diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzacxp" "b/data_all_eng_slimpj/shuffled/split2/finalzzacxp" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzacxp" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{S-intro}\n\n\n\n\nRotation and translation surfaces with constant mean curvature in\n$\\mathbb{H}^2 \\times \\mathbb{R}$ have been studied in details in \\cite{ST05, Sa08,\nST08} together with applications. We have studied rotation and\ntranslation minimal hypersurfaces with applications in\n\\cite{BS08a}.\\bigskip\n\nIn this paper, we consider constant non-zero mean curvature\nhypersurfaces in $\\mathbb{H}^n \\times \\mathbb{R}$.\\bigskip\n\n\nWe consider rotation $H$-hypersurfaces in Section\n\\ref{SS-dim3-Hrot}. For $H > \\frac{n-1}{n}$, we find the constant\nmean curvature sphere-like hypersurfaces obtained in \\cite{HH89} and\nthe Delaunay-like hypersurfaces obtained in \\cite{PR99}. When $0< H\n\\le \\frac{n-1}{n}$, we obtain complete simply-connected\nhypersurfaces ${\\mathcal S}_H$ which are entire vertical graphs above\n$\\mathbb{H}^n$, as well as some complete embedded or complete immersed\ncylinders which are bi-graphs (Theorems~\\ref{T-h3-r0} and\n\\ref{T-h3-rp}). When $H=\\frac{n-1}{n}$, the asymptotic behaviour of\nthe height function of these hypersurfaces is exponential, and it\nonly depends on the dimension when $n \\ge 3$. In\nSection~\\ref{S-appli}, we give geometric applications using the\nsimply-connected rotation $H$-hypersurfaces ${\\mathcal S}_H$ ($0 < H \\le\n\\frac{n-1}{n}$) mentioned above as barriers. We give existence and\ncharacterization of vertical $H$-graphs ($0 < H \\le \\frac{n-1}{n}$)\nover appropriate bounded domains (Proposition \\ref{P-appl-2}) as\nwell as symmetry and uniqueness results for compact hypersurfaces\nwhose boundary is one or two parallel submanifolds in slices\n(Theorems \\ref{T-appl-6} and \\ref{T-appl-7}). These results\ngeneralize the $2$-dimensional results obtained previously in\n\\cite{NSST08}.\\bigskip\n\nWe treat translation $H$-hypersurfaces in Section\n\\ref{SS-dim3-Htransl} (Theorem \\ref{T-tra-1}). When $n\\ge 3$ and\n$H=\\frac{n-1}{n}$, we in particular find a complete embedded\nhypersurface generated by a compact, simple, strictly convex\ncurve.\\bigskip\n\nWhen $0 < H <\\frac{n-1}{n}$, we obtain a complete non-entire\nvertical graph over the non-mean convex domain bounded by an\nequidistant hypersurface $\\Gamma$. This graph takes infinite\nboundary value data on $\\Gamma$ and it has infinite asymptotic\nboundary value data. \\bigskip\n\n\nThe authors would like to thank the Mathematics Department of\nPUC-Rio (PB) and the Institut Fourier -- Universit\\'{e} Joseph Fourier\n(RSA) for their hospitality. They gratefully acknowledge the\nfinancial support of CNPq, FAPERJ (in particular \\emph{Pronex} and\n\\emph{Cientistas do nosso Estado}), Acordo Brasil - Fran\\c{c}a,\nUniversit\\'{e} Joseph Fourier and R\\'{e}gion Rh\\^{o}ne-Alpes.\\bigskip\n\n\n\\bigskip\n\\section{Examples of $H$-hypersurfaces in $\\mathbb{H}^n \\times \\mathbb{R}$}\\label{S-examples}\n\n\nWe consider the ball model for the hyperbolic space $\\mathbb{H}^n$,\n\n$$\\mathbb{B} := \\ens{(x_1, \\ldots , x_n) \\in \\mathbb{R}^n}{x_1^2 + \\cdots + x_n^2 < 1},$$\n\nwith the hyperbolic metric $g_{\\mathbb{B}}$,\n\n$$g_{\\mathbb{B}} := 4 \\big( 1 - (x_1^2 + \\cdots + x_n^2) \\big)^{-2} \\big( dx_1^2 +\n\\cdots + dx_n^2\\big),$$\n\nand the product metric\n\n$$\\hat{g} = g_{\\mathbb{B}} + dt^2$$\n\non $\\mathbb{H}^n \\times \\mathbb{R}$. \\bigskip\n\n\n\n\\subsection{Rotation $H$-hypersurfaces in $\\mathbb{H}^n \\times \\mathbb{R}$}\n\\label{SS-dim3-Hrot} \\bigskip\n\n\nThe mean curvature equation for rotation hypersurfaces,\n\n\\begin{equation*}\\label{E-h3-rot1a}\nn H(\\rho) \\sinh^{n-1}(\\rho) = \\partial_{\\rho} \\Big(\n\\sinh^{n-1}(\\rho) \\dot{\\lambda}(\\rho) (1 +\n\\dot{\\lambda}^2(\\rho))^{-1\/2}\\Big)\n\\end{equation*}\\medskip\n\ncan be established using the flux formula, see Appendix\n\\ref{S-vflux}. We consider rotation hypersurfaces about $\\{0\\}\n\\times \\mathbb{R}$, where $\\rho$ denotes the hyperbolic distance to the axis\nand the mean curvature is taken with respect to the unit normal\npointing upwards. \\bigskip\n\nMinimal rotation hypersurfaces in $\\mathbb{H}^n \\times \\mathbb{R}$ have been\nstudied in \\cite{ST05} in dimension $2$ and in \\cite{BS08a} in\nhigher dimensions. In this Section we consider the case in which $H$\nis a non-zero constant. We may assume that $H$ is positive.\\bigskip\n\nIntegrating the above differential equation, we obtain the equation\nfor the generating curves of rotation $H$-hypersurfaces in $\\mathbb{H}^n\n\\times \\mathbb{R}$,\n\n\\begin{equation}\\label{E-h3-rot1}\n\\dot{\\lambda}(\\rho) \\big( 1+ \\dot{\\lambda}^2(\\rho)\\big)^{-1\/2}\n\\sinh^{n-1}(\\rho) = nH \\int_0^{\\rho} \\sinh^{n-1}(t) \\, dt + d\n\\end{equation}\n\nfor $H > 0$ and for some constant $d$. \\bigskip\n\nThis equation has been studied in \\cite{NSST08, ST05} in dimension\n$2$ (with a different constant $d$). \\bigskip\n\n\\noindent \\textbf{Notations.}~ For later purposes we introduce some\nnotations. \\bigskip\n\n\\noindent $\\bullet $~ For $m \\ge 0$, we define the function $I_m(t)$ by\n\\begin{equation}\\label{E-h3-rot2}\nI_m(t) := \\int_0^{t} \\sinh^m(r) \\, dr.\n\\end{equation} \\bigskip\n\n\\noindent $\\bullet $~ For $H>0$ and $d \\in \\mathbb{R}$, we define the functions,\n\n\\begin{equation}\\label{E-h3-4}\n\\left\\{%\n\\begin{array}{lll}\n M_{H,d}(t) &:=& \\sinh^{n-1}(t) - nH I_{n-1}(t) - d,\\\\\n P_{H,d}(t) &:=& \\sinh^{n-1}(t) + nH I_{n-1}(t) + d,\\\\\n Q_{H,d}(t) &:=& \\big[nH I_{n-1}(t) + d\\big] \\big[M_{H,d}(t) \\,\n P_{H,d}(t)\\big]^{-1\/2}, \\\\\n && \\text{when the square root exists.}\\\\\n\\end{array}%\n\\right.\n\\end{equation}\\bigskip\n\nWe see from (\\ref{E-h3-rot1}) that $\\dot{\\lambda}(t)$ has the sign\nof $n H I_{n-1}(t) +d$. It follows that $\\lambda$ is given, up to an\nadditive constant, by\n\n$$\\lambda_{H,d}(\\rho) = \\int_{\\rho_0}^{\\rho} \\frac{nH I_{n-1}(t) +\nd}{\\sqrt{\\sinh^{2n-2}(t) - \\big(nH I_{n-1}(t) + d\\big)^2}}\\, dt$$\n\nor, with the above notations,\n\n\\begin{equation}\\label{E-h3-rot3}\n\\lambda_{H,d}(\\rho) = \\int_{\\rho_0}^{\\rho} \\frac{nH\nI_{n-1}(t)+d}{\\sqrt{M_{H,d}(t) P_{H,d}(t)}}\\, dt =\n\\int_{\\rho_0}^{\\rho} Q_{H,d}(t) \\, dt\n\\end{equation}\n\nwhere the integration interval $[\\rho_0,\\rho]$ is contained in the\ninterval in which the square-root exists. The existence and\nbehaviour of the function $\\lambda_{H,d}$ depend on the signs of the\nfunctions $nH I_{n-1}(t) + d$, $M_{H,d}(t)$ and $P_{H,d}(t)$.\n\\bigskip\n\n\nUp to vertical translations, the rotation hypersurfaces about the\naxis $\\{0\\} \\times \\mathbb{R}$, with constant mean curvature $H>0$ with\nrespect to the unit normal pointing upwards, can be classified\naccording to the sign of $H - \\frac{n-1}{n}$ and to the sign of $d$.\nWe state three theorems depending on the value of $H$. \\bigskip\n\n\\newpage\n\n\\begin{thm}[Rotation $H$-hypersurfaces with $H=\\frac{n-1}{n}$]\\label{T-h3-r0}\n$ $\n \\begin{enumerate}\n \\item When $d=0$, the hypersurface ${\\mathcal S}_{\\frac{n-1}{n}}$ is a\n simply-connected entire vertical graph above $\\mathbb{H}^n \\times \\{0\\}$,\n tangent to the slice at $0$, generated by a strictly convex\n curve. The height function $\\lambda (\\rho)$ on ${\\mathcal S}_{\\frac{n-1}{n}}$\n grows exponentially.\n\n \\item When $d>0$, the hypersurface ${\\mathcal C}_{\\frac{n-1}{n}}$ is a complete\n embedded cylinder, symmetric with respect to the slice $\\mathbb{H}^n \\times \\{0\\}$.\n The parts ${\\mathcal C}_{\\frac{n-1}{n}}^{\\pm} := {\\mathcal C}_{\\frac{n-1}{n}} \\cap \\mathbb{H}^n\n \\times \\mathbb{R}_{\\pm}$ are vertical graphs above the exterior of a ball\n $B(0,a)$, for some constant $a > 0$ depending on $d$.\n The height function $\\lambda (\\rho)$ on ${\\mathcal C}_{\\frac{n-1}{n}}^{\\pm}$ grows\n exponentially. When $n=2$, the solution exists when $0< d<1$ only.\n\n \\item When $d<0$, the hypersurface ${\\mathcal D}_{\\frac{n-1}{n}}$ is complete and\n symmetric with respect to the slice $\\mathbb{H}^n \\times \\{0\\}$. It has\n self-intersections along a sphere in $\\mathbb{H}^n \\times \\{0\\}$. The parts\n ${\\mathcal D}_{\\frac{n-1}{n}}^{\\pm} := {\\mathcal D}_{\\frac{n-1}{n}} \\cap \\mathbb{H}^n\n \\times \\mathbb{R}_{\\pm}$ are vertical graphs above the exterior of a ball\n $B(0,a)$, for some constant $a > 0$ depending on $d$. The height\n function $\\lambda (\\rho)$ on ${\\mathcal D}_{\\frac{n-1}{n}}^{\\pm}$ grows exponentially.\n \\end{enumerate}\n\n The asymptotic behaviour of the height function when $\\rho$ tends to\n infinity is as follows.\n$$\n\\left\\{%\n\\begin{array}{l}\n\\text{For~ } n=2, ~\\lambda (\\rho) \\sim\n \\frac{e^{\\rho\/2}}{\\sqrt{1 - d}}.\\\\[8pt]\n\\text{For~ } n=3, ~\\lambda (\\rho) \\sim\n \\frac{1}{2 \\sqrt{2}} \\int^{\\rho} \\frac{e^{t}}{\\sqrt{t}}\\, dt.\\\\[8pt]\n\\text{For~ } n \\ge 4, ~\\lambda (\\rho) \\sim a(n) e^{b(n)t}, \\text{\n~for\nsome positive constants~ } a(n), b(n).\\\\\n\\end{array}%\n\\right.\n$$\n\\end{thm}\n\nThe generating curves are obtained by symmetries from the curves\n$(=)$ (standing for $H = \\frac{n-1}{n}$) which appear in\nFigures~\\ref{F-rot-1}-\\ref{F-rot-3}. \\bigskip\n\n\n\\begin{pb1-figs}\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{minipage}[c]{6.5cm}\n \\includegraphics[width=6.5cm]{f-rot-1.eps}\n \\caption[Case $d=0$]{Case $d=0$}\n \\label{F-rot-1}\n\\end{minipage}\\hfill\n\\begin{minipage}[c]{6.5cm}\n \\includegraphics[width=6.5cm]{f-rot-2.eps}\n \\caption[Case $d>0$]{Case $d>0$}\n \\label{F-rot-2}\n\\end{minipage}\\hfill\n\\end{center}\n\\end{figure}\\bigskip\n\\end{pb1-figs}\n\n\\begin{pb1-figs}\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{minipage}[c]{6.5cm}\n \\includegraphics[width=6.5cm]{f-rot-3.eps}\n \\caption[Case $d<0$]{Case $d<0$}\n \\label{F-rot-3}\n\\end{minipage}\\hfill\n\\end{center}\n\\end{figure}\\bigskip\n\\end{pb1-figs}\n\n\\textbf{Remark.}~ When $n=2$ the asymptotic growth depends on the\nvalue of the integration contant $d$. \\bigskip\n\n\n\\begin{thm}[Rotation $H$-hypersurfaces with $0 < H < \\frac{n-1}{n}$]\\label{T-h3-rp}\n$ $\n \\begin{enumerate}\n \\item When $d=0$, the hypersurface ${\\mathcal S}_{H}$ is a\n simply-connected entire vertical graph above $\\mathbb{H}^n \\times \\{0\\}$,\n tangent to the slice at $0$, generated by a strictly convex\n curve. The height function $\\lambda (\\rho)$ on ${\\mathcal S}_{H}$ grows linearly.\n\n \\item When $d>0$, the hypersurface ${\\mathcal C}_{H}$ is a complete embedded cylinder,\n symmetric with respect to the slice $\\mathbb{H}^n \\times \\{0\\}$. The parts\n ${\\mathcal C}_{H}^{\\pm} := {\\mathcal C}_{H} \\cap \\mathbb{H}^n \\times \\mathbb{R}_{\\pm}$ are vertical graphs above\n the exterior of a ball $B(0,a)$, for some constant $a > 0$ depending on\n $H$ and $d$. The height function $\\lambda (\\rho)$ on ${\\mathcal C}_{H}^{\\pm}$ grows linearly.\n\n \\item When $d<0$, the hypersurface ${\\mathcal D}_{H}$ is complete and symmetric with\n respect to the slice $\\mathbb{H}^n \\times \\{0\\}$. It has self-intersections along a\n sphere in $\\mathbb{H}^n \\times \\{0\\}$. The parts ${\\mathcal D}_{H}^{\\pm} := {\\mathcal D}_{H} \\cap\n \\mathbb{H}^n \\times \\mathbb{R}_{\\pm}$ are vertical graphs above\n the exterior of a ball $B(0,a)$, for some constant $a > 0$ depending on $H$\n and $d$. The height function $\\lambda(\\rho)$ on ${\\mathcal D}_{H}^{\\pm}$ grows linearly.\n \\end{enumerate}\n\nThe asymptotic behaviour of the height function when $\\rho$ tends to\ninfinity is given by\n$$\\lambda (\\rho) \\sim \\dfrac{\\frac{nH}{n-1}}{\\sqrt{1 - (\\frac{nH}{n-1})^2}} \\, \\rho.$$\n\\end{thm}\\bigskip\n\nThe generating curves are obtained by symmetries from the curves\n$(<)$ (standing for $H < \\frac{n-1}{n}$) which appear in\nFigures~\\ref{F-rot-1}-\\ref{F-rot-3}.\\bigskip\n\n\\newpage\n\n\\begin{thm}[Rotation $H$-hypersurfaces with $H > \\frac{n-1}{n}$]\\label{T-h3-rg}\n$ $\n \\begin{enumerate}\n \\item When $d=0$, the hypersurface ${\\mathcal K}_H$ is\n compact and diffeomorphic to an $n$-dimensional sphere. It\n is generated by a compact, simple, strictly convex curve.\n\n \\item When $d>0$, the hypersurface ${\\mathcal U}_H$ is\n complete, embedded and periodic in the $\\mathbb{R}$-direction. It looks like\n an unduloid and is contained in a domain of the form $B(0,b)\n \\setminus B(0,a) \\times \\mathbb{R}$, for some constants $0 < a < b$,\n depending on $H$ and $d$.\n\n \\item When $d<0$, the hypersurface ${\\mathcal N}_H$ is\n complete and periodic in the $\\mathbb{R}$-direction. It has\n self-intersections, looks like\n a nodoid and is contained in a domain of the form $B(0,b)\n \\setminus B(0,a) \\times \\mathbb{R}$, for some constants $0 < a < b$\n depending on $H$ and $d$.\n \\end{enumerate}\n\\end{thm}\\bigskip\n\n\nThe generating curves are obtained by symmetries from the curves\n$(>)$ (standing for $H > \\frac{n-1}{n}$) which appear in\nFigures~\\ref{F-rot-1}-\\ref{F-rot-3}. \\bigskip\n\n\n\n\n\n\\textbf{Remarks}\\bigskip\n\n1.~ Constant mean curvature rotation hypersurfaces with $H >\n\\frac{n-1}{n}$ were obtained in \\cite{HH89} and \\cite{PR99}.\\medskip\n\n2.~ The hypersurfaces ${\\mathcal S}_H$ and the upper (lower) halves of the\ncylinders ${\\mathcal C}_H$ in Theorems~\\ref{T-h3-r0} and \\ref{T-h3-rp} are\nstable (as vertical graphs).\n\\bigskip\n\n\n\\subsection{Proofs of Theorem \\ref{T-h3-r0} - \\ref{T-h3-rg}}\n\n\nThe proofs follow from an analysis of the asymptotic behaviour of\n$I_m(t)$ (Formula (\\ref{E-h3-rot2})) when $t$ goes to infinity and\nfrom an analysis of the signs of the functions $nH I_{n-1}(t) + d$,\n$M_{H,d}(t)$ and $P_{H,d}(t)$ (Formulas (\\ref{E-h3-4})), using the\ntables which appear below.\n\\bigskip\n\nWhen $d=0$, using (\\ref{E-h3-rot1}) one can show that\n$\\ddot{\\lambda} > 0$ and conclude that the generating curve is\nstrictly convex. When $d \\le 0$, the formula for $\\ddot{\\lambda}$\nalso shows that the curvature extends continuously at the vertical\npoints.\\bigskip\n\n\n\\noindent \\textbf{Proof of Theorem \\ref{T-h3-r0}}\\bigskip\n\nAssume $H = \\frac{n-1}{n}$. \\bigskip\n\nWhen $d=0$, the functions $M_{H,0}$ and $P_{H,0}$ are non-negative\nand vanish at $t=0$. Near $0$ we have $Q_{H,0}(t) \\sim Ht$ and hence\n$\\lambda_{H,0}(\\rho) = \\int_0^{\\rho} Q_{H,0}(t) \\, dt \\sim\n\\frac{H}{2}\\rho^2$.\n\\bigskip\n\nWhen $d>0$, the function $Q_{H,d}$ exists on an interval $]a_{H,d},\n\\infty[$ for some constant $a_{H,d} > 0$ and the integral\n$\\int_{a_{H,d}}^{\\rho}Q_{H,d}(t) \\, dt$ converges at\n$a_{H,d}$.\\bigskip\n\nWhen $d<0$, the function $Q_{H,d}$ exists on an interval\n$]\\alpha_{H,d}, \\infty[$ for some constant $\\alpha_{H,d}\n> 0$, changes sign from negative to positive, the integral\n$\\int_{\\alpha_{H,d}}^{\\rho}Q_{H,d}(t) \\, dt$ converges at\n$\\alpha_{H,d}$ and the curve has a vertical tangent at this point.\nThe generating curve can be extended by symmetry to a complete curve\nwith one self-intersection.