, namely\n$n_{c2}= (\\mu_2-V_2-2 g_2\\tilde n_2)\/\\bar g_2$. For $\\tilde n_j=\\tilde m_j=0$, they reduce to the usual TF equations at zero temperature.\nInspection of Eqs.(\\ref{TF1}) and (\\ref{TF2}) suggests that the stability of the mixture merely requires the conditions:\n\\begin{equation} \\label{Stab}\n\\bar g_1 >0, \\;\\;\\;\\;\\;\\, \\bar g_2 >0, \\;\\;\\;\\ \\text{and} \\;\\;\\;\\;\\;\\, \\Delta>1.\n\\end{equation}\nIn the limit $\\tilde m_j\/n_{cj} \\ll 1$, the conditions (\\ref{Stab}) reduce to the standard stability conditions at zero temperature namely $g_1 g_2>g_{12}^2$.\nFor $\\tilde m_j\/n_{cj} >1$, the system becomes strongly correlated. This means that at finite temperature, the stability criterion of the mixture requires \nthe inequality $-1<\\tilde m_j\/n_{cj} < 1$.\nIf $\\Delta<1$ and $g_{12} <0$, the gas is unstable whereas, for $g_{12} >0$, the two components do not overlap with each other (separated solutions).\nOne of the most important feature arising from our formula (\\ref{Stab}) is that when $\\tilde m_j$ is large, the mixture undergoes a transition from \nmiscible to immiscible phase. \n\nIn order to illustrate our approach, we consider the ${}^{133}$Cs-${}^{87}$Rb mixture confined in a spherical trap $V_j (r)= m_j \\omega_j^2 r^2\/2$ \nwith trap frequency $\\omega_{\\text{Cs}}=\\omega_{\\text{Rb}}= 2\\pi \\times 270$Hz. \nNotice that our theory can adequately treat all the existing mixtures.\nThe intraspecies scattering lengths are: $a_{\\text{Cs}}= 280\\, a_0$ and $a_{\\text{Rb}}=104\\, a_0$ with $a_0$ being the Bohr radius, \nthe interspecies scattering lengths can be adjusted by means of a Feshbach resonance,\nand particle numbers $N_{\\text{Cs}} = N_{\\text{Rb}} = 5 \\times 10^4$. The critical temperature for an ideal gas is about $T_c^0= 450$ nK.\n\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=0.5]{Densprof.eps}}\n \\caption{ (Color online) Top row: density profiles of an immiscible mixture of ${}^{87}$Rb (blue) and ${}^{133}$Cs (black) atoms at different temperatures for $\\Delta=0.9$.\nSolid lines: condensed density. Dashed lines: thermal cloud density. Dotted lines: anomalous density. \nThe thermal cloud and the anomalous densities have been amplified by 10 times for clarity.\nBottom row: same as top panels but for a miscible mixture for $\\Delta=2.5$.}\n\\label{DF} \n\\end{figure}\n\nFigure \\ref{DF} clearly shows that at low temperature, the Rb has a higher peak density and narrower width while the Cs atoms are pushed towards the outer\npart forming a shell structure around the Rb BEC [see Fig.\\ref{DF} (a)].\nSuch a symmetrical demixed phase, can be understood from the fact that the Rb sustains an extra confinement from the Cs shell surrounding it,\ni.e. originating from the coupling term $g_{12} n_{3-j}$ in Eq.(\\ref{RT:DH}). \nFor phase separation, the TF approximation becomes less satisfactory.\nIn this case, the Rb is located at the phase boundary, such that the density distribution varies fast in space and the\nkinetic terms cannot be omitted \\cite {Pu}.\n\nAt $T=0.5\\, T_c$, the two components start to overlap with each other\nand the overlap region is broadened with temperature [see Fig.\\ref{DF}(b)].\nAt $T \\geq 0.75\\,T_c$ where the binary condensates survive with significant thermal clouds,\nthe mixture becomes completely immiscible as is depicted in Fig.\\ref{DF} (c).\nThis suppression of the phase separation which has been predicted also by the HFB-Popov theory \\cite{Arko}, \ncan be attributed to the strong effects of thermal fluctuations.\nWe observe from the same figure that the anomalous density is larger than the noncondensed density at low temperature, it reaches its maximum at intermediate temperatures,\nand vanishes near the transition similar to the case of a single component. \nIndeed, this behavior remains valid irrespective of the mixture whether miscible or immiscible. \nAt higher temperature, both $\\tilde n$ and $\\tilde m$ have a Gaussian shape since the system becomes ultra-dilute \\cite {Boudj2}.\n\nFigure \\ref{DF} (d) shows that at $T=0.25\\,T_c$, both species overlap at the trap center. \nRemarkably, with an increase in temperature ($T=0.5\\, T_c$), the mixture becomes partially immiscible [see Fig.\\ref{DF} (e)].\nAs we have foreseen above, this phase transition is most likely due to the inclusion of the anomalous correlation \nwhich has a significant effect at this range of temperature.\nAt $T \\geq 0.75\\, T_c$, the mixture restores its miscibility due to the weakness of $\\tilde m$ [see Fig.\\ref{DF} (f)].\n\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=0.5]{Conddenst.eps}}\n \\caption{ (Color online) Condensed density for $\\Delta=0.9$ (left) and $\\Delta=2.5$ (right) at $T=0.5\\,T_c$.\nSolid lines: our predictions. Dotted lines: the results of the HFB-Popov theory. Parameters are the same as in Fig.\\ref{DF}.}\n\\label{DF1} \n\\end{figure}\n\nIn Fig.\\ref{DF1} we compare our results for the condensed density with the HFB-Popov calculations. As is clearly seen, the presence of the anomalous density \nleads to reduction of the condensed density and the degree of the overlap between the two condensates. \nThis is owing to the mutual interaction between condensed atoms on the one hand and the condensed atoms and noncondensed atoms on the other.\n \n\n\\subsection {Dynamics of spatial separation} \\label{TE}\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=0.8]{CoM.eps}}\n \\caption{ (Color online) Mean separation between the condensates (solid line) and thermal clouds (dotted lines) versus time in isotropic traps \nfor $z_0\/l_0=0.3$ and $\\Delta=2.5$ at $T=0.5 T_c$.}\n\\label{MSD} \n\\end{figure}\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=0.8]{CoM1.eps}}\n \\caption{ Mean separation between the condensates versus time in isotropic traps for ${}^{87}$Rb atoms, with \nthe scattering lengths are $ a_1: a_{12} :a_2 ::1.03:1:0.97$, with the average of the three being 55 \\AA \\,\\cite{Hall}. \nSolid line : our predictions. Dashed line: theoretical results of \\cite{Sinatra}. Plus: JILA experiment \\cite{Hall}. \nThese data were taken after 22 ms after switching off the trapping potential.}\n\\label{MSD1} \n\\end{figure}\n\nLet us consider the evolution of dual condensates in the presence of their own thermal clouds and anomalous components \nconfined in a spherically symmetric trap which at $t=0$ its centers for the first and second component are, respectively, displaced \nalong the $z$-axis by distances $\\pm z_0\/2$.\nThe separation is assumed to be small compared to the TF radii.\nThe time dependence of the mean separation between the two condensates is given by\n\\begin{equation} \\label{MdS}\nd_c (t) = \\int d {\\bf r} z \\left [n_{c1} ({\\bf r}, t) -n_{c2} ({\\bf r},t) \\right],\n\\end{equation}\nwhile the mean separation between the two thermal clouds reads\n\\begin{equation} \\label{Mdth}\nd_{th} (t) = \\int d {\\bf r} z \\left [ \\tilde n_1 ({\\bf r}, t) - \\tilde n_2 ({\\bf r},t) \\right].\n\\end{equation}\n\nThe numerical integration of our Eqs.