\\bigskip\n\nUsing the recurrence relations for the functions $I_m(t)$ one can\ndetermine their asymptotic behaviour at infinity and deduce the\nprecise exponential growth of the height function $\\lambda(\\rho)$.\n\n\\hfill \\hfill\\penalty10000\\copy\\qedbox\\par\\medskip\n\\bigskip\n\n\\noindent \\textbf{Proof of Theorem \\ref{T-h3-rp}}\\bigskip\n\nAssume $0 < H < \\frac{n-1}{n}$. \\bigskip\n\nWhen $d=0$, the functions $M_{H,0}$ and $P_{H,0}$ are non-negative\nand vanish at $t=0$. Near $0$ we have $Q_{H,0}(t) \\sim Ht$ and hence\n$\\lambda_{H,0}(\\rho) = \\int_0^{\\rho} Q_{H,0}(t) \\, dt \\sim\n\\frac{H}{2}\\rho^2$. \\bigskip\n\nWhen $d>0$, the function $Q_{H,d}$ exists on an interval $]a_{H,d},\n\\infty[$ for some constant $a_{H,d} > 0$ and the integral\n$\\int_{a_{H,d}}^{\\rho}Q_{H,d}(t) \\, dt$ converges at\n$a_{H,d}$.\\bigskip\n\nWhen $d<0$, the function $Q_{H,d}$ changes sign from negative to\npositive, exists on an interval $]\\alpha_{H,d}, \\infty[$ for some\nconstant $\\alpha_{H,d} > 0$, the integral\n$\\int_{\\alpha_{H,d}}^{\\rho}Q_{H,d}(t) \\, dt$ converges at\n$\\alpha_{H,d}$ and the generating curve has a vertical tangent at\nthis point. The generating curve can be extended by symmetry to a\ncomplete curve with one self-intersection.\\bigskip\n\nUsing the recurrence relations for the functions $I_m(t)$ one can\ndetermine their asymptotic behaviour at infinity and deduce the\nprecise linear growth of the height function $\\lambda(\\rho)$.\n\n\\hfill \\hfill\\penalty10000\\copy\\qedbox\\par\\medskip\n\\bigskip\n\n\\noindent \\textbf{Proof of Theorem \\ref{T-h3-rg}}\\bigskip\n\nAssume $H > \\frac{n-1}{n}$. \\bigskip\n\nWhen $d=0$, $Q_{H,0}(t)$ exists on some interval $]0,a_{H,0}[$ for\nsome positive $a_{H,0}$ and the integral $\\lambda_{H,0}(\\rho) =\n\\int_0^{\\rho} Q_{H,0}(t) \\, dt$ converges at $0$ and at $a_{H,0}$.\nThe generating curve has a horizontal tangent at $0$ and a vertical\ntangent at $a_H$. It can be extended by symmetries to a closed\nembedded convex curve.\n\\bigskip\n\nWhen $d>0$, the function $Q_{H,d}(t)$ exists on an interval\n$]b_{H,d},c_{H,d}[$ for some constants $0 < b_{H,d} < c_{H,d}$ and\nthe integral converges at the limits of this interval. The\ngenerating curve at these points is vertical. It can be extended by\nsymmetry to a complete embedded periodic curve (unduloid). \\bigskip\n\nWhen $d<0$, the function $Q_{H,d}(t)$ exists on an interval\n$]\\beta_{H,d},\\gamma_{H,d}[$ for some constants $0 < \\beta_{H,d} <\n\\gamma_{H,d}$, changes sign from negative to positive and the\nintegral converges at the limits of this interval. The generating\ncurve at these points is vertical. The generating curve can extended\nby symmetries to a complete periodic curve with self-intersections\n(nodoid).\n\n\\hfill \\hfill\\penalty10000\\copy\\qedbox\\par\\medskip\n\\bigskip\n\n\n\\textbf{Remark.}~ We note that the integrand $Q_{H,d}(t)$ in\n(\\ref{E-h3-rot3}) is an increasing function of $H$ for $t$ and $d$\nfixed. This fact provides the relative positions of the curves\n$\\lambda_{H,d}(\\rho)$ when $\\rho$ and $d$ are fixed. The curve\ncorresponding to $H > \\frac{n-1}{n}$ is above the curve\ncorresponding to $H = \\frac{n-1}{n}$ which is above the curve\ncorresponding to $H < \\frac{n-1}{n}$. See Figures \\ref{F-rot-1} to\n\\ref{F-rot-3}.\\bigskip\n\n\n\nThe above sketches of proof can be completed using the details\nbelow.\\bigskip\n\n\\noindent $\\bullet $~ We have the following relations for the functions $I_m$,\n\n\\begin{equation}\\label{E-h3-rot2a}\n\\left\\{%\n\\begin{array}{llll}\n m=0 & I_0(t) & = & t, \\\\\n m=1 & I_1(t) & = & \\cosh(t) - 1, \\\\\n m=2 & 2I_2(t) & = & \\sinh(t) \\cosh(t) - t, \\\\\n m=3 & 3 I_3(t) & = & \\sinh^2(t) \\cosh(t) - 2(\\cosh(t)-1), \\\\\n m\\ge 2 & m I_m(t) & = & \\sinh^{m-1}(t) \\cosh(t) - (m-1) I_{m-2}(t). \\\\\n\\end{array}\n\\right.\n\\end{equation}\\bigskip\n\nFor $m\\ge 5$, the asymptotic behavior of $I_m(t)$ near infinity is\ngiven by,\n\n\\begin{equation}\\label{E-h3-rot2b}\n\\left\\{%\n\\begin{array}{lll}\n m I_m(t) & = & \\sinh^{m-3}(t) \\cosh(t) \\big(\\sinh^2(t) - \\frac{m-1}{m-2}\\big)\n + O(e^{(m-4)t}),\\\\\n m I_m(t) & = & \\sinh^{m-1}(t) \\cosh(t) \\big( 1 + O(e^{-2t})\\big). \\\\\n\\end{array}\n\\right.\n\\end{equation}\\bigskip\n\nThe same holds for $m=4$ with remainder term $O(t)$ in the first\nrelation. \\bigskip\n\n\\noindent $\\bullet $~ The derivative of $P_{H,d}$ is positive for $t$ positive. The\nbehaviour of the function $P_{H,d}(t)$ is summarized in the\nfollowing table.\n\\bigskip\n\n\\begin{equation*}\n \\begin{array}{|c|ccc|}\n\\hline\nn\\ge 2 & & 0 < H &\\\\\n\\hline\n t & 0 & & \\infty \\\\\n\\hline\n \\partial_t P_{H,d} & & + & \\\\\n \\hline\n P_{H,d}(t) & d & \\nearrow & \\infty \\\\\n \\hline\n\\end{array}\n\\end{equation*}\\bigskip\n\n\n\\noindent $\\bullet $~ The derivative of $M_{H,d}$ is given by $\\partial_t M_{H,d}(t)\n= (n-1) \\sinh^{n-1}(t) \\big( \\coth(t) - \\frac{nH}{n-1}$\\big). For $H\n> \\frac{n-1}{n}$, we denote by $C_H$ the number such that\n$\\coth(C_H) = \\frac{nH}{n-1}$. The behaviour of the function\n$M_{H,d}(t)$ is summarized in the following tables.\n\n\\begin{equation*}\n \\begin{array}{|c|ccc||ccccc|}\n \\hline\n n=2 & & 0 < H \\le \\frac{1}{2} &&&& H > \\frac{1}{2}&&\\\\\n\\hline\n t & 0 & & \\infty & 0&&C_H&& \\infty \\\\\n\\hline\n \\partial_t M_{H,d} & & + & && +&0&-&\\\\\n\\hline\n M_{H,d}(t)& -d & \\nearrow &\n\\left\\{%\n\\begin{array}{lr}\n \\infty, & H < \\frac{1}{2} \\\\[4pt]\n 1-d, & H = \\frac{1}{2}\\\\\n\\end{array}\n\\right.\n & -d & \\nearrow & f_H(d) &\\searrow & -\\infty \\\\\n \\hline\n\\end{array}\n\\end{equation*}\\bigskip\n\n\n\\begin{equation*}\n \\begin{array}{|c|ccc||ccccc|}\n \\hline\nn\\ge 3 && 0 < H \\le \\frac{n-1}{n} && &&H > \\frac{n-1}{n}&&\\\\\n\\hline\n t & 0 & & \\infty &0&&C_H&& \\infty \\\\\n\\hline\n \\partial_t M_{H,d} & & + && &+& 0&-&\\\\\n\\hline\n M_{H,d}(t) & -d & \\nearrow & \\infty & -d & \\nearrow & f_H(d) &\\searrow & -\\infty \\\\\n\\hline\n\\end{array}\n\\end{equation*}\\bigskip\n\nwhere $f_H(d) := M_{H,d}(C_H) = \\sinh^{n-1}(C_H) - nH I_{n-1}(C_H)\n-d$. \\bigskip\n\n\n\nThe signs and zeroes of the functions $M_{H,d}(t)$ and $P_{H,d}(t)$\nwhen $d \\not = 0$ are summarized in the following charts, together\nwith the existence domain of the function $Q_{H,d}$.\\bigskip\n\nWhen $d > 0$, we have\n\n\n\\begin{equation*}\n \\begin{array}{|c|cccccc|}\n\\hline n=2 &&\n\\begin{array}{c}\n0 < H < \\frac{1}{2},\\\\\nH=\\frac{1}{2},\n\\end{array}\n&&\n\\begin{array}{c}\n0 < d \\\\\n0 \\frac{n-1}{n}\\\\\n0 < d < D_H\n\\end{array}\n&&&\\\\\n\\hline\nt & 0 & & b_{H,d} & C_{H} & c_{H,d} & & \\infty \\\\\n\\hline\nM_{H,d}& &-& 0 & + & 0& -& \\\\\n\\hline\nP_{H,d}& &+& & + & & +& \\\\\n\\hline\nQ_{H,d}& &\\not \\exists & +\\infty & \\exists & +\\infty & \\not \\exists & \\\\\n\\hline\n \\end{array}\n\\end{equation*}\\bigskip\n\n\n\nwhere $D_H := \\sinh^{n-1}(C_H) - nH I_{n-1}(C_H)$. \\bigskip\n\n\n\n\n\\newpage\n\nWhen $d<0$, we have the following tables.\\bigskip\n\n\\begin{equation*}\n \\begin{array}{|c|ccccccc|}\n\\hline n \\ge 2 &&&&\n\\begin{array}{c}\n0 < H \\le \\frac{n-1}{n}\\\\\nd < 0\n\\end{array}\n&&&\\\\\n\\hline\nt & 0 & & & \\alpha_{H,d} & & & \\infty \\\\\n\\hline\nM_{H,d}& && + & & +& & \\\\\n\\hline\nP_{H,d}& && - & 0 & +& & \\\\\n\\hline\nQ_{H,d}& & & \\not \\exists & - \\infty & \\exists & & \\\\\n\\hline\n \\end{array}\n\\end{equation*}\\bigskip\n\nNote that the function $Q_{H,d}$ changes sign from negative to\npositive when $t$ goes from $\\alpha_{H,d}$ to infinity.\\bigskip\n\n\\begin{equation*}\n \\begin{array}{|c|ccccccc|}\n\\hline n \\ge 2 &&&&\n\\begin{array}{c}\nH > \\frac{n-1}{n}\\\\\nd < 0\n\\end{array}\n&&&\\\\\n\\hline\nt & 0 & & \\gamma_{H,d} & & \\beta_{H,d} & & \\infty \\\\\n\\hline\nM_{H,d}& &+& + & +& 0& -& \\\\\n\\hline\nP_{H,d}& &-& 0 & + & +& +& \\\\\n\\hline\nQ_{H,d}& &\\not \\exists & -\\infty & \\exists & +\\infty & \\not \\exists & \\\\\n\\hline\n \\end{array}\n\\end{equation*}\\bigskip\n\n\nNote that the function $Q_{H,d}$ changes sign from negative to\npositive when $t$ goes from $\\gamma_{H,d}$ to $\\beta_{H,d}$.\\bigskip\n\n\n\n\n\n\\subsection{Translation invariant $H$-hypersurfaces\nin $\\mathbb{H}^n \\times \\mathbb{R}$}\\label{SS-dim3-Htransl}\n\\bigskip\n\n\\subsubsection{Translation hypersurfaces}\n\\bigskip\n\n\\noindent $\\bullet $~ \\textbf{Definitions and Notations.}~ We consider $\\gamma$ a\ngeodesic through $0$ in $\\mathbb{H}^n$ and the totally geodesic vertical\nplane $\\mathbb{V} = \\gamma \\times \\mathbb{R} = \\ens{(\\gamma (\\rho),t)}{(\\rho ,t) \\in\n\\mathbb{R} \\times \\mathbb{R}}$ where $\\rho$ is the signed hyperbolic distance to $0$\non $\\gamma$.\n\\bigskip\n\nTake $\\mathbb{P}$ a totally geodesic hyperplane in $\\mathbb{H}^n$, orthogonal to\n$\\gamma$ at $0$. We consider the hyperbolic translations with\nrespect to the geodesics $\\delta$ through $0$ in $\\mathbb{P}$. We shall\nrefer to these translations as translations with respect to $\\mathbb{P}$.\nThese isometries of $\\mathbb{H}^n$ extend ``slice-wise'' to isometries of\n$\\mathbb{H}^n \\times \\mathbb{R}$. \\bigskip\n\nIn the vertical plane $\\mathbb{V}$, we consider the curve $c(\\rho) := \\big(\n\\tanh(\\rho \/2), \\mu(\\rho)\\big)$. \\bigskip\n\nIn $\\mathbb{H}^n \\times \\{\\mu(\\rho)\\}$, we translate the point $c(\\rho)$ by\nthe translations with respect to $\\mathbb{P}\\times \\{\\mu(\\rho)\\}$ and we\nget the equidistant hypersurface $\\mathbb{P}_{\\rho}$ passing through\n$c(\\rho)$, at distance $\\rho$ from $\\mathbb{P}\\times \\{\\mu(\\rho)\\}$. The\ncurve $c$ then generates a \\emph{translation hypersurface} $M =\n\\cup_{\\rho}\\mathbb{P}_{\\rho}$ in $\\mathbb{H}^n \\times \\mathbb{R}$.\\bigskip\n\n\\noindent $\\bullet $~ \\textbf{Principal curvatures.}~ The principal directions of\ncurvature of $M$ are the tangent to the curve $c$ in $\\mathbb{V}$ and the\ndirections tangent to $\\mathbb{P}_{\\rho}$. The corresponding principal\ncurvatures with respect to the unit normal pointing upwards are\ngiven by\n\n\\begin{equation*}\\label{E-tra-1}\n\\left\\{%\n\\begin{array}{lll}\n k_{\\mathbb{V}} & = & \\ddot{\\mu}(\\rho) \\big( 1 + \\dot{\\mu}^2(\\rho) \\big)^{-3\/2}, \\\\\n k_{\\mathbb{P}} & = & \\dot{\\mu}(\\rho) \\big( 1 + \\dot{\\mu}^2(\\rho) \\big)^{-1\/2}\n \\tanh(\\rho). \\\\\n\\end{array}%\n\\right.\n\\end{equation*}\\bigskip\n\nThe first equality comes from the fact that $\\mathbb{V}$ is totally geodesic\nand flat. The second equality follows from the fact that\n$\\mathbb{P}_{\\rho}$ is totally umbilic and at distance $\\rho$ from\n$\\mathbb{P}\\times \\{\\mu(\\rho)\\}$ in $\\mathbb{H}^n \\times \\{\\mu(\\rho)\\}$. \\bigskip\n\n\n\\noindent $\\bullet $~ \\textbf{Mean curvature.}~ The mean curvature of the\ntranslation hypersurface $M$ associated with $\\mu$ is given by\n\n\n\n\\begin{equation}\\label{E-tra-3}\nn H(\\rho) \\cosh^{n-1}(\\rho) = \\partial_{\\rho} \\Big(\n\\cosh^{n-1}(\\rho) \\dot{\\mu}(\\rho) \\big( 1 + \\dot{\\mu}^2(\\rho)\n\\big)^{-1\/2} \\Big).\n\\end{equation}\\bigskip\n\n\n\\subsubsection{Constant mean curvature translation hypersurfaces}\n\\bigskip\n\n\nWe may assume that $H \\ge 0$. The generating curves of translation\nhypersurfaces with constant mean curvature $H$ are given by the\ndifferential equation\n\n\\begin{equation}\\label{E-tra-4}\n\\dot{\\mu}(\\rho) \\big( 1 + \\dot{\\mu}^2(\\rho) \\big)^{-1\/2}\n\\cosh^{n-1}(\\rho) = n H \\int_0^{\\rho} \\cosh^{n-1}(t) \\, dt + d\n\\end{equation}\n\nfor some integration constant $d$.\\bigskip\n\nMinimal translation hypersurfaces have been studied in \\cite{Sa08,\nST08} in dimension $2$ and in \\cite{BS08a} in higher dimensions.\nConstant mean curvature ($H \\not = 0$) translation hypersurfaces\nhave been treated in \\cite{Sa08} in dimension $2$. The purpose of\nthe present section is to investigate the higher dimensional\ntranslation $H$-hypersurfaces.\\bigskip\n\n\n\\textbf{Notations.}~ For later purposes, we introduce some\nnotations.\\bigskip\n\n\\noindent $\\bullet $~ For $m \\ge 0$, we define the functions\n\n\\begin{equation}\\label{E-tra-5a}\nJ_m(r) := \\int_0^r \\cosh^m(t) \\, dt.\n\\end{equation}\\bigskip\n\n\n\\noindent $\\bullet $~ For $H > 0$ and $d \\in \\mathbb{R}$, we introduce the functions,\n\n\\begin{equation}\\label{E-tra-11}\n\\left\\{%\n\\begin{array}{lll}\nR_{H,d}(t) & = & \\cosh^{n-1}(t) - nH J_{n-1}(t) -d ,\\\\\nS_{H,d}(t) & = & \\cosh^{n-1}(t) + nH J_{n-1}(t) +d ,\\\\\nT_{H,d}(t) & = & \\big[ nH J_{n-1}(t) + d \\big]\n\\big[R_{H,d(t)} S_{H,d}(t) \\big]^{-1\/2}.\\\\\n\\end{array}%\n\\right.\n\\end{equation}\\bigskip\n\n\n\nWe note from (\\ref{E-tra-4}) that $\\dot{\\mu}(t)$ has the sign of\n$nHJ_{n-1}(t) +d$. It follows that $\\mu$ is given (up to an additive\ncontant) by\n\n\\begin{equation*}\\label{E-tra-8a}\n\\mu_{H,d} (\\rho) = \\int_{\\rho_0}^{\\rho} \\big[ nH J_{n-1}(t) + d\n\\big] \\big[ \\cosh^{2n-2}(t) - \\big( nH J_{n-1}(t) + d \\big)^2\n\\big]^{-1\/2} \\, dt\n\\end{equation*}\n\nor, using the above notations,\n\n\n\\begin{equation}\\label{E-tra-8}\n\\mu_{H,d} (\\rho) = \\int_{\\rho_0}^{\\rho} \\big[ nH J_{n-1}(t) + d\n\\big] \\big[R_{H,d(t)} \\, S_{H,d}(t) \\big]^{-1\/2} \\, dt =\n\\int_{\\rho_0}^{\\rho} T_{H,d}(t) \\, dt \\, ,\n\\end{equation}\n\nwhere the integration interval $[\\rho_0, \\rho]$ is contained in the\ninterval in which the square root exists. The existence and\nbehaviour of the function $\\mu_{H,d}$ depend on the signs of the\nfunctions $nH J_{n-1}(t) + d$, $R_{H,d}(t)$ and\n$S_{H,d}(t)$.\\bigskip\n\n\nFor $H=\\frac{n-1}{n}$, we give a complete description of the\ncorresponding translation $H$-hypersurfaces. For $0 < H <\n\\frac{n-1}{n}$, we prove the existence of a complete non-entire\n$H$-graph with infinite boundary data and infinite asymptotic\nbehaviour. The other cases can be treated similarly using the tables\nbelow. \\bigskip\n\n\n\\begin{thm}[Translation $H$-hypersurfaces, with $n\\ge 3$ and $H =\n\\frac{n-1}{n}$]\\label{T-tra-1}\n$ $\n\\begin{enumerate}\n \\item When $d=0$, ${\\mathcal T}_0$ is a complete embedded smooth\n hypersurface generated by a compact, simple, strictly convex curve.\n The hypersurface is symmetric with respect\n to a horizontal hyperplane and the parts above and below this hyperplane\n are vertical graphs. The hypersurface also admits a vertical\n symmetry. The asymptotic boundary of ${\\mathcal T}_0$ is topologically\n a cylinder.\n\n \\item When $0 < d < 1$, the hypersurface ${\\mathcal T}_d$ is similar to\n ${\\mathcal T}_0$ except that it is not smooth.\n\n\n \\item When $d \\le -1$, ${\\mathcal T}_d$ is a smooth complete immersed\n hypersurface with self-intersections and horizontal symmetries.\n The asymptotic boundary of ${\\mathcal T}_d$ is topologically a cylinder.\n\n \\item When $-1 < d < 0$, the hypersurface ${\\mathcal T}_d$ looks like\n ${\\mathcal T}_{-1}$ except that it is not smooth.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{pb1-figs}\n\\begin{figure}[h]\n\\begin{center}\n\\begin{minipage}[c]{6.5cm}\n \\includegraphics[width=6.5cm]{f-tra-3a.eps}\n \\caption[$n \\ge 3, H =\\frac{n-1}{n}, d = 0$]{$n \\ge 3, H =\\frac{n-1}{n}, d = 0$}\n \\label{F-tra-3a}\n\\end{minipage}\\hfill\n\\begin{minipage}[c]{6.5cm}\n \\includegraphics[width=6.5cm]{f-tra-3c.eps}\n \\caption[$n \\ge 3, H =\\frac{n-1}{n}, d < -1$]{$n \\ge 3, H =\\frac{n-1}{n}, d < -1$}\n \\label{F-tra-3c}\n\\end{minipage}\\hfill\n\\end{center}\n\\end{figure}\n\\end{pb1-figs}\\bigskip\n\n\\textbf{Remark.}~ When $d \\ge 1$, the differential equation\n(\\ref{E-tra-4}) does not have solutions.\n\n\n\\begin{pb1-figs}\n\\begin{figure}[h]\n\\begin{center}\n\\begin{minipage}[c]{6.5cm}\n \\includegraphics[width=6.5cm]{f-tra-3d.eps}\n \\caption[$n \\ge 3, H =\\frac{n-1}{n}, d = - 1$]{$n \\ge 3, H =\\frac{n-1}{n}, d = - 1$}\n \\label{F-tra-3d}\n\\end{minipage}\\hfill\n\\begin{minipage}[c]{6.5cm}\n \\includegraphics[width=6.5cm]{f-tra-3b.eps}\n \\caption[$n \\ge 3, H =\\frac{n-1}{n}, 0 < d < 1$]{$n \\ge 3,\n H =\\frac{n-1}{n}, 0 < d < 1$}\n \\label{F-tra-3b}\n\\end{minipage}\\hfill\n\\end{center}\n\\end{figure}\n\\end{pb1-figs}\n\n\\begin{pb1-figs}\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{minipage}[c]{6.5cm}\n\\includegraphics[width=6.5cm]{f-tra-3e.eps}\n \\caption[$n \\ge 3, H =\\frac{n-1}{n}, -1 < d < 0$]{$n \\ge 3,\n H =\\frac{n-1}{n}, -1 < d < 0$}\n \\label{F-tra-3e}\n\\end{minipage}\\hfill\n\\begin{minipage}[c]{6.5cm}\n\\includegraphics[width=6.5cm]{f-tra-33.eps}\n \\caption[$n \\ge 2, H <\\frac{n-1}{n}$]{$n \\ge 2,\n H < \\frac{n-1}{n}$}\n \\label{F-tra-33}\n\\end{minipage}\\hfill\n\\end{center}\n\\end{figure}\\bigskip\n\\end{pb1-figs}\n\n\n\\begin{thm}[Complete $H$-graph with infinite boundary data]\\label{T-tra-2}\n$ $\\\\[2pt]\nThere exists a complete translation hypersurface ${\\mathcal T}_{H}$, with $0\n< H < \\frac{n-1}{n}$, such that\n\\begin{enumerate}\n\\item ${\\mathcal T}_H$ is a complete monotone vertical $H$-graph over the non mean\nconvex side of an equidistant hypersurface $\\Gamma\\subset \\mathbb{H}^n$\nwith mean curvature $\\frac{nH}{n-1},$\n\\item ${\\mathcal T}_H$ takes infinite boundary value data on $\\Gamma$ and infinite\nasymptotic boundary data.\n\\end{enumerate}\n\\end{thm}\\bigskip\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{T-tra-1}}\n\n\\bigskip\n\nThe proof of Theorem \\ref{T-tra-1} follows from an analysis of the\nasymptotic behaviour of the functions $J_m(t)$ (Formula\n(\\ref{E-tra-5a})) when $t$ goes to infinity and from an analysis of\nthe signs of the functions $R_{H,d}$ and $S_{H,d}$ (Formulas\n(\\ref{E-tra-11})) depending on the signs of $H - \\frac{n-1}{n}$ and\n$d$.\\bigskip\n\n\n\\noindent $\\bullet $~ We have the relations\n\n\\begin{equation}\\label{E-tra-10}\n\\left\\{%\n\\begin{array}{lll}\nJ_0(t) & = & t ,\\\\\nJ_1(t) & = & \\sinh(t) ,\\\\\n2J_2(t) & = & \\sinh(t) \\cosh(t) + t ,\\\\\n3J_3(t) & = & \\sinh(t) \\cosh^2(t) + 2 J_1(t) ,\\\\\nmJ_m(t) & = & \\sinh(t) \\cosh^{m-1}(t) + (m-1) J_{m-2}(t), \\text{ ~for~ } m\\ge 3.\\\\\n\\end{array}%\n\\right.\n\\end{equation}\\bigskip\n\n\nThese relations give us the asymptotic behaviour of the functions\n$J_m(t)$ when $t$ tends to infinity. In particular,\n\n$$\nm J_m(t) = \\sinh(t) \\cosh^{m-1}(t) + \\frac{m-1}{m-2} \\sinh(t)\n\\cosh^{m-3}(t) + O(e^{(m-4)t}), \\text{ ~for~ } m\\ge 5\n$$\n\nwith the remainder term replaced by $O(t)$ when $m=4$.\\bigskip\n\n\n\n\\noindent $\\bullet $~ \\textbf{The function $S_{H,d}(t)$}\\bigskip\n\nFor all $H > 0$, the function $S_{H,d}$ increases from $1+d$ to $+\n\\infty$. Its behaviour is summarized in the following table.\n\n\\begin{equation}\\label{E-tra-12s}\n\\begin{array}{|c|ccccc|}\n \\hline\n & \\text{Case} & 0 \\frac{n-1}{n} & & \\\\\n \\hline\n t & 0 & & & & + \\infty \\\\\n \\hline\n R_{H,d}(t) & 1-d & & \\searrow & & -\\infty \\\\\n \\hline\n\\end{array}\n\\end{equation}\\bigskip\n\n\n\n\\textbf{Proof of Theorem \\ref{T-tra-1}, continued}\\bigskip\n\nWe now investigate the behaviour of the solution $\\mu$ to Equation\n(\\ref{E-tra-4}) when $n\\ge 3$ and $H=\\frac{n-1}{n}$ (for $n=2$, see\n\\cite{Sa08}).\\bigskip\n\n\n\n\\noindent According to Table (\\ref{E-tra-12s}), the function $S_{H,d}$\nincreases from $1+d$ to $+ \\infty$ and we have to consider two\ncases, \\emph{(i)} $d \\ge - 1$, in which case $S_{H,d}$ is always\nnon-negative and \\emph{(ii)} $d < -1$, in which case $S_{H,d}$ has\none zero $\\alpha_{H,d}$ such that $$\\cosh^{n-1}(\\alpha_{H,d}) + nH\nJ_{n-1}(\\alpha_{H,d}) + d = 0.$$\n\n\\noindent According to Table (\\ref{E-tra-12r}), the function $R_{H,d}$\ndecreases from $1-d$ to\n$\\left\\{%\n\\begin{array}{cc}\n- \\infty, & n \\ge 3 \\\\\n-d, & n=2 \\\\\n\\end{array} \\right. $, depending on the value of $n$. It follows\nthat we have two cases, \\emph{(i)} $d \\ge 1$, in which case the\nfunction $R_{H,d}$ is always non-positive and \\emph{(ii)} $d < 1$,\nin which case it has one zero $c_{H,d}$ for $n \\ge 3$. When it\nexists, the zero $c_{H,d}$ satisfies $$\\cosh^{n-1}(c_{H,d}) - nH\nJ_{n-1}(c_{H,d}) - d = 0.$$\n\\bigskip\n\nLooking at the equations defining $\\alpha_{H,d}$ and $c_{H,d}$ we\nsee that $\\alpha_{H,d} < c_{H,d}$ when they both exist.\\bigskip\n\n\n\nThe behaviour of the function $\\mu$ is described in the following\ntables, see also Figures~\\ref{F-tra-3a} to \\ref{F-tra-3e}.\\bigskip\n\n\\begin{equation}\\label{E-tra-15-1}\n\\begin{array}{|c|ccccccc|}\n\\hline\n \\textbf{Case 1} & & H=\\frac{n-1}{n} & & d < -1 & & n\\ge 3 & \\\\\n\\hline\n t & 0 & & \\alpha_{H,d} & & c_{H,d} & & + \\infty \\\\\n\\hline\n R_{H,d} & & + & + & + & 0 & - & \\\\\n\\hline\n S_{H,d} & & - & 0 & + & + & + & \\\\\n\\hline\n T_{H,d} & & \\not \\exists & - \\infty & \\exists & + \\infty\n & \\not \\exists & \\\\\n\\hline\n\\end{array}\n\\end{equation}\\bigskip\n\nThe function $\\mu$ is given by\n\n$$\\mu(\\rho) = \\int_{\\rho_0}^{\\rho} T_{H,d}(t) \\, dt$$\n\nfor $\\rho_0, \\rho \\in [\\alpha_{H,d}, c_{H,d}]$ and the integral\nexists at both limits. Note that the integrand is negative near the\nlower limit while it is positive near the upper limit. \\bigskip\n\nWhen $d=0$, using (\\ref{E-tra-4}) one can show that $\\ddot{\\mu}\n> 0$ and conclude that the generating curve is strictly convex. The\nformula for $\\ddot{\\mu}$ also shows that the curvature extends\ncontinuously at the vertical points.\\bigskip\n\n\nThe generating curve can be extended by symmetry and periodicity to\ngive rise to a complete immersed hypersurface with\nself-intersections.\\bigskip\n\n\n\\begin{equation}\\label{E-tra-15-2}\n\\begin{array}{|c|ccccccc|}\n\\hline\n \\textbf{Case 2} & & H=\\frac{n-1}{n} & & -1 \\le d < 1 & & n\\ge 3 & \\\\\n\\hline\n t & 0 & & & & c_{H,d} & & + \\infty \\\\\n\\hline\n R_{H,d} & & & + & & 0 & - & \\\\\n\\hline\n S_{H,d} & & & + & & + & + & \\\\\n\\hline\n T_{H,d} & & & \\exists & & + \\infty\n & \\not \\exists & \\\\\n\\hline\n\\end{array}\n\\end{equation}\\bigskip\n\n\nThe function $\\mu$ is given by\n\n$$\\mu(\\rho) = \\int_{0}^{\\rho} T_{H,d}(t) \\, dt$$\n\nfor $\\rho_0, \\rho \\in [0, c_{H,d}]$ and the integral exists at both\nends. Note that the integrand has the sign of $d$ near $0$, with\n$\\dot{\\mu}(0) = d\/\\sqrt{1-d^2}$ ; it is positive near the upper\nbound with $\\dot{\\mu}(c_{H,d}) = + \\infty$. \\bigskip\n\nWhen $d=-1$, the original curve has a vertical tangent at $0$. It\ncan be extended by symmetry and periodicity to give rise to a\ncomplete immersed hypersurface with self-intersections. \\bigskip\n\nWhen $d=0$, the curve has a horizontal tangent and is strictly\nconvex (use (\\ref{E-tra-4})). It can be extended by symmetry as a\ntopological circle and gives rise to a complete embedded surface.\n\\bigskip\n\n\nWhen $d \\ge 1$, Equation (\\ref{E-tra-4}) has no solution.\\bigskip\n\n\n\\subsection{Proof of Theorem \\ref{T-tra-2}}\n\nGiven $n$ and $H$, such that $0 0$ for $t > t_H$ and hence the\nquantity $nH J_{n-1}(t) +d_H$ does not change sign for $t>t_H$ and\nthe same is true for $T_{H,d_H}(t)$.\\bigskip\n\n\nTaking (\\ref{E-tra-11}) into account, we choose $\\rho_0 > t_H$ and\ndefine the generating curve by Formula (\\ref{E-tra-8}).\\bigskip\n\n\nWe conclude that $\\mu(\\rho)$ is well-defined and strictly increasing\nfor $\\rho>t_H$. Moreover, $\\mu(\\rho)$ goes to $-\\infty$, if\n$\\rho\\rightarrow t_H^+.$ Notice that the mean curvature of the\nequidistant hypersurface at distance $t_H$ to $\\mathbb{P}$ is $\\tanh\n(t_H)=\\frac{n H}{n-1}$, by the choice of $t_H$.\\bigskip\n\n\n Now recall that if $ 0 1,\\\\\n\\end{array}%\n\\right.\n\\end{equation}\n\nwhere the principal curvatures are taken with respect to the unit\nnormal to $\\partial \\Omega $ pointing inwards.\\bigskip\n\n\nGiven a hypersurface $\\Gamma$ satisfying Properties\n(\\ref{E-appl-gam}), there exists some radius $R$ such that for any\npoint $p$, the ball $B_{p,R} \\subset \\mathbb{H}^n$ with radius $R$ is\ntangent to $p$ at $\\Gamma$ and $\\Gamma \\subset B_{p,R}$. We denote\nby\n\n\\begin{equation}\\label{E-appl-gam2}\n{\\mathcal S}_{p,+} \\text{ ~and~ } {\\mathcal S}_{p,-}\n\\end{equation}\n\nthe two hypersurfaces in ${\\mathcal R}$ passing through the sphere $\\partial\nB_{p,R}$ and symmetric with respect to the slice $\\mathbb{H}^n \\times\n\\{0\\}$.\\bigskip\n\n\nWe first prove an existence result for a Dirichlet problem.\n\\bigskip\n\n\\begin{prop}\\label{P-appl-2}\nLet $\\Omega \\subset \\mathbb{H}^n \\times \\{0\\}$ be a bounded domain with\nsmooth boundary $\\Gamma$ satisfying (\\ref{E-appl-gam}). Then, for\nany $H, 0 < H \\le \\frac{n-1}{n}$, there exists a vertical graph\n$M_{\\Gamma}$ over $\\Omega$ in $\\mathbb{H}^n \\times \\mathbb{R}$, with constant mean\ncurvature $H$ with respect to the upward pointing normal. This means\nthat there exists a function $u : \\Omega \\to \\mathbb{R}$, smooth up to the\nboundary, such that $u|_{\\Gamma} = 0$, and whose graph\n$\\ens{(x,u(x))}{x \\in \\Omega}$ has constant mean curvature $H$ with\nrespect to the unit normal pointing upwards.\n\\end{prop}\\bigskip\n\n\\textbf{Remark.} ~The graph $M_{\\Gamma}$ having positive mean\ncurvature with respect to the upward pointing normal, must lie below\nthe slice $\\mathbb{H}^n \\times \\{0\\}$. The symmetric $\\check{M}_{\\Gamma}$\nwith respect to the slice lies above the slice and has positive mean\ncurvature with respect to the normal pointing downwards. \\bigskip\n\n\\textbf{Proof of Proposition \\ref{P-appl-2}}\\bigskip\n\n\\noindent $\\bullet $~ We first consider the case $H = \\frac{n-1}{n}$.\\bigskip\n\nBy our assumption on $\\Gamma$, using the hypersurfaces\n(\\ref{E-appl-gam2}) and the Convex hull lemma, Proposition\n\\ref{P-appl-1}, any solution to our Dirichlet problem must be\ncontained in ${\\mathcal C}({\\mathcal S}_{p,-}) \\cap {\\mathcal C}({\\mathcal S}_{p,+})$. This provides a\npriori height estimates and boundary gradient estimates on the\nsolution. \\bigskip\n\nWe could use \\cite{Spr08} and classical elliptic theory \\cite{GT83},\nto get existence for our Dirichlet problem when $H=\\frac{n-1}{n}$.\nWe shall instead apply \\cite{Spr08} directly. Indeed, in our case,\nthe mean curvature $H_{\\Gamma}$ of $\\Gamma$ satisfies $H_{\\Gamma} >\n1 = H \\frac{n}{n-1}$, and the Ricci curvature of $\\mathbb{H}^n$ satisfies\n$\\mathrm{Ric} = - (n-1) \\ge - \\frac{n^2}{n-1}H^2$. Theorem 1.4 in\n\\cite{Spr08} states that under theses assumptions there exists a\nvertical graph over $\\Omega$ with boundary $\\Gamma$ and constant\nmean curvature $H=\\frac{n-1}{n}$.\\bigskip\n\n\n\\noindent $\\bullet $~ We now consider the case $0 < H \\le \\frac{n-1}{n}$. \\bigskip\n\nWe use the graphs constructed previously as barriers to obtain a\npriori height estimates and apply the interior and global gradient\nestimates of \\cite{Spr08} to conclude. \\bigskip\n\nWe consider the Dirichlet problem $(P_t)$ for $0 \\le t \\le 1$,\n\n\\begin{equation*}\\label{E-appl-5}\n\\left\\{%\n\\begin{array}{cccc}\n\\mathrm{div}\\big(\\dfrac{\\nabla u}{W}\\big) & = &\nt \\, (n-1) & \\text{in~ }\\Omega\\\\[5pt]\nu & = & 0 & \\text{on~ } \\Gamma \\\\\n\\end{array}%\n\\right.\n\\end{equation*}\n\n\nwhere $u\\in C^2(\\Omega)$ is the height function, $\\nabla u$ its\ngradient and $W=(1 +|\\nabla u|^2)^{1\/2}$, and where the gradient and\nthe divergence are taken with respect to the metric on $\\mathbb{H}^n$. This\nis the equation for \\emph{vertical} $H$-graphs in $\\mathbb{H}^n \\times \\mathbb{R}$.\nIt is elliptic of divergence type. \\bigskip\n\nBy the first step, we have obtained the solution $u_1$ for the\nDirichlet problem $(P_1)$. The solution for $(P_0)$ is the trivial\nsolution $u_0=0$. By the maximum principle, using the fact that\nvertical translations are positive isometries for the product\nmetric, and the existence of the solutions $u_1$ and $u_0$, we have\nthat any $C^1(\\overline{\\Omega})$ solution $u_t$ of the Dirichlet\nproblem $(P_t)$ stays above $u_1$ and below $u_0.$ This yields a\npriori height and boundary gradient estimates, independently of $t$\nand $u_t$. Global gradient estimates follow Theorem 1.1 and Theorem\n3.1 in \\cite{Spr08}. We have therefore $C^1(\\overline{\\Omega})$ a\npriori estimates independently of $t$ and $u_t$. The existence of\nthe solution $u_t$ for $00$. Let $M$ be a\ncompact connected embedded $H$-hypersurface such that $\\partial M =\n\\Gamma_{+} \\cup \\Gamma_{-}$, with $0 < H \\le \\frac{n-1}{n}$. Assume\nthat $2a \\ge \\frac{\\pi}{n-1}$.\n\\begin{enumerate}\n \\item Assume that $\\Gamma$ is symmetric with respect to a hyperbolic\n hyperplane $P$ and that each connected component of $\\Gamma\\setminus P$\n is a graph above $P$. Then $M$ is symmetric with respect to the vertical\n hyperplane $P\\times \\mathbb{R}$ and each connected component of $M\\setminus\n P\\times \\mathbb{R}$ is a horizontal graph.\n \\item Assume that $\\Gamma$ is an $(n-1)$-sphere. Then $M$ is part of\n the complete embedded rotation hypersurface given by\n Theorem~\\ref{T-h3-r0} and \\ref{T-h3-rp} and containing $\\Gamma$.\n It follows that $M$ is symmetric with respect to the slice $\\mathbb{H}^n \\times\n \\{0\\}$ and the parts of $M$ above and below the slice of symmetry are\n vertical graphs.\n\\end{enumerate}\n\\end{thm}\\bigskip\n\n\\textbf{Proof of Theorem \\ref{T-appl-7}}.\\bigskip\n\n\\noindent Let $\\Omega_{+} = \\Omega \\times \\{a\\}$ and $\\Omega_{-} = \\Omega\n\\times \\{-a\\}$. By the Convex hull Lemma, Proposition\n\\ref{P-appl-1}, using the hypersurfaces given by (\\ref{E-appl-gam2})\nwe have that $M \\cap \\overline{\\mathrm{ext}(\\Omega_{+})} = \\Gamma_+$\nand $M \\cap \\overline{\\mathrm{ext}(\\Omega_{-})} = \\Gamma_-$.\\bigskip\n\n\\noindent We claim that $M \\cap (\\overline{\\Omega} \\times \\mathbb{R}) = \\Gamma_+\n\\cap \\Gamma_-$. Let $M_{\\Gamma,a}$ be the graph above\n$\\overline{\\Omega_+}$ contained in $\\mathbb{H}^n \\times [a, \\infty[$ and\n$M_{\\Gamma,-a}$ be the graph below $\\overline{\\Omega_-}$ contained\nin $\\mathbb{H}^n \\times ]-\\infty, a]$, given by Theorem~\\ref{T-appl-6}.\n\\bigskip\n\nConsider $\\widetilde{M} = M_{\\Gamma,a} \\cap M \\cap M_{\\Gamma,-a}$\noriented by the mean curvature vector of $M$ by continuity. Take the\nfamily of (minimal) catenoids symmetric with respect to $\\mathbb{H}^n\n\\times \\{0\\}$ with rotation axis some $\\{\\bullet \\} \\times \\mathbb{R}$.