(\\ref{MdS}) and (\\ref{Mdth}) shows that at finite temperature, \nthe relative motion of the centers-of-mass of the BECs and thermal clouds in a miscible mixture is strongly damped in particular at long time scales (see Fig.\\ref{MSD}).\nSuch a damping, which has also been predicted by the ZNG theory \\cite{Lee,Lee1}, is caused by condensate-condensate, \ncondensate-thermal, and thermal-thermal interactions. \nThe intra- and inter-component anomalous pair correlations may play also a crucial role for the appearance of the aforementioned \ndamping of oscillations especially at $T \\approx 0.5\\,T_c$.\nAt fixed temperature, the damping of the oscillations of the mean separation between the condensates, and the thermal clouds becomes more and more strong\nfor a large displacement, $z_0$, regime.\nOne can expect that the same behavior persists in the immiscible mixture but with larger oscillation amplitudes.\n\n\nTo better understand the impact of the pair anomalous density on the damping mechanism, we compare \nour predictions for the relative motion of the centers-of-mass of the two BECs with the experimental measurements of \\cite{Hall} \nand the theoretical results of \\cite{Sinatra} based on the GP equation.\nAs is clearly visible from Fig.\\ref{MSD1}, the curves of our TDHFB model improve the theoretical result of Ref. \\cite{Sinatra} especially at large time scale. \nThis correction makes our theory in good agreement with the JILA experiment \\cite{Hall}.\nThe difference between the two models may be justified in terms of the significant contribution of the anomalous density which causes a huge loss\nof atoms during the oscillation even at zero temperature.\n\n\n\\section {Homogeneous Bose-Bose mixture } \\label{HM}\n\n\n\nIn this section we analyze the elementary excitations in a homogeneous mixed Bose gas, where $V_j(r)=0$ using the generalized RPA \\cite{Boudj1, Giorg, Xia}.\nThis latter consists of imposing small fluctuations of the condensates, the noncondensates, and the anomalous components, respectively, as: \n$\\Phi_j = \\sqrt{n_{cj}}+\\delta \\Phi_j $, $\\tilde n_j=\\tilde n_j+\\delta \\tilde n_j$, and $\\tilde m_j=\\tilde m_j+\\delta \\tilde m_j$, \nwhere $\\delta \\Phi_j \\ll \\sqrt{n_{cj}}$, $\\delta \\tilde n_j \\ll \\tilde n_j$, and $\\delta \\tilde m_j \\ll \\tilde m_j$. \nThus, we obtain the TDHFB-RPA equations \n\\begin{align} \ni\\hbar \\delta \\dot \\Phi_j & = \\left[ h_j^{sp}+ 2\\bar g_j n_{cj}+2g_j \\tilde n_j + g_{12} n_{3-j} \\right] \\delta \\Phi_j \\label {RPA1} \\\\ \n &+\\bar g_j n_{cj} \\delta \\Phi_j^*+ 2g_j \\sqrt{n_{cj}} \\delta \\tilde n_j + g_{12}\\sqrt{n_{c{3-j}}} \\delta \\tilde n_{3-j} \\nonumber \\\\\n&+ g_{12} \\sqrt{ n_{cj} n_{c{3-j}} } (\\delta \\Phi_{3-j}+\\delta \\Phi_{3-j}^*), \\nonumber \n\\end{align}\nand \n\\begin{align} \ni\\hbar \\delta \\dot{\\tilde m}_j &= 4\\left[ h_j^{sp}+2g_j n_j+G_j (2\\tilde n_j +1)+g_{12} n_{3-j} \\right] \\delta\\tilde m_j \\label {RPA2} \\\\ \n&+ 8g_j \\tilde m_j \\left[ \\sqrt{ n_{cj}} (\\delta \\Phi_j+ \\delta \\Phi_j^*)+ \\delta \\tilde n_j+ (G_j\/g_j) \\delta \\tilde n_j \\right] \\nonumber\\\\\n&+ g_{12}\\tilde m_j \\left[\\sqrt{ n_{c{3-j}}} (\\delta \\Phi_{3-j}+\\delta \\Phi_{3-j}^*) +\\delta \\tilde n_{3-j} \\right]. \\nonumber\n\\end{align}\nHere we recall that $\\delta \\tilde n_j$ and $\\delta \\tilde m_j$ are related with each other through (\\ref{Inv1}).\nRemarkably, Eqs.(\\ref{RPA1}) and (\\ref{RPA2}) contain a class of terms beyond second order.\nThey can be regarded as a natural extension of the HFB-RPA \\cite{Xia} theory developed for the single component BEC. \nIf one neglects the anomalous density, the TDHFB-RPA equations reduce to the HFB-Popov-RPA equations.\n\nSince we restrict ourselves to second order in the coupling constants, one must retain in Eqs.(\\ref{RPA1}) and (\\ref{RPA2}) \nonly the terms which describe the coupling to the condensate and neglect all terms associated with \nfluctuations $\\delta \\tilde n$ and $\\delta \\tilde m$ \\cite {Giorg}.\nIn fact, this assumption is relevant to ensure the gaplessness of the spectrum.\nWritting the field fluctuations associated to the condensate in the form \n$\\delta \\Phi_j ({\\bf r},t)= u_{jk} e^{i {\\bf k \\cdot r}-i\\varepsilon_k t\/\\hbar}+v_{jk} e^{i {\\bf k \\cdot r}+i\\varepsilon_k t\/\\hbar}$, \nwe obtain the second order coupled TDHFB-de Gennes equations for the quasiparticle amplitudes $u_{kj}$ and $v_{kj}$ :\n\\begin{equation} \\label{BdG}\n\\begin{pmatrix} \n{\\cal L}_1 & {\\cal M}_1 & {\\cal A} & {\\cal A}\n\\\\\n {\\cal M}_1 & {\\cal L}_1 & {\\cal A} & {\\cal A}\n\\\\\n {\\cal A} & {\\cal A}& {\\cal L}_2 & {\\cal M}_2\n\\\\\n {\\cal A} & {\\cal A} & {\\cal M}_2 & {\\cal L}_2\n\\end{pmatrix}\\begin{pmatrix} \nu_{1k} \\\\ v_{1k} \\\\ u_{2k} \\\\ v_{2k} \n\\end{pmatrix}=\\varepsilon_k \\begin{pmatrix} \nu_{1k} \\\\ -v_{1k} \\\\ u_{2k} \\\\ -v_{2k}\n\\end{pmatrix},\n\\end{equation} \nwhere $\\int d {\\bf r} [u_j^2( {\\bf r})- v_j^2({\\bf r})]=1$, \n${\\cal L}_j = E_k+ 2 \\bar g_j n_{cj}+ 2 g_j \\tilde n_j + g_{12} n_{3-j} -\\mu_j$, ${\\cal M}_j= \\bar g_j n_{cj}$, \n${\\cal A}=g_{12}\\sqrt{n_{c1} n_{c2} }$, $\\varepsilon_k$ is the Bogoliubov excitation energy and \n$E_k=\\hbar^2k^2\/2m$ is the kinetic energy which is the same for both species since we consider equal masses ($m_1=m_2=m$). \nFor $g_{12}=0$, Eqs.(\\ref{BdG}) coincide with the finite temperature second-order equations obtained by Shi and Griffin using diagrammatic methods \\cite{Griffin}\nand with the finite temperature time-dependent mean-field scheme proposed by Giorgini \\cite{Giorg}.\nAt zero temperature they correspond to the well-known second-order Beliaev's results \\cite{Beleav} discussed in a single component Bose condensed gas over six decades ago,\nwhile at high temperature our second-order coupled TDHFB-de Gennes equations reproduce those derived by Fedichev and Shlyapnikov \\cite{FedG} \nemploying Green's function perturbation scheme.\\\\\nThe chemical potentials turn out to be given as\n\\begin{equation} \\label{EoS}\n\\mu_j= \\bar g_j n_{cj} + 2 g_j \\tilde n_j + g_{12} n_{3-j},\n\\end{equation}\nInserting Eq.(\\ref{EoS}) into (\\ref{BdG}), one obtains the following Bogoliubov spectrum composed of two branches:\n\\begin{equation} \\label {Bog}\n\\varepsilon_{k+}= \\sqrt{E_k^2+2E_k \\mu_+}\\,, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\varepsilon_{k-}= \\sqrt{E_k^2+2E_k \\mu_-}\\,, \n\\end{equation}\nwhere\n\\begin{equation} \\label {Chmp}\n\\mu_{+,-}= \\frac{\\bar g_1 n_{c1}} {2} f_{+,-} (\\Delta, \\alpha),\n\\end{equation}\nwhere $f_{+,-} (\\Delta, \\alpha)= 1 + \\alpha \\pm \\sqrt{ (1-\\alpha)^2 +4 \\Delta ^{-1}\\alpha }$ and\n$\\alpha= \\bar g_2 n_{c2}\/\\bar g_1 n_{c1}$.\\\\\nIn the limit $k \\rightarrow 0$, we have $\\varepsilon_{jk}= \\hbar c_j k$ where $c_j= \\sqrt{\\bar g_j n_{cj} \/m_j}$ is the sound velocity of a single condensate. \nThe total dispersion is phonon-like in this limit\n\\begin{equation} \\label{sound}\n\\varepsilon_{k (+,-)}= \\hbar c_{+,-} k,\n\\end{equation} \nwhere the sound velocities $c_{+,-}$ are\n \\begin{equation} \\label{sound1}\nc_{+,-} ^2=\\frac{1}{2} \\left[ c_1^2+c_2^2 \\pm \\sqrt{ \\left( c_1^2-c_2^2\\right) ^2 + 4 \\Delta^{-1} c_1^2 c_2^2} \\right].