\nComing from infinity with such catenoids, using the assumption that\n$2a \\ge \\frac{\\pi}{n-1}$ and the fact that the catenoids have height\n$< \\frac{\\pi}{n-1}$, we see that one catenoid will eventually touch\n$\\widetilde{M}$ at some interior point in $M$. This implies that the\nnormal to $M$ at this point is the same as the normal to the\ncatenoid at the same point (maximum principle) and hence that the\nnormal to $M$ points inside $\\widetilde{M}$.\\bigskip\n\nAssume that $M \\cap (\\Omega \\times \\{a\\}) \\not = \\emptyset$ (\\emph{resp. }\nthat $M \\cap (\\Omega \\times \\{-a\\}) \\not = \\emptyset$). Then at the\nhighest point of $M$ the normal would be pointing upwards (\\emph{resp. }\ndownwards) and we would get a contradiction with the maximum\nprinciple by considering the horizontal slice (a minimal\nhypersurface) at this point.\\bigskip\n\nFinally, $M \\cap (\\overline{\\Omega} \\times \\mathbb{R}) = \\Gamma_+ \\cap\n\\Gamma_-$ and the normal to $M$ points inside $M \\cup \\Omega_+ \\cup\n\\Omega_-$.\\bigskip\n\n\\noindent To conclude, we use Alexandrov Reflection Principle in vertical\nhyperplanes $P_t\\times \\mathbb{R}$ in ambient space, obtained by applying\nhorizontal translations along the horizontal geodesic orthogonal to\n$P,$ to the hyperplane $P\\times \\mathbb R$ of symmetry of $\\Gamma$.\nWe conclude that $M$ is symmetric about $P\\times \\mathbb{R}$ and that each\nconnected component of $M\\setminus P\\times \\mathbb{R}$ is a horizontal\ngraph. This complete the proof of the first statement in the\ntheorem. \\bigskip\n\nIf $\\Gamma$ is spherical then $M$ is a rotation hypersurface. As the\nmean curvature vector points into the region of ambient space that\ncontains the axes, by the geometric classification of the rotation\n$H$-hypersurfaces with constant mean curvature $H \\le (n-1)\/n$ given\nby Theorems~\\ref{T-h3-r0} and \\ref{T-h3-rp}, it follows that $M$ is\npart of a complete embedded rotation hypersurface $\\overline{M}$. It\nfollows that $\\overline{M}$ has a slice of symmetry at $\\mathbb{H}^n \\times\n\\{0\\}$ and each connected component of $\\overline{M}$ above and\nbelow $t=0$ is a complete vertical graph over the exterior of a\nround ball in $t=0$.\n\n\\hfill \\hfill\\penalty10000\\copy\\qedbox\\par\\medskip \\bigskip\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThermoelectric response of a hybrid junction between two normal metals in the mesoscopic regime has been discussed extensively both theoretically and experimentally \\cite{sivan_PRB_33_551,staring_EPL_22_57, moller_PRL_81_5197, scheibner_PRL_95_176602,ludoph_PRB_59_12290, reddy_AAAS_315_1568, widawsky_NL_12_354, thierschmann_NN_10_854, shankouri_ARMR_41_399, sanchez_PRB_83_085428, beenakker_PRB_46_9667, entin_PRB_82_115314, sanchez_PRB_84_201307, jordan_PRB_87_075312, brandner_PRL_110_070603}. Whereas, analogous situation comprising of a junction of superconductors is a less explored topic though discussion of thermoelectric response of superconductor has a long history. Such set-ups are of great importance because of the possibility of its applications in improving the efficiency of thermoelectric generator by strongly suppressing Ohmic losses \\cite{kolenda_PRL_116_097001, kolenda_PRB_95_224505, shimizu_NC_10_825, tan_NC_12_138, fornieri_NN_12_944, giazotto_RMP_78_217}. \n\nIn 1944, Ginzberg \\cite{ginzburg_JPUSSR_8_148, ginzburg_RMP_76_981} showed that a temperature gradient in a bulk superconductor leads to a finite normal current response, though this current gets completely cancelled by a counter flow of supercurrent in a homogeneous isotropic superconductor which make it impossible to detect the thermoelectric response in isolation. This fact lead him to theoretically explore the possibilities of anisotropic and inhomogeneous superconductor for the detection of the thermoelectric effect. Since then, various theoretical study\\cite{ginzburg_SST_4_S1, ginzburg_PCS_235_3129, marinescu_PRB_55_11637, galperin_ZETF_66_1387, galperin_PRB__65_064531, virtanen_APA_89_625} has been conducted exploring possibilities of detection of thermoelectric response of superconductors in anisotropic and inhomogeneous situations. Experimental study in this direction goes back all the way to 1920's \\cite{Falco1981book, borelius_PKNAW_34_1365, burton_Nature_136_141, keesom_Physica_5_437, casimir_Physica_13_33, pullan_PRSLSA_217_280, harlingen_PRB_21_1842, kartsovnik_JETPL_33_7, fornieri_NN_11_258} and this topic has been revisited in the recent past in an interesting work by Shelly et.al.\\cite{connor_SA_2_e1501250}. The discovery of Josephson effect \\cite{josephson_PL_1_251} in 1962 provided a natural setting for exploring thermoelectric response for a inhomogeneous superconductor. Later in 1997, Guttman and Bergman made an attempt to theoretically explore the thermoelectric response of a JJ in a tunnel Hamiltonian approach \\cite{guttman_PRB_55_12691}.\n\nPershoguba and Glazman \\cite{pershoguba_PRB_99_134514} have carried out an elaborate study on the possibility of generating thermometric current across a junction between two quasi-one dimensional superconductors, which goes beyond the tunneling limit and also discussed the relevance of the odd and the even part of the Josephson current as a function of the superconducting phase bias $\\phi_{12}$ owing to scattering in junction region which breaks the $\\omega \\rightarrow -\\omega$ symmetry. In this regard, the helical edge state of two dimensional topological insulators pose an interesting and cleane testing ground for such theoretical study which hosts one-dimensional Josephson junction\\cite{hart_NP_10_638}. Thermal response of quantum hall edge has already being studies in experiment\\cite{banerjee_Nat_545_75} and hence an similar experimental set-up involving the spin Hall edge may not be far in the future. Recent theoretical studies have explored the possibility of inducing thermoelectric effect in helical edge state-based Josephson junction involving either an anisotropic ferromagnetic barrier\\cite{gresta_PRL_123_186801, marchegiani_APL_117_212601} or a three-terminal geometry\\cite{blasi_PRL_124_227701, blasi_PRB_103_235434, blasi_PRB_102_241302}. In this work we show that thermoelectric effect can exist in the HES of QSH even in a simplest case of two terminal ballistic JJ owing to breaking of the $\\omega \\rightarrow -\\omega$ symmetry of the quasi-particle transmission probabilities across the junction at finite length. We argue that this is generic to ballistic JJ and is not specific to HES. Lastly, it is worth noting that the use of thermal transport for probing quantum states has been much in pursuit in contemporary science\\cite{li_MRSB_45_348} and hence such a discussion is quite timely.\n\nThe paper is organized as follows. In Section \\ref{system_HES} we described the JJ based on HES of a 2D QSH state and in Section \\ref{left_righth_symmetry} we discussed how a long ballistic JJ can break the $\\omega \\rightarrow -\\omega$ symmetry and hence resulting in thermoelectric response which also survives in presence of disorder. In Section \\ref{even_and_odd_section} we extend our discussion to the odd-in-$\\phi_{12}$ and even-in-$\\phi_{12}$ part of the thermoelectric conductance and shown that minimal breaking of the $\\omega \\rightarrow -\\omega$ symmetry is not enough to induce an even-in-$\\phi_{12}$ contribution.We have also argued that the presence of thermoelectric response through the breaking of $\\omega \\rightarrow -\\omega$ symmetry is not unique to HES, rather it is a generic property of a JJ.\n\n\n\n\n\n\\section{Ballistic Josephson junction in helical edge state}\n\\label{system_HES}\n\n\\begin{figure}[]\n\t\\includegraphics[width=0.4\\textwidth]{set_up_BW.eps}\n\t\\caption{Schematic of the Josephson junction set-up in a Helical edge state.}\n\t\\label{set_up}\n\\end{figure}\n\n\\begin{figure*}[t!]\n\t\\includegraphics[width=\\textwidth]{ABS.eps}\n\t\\caption{(a) Pictorial representation of below the gap $(\\omega<\\Delta_0)$ tunneling of Cooper pairs (CP) form left(right) to right(left) via two different Andreev bound states $\\omega_0^{21}$ (indicated by blue lines) and $\\omega_0^{12}$ (indicated by orange lines). The dotted lines represent the fact that tunnelling of the quasielectrons (quasiholes) above the gap $(\\omega>\\Delta_0)$ across the junction are in correspondence with distinct bound states, as can be noted from the poles of the quasielectron (quasihole) transmission probabilities $\\mathcal{T}_{ee}^{21}$ and $\\mathcal{T}_{ee}^{12}$ (or $\\mathcal{T}_{hh}^{12}$ and $\\mathcal{T}_{hh}^{21}$). (b) Two types of Andreev bound states as a function of superconducting phase difference $\\phi_{12}$, are plotted for different values of junction lengths where $\\xi=\\hbar v_F\/\\Delta_0$ is the superconducting coherence length. \n\t(c) Density plot for the thermoelectric coefficient $\\kappa^{21}$ of a ballistic JJ based on the edge states of a quantum spin Hall insulator in proximity to a s-wave superconductor, as a function of superconducting phase bias $\\phi_{12}$ and junction length $L$. The average temperature of the junction is considered to be $k_BT=0.5 \\Delta_0$.}\n\t\\label{ABSeps}\n\\end{figure*}\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.9 \\textwidth]{Transmission_mod.eps}\n\t\\caption{Different transmission probabilities of the quasiparticles through a ballistic Josephson junction based on the helical edge state of a quantum spin Hall insulator, in the space of energy ($\\omega$) and superconducting phase-difference ($\\phi_{12}$) for different values of junction length ($L$). A clear asymmetry between the electron and hole transmission probability from left (right) to right(left) develops as we increase the length of the junction. The plot in energy window $(\\vert\\omega\\vert<\\Delta_0)$ signifies the evolution of the pole (location of ABS) of the transmission amplitude as a function of $\\phi_{12}$.}\n\t\\label{transmission}\n\\end{figure*}\n\n\nWe first consider a JJ based on 1D Dirac fermions in proximity to a s-wave superconductor, realized in a HES of QSH insulator\\cite{fu_PRB_79_161408, fu_PRL_100_096407, calzona_arxiv_1909_06280} because of its algebraic simplicity. Later we will also explore the case of quadratic dispersion. The junction is considered to be of length $L$ laying over the region $|x|\\Delta_0$ ) across the JJ. It is straightforward to match the plane wave solutions of the BdG equation to obtain the transmission probabilities across the JJ (from $S1$ to $S2$) as described by Eq. \\ref{DiracHamiltonian} are given by (see Appendix \\ref{Appendix_clean_junction})\n\\begin{align}\n\t\\mathcal{T}_{ee}^{21} &= \\mathcal{T}_{hh}^{12} &=\\dfrac{\\omega^2-\\Delta_0^2}{\\omega^2-\\Delta_0^2 \\cos^2 \\left( \\frac{k_e-k_h}{2}L-\\frac{\\phi_{12}}{2} \\right)},\t\\label{T_ee^21}\\\\\n\t\\mathcal{T}_{hh}^{21} &= \\mathcal{T}_{ee}^{12} &=\\dfrac{\\omega^2-\\Delta_0^2}{\\omega^2-\\Delta_0^2 \\cos^2 \\left( \\frac{k_e-k_h}{2}L+\\frac{\\phi_{12}}{2} \\right)}, \\label{T_hh^21}\n\\end{align}\nwhile $\\mathcal{T}_{he}^{21}=\\mathcal{T}_{eh}^{21}=\\mathcal{T}_{he}^{12}=\\mathcal{T}_{eh}^{12}=0$. Quasiparticle transmission probabilities through a ballistic JJ is shown in FIG. \\ref{transmission} for two different lengths of the junction. Here $\\mathcal{T}_{q'q}^{ji}$ denote the transmission probability of an $q$-like QP ($q=e,h$) from lead $Si$ to a $q'$-like QP in lead $Sj$. Note that, the tunneling of an electron- (hole-) like QP from S1 to S2 (S2 to S1) is in correspondence with the ABS having energy $\\omega_0^{21}$ while the tunneling of a hole- (electron-) like QP from S1 to S2 (S2 to S1) is in correspondence with the ABS having energy $\\omega_0^{12}$ [See Fig. \\ref{ABSeps}(a)] which is apparent from the fact that the poles of the transmission amplitudes for these two processes coincides with the corresponding ABS energies. Within linear response theory, thermoelectric coefficient of a JJ can be defined in terms of the transmission probabilities as\\cite{pershoguba_PRB_99_134514}\n\\begin{align}\n\t\\kappa^{21} = \\left[\\dfrac{e}{h}\\int_{\\Delta_0}^{\\infty} d\\omega \\dfrac{\\omega}{\\sqrt{\\omega^2-\\Delta^2}} [i_{e}^{21}-i^{21}_{h}] \\dfrac{d\\mathrm{f}(\\omega,T)}{dT}\\right]_{T=T_{\\text{avg}}}\t\\label{thermoelectricCoefficient}\n\\end{align}\nwhere $i^{21}_{e}=(\\mathcal{T}_{ee}^{21}-\\mathcal{T}_{he}^{21})$, $i^{21}_{h}=(\\mathcal{T}_{hh}^{21}-\\mathcal{T}_{eh}^{21})$, $e$ is the electronic charge, $\\mathrm{f (\\omega,T)}$ is the Fermi distribution function at temperature $T$ and $T_{\\text{avg}}$ is the average temperature of the junction. Note that, in the limit $L \\rightarrow 0$, $\\kappa^{21}$ is zero.\n\nThe integration in Eq. (\\ref{thermoelectricCoefficient}) can be done numerically and $\\kappa^{21}$ can be obtained as a function of superconducting phase difference $\\phi_{12}$ and junction length $L$ which is plotted as a density plot in FIG. \\ref{ABSeps}(c). In case of HES, owing to its linear dispersion, the value of the overall chemical potential $\\mu$ does not effect the calculations for the ballistic case.\n\nTo obtain an estimate of the extremum values of thermoelectric conductance for a single channel ballistic junction, we perform a numerical scan over the parameter space of $\\phi_{12}$ and $L$ for a given temperature of $k_BT_{\\text{avg}}=0.5\\Delta_0$. We found that the maximum of (minimum of) $|\\kappa^{21}_e| \\approx\\,0.3438 \\,ek_B\/h$ $(\\approx 1.477 nA\/K)$ is obtained at a junction length $L \\approx 0.555 \\xi$ for $\\phi_{12}\\approx 0.353 \\pi$ (minimum at $\\phi_{12} \\approx 1.647 \\pi$).\n\n\\begin{figure*}[t]\n\t\\includegraphics[width=1.0 \\textwidth]{Even_Odd.eps}\n\t\\caption{Thermoelectric conductance of a Josephson junction based on helical edge state of quantum spin Hall insulator in presence of four random scattering centers. (a) Total thermoelectric conductance (b) the part of thermal conductance that is even in $\\phi_{12}$ (c) the part of conductance that is odd in $\\phi_{12}$, for single random configuration of scattering centers and after averaging over different numbers of random configurations.}\n\t\\label{even_odd}\n\\end{figure*}\n\n\nAs we can see from FIG. \\ref{transmission}, for a ballistic JJ, in general for any given value of $\\phi_{12}$ and at an energy $\\omega>\\Delta_0$ the quasiparticle transmission probabilities $\\mathcal{T}_{ee}^{21}$ and $\\mathcal{T}_{hh}^{21}$ are different if the length of the junction $L$ is comparable to the superconducting coherence length (i.e. when we are not in the short junction limit). Note that, the difference between these quantities at a given $\\omega$ is maximum in the neighborhood of $\\omega=\\Delta_0$ and it decreases as we go higher in $\\omega$, although non-monotonically. Additionally, one must notice, the thermoelectric effect identically vanishes both at $\\phi_{12}=0$ and $\\pi$ which are the time reversal symmetric points (See FIG. \\ref{ABSeps}(c)).