\n\\end{equation}\nFor $g_{12}^2> \\bar g_1\\, \\bar g_2$, the spectrum (\\ref{Bog}) becomes unstable and thus, the two condensates spatially separate.\nWe can see that the sound velocity $c_{+,-} \\rightarrow 0$ as $T\\rightarrow T_c$ since $n_{cj}=\\tilde m_j=0$ near the transition, \nwhich means that the phonons in the TDHFB theory are the soft modes of the Bose-condensed mixture. \n\n\n\\subsection{Quantum and thermal fluctuations}\n\nA straightforward calculation using Eq.(\\ref{Inv1}) permits us to rewrite the normal and anomalous densities in terms of $\\sqrt{I_k}$ \\cite{Boudj,Boudj4}\n\\begin{equation}\\label {nor}\n\\tilde n_j=\\frac{1}{2}\\int \\frac{d \\bf k} {(2\\pi)^3} \\left[\\frac{E_k+ \\mu_{+,-}} {\\varepsilon_{k (+,-)}} \\sqrt{I_{kj}}-1\\right],\n\\end{equation}\nand\n\\begin{equation}\\label {anom}\n\\tilde m_j=-\\frac{1}{2}\\int \\frac{d \\bf k} {(2\\pi)^3} \\frac{ \\mu_{+,-} } {\\varepsilon_{k (+,-)}} \\sqrt{I_{kj}}.\n\\end{equation}\n\nAt $T=0$, the total depletion $\\tilde n=\\tilde n_1+\\tilde n_2$ can be calculated via the integral (\\ref {nor})\n\\begin{equation}\\label {nor1}\n\\tilde n= \\frac{1}{6 \\sqrt{2} \\pi^2} \\left(\\frac{1}{\\xi_+^3}+\\frac{1}{\\xi_-^3} \\right),\n\\end{equation}\nwhere $\\xi_{+,-}=\\hbar\/\\sqrt{m \\mu_{+,-}}$.\\\\\nAs remarked in integral (\\ref{anom}), dimensional analysis suggests that we face the ultraviolet divergences in the expression of $\\tilde m$ as anticipated above.\nThis problem can be cured by means of the dimensional regularization \\cite{Boudj, Zin, Klein, Anders} which follows from perturbation theory of scattering.\nIt gives asymptotically exact results at weak interactions (for more details, see Appendix A of \\cite{Boudj}).\nThis yields for the total anomalous density $\\tilde m=\\tilde m_1+\\tilde m_2$:\n\\begin{equation}\\label {anom1} \n\\tilde m= \\frac{1}{2\\sqrt{2} \\pi^2} \\left(\\frac{1}{\\xi_+^3}+\\frac{1}{\\xi_-^3} \\right).\n\\end{equation}\nImportantly, the above expressions of the noncondensed and anomalous densities are proportional to $g_j^2$ and $g_{12}^2$.\nFor $g_{12}=0$, Eqs.(\\ref {nor1}) and (\\ref{anom1}) recover those obtained by the second-order Beliaev theory \\cite{Beleav} and the perturbative\ntime-dependent mean-field scheme \\cite{Giorg}.\n\n\nFrom now onward, we assume that $\\tilde m\/ n_c \\ll 1$, this condition is valid at low\ntemperature and necessary for the diluteness of the system \\cite{Yuk, Boudj2015}.\nTherefore, the condensate depletion (\\ref{nor1}) reduces to \n\\begin{equation}\\label {norT}\n\\tilde n= \\frac{1}{2 \\sqrt{2}} \\tilde n_1^0 \\left [f_+^{3\/2} (\\Delta, \\alpha)+f_-^{3\/2} (\\Delta, \\alpha) \\right],\n\\end{equation}\nwhere $\\tilde n_1^0=(8\/3) n_{c1} \\sqrt{n_{c1} a_1^3\/\\pi}$ is the single condensate depletion (type-1).\nThe depletion (\\ref{norT}) is formally similar to that obtained by Tommasini et \\textit {al}. \\cite{Tom} \nusing the Bogoliubov theory, with only $n_{cj}$ appearing as a corrected parameter instead of the total density $n_j$. \nAt $T=0$ and for fixed density $\\tilde n_1^0$, the noncondensed fraction is proportional to $\\Delta$ and $\\alpha$\nsignaling that the number of excited atoms increases with $\\Delta$ and $\\alpha$ as is displayed in Fig.\\ref{dd}.