\n\n\\section{Even-in-$\\phi_{12}$ and Odd-in-$\\phi_{12}$ part of the thermoelectric response and the effect of disorder}\n\\label{even_and_odd_section}\nPresence of scatter within the junction region, which breaks the $\\omega \\rightarrow -\\omega$ symmetry, not only leads to a finite thermoelectric conductance, but also results in deviation from thermoelectric conductance being odd in $\\phi_{12}$\\cite{pershoguba_PRB_99_134514}. As discussed above, a JJ of finite length also breaks the $\\omega \\rightarrow -\\omega$ symmetry hence it is curious if this minimal symmetry breaking can result in such a deviation, i.e. the thermoelectric response can be written as a liner sum of an even-in-$\\phi_{12}$ part and an odd-in-$\\phi_{12}$ part.\n\nIt is straightforward to check that the expression for thermoelectric conductance in the ballistic limit, obtained from Eq. \\ref{T_ee^21}, \\ref{T_hh^21} and \\ref{thermoelectricCoefficient} is an odd function of $\\phi_{12}$, independent of the length of the junction. This implies that the breaking of $\\omega \\rightarrow -\\omega$ symmetry via $k_e\\neq k_h$ (as discussed in the previous section) does not lead to any contribution to the thermoelectric response which is even in $\\phi_{12}$. Further, we calculate the thermoelectric conductance in presence of a single localized scatterer which is positioned at an arbitrary point within the junction region and we assume that the scattering matrix corresponding to the scatterer has no energy dependence. The expression for the thermoelectric conductance in this case is given below,\n\\begin{widetext}\n\\begin{equation}\n \\kappa^{21} = \\left[\\dfrac{e}{h}\\int_{\\Delta_0}^{\\infty} d\\omega \\dfrac{\\omega}{\\sqrt{\\omega^2-\\Delta^2}} \\left[ \\dfrac{4 \\tau \\left((1-\\tau) \\sin{((k_e-k_h)L(m-n))}+ \\sin{((k_e-k_h)L)} \\right)\\sin{\\phi_{12}} \\sinh2{\\theta}}{\\Omega \\Omega^*} \\right] \\dfrac{d\\mathrm{f}(\\omega,T)}{dT}\\right]_{T=T_{\\text{avg}}},\n \\label{m_n_junction}\n\\end{equation}\n\\end{widetext}\nwhere, $\\Omega=(1-\\tau) \\cos{\\left((k_e-k_h)L(m-n)\\right)} + \\cos{\\left( (k_e-k_h)L-2i\\theta \\right)}-\\tau \\cos{\\phi_{12}}$, $\\theta=\\text{arccosh}{\\omega\/\\Delta_0}$, $\\tau$ is the normal state transmission probability across the scatterer and the position of the scattering center divides the junction region in the ratio $m:n$ ($m,n\\leq 1$ and $m+n=1$). All other notations have their usual meanings as discussed before. Eq. \\ref{m_n_junction} clearly shows that the thermoelectric response in this case also, is odd in $\\phi_{12}$. Hence, our study establishes the fact that the minimal breaking of $\\omega \\rightarrow - \\omega$ symmetry for a finite length ballistic junction (or in presence of a single scatterer which does not break the $\\omega \\rightarrow -\\omega$ symmetry) is sufficient to induce thermoelectric response across the JJ, though it is not enough to induce an even-in-$\\phi_{12}$ contribution to the thermoelectric conductance.\n\n\\begin{figure*}[t]\n\t\\includegraphics[width=1.0 \\textwidth]{multiple_disorders.eps}\n\t\\caption{Disordered averaged mean value (left figure) and the variance (right figure) of the thermoelectric coefficient $\\kappa^{21}$ of a S-TI-S junction based on the edge states of a quantum spin Hall insulator with proximity to a s-wave superconductor, are plotted as a function of superconducting phase difference $\\phi_{12}$ . The average temperature of the junction is considered to be $k_BT=0.5 \\Delta_0$ and the overall chemical potential to be $\\mu=10\\Delta_0$. Length of the junction is considered to be $L=0.555 \\xi$ where $\\xi$ is the superconducting coherence length. Average is done over 500 disorder configurations. The middle plot show that, as we increase the number of scatterers, the curves for thermoelectric conductance tend to a $\\sin{(\\phi_{12})}$ curves (solid lines) with an amplitude (Max($\\kappa_e^{12}$)-Min($\\kappa_e^{21}$))\/2.}\n\t\\label{multipleDisorders}\n\\end{figure*}\n\nNow, if we consider a situation comprising of more than one such scatterer, then the effective scattering matrix describing the collection of scatterers will become energy dependent and in general will also break the $\\omega \\rightarrow -\\omega$ symmetry, resulting in an even-in-$\\phi_{12}$ contribution to the thermoelectric conductance as expected \\cite{pershoguba_PRB_99_134514}. The even-in-$\\phi_{12}$ part of the thermoelectric conductance is proportional to $(\\tau_{\\omega} -\\tau_{-\\omega})$, where $\\tau_{\\omega}$ is the normal state transmission probability across the junction at an energy $\\omega$, and thus can vary drastically (both in amplitude and in sign) for different disorder configurations for a given $\\phi_{12}$. Hence, averaging over random configurations results in vanishingly small values of the even part. Next we perform a numerical calculation to analyze the effect of averaging over a large number of disorder configurations in presence of multiple scatterers. To begin with, we consider four scattering centers represented by four energy-independent scattering matrices placed at random positions inside the junction region. Transmission probabilities of the scattering matrices are chosen randomly from a one-sided Gaussian distribution with a mean of $95\\%$ and standard deviation of $5\\%$. All the phase freedom of the disorders have been chosen randomly from a Gaussian distribution with a mean of $0$ and standard deviation $0.05 \\pi$. We have fixed the length of the junction to be $L=0.555 \\xi$, the value at which we get maximum thermoelectric conductance (which occurs for $\\phi=0.353 \\pi$) for a ballistic JJ. It can be seen clearly from FIG. \\ref{even_odd} that averaging over as-small-as 10 configurations already shows a convergence towards an odd-in-$\\phi_{12}$ behaviour while the even-in-$\\phi_{12}$ part is strongly suppressed. It is interesting to note that, in absence of an averaging (corresponding to a fixed quenched disorder configuration), for certain range of values of $\\phi_{12}$, the even part can be the dominant contribution in the net thermal conductance (See FIG. \\ref{even_odd}).\n\nNow we extend the numerical analysis to a larger number of scattering centers. The scattering centers are modeled as before and the length of the junction is fixed at $L=0.555 \\xi$. For a given number of scattering centers, disorder average is done over 500 configurations where we have checked that beyond this, there is negligible variation of the result. The mean and the variance of the thermoelectric conductances are plotted as a function of the superconducting phase difference $\\phi_{12}$ in FIG.\\ref{multipleDisorders}. Note that, in presence of a single scatterer, the variance of the thermoelectric conductance is smallest because in this case there is no even-in-$\\phi_{12}$ part of the thermoelectric conductance. We have also observed that, in general, the variance is relatively lower in the neighborhood of $\\phi_{12}=\\pi$ rather than in the neighborhood of $\\phi_{12}=0$ or $2\\pi$. To conclude, the plot for thermoelectric conductance after averaging tend to reduce to the universal sinusoidal dependence of $\\phi_{12}$ as the number of scatterers within the junction region increases (see the middle figure of FIG. \\ref{multipleDisorders}). This is due to the fact that, with increasing opacity of the JJ, the $\\phi_{12}$ sensitivity of the thermoelectric conductance via the poles of the quasiparticle transmission probabilities decreases and the major contribution comes from the explicit $\\sin{(\\phi_{12})}$ factor in the numerator.\n\n\n\\begin{figure*}[t]\n\n\t\\includegraphics[width=0.7 \\textwidth]{DensityPlotInTauAndPhi.eps}\n\t\\caption{The maximum possible thermoelectric coefficient of a JJ with (left) s-wave and (right) p-wave superconductivity for a given junction length and normal state reflection probability $r$ (as calculated wit the analytic approximation $\\mu>>\\Delta_0,k_BT$). Note that the scatterer is assumed to be energy independent and is placed at the middle of the junction. The parameters are assumed to be $\\mu=100\\Delta_0$ and $k_BT=0.5\\Delta_0$.}\n\t\\label{max_thermo}\n\\end{figure*}\n\n\\section{Discussion}\nOccurrence of thermoelectric effect through the breaking of $\\omega \\rightarrow -\\omega$ symmetry for a ballistic long JJ is not specific to the HES. 1D JJ with quadratic dispersion and with s-wave or p-wave superconductivity should also demonstrate such a response. Of course, in the high doping limit, the thermoelectric coefficient should reduce to the results obtained in the paper when linearized about the Fermi energy. Thus, the thermoelectric response is a generic property of any ballistic JJ with junction length of the order of the superconducting coherence length. However, with the increasing opacity of the JJ for junction length less than the superconducting coherence length, the ABS energies tend to move towards the zero energy for p-wave superconductivity due to the presence of Majorana fermions. Whereas, for a JJ with s-wave superconductivity within the same limit, the ABS energies tend to move towards the continuum with increasing opacity of the junction. This fact manifests itself in the thermoelectric conductance via the poles of the quasiparticle transmission probabilities. Also, for JJ with junction length longer than the superconducting coherence length, the states from the continuum spectrum tend to leak into the superconducting gap, thereby changing the details of the thermoelectric coefficient.\n\nFurther, to check if the s-wave or the p-wave leads to a larger thermoelectric coefficient for a given junction length we perform an analysis where we have placed an energy-independent scatterer at the middle of a JJ, and plotted the maximum possible thermoelectric conductance (scanned over all values of $\\phi_{12}$) for a given junction length $L$ and given transparency of the scatterer (normal state transmission probability $\\tau$) as shown in FIG. \\ref{max_thermo}. We have performed this study within the approximation $\\mu>>\\Delta_0, k_BT$ (See Appendix \\ref{Appendix_middle_scatterer}). From these results we can conclude that in general, there is no distinguishable pattern in the thermoelectric coefficient for the case of s-wave and p-wave superconductivity.\n\nAs far as the possible strategy for the measurement of the thermoelectric current is concerned, it cannot be measured in isolation as it will always be accompanied by the finite temperature Josephson current. However, there may be ways to measure the thermoelectric coefficient indirectly. For example, consider a situation where a JJ is initially maintained at an equilibrium temperature $T$. The current that is obtained, is totally the Josephson current $\\mathtt{S}_{(T,T)}=I_J$, where the first (second) subscript corresponds to the temperature of the left (right) lead S1 (S2). Now, if $S1$ is raised to temperature $T+\\Delta T$, then the corresponding total current will be a sum of the Josephson current and the thermoelectric current $\\mathtt{S}_{(T+\\Delta T, T)}=I_J-\\Delta I_J + \\kappa^{21}_e \\Delta T$, where $\\Delta I_J$ is the variaation in the Josephson current due temperature bias. Next, consider the situation where $S1$ is kept at temperature $T$ while $S2$ is raised to temperature $T+\\Delta T$, then the corresponding total current will be $\\mathtt{S}_{(T, T+\\Delta T)}=I_J-\\Delta I_J - \\kappa^{21}_e \\Delta T$. Now if, $(2\\mathtt{S}_{(T,T)}-(\\mathtt{S}_{(T+\\Delta T, T)}+\\mathtt{S}_{(T, T+\\Delta T)}))\/2\\mathtt{S}_{(T,T)}<< 1$ then a measurement of the ratio, $(\\mathtt{S}_{(T+\\Delta T, T)}-\\mathtt{S}_{(T, T+\\Delta T)})\/2\\Delta T$ will provide the thermoelectric coefficient. Note that, a similar strategy involving $\\phi_{12}\\rightarrow -\\phi_{12}$ rather than involving $\\Delta T \\rightarrow -\\Delta T$ is difficult to implement due to the presence of even-in-$\\phi_{12}$ part of the thermoelectric coefficient.\n\n\\textit{\\underline{Acknowledgment}}: A.M. thanks Vivekananda Adak for helpful discussions. We thank Chris Olund, Sergey Pershoguba and Erhai Zhao for useful communication over email. A.M. acknowledges Ministry of Education, India for funding. S.D. would like to acknowledge the MATRICS\ngrant (MTR\/ 2019\/001 043) from the Science and Engineering Research Board (SERB) for funding.\n\n\n\\section{Matrix formalism}\n\\label{matrixFormalism}\nTo have a clear physical insight into different semi-classical paths that give rise to degeneracy-lifted ABS and the thermoelectric response of a JJ, we shall be using the matrix method as discussed by A. Kundu et. al. \\cite{kundu_PRB_82_155441}.\n\nLet $\\Psi_{qp[N]}^{e+(-)}$ and $\\Psi_{qp[N]}^{h+(-)}$ denote forward (backward) moving electron-like QP and forward (backward) moving hole-like QP respectively within the superconducting lead $S_i$ having superconducting phase $\\phi_i$ $(i \\in \\{1,2\\})$ [within the normal region]. These wave functions can be explicitly calculated using the BdG Hamiltonian (\\ref{DiracHamiltonian}) in the main text or the BdG Hamiltonian with quadratic dispersion and with s-wave or p-wave superconductivity\n\\begin{align}\n\t\\mathcal{H}_{\\eta}= \\left( -\\dfrac{\\hbar^2}{2m}\\dfrac{\\partial^2}{\\partial x^2}-\\mu \\right) \\tau_z + \\Delta^{\\eta}(x) (\\cos \\phi_r \\tau_x - \\sin \\phi_r \\tau_y);\t\\label{HamiltonianSP}\n\\end{align}\nwhere $\\Delta^{\\eta}(x)=\\Delta_0 [ \\Theta(-x-L\/2)+\\Theta(x-L\/2) ]\\mathit{f}(\\eta)$, $\\mathit{f}(\\eta)=(-i\\partial_x\/k_F)^{(1-\\eta)\/2}$, $\\eta=\\pm 1$ for s-wave and p-wave superconductivity respectively and $p_F=\\hbar k_F=\\sqrt{2m \\mu}$ is the Fermi momentum.\n\nWe first consider two reflection matrices $\\mathbb{R}^{\\gamma}$, $\\gamma \\in \\{1,2\\}$, which describe both Andreev and normal reflections at the normal-superconducting junctions.\n\\begin{equation*}\n\t\\begin{aligned}[c]\n\t\t\\mathbb{R}^1 \\Psi^{e+}_N &= r^1_{Ahe} \\Psi^{h-}_N + r^1_{Nee} \\Psi^{e-}_N,\\\\\n\t\t\\mathbb{R}^1 \\Psi^{h-}_N &= r^1_{Aeh} \\Psi^{e+}_N + r^1_{Nhh} \\Psi^{h+}_N,\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t\\mathbb{R}^2 \\Psi^{e-}_N &= r^2_{Ahe} \\Psi^{h+}_N + r^2_{Nee} \\Psi^{e+}_N,\\\\\n\t\t\\mathbb{R}^2 \\Psi^{h+}_N &= r^2_{Aeh} \\Psi^{e-}_N + r^2_{Nhh} \\Psi^{h-}_N,\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t\\mathbb{R}^1 \\Psi^{e-}_N &= \\mathbb{R}^1 \\Psi^{h+}_N = \\mathbb{R}^2 \\Psi^{e+}_N = \\mathbb{R}^2 \\Psi^{h-}_N = 0,\n\t\\end{aligned}\n\\end{equation*}\nwhere $r^{\\gamma}_{Aqq'}$ and $r^{\\gamma}_{Nqq'}$ respectively describe the amplitudes of different Andreev reflections and normal reflections.\n\nTo consider the propagation of the wave functions through a length $l$ within the normal region, we consider two propagation matrices, $\\mathbb{T}^{\\gamma}$ $(\\gamma \\in \\{1,2\\})$, such that\n\\begin{equation*}\n\t\\begin{aligned}[c]\n\t\t&\\mathbb{T}^1(l) \\Psi^{e+}_N|_{x} = \\Psi^{e+}_N|_{x+l},\\\\\n\t\t&\\mathbb{T}^1(l) \\Psi^{h-}_N|_{x} = \\Psi^{h-}_N|_{x-l},\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t&\\mathbb{T}^2(l) \\Psi^{e-}_N|_{x} = \\Psi^{e-}_N|_{x-l},\\\\\n\t\t&\\mathbb{T}^2(l) \\Psi^{h+}_N|_{x} = \\Psi^{h+}_N|_{x+l},\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t\t&\\mathbb{T}^1(l) \\Psi^{e-}_N = \\mathbb{T}^1(l) \\Psi^{h+}_N = \\mathbb{T}^2(l) \\Psi^{e+}_N = \\mathbb{T}^2(l) \\Psi^{h-}_N =0.