\\\\\nThe anomalous density of the mixture (\\ref{anom1}) simplifies\n\\begin{equation}\\label {anomT} \n\\tilde m=\\frac{1}{2 \\sqrt{2}} \\tilde m_1^0 \\left [f_+^{3\/2} (\\Delta, \\alpha)+f_-^{3\/2} (\\Delta, \\alpha) \\right],\n\\end{equation}\nwhere $\\tilde m_1^0= 8 n_{c1} \\sqrt{n_{c1} a_1^3\/\\pi}$ is the anomalous density of a single component.\nTo the best of our knowledge, Eq.(\\ref {anomT}) has never been derived in the literature.\nIt shows that $\\tilde m$ is larger than $\\tilde n$ similarly to the case of a single component. \nThis indicates that the anomalous density is significant even at zero temperature in Bose-Bose mixtures.\nWe see also that $\\tilde m$ is increasing with $(n_{c1} a^3)^{1\/2}$, $\\Delta$ and $\\alpha$.\nIf the interspecies and intraspecies interactions were strong enough, the pair anomalous density becomes important results in\na large fraction of the total atoms would occupy the excited states.\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=0.8]{depM.eps}}\n \\caption{(Color online) Condensed depletion from Eq.(\\ref{norT}) as a function of $\\Delta$ for different values $\\alpha$.}\n\\label{dd} \n\\end{figure}\n\nAt temperatures $T\\ll g_1 n_{c1} $, the main contribution to integrals (\\ref{nor}) and (\\ref{anom}) comes from the long-wavelength region\nwhere the spectrum takes the form (\\ref{sound}).\nThen the use of the integral $\\int_0^ {\\infty} {x^{2j-1}\\left[\\coth(\\alpha x)-1\\right]dx}=\\pi^{2j}|B_{2j}|\/ 2j\\alpha^{2j}$ \\cite{Yuk},\nwhere $B_{2j}$ are the Bernoulli number, allows us to obtain the following expressions for the thermal\ncontribution of the noncondensed and anomalous densities:\n\\begin{align} \\label {thfluc}\n\\tilde{n}_{th} &=|\\tilde{m}_{th} | = \\frac{2 \\sqrt{2} }{3} n_{c1} \\sqrt{\\frac{ n_{c1} a^3}{\\pi}} \\left(\\frac{\\pi T} {n_{c1} g_1}\\right)^2 \\\\\n&\\times \\left [f_+^{-1\/2} (\\Delta, \\alpha)+f_-^{-1\/2} (\\Delta, \\alpha) \\right]. \\nonumber\n\\end{align}\nEquation (\\ref{thfluc}) shows clearly that $\\tilde n$ and $\\tilde m$ are of the same order of magnitude at low temperature and only their signs are opposite. \nA comparaison between Eqs.(\\ref{norT}), (\\ref{anomT}) and (\\ref{thfluc}) shows that at $T\\ll gn_c$, thermal fluctuations are smaller than\nthe quantum fluctuations.\n\nLet us now discuss some relevant cases predicted by Eqs.(\\ref{norT}) and (\\ref{anomT}), \nfor a balanced mixture where $n_1=n_2=n$, and $g_1=g_2=g_{12}$. Hence, the noncondensed and the anomalous densities reduce respectively to\n$\\tilde n= (4 \/\\sqrt{2}) \\, \\tilde n^0$ and $\\tilde m= (4 \/\\sqrt{2}) \\, \\tilde m^0$, whereas at low temperature, the thermal depletion and the anomalous density \nturn out to be given as $\\tilde{n}_{th} =\\tilde{m}_{th} =(2\/3) n_{c1} \\sqrt{n_{c1} a_1^3\/\\pi} \\left(\\pi T\/n_{c1} g_1\\right)^2$. \nNear the phase separation where $ g_{12}^2 \\rightarrow \\bar g_1 \\bar g_2$, \nthe condensate depletion and the anomalous density become, respectively $\\tilde n= (1 + \\alpha)^{3\/2} \\tilde n_1^0 $ and $\\tilde m= (1 + \\alpha)^{3\/2} \\tilde m_1^0 $.\nAt low temperature, the lower branch has the free-particle dispersion law: $\\varepsilon_{k-} = E_k$ \\cite {CFetter} \nwhile the upper branch is phonon-like $\\varepsilon_{k+} =\\hbar c_1 (1 + \\alpha)^{1\/2} k$. \nTherefore, the thermal depletion has a distinct temperature dependence as\n\\begin{align} \\label {thfluc1}\n\\tilde n_{th} & = \\frac{2 }{3} n_{c1} \\sqrt{\\frac{ n_{c1} a_1^3}{\\pi}} (1 + \\alpha)^{-1\/2} \\left(\\frac{\\pi T} {n_{c1} g_1}\\right)^2 \\\\\n&+ \\left(\\frac{m T} {2\\pi \\hbar^2}\\right)^{3\/2} \\zeta (3\/2), \\nonumber \n\\end{align}\nwhere $\\zeta (3\/2)$ is the Riemann Zeta function. \nThe second term in (\\ref{thfluc1}) is the density of noncondensed atoms in a noninteracting gas. \nThis reveals that the component associated with lower branch becomes ultradilute.