\n\t\\end{aligned}\n\\end{equation*}\n\nFor energies above the superconducting gap, two tunneling matrices at the two boundaries, $\\mathbb{T}^{L,R}_B$, are defined as\n\\begin{equation*}\n\t\\begin{aligned}[c]\n\t\t&\\mathbb{T}_B^L \\Psi_{qp}^{e+}[\\phi_1] = t_e \\Psi^{e+}_N + t_{Ae} \\Psi^{h+}_N,\\\\\n\t\t&\\mathbb{T}_B^L \\Psi_{qp}^{h+}[\\phi_1] = t_h \\Psi^{h+}_N + t_{Ah} \\Psi^{e+}_N,\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t&\\mathbb{T}_B^R \\Psi^{e+}_N = t_e^{qp} \\Psi_{qp}^{e+}[\\phi_2] + t_{Ae}^{qp}\\Psi_{qp}^{h+}[\\phi_2],\\\\\n\t\t&\\mathbb{T}_B^R \\Psi^{h+}_N = t_h^{qp} \\Psi_{qp}^{h+}[\\phi_2] + t_{Ah}^{qp} \\Psi_{qp}^{e+}[\\phi_2],\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t&\\mathbb{T}_B^L \\Psi_{qp}^{e-} = \\mathbb{T}_B^L \\Psi_{qp}^{h-} \\\\\n\t\t&= \\mathbb{T}_B^R \\Psi^{e-}_N= \\mathbb{T}_B^R \\Psi^{h-}_N = 0.\n\t\\end{aligned}\n\\end{equation*}\n\nWe also consider scattering matrices within the normal region to account for the disorders,\n\\begin{equation*}\n\t\\begin{aligned}[c]\n\t\t\\mathscr{T}^e\n\t\t\\begin{bmatrix}\n\t\t\t\\Psi^{e+}|_{-x}\t\\\\\n\t\t\t\\Psi^{e-}|_{+x}\n\t\t\\end{bmatrix}\n\t\t&=\n\t\t\\begin{bmatrix}\n\t\t\t\\Psi^{e+}|_{+x}\t\\\\\n\t\t\t\\Psi^{e-}|_{-x}\n\t\t\\end{bmatrix}\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t\\mathscr{T}^h\n\t\t\\begin{bmatrix}\n\t\t\t\\Psi^{h-}|_{+x}\t\\\\\n\t\t\t\\Psi^{h+}|_{-x}\n\t\t\\end{bmatrix}\n\t\t&=\n\t\t\\begin{bmatrix}\n\t\t\t\\Psi^{h-}|_{-x}\t\\\\\n\t\t\t\\Psi^{h+}|_{+x}\n\t\t\\end{bmatrix}\n\t\\end{aligned}\n\\end{equation*}\nNote that, the matrices $\\mathscr{T}^e$ and $\\mathscr{T}^h$ are related by the particle-hole symmetry of the corresponding BdG Hamiltonian.\n\nExplicit expressions of the reflection matrices $\\mathbb{R}^{\\gamma}$ and tunneling matrices $\\mathbb{T}^{L,R}_B$ can be obtained by demanding the continuity of the wave functions across the boundaries in case of JJ based on HES or by using the following boundary conditions in case of JJ with quadratic dispersion\\cite{tinyukova2019andreev}\n\\begin{align}\n\t\\dfrac{\\hbar^2}{2m} \\tau_z \\left[ \\partial_x^{(\\beta)} \\Psi_S^{\\pm} - \\partial_x^{(\\beta)} \\Psi_N^{\\pm} \\right] +i \\beta \\left(\\dfrac{1-\\eta}{2}\\right) \\dfrac{\\Delta_0}{k_F} \\left[ \\cos \\phi_{\\pm} \\tau_x - \\sin \\phi_{\\pm} \\tau_y \\right] \\Psi_S^{\\pm}=0\n\\end{align}\nwhere $\\beta \\in \\{0,1\\}$; $\\eta=1$ for s-wave and $\\eta=-1$ for p-wave superconductivity; $\\phi_{+}=\\phi_2$ and $\\phi_{-}=\\phi_1$; $\\Psi_S$ and $\\Psi_N$ are the wave functions in the superconducting and normal regions respectively.\n\n\\section{Clean junction}\n\\label{Appendix_clean_junction}\nAndreev bound states are the result of multiple Andreev reflections. There are two ways in which Andreev bound state can be formed as discussed in the main text. We shall describe the same processes here with the help of matrix formalism discussed in \\ref{matrixFormalism}.\n\n\\textit{(i) Tunneling of a Cooper pair from left to right:} An electron-like quasiparticle starts at $x=-L\/2$ (i.e. $\\Psi^{e+}_N|_{x=-L\/2}$) and propagates through the normal region and reaches at $x=L\/2$ (i.e. $\\Psi^{e+}_N|_{x=L\/2}=\\mathbb{T}^{1} \\Psi^{e+}_N|_{x=-L\/2}$). It Andreev reflects back as a hole with uni-modular amplitude $r^1_{Ahe}$ (i.e. $r^1_{Ahe}\\Psi^{h-}_N= \\mathbb{R}_A^{1} \\Psi^{e+}_N$) by creating a Cooper pair in the superconducting lead 2 (S2). The reflected hole then travels through the normal region and reaches at $x=-L\/2$ (i.e. $\\Psi^{h-}_N|_{x=-L\/2} = \\mathbb{T}^{1} \\Psi^{h-}_N|_{x=L\/2}$). It then again Andreev reflects as an electron with uni-modular amplitude $r^1_{Aeh}$ (i.e. $r^1_{Aeh} \\Psi^{e+}_N = \\mathbb{R}_A^{1} \\Psi^{h-}_N$) by annihilating a Cooper pair in the superconducting lead 1 (S1). Now for $\\omega \\leq \\Delta_0$, matrices $\\mathbb{R}^{\\gamma}$ and $\\mathbb{T}^{\\gamma}$ are unitary, so it must be\n\\begin{align}\n\t\\Psi^{e+}_N|_{x=-L\/2} = (\\mathbb{R}^{1}\\mathbb{T}^{1}\\mathbb{R}^{1}\\mathbb{T}^{1}) \\Psi^{e+}_N|_{x=-L\/2}.\t\\label{conditionABSfirstType}\n\\end{align}\nThe corresponding Andreev bound state energy can be obtained by solving the determinant condition\n\\begin{align}\n\t\\text{det.}(\\mathbb{I}_{4\\times 4}-\\mathbb{R}^{1}\\mathbb{T}^{1}\\mathbb{R}^{1}\\mathbb{T}^{1}) =0,\n\\end{align}\nwhich gives the ABS energy $\\omega_0^{21}$.\n\n\\textit{(ii) Tunneling of a Cooper pair from right to left:} If a right-moving hole-like quasiparticle starts from $x=-L\/2$ (i.e. $\\Psi^{h+}_N|_{x=-L\/2}$) and completes the cycle after two Andreev reflections, it can transfer a Cooper pair from S2 to S1\n\\begin{align}\n\t\\Psi^{h+}_N|_{x=-L\/2} = (\\mathbb{R}^2\\mathbb{T}^2\\mathbb{R}^2\\mathbb{T}^2) \\Psi^{h+}_N|_{x=-L\/2}.\t\\label{conditionABSsecondType}\n\\end{align}\nThe corresponding Andreev bound state energy can be obtained by solving the equation\n\\begin{align}\n\t\\text{det.}(\\mathbb{I}_{4\\times 4}-\\mathbb{R}^{2}\\mathbb{T}^{2}\\mathbb{R}^{2}\\mathbb{T}^{2}) =0,\n\\end{align}\nwhich gives the ABS energy $\\omega_0^{12}$.\n\nNow, tunneling of a quasiparticle with energy $\\omega>\\Delta_0$ from S1 to S2 can be understood in terms of the matrices $\\mathbb{R}^{\\gamma}$, $\\mathbb{T}^{\\gamma}$ and $\\mathbb{T}_B^{(L,R)}$.\n\n\\textit{(i) Tunneling of an electron (hole)-like quasiparticle from left (right) to right (left):} For a clean junction, an incident electron-like quasiparticle in S1 (i.e. $\\Psi_{qp}^{e+}[\\phi_1]$) can tunnel into S2 as a electron-like quasiparticle (i.e. $\\Psi_{qp}^{e+}[\\phi_2]$) either directly or by any even number of Andreev reflections. Mathematically,\n\\begin{align}\n\t\\chi_{ee}^{21} \\Psi_{qp}^{e+}[\\phi_2]&=\\mathbb{T}_B^R (\\mathbb{T}^1 + \\mathbb{T}^1 \\mathbb{R}^1 \\mathbb{T}^1 \\mathbb{R}^1 \\mathbb{T}^1 + ...)\\mathbb{T}_B^L \\Psi_{qp}^{e+}[\\phi_1] = \\mathbb{T}_B^R \\mathbb{T}^1(\\mathbb{I}-\\mathbb{R}^1 \\mathbb{T}^1 \\mathbb{R}^1 \\mathbb{T}^1)^{-1} \\mathbb{T}_B^L \\Psi_{qp}^{e+}[\\phi_1].\t\\label{electronTransmission}\n\\end{align}\nIt is clear from Eq. (\\ref{electronTransmission}) and (\\ref{conditionABSfirstType}) that the tunneling of an electron-like quasiparticle from S1 to S2 is in correspondence with the Andreev bound state having energy $\\omega_0^{21}$. Solving Eq. (\\ref{electronTransmission}) we can calculate $\\chi_{ee}^{21}$ and hence $\\mathcal{T}_{ee}^{21}$.\n\n\\textit{(ii) Tunneling of an hole (electron)-like quasiparticle from left (right) to right (left):} Similarly, tunneling of a hole-like quasiparticle from S1 to S2 can be mathematically expressed as\n\\begin{align}\n\t\\chi_{hh}^{21} \\Psi_{qp}^{h+}[\\phi_2]&=\\mathbb{T}_B^R (\\mathbb{T}^2 + \\mathbb{T}^2 \\mathbb{R}^2 \\mathbb{T}^2 \\mathbb{R}^2 \\mathbb{T}^2 + ...)\\mathbb{T}_B^L \\Psi_{qp}^{h+}[\\phi_1]= \\mathbb{T}_B^R \\mathbb{T}^2(\\mathbb{I}-\\mathbb{R}^2 \\mathbb{T}^2 \\mathbb{R}^2 \\mathbb{T}^2)^{-1} \\mathbb{T}_B^L \\Psi_{qp}^{h+}[\\phi_1].\t\\label{holeTransmission}\n\\end{align}\nA comparison between Eq. (\\ref{holeTransmission}) and (\\ref{conditionABSsecondType}) clearly indicates the fact that the tunneling of a hole-like quasiparticle from S1 to S2 is in correspondence with the Andreev bound state having energy $\\omega_0^{12}$. Solving Eq. (\\ref{holeTransmission}) we can calculate $\\chi_{hh}^{21}$ and hence $\\mathcal{T}_{hh}^{21}$.\n\n\\section{Significance of the quantity $(k_e-k_h)L\/2$}\n\\label{ApproximationkL}\nWe have assumed the doping of the junction is sufficiently high, so let us retain the expressions of $k_e$ and $k_h$ up to the first order of $\\omega\/\\mu$ for quadratic dispersion relation,\n\\begin{align}\n\tk_e =\\dfrac{\\sqrt{2m}}{\\hbar}\\sqrt{\\mu+\\omega} \\approx \\dfrac{\\sqrt{2m \\mu}}{\\hbar} \\left( 1+\\dfrac{\\omega}{2\\mu} \\right)\t\\text{\t;\t}\tk_h =\\dfrac{\\sqrt{2m}}{\\hbar}\\sqrt{\\mu-\\omega} \\approx \\dfrac{\\sqrt{2m \\mu}}{\\hbar} \\left( 1-\\dfrac{\\omega}{2\\mu} \\right)\n\\end{align}\nNow, we shall consider the length of the junction $L$ to be finite compare to the superconducting coherence length $\\xi=\\hbar \\sqrt{2\\mu\/m}\/\\Delta_0$ so let $L=\\mathrm{x}\\xi$. Now,\n\\begin{align}\n\t\\dfrac{k_e-k_h}{2}L\n\t\\approx \\dfrac{1}{2} \\dfrac{\\sqrt{2m \\mu}}{\\hbar} \\left[ \\left( 1+\\dfrac{\\omega}{2\\mu} \\right)-\\left( 1-\\dfrac{\\omega}{2\\mu} \\right) \\right] \\mathrm{x}\\xi\n\t\\approx \\dfrac{1}{2} \\dfrac{\\sqrt{2m \\mu}}{\\hbar} \\dfrac{\\omega}{\\mu} \\left( \\mathrm{x} \\dfrac{\\hbar}{\\Delta_0} \\sqrt{\\dfrac{2\\mu}{m}} \\right)\t\n\t\\approx \\mathrm{x} \\dfrac{\\omega}{\\Delta_0}.\n\\end{align}\nThus, even for large enough doping, the quantity $(k_e-k_h)L\/2$ is of the order of $\\omega\/\\Delta_0$, and thus cannot be neglected.\n\nFor linear dispersion relation, $k_e =\\frac{\\mu+\\omega}{\\hbar v_F}$ and $k_h =\\frac{\\mu-\\omega}{\\hbar v_F}$, hence, here also $\\frac{k_e-k_h}{2}L \\approx \\mathrm{x} \\frac{\\omega}{\\Delta_0}$.\n\n\\section{Presence of a scatterer in the middle of the junction}\n\\label{Appendix_middle_scatterer}\nStarting with an initial state $\\left( (\\Psi^{e+}_N|_{x=-L\/2}), (\\Psi^{e-}_N|_{x=L\/2}) \\right)^T$, it will come back to the same state after a electron scattering followed by an Andreev reflection, a hole scattering and another Andreev reflection. For $\\omega<\\Delta_0$, these matrices all being unitary, it must be\n\\begin{align}\n\t\\begin{bmatrix}\n\t\t\\Psi^{e+}_N|_{x=-L\/2}\t\\\\\n\t\t\\Psi^{e-}_N|_{x=L\/2}\n\t\\end{bmatrix}\n\t=\n\t\\mathbb{R}^A_P \\mathscr{T}^h_P \\mathbb{R}^A_P \\mathscr{T}^e_P\n\t\\begin{bmatrix}\n\t\t\\Psi^{e+}_N|_{x=-L\/2}\t\\\\\n\t\t\\Psi^{e-}_N|_{x=L\/2}\n\t\\end{bmatrix}\n\\end{align}\nwhere we have defined $\\mathbb{M}_P=\\mathbb{M} \\mathbb{T}^P$. Note that, in the absence of barrier i.e. at $\\mathscr{T}^e=\\mathscr{T}^h=\\mathbb{I}$, all the matrices $\\mathbb{T}^P$, $\\mathbb{R}^A$, $\\mathscr{T}^e$ and $\\mathscr{T}^h$ are block diagonal and the aforesaid two types of ABS ($\\omega_0^{21}$ and $\\omega_0^{12}$) do not interfere. In presence of barrier, finite backscattering (off-diagonal blocks of $\\mathscr{T}^e$ and $\\mathscr{T}^h$) gives rise to the interference between the two types of ABS ($\\omega_0^{21}$ and $\\omega_0^{12}$).\n\nABS energies, in presence of barrier can be obtained by solving the equation\n\\begin{align}\n\t\\text{det}.\\left( \\mathbb{I}_{4\\times 4}-\\mathbb{R}^A_P \\mathscr{T}^h_P \\mathbb{R}^A_P \\mathscr{T}^e_P \\right) =0\n\t\\label{EQ63}\n\\end{align}\nNote that, if we had started with the initial state $\\left( (\\Psi^{h-}|_{x=L\/2}), (\\Psi^{h+}|_{x=-L\/2}) \\right)^T$ then Eq. (\\ref{EQ63}) would have looked like\n\\begin{align}\n\t\\text{det}.\\left( \\mathbb{I}_{4\\times 4}-\\mathbb{R}^A_P \\mathscr{T}^e_P \\mathbb{R}^A_P \\mathscr{T}^h_P \\right) =0\t\\label{EQ64}\n\\end{align}\nIt turns out, the ABS energies, as obtained from (\\ref{EQ63}) or (\\ref{EQ64}) are same.\n\n\nFor energies $\\omega>\\Delta_0$, we define the following matrices\n\\begin{equation*}\n\t\\begin{aligned}[c]\n\t\t\\mathbb{T}^L =\n\t\t\\begin{bmatrix}\n\t\t\t\\mathbb{T}_B^L\t&0\t\\\\\n\t\t\t0\t&\\mathbb{T}_B^L\n\t\t\\end{bmatrix}\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t\\mathbb{T}^R_e =\n\t\t\\begin{bmatrix}\n\t\t\t\\mathbb{T}_B^R\t&0\t\\\\\n\t\t\t0\t&0\n\t\t\\end{bmatrix}\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t\\mathbb{T}^R_h =\n\t\t\\begin{bmatrix}\n\t\t\t0\t&0\t\\\\\n\t\t\t0\t&\\mathbb{T}_B^R\n\t\t\\end{bmatrix}\n\t\\end{aligned}\n\\end{equation*}\nWith this, tunneling of a QP from S1 to S2 can be understood as follows:\n\n\\textbf{(i)} An incident electron-like QP in S1 $\\left( (\\Psi_{qp}^{e+}[\\phi_1]),(0) \\right)^T$ can tunnel into S2 as an electron-like QP $\\left( (\\Psi_{qp}^{e+}[\\phi_2]),(0) \\right)^T$ either directly or by any even number of Andreev reflections whereas tunneling of an electron-like QP from S1 into S2 as an hole like QP $\\left( (0),(\\Psi_{qp}^{h+}[\\phi_2]) \\right)^T$ must be mediated by an odd number of Andreev reflections.\n\\begin{equation*}\n\t\\begin{aligned}[c]\n\t\t\\chi_{ee}^{21} \n\t\t\\begin{bmatrix}\n\t\t\t\\Psi_{qp}^{e+}[\\phi_2] \t\\\\\n\t\t\t0\n\t\t\\end{bmatrix}\n\t\t&=\\mathbb{T}_e^R \\mathbb{T}^P \\mathscr{T}^e_P (\\mathbb{B}^e)^{-1} \\mathbb{T}^L\n\t\t\\begin{bmatrix}\n\t\t\t\\Psi_{qp}^{e+}[\\phi_1] \t\\\\\n\t\t\t0\n\t\t\\end{bmatrix}\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t\\chi_{he}^{21}\n\t\t\\begin{bmatrix}\n\t\t\t0\t\\\\\n\t\t\t\\Psi_{qp}^{h+}[\\phi_2]\n\t\t\\end{bmatrix}\n\t\t&= \\mathbb{T}_h^R \\mathbb{T}^P \\mathscr{T}^h_P \\mathbb{R}^A_P \\mathscr{T}^e_P (\\mathbb{B}^e)^{-1}\\mathbb{T}^L\n\t\t\\begin{bmatrix}\n\t\t\t\\Psi_{qp}^{e+}[\\phi_1]\t\\\\\n\t\t\t0\n\t\t\\end{bmatrix}\n\t\\end{aligned}\n\\end{equation*}\nwhere $\\mathbb{B}^e=\\mathbb{I}_{4\\times 4}-\\mathbb{R}^A_P \\mathscr{T}_P^{h}\\mathbb{R}^A_P\\mathscr{T}_P^{e}$. Solving above equations we can calculate $\\chi_{ee}^{21}$ and $\\chi_{he}^{21}$ and hence we $\\mathcal{T}_{ee}^{21}$ and $\\mathcal{T}_{he}^{21}$.\n\n\\textbf{(ii)} Similarly, tunneling of a hole-like QP from S1 $\\left( (0),(\\Psi_{qp}^{h+}[\\phi_1]) \\right)^T$ into S2 as an hole-like QP $\\left( (0),(\\Psi_{qp}^{h+}[\\phi_2]) \\right)^T$ can be mediated directly or by any even number of Andreev reflections whereas tunneling of a hole-like QP from S1 into S2 as an electron-like QP $\\left( (\\Psi_{qp}^{e+}[\\phi_2]),(0) \\right)^T$ must be mediated by an odd number of Andreev reflections.\n\\begin{equation*}\n\t\\begin{aligned}[c]\n\t\t\\chi_{hh}^{21} \n\t\t\\begin{bmatrix}\n\t\t\t0\t\\\\\n\t\t\t\\Psi_{qp}^{h+}[\\phi_2]\n\t\t\\end{bmatrix}\n\t\t&=\\mathbb{T}_h^R \\mathbb{T}^P \\mathscr{T}^h_P (\\mathbb{B}^h)^{-1} \\mathbb{T}^L\n\t\t\\begin{bmatrix}\n\t\t\t0\t\\\\\n\t\t\t\\Psi_{qp}^{h+}[\\phi_1]\n\t\t\\end{bmatrix}\n\t\\end{aligned}\n\t\\qquad\n\t\\begin{aligned}[c]\n\t\t\\chi_{eh}^{21}\n\t\t\\begin{bmatrix}\n\t\t\t\\Psi_{qp}^{e+}[\\phi_2]\t\\\\\n\t\t\t0\n\t\t\\end{bmatrix}\n\t\t&= \\mathbb{T}_e^R \\mathbb{T}^P \\mathscr{T}^e_P \\mathbb{R}^A_P \\mathscr{T}^h_P (\\mathbb{B}^h)^{-1}\\mathbb{T}^L\n\t\t\\begin{bmatrix}\n\t\t\t0\t\\\\\n\t\t\t\\Psi_{qp}^{h+}[\\phi_1]\n\t\t\\end{bmatrix}\n\t\\end{aligned}\n\\end{equation*}\nwhere $\\mathbb{B}^h=\\mathbb{I}_{4\\times 4}-\\mathbb{R}^A_P \\mathscr{T}_P^{e}\\mathbb{R}^A_P\\mathscr{T}_P^{h}$. Solving above equations we can calculate $\\mathcal{T}_{hh}^{21}$ and $\\mathcal{T}_{eh}^{21}$.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Proofs} \n\\label{sec:appendix_more_dis}\n\n\\printProofs \n\n \\begin{thm} \\label{thm:lyap2} \n Assume $$\\d Z_t = \\eta(Z_t, t) \\dt + \\sigma(\\X_t, t) \\dW_t,~~~~~~~t\\in[0,\\T].