\nNotice that a similar temperature dependence distinction was obtained earlier by Colson and Fetter \\cite {CFetter} for ${}^4$He-${}^6$He mixture. \nSuch a distinction in the temperature dependence cannot occur in $\\tilde m_{th}$ where the term $\\propto T^{3\/2}$ \nis absent since the anomalous density itself does not exist in an ideal gas \\cite{Boudj, Griffin}.\n\n\n\n\\subsection{Thermodynamics}\n\nCorrections to the EoS of the mixture due to quantum and thermal fluctuations can be derived from Eq.(\\ref{EoS}). \nCombining Eqs.(\\ref{norT}), (\\ref{anomT}) and (\\ref{thfluc}) gives \n\\begin{align} \\label{EoS1}\n\\delta \\mu &=\\frac{\\mu_1^0 }{2 \\sqrt{2}} \\bigg\\{ \\left [f_+^{3\/2} (\\Delta, \\alpha)+f_-^{3\/2} (\\Delta, \\alpha) \\right] \\\\\n&+\\frac{1}{2} \\left [f_+^{-1\/2} (\\Delta, \\alpha)+f_-^{-1\/2} (\\Delta, \\alpha) \\right] \\left(\\frac{\\pi T} {n_{c1} g_1}\\right)^2 \\bigg\\}, \\nonumber\n\\end{align}\nwhere $\\mu_1^0= (32\/3) g_1 n_{c1} \\sqrt{n_{c1} a_1^3\/\\pi}$ is the zero temperature chemical potential of a single condensate.\nAt zero temperature and for $g_{12}=0$, Eq.(\\ref{EoS1}) reduces to the seminal Lee-Huang-Yang (LHY) corrected EoS \\cite{LHY} for \none component BEC.\n\n\n\nAt finite temperature, the grand-canonical ground state energy can be calculated using the thermodynamic relation\n$E=E_0+\\delta E=-T^2 \\left( \\frac{\\partial} {\\partial T } \\frac{F}{T} \\right) |_{V,\\mu}$ where the free energy is given by \n$F=E+T\\sum_{\\bf k}\\ln[1-\\exp(-\\varepsilon_{k(+,-)}\/T)]$, and \n$E_0=(g_j\/2) \\sum_j ( n_{cj}^2+ 4 n_{cj} \\tilde n_j +2\\tilde n_j^2 +|\\tilde m_j|^2 + 2 n_{cj} \\tilde m_j) + g_{12} n_1 n_2$. \nWhen $\\tilde m_j\/n_{cj} \\ll1$ and $\\tilde n_j\/n_{cj} \\ll1$, one has $E_0=(g_j\/2) \\sum_j n_{cj}^2 + g_{12} n_{c1} n_{c2}$. \nThe shift to the ground state energy due to quantum and thermal fluctuations is defined as \n\\begin{align} \\label{CGSE}\n\\delta E&= \\frac{1}{2} \\sum_j \\bigg[\\sum_k \\left[\\varepsilon_{k(+,-)} -( E_k+\\mu_{+,-}) \\right] \\\\\n&+\\sum_k \\varepsilon_{k(+,-)} \\left(\\sqrt{I_{kj}}-1 \\right) \\bigg]. \\nonumber\n\\end{align}\nThe first term on the r.h.s of (\\ref{CGSE}) which represents the energy corrections due to quantum fluctuations is \nultraviolet divergent. To circumvent such a divergency, we will use the standard dimensional regularization.\nThe second term accounts for the thermal fluctuation contributions to the energy. The main contribution to it comes from the phonon region.\nAfter some algebra, we obtain\n\\begin{align} \\label{GSE}\n\\delta E&=\\frac{1}{4\\sqrt{2}} E_1^0 \\bigg\\{ \\left [f_+^{5\/2} (\\Delta, \\alpha)+f_-^{5\/2} (\\Delta, \\alpha) \\right] \\\\\n&+\\frac{1}{2\\sqrt{2}} \\left [f_+^{-3\/2} (\\Delta, \\alpha)+f_-^{-3\/2} (\\Delta, \\alpha) \\right] \\left(\\frac{\\pi T} {n_{c1} g_1}\\right)^4 \\bigg\\}, \\nonumber\n\\end{align}\nwhere $E_1^0\/V= (64\/15) g_1 n_{c1}^2 \\sqrt{ n_{c1} a_1^3\/\\pi}$ is the zero-temperature single condensate ground state energy which can be obtained also\nby integrating the chemical potential with respect to the density.\nThe same result could be obtained within the renormalization of coupling constants which consists of\nadding $ g_j ^2 \\int d {\\bf k} \/E_k$ and $g_{12} ^2 \\int d {\\bf k} \/E_k$ \\cite{Larsen, Petrov} to the r.h.s of Eq.(\\ref{CGSE}).