$$ \nWe have $Z_\\T \\in A$ with probability one \nif there exists a function $U\\colon \\RR^d \\times [0,1] \\to \\RR$ such that \n\n1) $U(\\cdot, t) \\in C^2(\\RR^d)$ and $U(z, \\cdot) \\in C^1([0,\\T]);$ \n \n 2) $U(z, \\T) \\geq 0$, $z\\in \\RR^d$, and $U(z, \\T) = 0$ implies that $z \\in A$, where $A$ is a measurable set in $\\RR^d$; \n \n 3) There exists a sequence $\\{\\alpha_t$, $\\beta_{t}, \\gamma_t \\colon t\\in[0, \\T] \\}$, such that for %\n $t\\in[0,\\T]$, \n \\bb \n \\E[\\dd_z U(Z_t, t) \\tt \\eta(Z_t, t)] & \\leq - \\alpha_t \\E[U(Z_t, t)] + \\beta_{t}, \\\\ \n \\E[\\partial_t U(Z_t, t) + \\frac{1}{2} \\trace(\\dd_z^2 U(Z_t, t) \\sigma^2(Z_t, t))] & \\leq \\gamma_t; \n\\ee \n\n4) Define $\\zeta_t = \\exp(\\int_0^t \\alpha_s \\d s)$. We assume \n\\bbb \\label{equ:zeta}\n\\lim_{t\\uparrow T} \\zeta_t = +\\infty, ~~~~ \n\\lim_{t\\uparrow T} \\frac{\\zeta_t}{\\int_0^t \\zeta_s (\\beta_s + \\gamma_s) \\d s} = +\\infty. \n\\eee \n\n\\end{thm} \n \n \\begin{proof} \n Following $\\d Z_t = \\eta(Z_t, t) \\dt + \\sigma(\\X_t, t) \\dW_t$, we have by Ito's Lemma, \n \\bb \n \\d U(Z_t, t) \n & = \\dd U(Z_t, t) \\tt (\\eta(Z_t, t) \\dt + \\sigma(\\X_t, t) \\dW_t) + \n \\partial_t U(Z_t, t) \\dt + \n \\frac{1}{2} \\trace(\\dd^2 U(Z_t, t)\\sigma^2(\\X_t, t)) \\dt,\n \\ee \n for $t\\in[0,T]$. \n Taking expectation on both sides, \n $$\n \\frac{\\d}{\\d t} \\E(U(Z_t)) = \n \\E[\\dd_z U(Z_t, t) \\tt \\eta(Z_t, t)] + \n \\E\\left [\\partial_t U(Z_t, t) + \\frac{1}{2} \\trace(\\dd^2 U(Z_t, t)\\sigma^2(\\X_t, t)) \\right].\n $$\n Let $u_t = \\E[U(Z_t, t)]$.\n By the assumption above, we get \n $$\n \\dot u_t \\leq -\\alpha_t u_t + \\beta_t + \\gamma_t. \n $$\nFollowing Gr\u00f6nwall's inequality (see Lemma~\\ref{lem:gronwall} below), we have $\\E[U(Z_\\T, \\T)] = u_\\T = \\lim_{t\\uparrow \\T} u_t \\leq 0$ if \\eqref{equ:zeta} holds. Because $U(z, \\T) \\geq 0$, this suggests that $U(Z_\\T, \\T) = 0$ and hence $Z_\\T \\in A$ almost surely. \n\\end{proof} \n\n\\begin{lem}\\label{lem:gronwall} \nLet $u_t\\in \\RR$ and $\\alpha_t, \\beta_t\\geq 0$, and \n $\\frac{\\d}{\\dt } u_t \\leq - \\alpha_t u_t + \\beta_t$, $t \\in [0,T]$ for $T>0$. \nWe have \n\\bb \nu_t \\leq \\frac{1}{\\zeta_t} (\\zeta_0 u_0 + \\int_0^t \\zeta_s \\beta_s \\d s), \n&&\\text{where}&& \\zeta_t = \\exp(\\int_0^t \\alpha_s \\d s).\n\\ee \nTherefore, we have $\\lim_{t\\uparrow T} u_t \\leq 0$ if \n$$\n\\lim_{t\\uparrow T} \\zeta_t = +\\infty, ~~~~ \n\\lim_{t\\uparrow T} \\frac{\\zeta_t}{\\int_0^t \\zeta_s \\beta_s \\d s} = +\\infty. \n$$\n\\end{lem}\n\\begin{proof} \n Let $v_t = \\zeta_t u_t$, where $\\zeta_t =\\exp(\\int_0^t \\alpha_s \\d s)$ so $\\dot \\zeta_t = \\zeta_t \\alpha_t$. Then \n $$\\frac{\\d}{\\dt} v_t =\\dot \\zeta_t u_t + \\zeta_t \\dot u_t \\leq (\\dot \\zeta_t - \\zeta_t \\alpha_t ) u_t + \\zeta_t \\beta_t = \\zeta_t \\beta_t. \n $$\n So \n $$\n v_t \\leq v_0 + \\beta \\int_0^t \\gamma_s \\d s, \n $$\n and hence \n $$\n u_t \\leq \\frac{1}{\\zeta_t} (\\zeta_0 u_0 + \\int_0^t \\zeta_s \\beta_s \\d s). \n $$\n To make $\\lim_{t\\uparrow T} u_t \\leq 0$, we want \n$$\n\\lim_{t\\uparrow T} \\zeta_t = +\\infty, ~~~~ \n\\lim_{t\\uparrow T} \\frac{\\zeta_t}{\\int_0^t \\zeta_s \\beta_s \\d s} = +\\infty. \n$$\n \\end{proof} \n \n\n\\begin{cor}\nLet $\\d \\Z_t = \\frac{x-\\Z_t}{\\T-t} + \\varsigma_t\\d W_t$ with law $\\Q$. This uses the drift term of Brownian bridge, but have a time-varying diffusion coefficient $\\varsigma_t\\geq0$. \nAssume $\\sup_{t\\in[0,T]}\\varsigma_t <\\infty$. Then $\\Q(Z_\\T = z) = 1$. \n\\end{cor}\n\\begin{proof}\nWe verify the conditions in Theorem~\\ref{thm:lyap2}. \nDefine $U(z,t) = \\norm{x-z}^2\/2$, and $\\eta(z,t) = \\frac{x-\\Z_t}{\\T-t}$. We have \n$\\eta(z,t) \\tt \\dd U(z,t) = - U(z,t)\/(T-t)$. So $\\alpha_t = 1\/(T-t)$. \n\nAlso, $\\partial_t U(z,t)+\\frac{1}{2} \\trace(\\varsigma_t^2 \\dd_z ^2 U(z,t)) = \\frac{1}{2} \\diag(\\varsigma_t^2 I_{d\\times d}) = \\frac{d}{2} \\varsigma_t^2 \\defeq \\beta_t \\leq C <\\infty$. \n\n\nThen $\\zeta_t = \\exp(\\int_0^t \\alpha_s \\d s) = \\frac{\\T}{\\T-t} \\to +\\infty$ as $t\\uparrow T$. \n\nAlso, $\\int_0^t \\zeta_s \\beta_s \\d s \\leq C \\int_0^t \\zeta_s \\d s = C T(\\log(T) - \\log (T-t))$. \nSo \n$$\n\\lim_{t\\uparrow T} \\frac{\\zeta_t}{\\int_0^t \\zeta_s \\beta_s \\d s} \n\\geq \\lim_{t\\uparrow T} \\frac{\\frac{\\T}{\\T-t}}{ C T(\\log(T) - \\log (T-t))} = +\\infty . \n$$\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\nUsing Girsanov theorem, we \n show that introducing arbitrary non-singular changes (as defined below) on the drift and initialization of a process does not change its bridge conditions. \n\\renewcommand{\\breve}{\\tilde}\n \\begin{pro} \\label{thm:perturb}\nConsider the following processes \n\\bb \n& \\Q \\colon ~~~~ Z_t = b_t(Z_t) \\dt + \\sigma_t (Z_t) \\d W_t, ~~~ Z_0 \\sim \\mu_0\\\\ \n&\\tilde \\Q \\colon ~~~~ Z_t = (b_t(Z_t) + \\sigma_t(Z_t) f_t(Z_t)) \\dt + \\sigma_t (Z_t) \\d W_t, ~~~ Z_0 \\sim \\tilde \\mu_0. \n\\ee \nAssume we have $\\KL(\\mu_0~||~\\breve \\mu_0) <+\\infty$ and $\\E_{\\Q}[\\int_0^T \\norm{f_t(Z_{t})}^2] < \\infty$. Then for any event $A$, we have $ \\Q(Z\\in A) = 1$ if and only if $\\breve \\Q(Z\\in A) = 1$. %\n \\end{pro} \n \\begin{proof} \n Using Girsnaov theorem \\cite{Oksendal2013}, we have \n $$\n \\KL(\\Q~||~\\tilde \\Q) = \n \\KL(\\mu_0~||~ \\tilde \\mu_0) + \\frac{1}{2} \n \\E_{\\Q}\\left [\\int_0^\\t \\norm{f_t(Z_t)}_2^2\\dt \\right]. \n $$\n Hence, we have $\\KL(\\Q~||~\\tilde \\Q) < +\\infty$. This implies that $\\Q$ and $\\tilde \\Q$ has the same support. Hence $\\Q( Z \\in A) = 1$ iff $\\tilde \\Q(Z\\in A) = 1$ for any measurable set $A$. \n \\end{proof}\n\nThis gives an immediate proof of the following result that we use in the paper. %\n\n\\begin{cor}\n\\label{app:cor5}\nConsider the following two processes: \n\\bb %\n\\Q^{x, \\mathrm{bb}}: && \n\\d Z_t = \\left ( \\sigma_t^2\\frac{x-Z_t}{\\beta_\\t - \\beta_t} \\right) \\dt + \\sigma_t \\d W_t,~~~~ Z_0 \\sim \\mu_0, \\\\ \n\\Q^{x, \\mathrm{bb}, f}: && \n\\d Z_t = \\left ( \\sigma_t f_t(Z_t) + \\sigma_t^2\\frac{x-Z_t}{\\beta_\\t - \\beta_t} \\right) \\dt + \\sigma_t \\d W_t,~~~~ Z_0 \\sim \\mu_0.\n\\ee \nAssume $\\E_{\\Q^{x, \\mathrm{bb}, f}} [\\norm{f_t(Z_t)}^2] <+\\infty$ and $\\sigma_t > 0$ for $t\\in[0,+\\infty)$. Then $\\Q^{x, \\mathrm{bb}, f}$ is a bridge to $x$. \n\\end{cor} \n\n\n\n\n \n\\section{Experiment}\n\\label{sec:exp}\nWe verify the advantages of our proposed method (Bridge with Priors) in several different domains.\nWe first compare our method with advanced generators (\\emph{e.g.}, diffusion model, normalizing flow, etc.) on molecule generation tasks.\nWe then implement our method on point cloud generations, which targets producing generated samples in a higher quality.\nWe directly compare the performance and also analyze the difference between our energy prior and other energies we discuss in Section \\ref{sec:method}.\n\n\n\\subsection{Force Guided Molecule Generation}\n\nTo demonstrate the efficiency and effectiveness of our bridge processes and physical energy, we conduct experiments on molecule and macro-molecule generation experiments.\nWe follow \\cite{luo2022equivariant} in settings and observe that our proposed prior bridge processes consistently improve the state-of-the-art performance. Diving deeper, we analyze the impact of different energy terms and hyperparameters.\n\n\\textbf{Metrics.} Following \\cite{hoogeboom2022equivariant,satorras2021n}, we use the atom and molecular stability score to \nmeasure the model performance.\nThe atom stability is the proportion of atoms that have the right valency while the molecular stability stands for the proportion of generated\nmolecules for which all atoms are stable.\nFor visualization, we use the distance between pairs of atoms and the atom types to predict bond types, which is a common practice.\nTo demonstrate that our force does not only memorize the data in the dataset, we further calculate and report the RDKit-based \\cite{landrum2013rdkit} novelty score.\nwe extracted 10,000 samples to calculate the above metrics.\n\n\\begin{table}[h]\n \\centering\n \\caption{\\small{Results of our method and several baselines on QM9 and GEOM-DRUG. For QM9, we additionally report the `Novelty' score evaluated by RDKit \\cite{landrum2013rdkit} to show that our method can generate novel molecules. \\RV{We evaluate the percentage of valid and unique molecules out of 12000 generated molecules.}}\n }\n \\scalebox{0.7}{\n \\begin{tabular}{l|cccc|cc}\n \\hline\n & \\multicolumn{4}{c|}{QM9} & \\multicolumn{2}{c}{GEOM-DRUG} \\\\\n \\hline\n & Atom Sta (\\%) $\\uparrow$ & Mol Sta (\\%) $\\uparrow$ & Novelty (\\%) $\\uparrow$ & \\RV{Valid + Unique $\\uparrow$} & Atom Sta (\\%) $\\uparrow$ & Mol Sta (\\%) $\\uparrow$ \\\\\n \\hline\n EN-Flow \\cite{satorras2021n} & 85.0 & 4.9 & 81.4 & \\RV{0.349} & 75.0 & 0.0 \\\\\n GDM \\cite{hoogeboom2022equivariant} & 97.0 & 63.2 & 74.6 & \\RV{-} & 75.0 & 0.0 \\\\\n E-GDM \\cite{hoogeboom2022equivariant} & \\bf{98.7$\\pm$0.1} & 82.0$\\pm$0.4 & 65.7$\\pm$0.2 & \\RV{0.902} & 81.3 & 0.0 \\\\\n \\hline \n Bridge & \\bf{98.7$\\pm$0.1} & 81.8$\\pm$0.2 & 66.0$\\pm$0.2 & \\RV{0.902} & 81.0$\\pm$0.7 & 0.0 \\\\\n Bridge + Force \\eqref{eq:our_energy} & \\bf{98.8$\\pm$0.1} & \\bf{84.6$\\pm$0.3} & 68.8$\\pm$0.2 & \\RV{0.907} & \\bf{82.4$\\pm$0.8} & 0.0 \\\\\n \\hline\n \\end{tabular}}\n \\label{tab:mol_main}\n\\end{table}\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=1.0\\textwidth]{text\/figures\/mol6.png}\n \\vspace{-2\\baselineskip}\n \\caption{\\small{Examples of molecules generated by our method on QM9 and GEOM-DRUG.}}\n \n \\label{fig:mol_example}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{text\/figures\/traj32.png}\n \n \\caption{\\small{An example of generation trajectory following $\\P^\\theta$ of our method, trained on GEOM-DRUG.}} \n \n \\label{fig:mol_traj}\n\\end{figure}\n\n\n\\begin{table}[h]\n \\centering\n \\caption{\\small{We compare w. and w\/o force results with different discretization time steps.}\n }\n \\scalebox{0.72}{\n \\begin{tabular}{l|cc|cc|cc}\n \\hline \n & \\multicolumn{6}{c}{Time Step} \\\\\n \\hline\n & \\multicolumn{2}{c|}{50} & \\multicolumn{2}{c|}{100} & \\multicolumn{2}{c}{500} \\\\\n \\hline\n & Atom Stable (\\%) & Mol Stable (\\%) & Atom Stable (\\%) & Mol Stable (\\%) & Atom Stable (\\%) & Mol Stable (\\%) \\\\\n \\hline\n EGM & 97.0$\\pm$0.1 & 66.4$\\pm$0.2 & 97.3$\\pm$0.1 & 69.8$\\pm$0.2 & \\bf{98.5$\\pm$0.1} & \\bf{81.2$\\pm$0.1} \\\\\n Bridge + Force \\eqref{eq:our_energy} & \\bf{97.3$\\pm$0.1} & \\bf{69.2$\\pm$0.2} & \\bf{97.9$\\pm$0.1} & \\bf{72.3$\\pm$0.2} & \\bf{98.7$\\pm$0.1} & \\bf{83.7$\\pm$0.1} \\\\\n \\hline\n \\end{tabular}}\n \\label{tab:mol_timestep}\n \n\\end{table}\n\n\\textbf{Dataset Settings}\nQM9 \\cite{ramakrishnan2014quantum} molecular properties and atom coordinates for 130k small molecules with up to 9 heavy atoms with 5 different types of atoms. This data set contains small amino acids, such as GLY, ALA, as well as nucleobases cytosine, uracil, and thymine. We follow the common practice in \\cite{hoogeboom2022equivariant} to split the train, validation, and test partitions, with 100K, 18K, and 13K samples.\nGEOM-DRUG \\cite{axelrod2022geom} is a dataset that contains drug-like molecules. It features 37 million molecular conformations annotated by energy and statistical weight for over 450,000 molecules. \nEach molecule contains 44 atoms on average, with 5 different types of atoms.\nFollowing \\cite{hoogeboom2022equivariant,satorras2021n}, we retain the 30 lowest energy conformations for each molecule.\n\n\\textbf{Training Configurations.}\nOn QM9, we train the EGNNs with 256 hidden features and 9 layers for 1100 epochs, a batch size 64, and a constant learning rate $10^{-4}$, which is the default training configuration. \nWe use the polynomial noise schedule used in \\cite{hoogeboom2022equivariant} which linearly decay from $10^{-2} \/ T$ to 0. We linearly decay $\\alpha$ from $10^{-3} \/ T$ to 0 \\emph{w.r.t.} time step. \nWe set $k=5$ \\eqref{eq:our_energy} by default.\nOn GEOM-DRUG, we train the EGNNs with 256 hidden features and 8 layers with batch size 64, a constant learning rate $10^{-4}$, and 10 epochs.\nIt takes approximately 10 days to train the model on these two datasets on one \\texttt{Tesla V100-SXM2-32GB} GPU.\nWe provide E(3) Equivariant Diffusion Model (EDM) \\cite{hoogeboom2022equivariant} and E(3) Equivariant Normalizing Flow (EN-Flow) \\cite{satorras2021n} as our baselines. Both two are trained with the same configurations as ours.\n\n\n\n\\textbf{Results: Higher Quality and Novelty.}\nWe summarize our experimental results in Table \\ref{tab:mol_main}. We observe that \\textbf{(1)} our method generates molecules with better qualities than the others. On QM9, we notice that we improve the molecule stability score by a large margin (from $82.0$ to $84.6$) and slightly improve the atom stability score (from $98.7$ to $98.8$). \nIt indicates that with the informed prior bridge helps improves the \nquality of the generated molecules. \n\\textbf{(2)} Our method achieves a better novelty score. Compared to E-GDM, we improve the novelty score from $65.7$ to $68.8$. This implies that our introduced energy does not hurt the novelty when the statistics are estimated over the training dataset. Notice that although the GDM and EN-Flow achieve a better novelty score, the sample quality is much worse. The reason is that, due to the metric definition, low-quality out-of-distribution samples lead to high novelty scores.\n\\textbf{(3)} On the GEOM-DRUG dataset, the atom stability is improved from $81.3$ to $82.4$, which shows that our method can work for macro-molecules.\n\\textbf{(4)} We visualize and qualitatively evaluate our generate molecules. Figure \\ref{fig:mol_traj} displays the trajectory on GEOM-DRUG and Figure \\ref{fig:mol_example} shows the samples on two datasets. \n\\textbf{(5)} Bridge processes and E-GDM obtain comparable results on our tested benchmarks.\n\\RV{\\textbf{(6)} \nThe computational load added by introducing prior bridges is small. \nCompared to EGM, we only introduce 8\\% additional cost in training and 3\\% for inference.}\n\n\n\n\n\\textbf{Result: Better With Fewer Time Steps.}\nWe display the performance of our method with fewer time steps in Table \\ref{tab:mol_timestep}. We observe that\n\\textbf{(1)} with fewer time steps, the baseline EGM method gets worse results than 1000 steps in Table \\ref{tab:mol_main}. \n\\textbf{(2)} with 500 steps, our method still keeps a consistently good performance. \\textbf{(3)} with even fewer 50 or 100 steps, our method yields a worse result than 1000 steps in Table \\ref{tab:mol_main}, but still outperforms the baseline method by a large margin. \n\n\n\\begin{table}[t]\n \\centering\n \\caption{\\small{We compare EGM models trained with different force mentioned in Section \\ref{sec:method}.}\n }\n \\scalebox{.78}{\n \\begin{tabular}{l|cc||l|cc}\n \\hline \n Method & Atom Stable (\\%) & Mol Stable (\\%) & Method & Atom Stable (\\%) & Mol Stable (\\%) \\\\\n \\hline\n Force \\eqref{eq:our_energy}, $k=7$ & \\bf{98.8$\\pm$0.1} & \\bf{84.5$\\pm$0.2} & Force \\eqref{eq:our_amber_energy} & 98.7$\\pm$0.1 & 83.1$\\pm$0.2 \\\\ \n Force \\eqref{eq:our_energy}, $k=5$ & \\bf{98.8$\\pm$0.1} & \\bf{84.6$\\pm$0.3} & Force \\eqref{eq:our_amber_energy} w\/o. bond & 98.7$\\pm$0.1 & 82.5$\\pm$0.1 \\\\\n Force \\eqref{eq:our_energy}, $k=3$ & \\bf{98.8$\\pm$0.1} & 83.9$\\pm$0.3 & Force \\eqref{eq:our_amber_energy} w\/o. angle & 98.7$\\pm$0.1 & 82.4$\\pm$0.2 \\\\\n Force \\eqref{eq:our_energy}, $k=1$ & \\bf{98.8$\\pm$0.1} & 82.7$\\pm$0.3 & Force \\eqref{eq:our_amber_energy} w\/o. Long-range & 98.7$\\pm$0.1 & 82.7$\\pm$0.2 \\\\\n \\hline\n \\end{tabular}}\n \\label{tab:mol_force}\n\\end{table}\n\n\\textbf{Ablation: Impacts of Different Energies.