\n\nIn the case $g_{12}=0$, we read off from (\\ref{GSE}) that $\\delta E$ reduces to the ground-state energy of a single Bose gas. \nAt $T=0$ and for $n_c=n$, $\\delta E$ becomes identical to the Larsan's formula \\cite{Larsen}.\nEquation (\\ref{GSE}) is a finite-temperature extension of that recently obtained by Cappellaro et \\textit{al}. \\cite{Capp} for a balanced mixture using \nthe functional integration formalism within a regularization of divergent Gaussian fluctuations.\nIndeed, the resulting ground-state energy is appealing since it furnishes an extra repulsive term proportional to $n_c^{5\/2}+ n_c^{-3\/2} T^4$\nbalancing the attractive mean-field term, allowing quantum and thermal fluctuations to stabilize mixture droplets at finite temperature.\nQuantum stabilization and the related droplet nucleation was proposed in Bose-Bose mixtures with \\cite{Petrov, PetAst, Cab, Sem} and without Rabi coupling \\cite{Capp} \nas well as applied on dipolar condensates \\cite {Pfau,Wach, Bess, Chom}.\nThe finite-temperature generalization of these LHY corrections to the case of a dipolar Bose gas has been also analyzed in our recent work \\cite{BoudjDp}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion and outlook} \\label{concl}\n\nIn this paper we have systematically studied effects of quantum and thermal fluctuations on the dynamics and the collective excitations of \na two-component Bose gas utilizing the TDHFB theory.\nWe revealed that our approach is able to capture the qualitative evolution of two-component BECs at finite temperature.\nThe impact of the anomalous fluctuations on the miscibility criterion for the mixture was discussed.\n\nWithin an appropriate numerical method, we elucidated the behavior of the condensed, the noncondensed and the anomalous densities in terms of temperature \nfor both miscible and immiscible mixtures under spherical harmonic confinement.\nWe demonstrated in particular that the mixture undergoes a transition from miscible to immiscible regime \nowing to the predominant contribution of anomalous fluctuations notably at $T\\simeq 0.5 T_c$.\nWe found that such fluctuations are also the agent responsible for the strong damping of the relative motion of the centers-of-mass of the condensed and thermal components.\nAt zero temperature, the TDHFB results correct the existing theoretical models, making the theory in good agreement with JILA experiment \\cite{Hall}.\n\nWe linearized our TDHFB equations using the RPA for a weakly interacting uniform Bose-Bose mixture.\nThis method is a finite-temperature extension of the famous second-order Beliaev approximation \\cite {Beleav}.\nThe TDHFB-RPA theory provides us with analytical machinery powerful enough to calculate quantum and thermal \nfluctuation corrections to the excitations, the sound velocity, the EoS and the ground-state energy. \nWe compared the theory with previous theoretical treatments and excellent agreement has been found in the limit $\\tilde m\/n_c \\ll 1$.\nOne should stress that the results of our TDHFB-RPA technique can be generalized to the case of a harmonically trapped mixture using the local density approximation. \n\nThe findings of this work are appealing for investigating the properties of mixture droplets at finite temperature.\nAn important topic for future work is to look at how thermal fluctuations manifest themselves in dipolar Bose-Bose mixtures.\n\n\n\n\\section{Acknowledgements}\nWe are indebted to Eugene Zaremba and Hui Hu for stimulating discussions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}