}\nWe apply several energies we discuss in Section \\ref{sec:method}, and compare them on the QM9 dataset.\n\\textbf{(1)} We notice that our energy \\eqref{eq:our_energy} gets better performance with larger $k$ when $k \\leq 5$. $k=7$ achieves comparable performance as $k=5$.\nLarger $k$ also requires more computation time, which yields a trade-off between performance and efficiency.\n\\textbf{(2)} For \\eqref{eq:our_amber_energy}, once removing a typical term, the performance drops.\n\\textbf{(3)} In all the cases, applying additional forces outperforms the bridge processes baseline w\/o. force.\n\n\\subsection{Force Guided Point Cloud Generation}\n\n\nWe apply uniformity-promoting priors \nto point cloud generation. %\nWe apply our method based on the diffusion model for point cloud generation introduced by point cloud diffusion model ~\\cite{luo2021diffusion} and compare it with the original diffusion model as well as the case of bridge processes w\/o. force prior.\nWe observe that our method yields better results in various evaluation metrics under different setups.\n\n\n\\begin{table}[t!]\n \\centering\n \\caption{ \\small{Point cloud generation results. CD is multiplied by $10^3$, EMD is multiplied by $10$.}\n }\n \\scalebox{0.87}{\n \\begin{tabular}{l|l|cccc|cccc}\n \\hline\n & & \\multicolumn{4}{c|}{10 Steps} & \\multicolumn{4}{c}{100 Steps} \\\\\n \\hline\n & & \\multicolumn{2}{c}{MMD $\\downarrow$} & \\multicolumn{2}{c|}{COV $\\uparrow$} & \\multicolumn{2}{c}{MMD $\\downarrow$} & \\multicolumn{2}{c}{COV $\\uparrow$} \\\\\n \\cline{3-10}\n & & CD & EMD & CD & EMD & CD & EMD & CD & EMD \\\\\n \\hline\n \\multirow{4}{*}{Chair} & Diffusion \\cite{luo2021diffusion} & 14.01 & 3.23 & 32.72 & 29.36 & 12.32 & 1.79 & 47.41 & \\textbf{47.59} \\\\\n & Bridge & 13.04 & 2.14 & 46.01 & 42.59 & 12.47 & 1.85 & 47.83 & 47.13 \\\\\n \\cline{2-10}\n & + Riesz & 12.84 & 1.95 & 47.21 & 44.31 & 12.31 & 1.82 & 48.14 & 47.42 \\\\\n & + Statistic & \\textbf{12.65} & \\textbf{1.84} & \\textbf{47.58} & \\textbf{45.23} & \\textbf{12.25} & \\textbf{1.78} & \\textbf{48.39} & 47.56 \\\\\n \\hline\n \\hline\n \\multirow{4}{*}{Airplane} & Diffusion \\cite{luo2021diffusion} & 3.71 & 1.31 & 43.12 & 39.94 & 3.28 & \\textbf{1.04} & \\textbf{48.74} & 46.38 \\\\\n & Bridge & 3.44 & 1.24 & 46.90 & 43.46 & 3.37 & 1.08 & 47.11 & 46.17 \\\\\n \\cline{2-10}\n & + Riesz & 3.39 & 1.20 & \\textbf{47.11} & 43.12 & \\textbf{3.24} & 1.09 & 48.62 & 46.23 \\\\\n & + Statistic & \\textbf{3.30} & \\textbf{1.12} & \\textbf{47.02} & \\textbf{44.67} & \\textbf{3.24} & 1.06 & 48.53 & 46.73 \\\\\n \\hline\n \\end{tabular}}\n \n \\label{tab:pc_main}\n\\end{table}\n\n\\textbf{Dataset.} \nWe use the ShapeNet~\\cite{chang2015shapenet} dataset \nfor point cloud generation. \nShapeNet contains 55 categories. \nWe select Airplane and Chair,\nwhich are the two most common categories to evaluate in recent point cloud generation works~\\cite{cai2020learning,luo2021diffusion,yang2019pointflow,zhou20213d}. We construct the point clouds following the setup in ~\\cite{luo2021diffusion}, split the train, valid and test dataset in $80\\%$, $15\\%$ and $5\\%$ and samples 2048 points uniformly on the mesh surface.\n\n\n\n\\textbf{Evaluation Metric.} We evaluate the generated shape quality in two aspects following the previous works, including the minimum matching distance (MMD) and coverage score (COV). These scores are the two most common practices in the previous works. We use Chamfer Distance (CD) and Earth Mover's Distance (EMD) as the distance metric to compute the MMD and COV.\n\n\\textbf{Experiment Setup.} We train the model with two different configurations. The first one uses exactly the same experiment setup configuration introduced in~\\cite{luo2021diffusion}. Thus, we use the same model architecture and train the model in 100 diffuse steps with a learning rate $2 \\times 10^{-3}$, batch size 128, and linear noise schedule from $0.02$ to $10^{-4}$. We initial $\\alpha$ with 0.1 and jointly learn it with the network. For the second setup, to evaluate the better converge speed of our method, we decrease the diffuse step from 100 to 10 with other settings the same. For the diffusion model baseline, we reproduce the number by directly using the pre-trained model checkpoint and testing it on the test set provided by the official codebase.\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=1.0\\textwidth]{text\/figures\/pcv.pdf}\n \n \\caption{\\small{From left to right are examples of point clouds generated by \\cite{luo2021diffusion}, our method with uniformative bridge ($f_t=0$), \n bridge with Riesz energy ($f_t=-\\dd E_{\\mathrm{Riesz}}$) and with KNN energy ($f_t = -\\dd E_{knn}$). \n We see that the Riesz and KNN energies \n yield more uniformly distributed points. \n Riesz energy sometimes creates additional outlier points due to its repulsive nature. \n }}\n\n \\label{fig:pc}\n\\end{figure}\n\n\\textbf{Result.} We show our experimental result in Table~\\ref{tab:pc_main}. We see that\n\\textbf{(1)} In the 10 steps setup, all variants of our approach are clearly stronger than the diffusion model. \nWith force added, our method with physical prior achieves nearly the same performance as the 100 steps setup.\n\\textbf{(2)} In the 100 steps setup, adding energy potential as prior improves the bridge process performance and further let it beat the diffusion model baseline.\\textbf{(3)}\nSince the test points are uniformly sampled on the surface, a better score indicates a closer point distribution to the reference set. Further, when compare with Riesz energy \\eqref{eq:riesz_energy}, statistic gap energy \\eqref{eq:knn_energy} performs better. \nOne explanation is the Riesz energy pushes the points to some outlier position in sample generate samples, while statistic gap energy is more robust. We also show visualization samples in Figure~\\ref{fig:pc}. \n\n\n\n\n\n\n\n\\section{Introduction}\nAs exemplified by the success of AlphafoldV2 \\cite{jumper2021highly}\nin solving protein folding,\ndeep learning techniques\nhave been creating new frontiers\non molecular sciences \\cite{wang2017accurate}.\nIn particular, the problem of building deep generative models\nfor molecule design has attracted increasing interest with\n a magnitude of applications in physics, chemistry, and drug discovery \\cite{alcalde2006environmental,anand2022protein,lu2022machine}.\nRecently, diffusion-based generative model have been applied to molecule generation problems \\cite{de2021diffusion,hoogeboom2022equivariant} and obtain superior performance.\nThe idea of these methods is to corrupt the data with diffusion noise and learn a neural diffusion model to revert the corruption process to\ngenerate meaningful data from noise.\nA key challenge in deep generative models for molecule and 3D point generation is to efficiently incorporate strong prior information to reflect the physical and problem-dependent statistical properties of the problems at hand.\nIn fact, a recent fruitful line of research \\cite{du2022molgensurvey,liu2021pre, satorras2021n, gong2022fill}\nhave shown promising results by introducing inductive bias\ninto the design of model architectures to reflect physical constraints such as SE(3) equivariance.\nIn this work, we present a different paradigm\nof prior incorporation tailored to diffusion-based generative models,\nand leverage it to yield substantial improvement in both\n1) high-quality and stable molecule generation and 2) uniformity-promoted point cloud generation.\nOur contributions are summarized as follows.\n\n\\textbf{Prior Guided Learning of Diffusion Models.}\nWe introduce a simple and flexible framework for injecting informative problem-dependent prior and physical information when learning diffusion-based generative models.\nThe idea is to elicit and inject prior information regarding how the diffusion process should look like for generating each given data point,\nand train the neural diffusion model to imitate the prior processes.\nThe prior information is presented in the form of diffusion bridges\nwhich are diffusion processes that are guaranteed to generate each data point at the fixed terminal time.\nWe provide a general Lyapunov approach for constructing and determining bridges and leverage it to develop a way to systematically incorporate prior information into bridge processes.\n\n\\textbf{Physics-informed Molecule Generation.} We apply our method to molecule generation. We propose a number of energy functions for incorporating physical and statistical prior information. Compared with existing physics-informed\nmolecule\ngeneration methods \\cite{de2021diffusion, gnaneshwar2022score,luo2022equivariant, gnaneshwar2022score},\nour method modifies the training process, rather than imposing constraints on the model architecture.\nExperiments show that our method achieves current state-of-the-art generation quality and stability on multiple test benchmarks of molecule generation.\n\n\\textbf{Uniformity-promoting Point Generation.}\nA challenging task in physical simulation,\ngraphics, 3D vision is to generate\npoint clouds for representing real objects ~\\cite{achlioptas2018learning,cai2020learning,luo2021diffusion,yang2019pointflow}. A largely overlooked problem of existing approaches is that they tend to generate\nunevenly distributed points, which lead to unrealistic shapes and make the subsequent processing and applications, such as mesh generation, challenging and inefficient.\nIn this work, we leverage our framework to introduce uniformity-promoting forces into the prior bridge of diffusion generative models. This yields a simple and efficient approach to generating regular and realistic point clouds in terms of both shape and point distribution.\n\n\n\\section{Conclusion and Limitations}\n\\vspace{-2mm}\nWe propose a framework to inject informative priors into\nlearning neural parameterized diffusion models, with applications to both\nmolecules and 3D point cloud generation.\nEmpirically, we demonstrate that our method has the advantages such as better generation quality, less sampling time and easy-to-calculate potential energies.\nFor future works, we plan to 1) study the relation between different types of forces for different domain of molecules, 2) study how to generate valid proteins in which the number of atoms is very large,\nand 3) apply our method to more realistic applications such as antibody design or hydrolase engineering.\n\nIn both energy functions in \\eqref{eq:our_amber_energy} and \\eqref{eq:our_energy}, we do not add torsional angle related energy \\cite{jing2022torsional} mainly because it is hard to verify whether four atoms are bonded together\nduring the stochastic process.\nWe plan to study how to include this for better performance in future works.\n\nAnother weakness of deep diffusion bridge processes are their computation time. Similar to previous diffusion models \\cite{luo2022equivariant}, it takes a long time to train a model.\nWe attempted to speed the training up by using a large batch size (\\emph{e.g.}, 512, 1024) but found a performance drop. An important future direction is to study methods to distribute and accelerate the training. %\n\n\\paragraph{Acknowledgements}\nAuthors are supported in part by CAREER-1846421, SenSE-2037267, EAGER-2041327, and Office of Navy Research, and NSF AI Institute for Foundations of Machine Learning (IFML). We would like to thank the anonymous reviewers and the area chair for their thoughtful comments and efforts towards improving our manuscript.\n\n\n\n\n\\section{Method}\n\\label{sec:method}\n\n\n\n\nWe first introduce the definition of diffusion generative models and discuss how to learn these models with prior bridges.\nAfter introducing the training algorithm for deep diffusion generative models, we discuss the energy functions that we apply to molecules and point cloud examples.\n\n\\subsection{Learning Diffusion Generative Models with Prior Bridges} \n\n\\textbf{Problem Definition.}\nWe aim at learning a generative model given a dataset $\\{x\\datak\\}_{k=1}^n$ drawn from an unknown distribution $\\tg$ on $\\RR^d$. \nA diffusion model on time interval $[0,\\t]$ is\n$$\n\\P^\\theta \\colon ~~~~~\\d Z_t = s_t^\\theta(Z_t) \\dt + \\sigma_t(Z_t) \\d W_t,\n~~~ \\forall t\\in[0,\\t], ~~ \n~~~ Z_0 \\sim \\mu_0, \n$$\nwhere $W_t$ is a standard Brownian motion; \n$\\sigma_t\\colon \\RR^d\\to \\RR^{d\\times d}$ \nis a positive definition covariance coefficient; \n$s^\\theta_t\\colon \\RR^d\\to \\RR^{d}$ is parameterized as a neural network with parameter $\\theta$, and $\\mu_0$ is the initialization. Here we \nuse $\\P^\\theta$ to denote the distribution of the whole trajectory $Z=\\{Z_t\\colon t\\in[0,1] \\}$, and $\\P^\\theta_t$ the marginal distribution of $Z_t$ at time $t$. \nWe want to learn the parameter $\\theta$ such that the distribution $\\P_\\t^\\theta$ of the terminal state $Z_\\t$ equals the data distribution $\\tg$. \n\n\\textbf{Learning Diffusion Models.}\nThere are an infinite number of diffusion processes $\\P^\\theta$ that yield the same terminal distribution but have different distributions of latent trajectories $Z$. \nHence, it is important to \ninject problem-dependent prior information \ninto the learning procedure to obtain a model $\\P^\\theta$ that simulate the data for the problem at hand fast and accurately. \nTo achieve this, we elicit an \n\\emph{imputation} process $\\Q^x$ for each $x\\in \\RR^d$, \nsuch that a draw $Z\\sim \\Q^x$ \nyields trajectories that 1) are consistent with $x$ in that $\\Z_\\t = x$ deterministically, and \n2) reflect important physical and statistical prior information on the problem at hand.\n\nFormally, if $\\Q^x(\\Z_\\t = x) = 1$, we call that $\\Q^x$ is a bridge process pinned at end point $x$, or simply an $x$-bridge. Assume we first generate a data point $x\\sim \\tg$, and then draw a bridge $Z\\sim \\Q^x$ pinned at $x$, then the distribution of $Z$ is a mixture of $\\Q^x$ with $x$ drawn from the data distribution: $\\Q^{\\tg} := \\int \\Q^x(\\cdot) \\tg(\\dx)$. \n\nA key property of $\\Q^\\tg$ is that its terminal distribution equals the data distribution, i.e., $\\Q_\\t^\\tg = \\tg$. Therefore, we can learn the diffusion model $\\P^\\theta$ by fitting the trajectories drawn from $\\Q^\\tg$ with the ``backward'' procedure above. This can be formulated by maximum likelihood or equivalently minimizing the KL divergence: \n$$\n\\min_{\\theta}\\left \\{\\L(\\theta)\\defeq \\KL(\\Q^\\tg~||~ \\P^\\theta) \\right\\}.\n$$\nFurthermore, assume that the bridge $\\Q^x$ is a diffusion model of form \n\\begin{align}\n \\Q^x \\colon ~~~~\n \\d Z_t = b_t(Z_t ~|~ x) \\dt + \\sigma_t(Z_t) \\d W_t, ~~~ \\Z_0 \\sim \\mu_0,\n \\label{equ:Qx}\n\\end{align}\nwhere $b_t(Z_t~|~x)$ is an $x$-dependent drift term need to carefully designed to both satisfy the bridge condition and incorporate important prior information (see Section~\\ref{sec:lyap}). \nAssuming this is done, using Girsanov theorem \\cite{mao2007stochastic}, \nthe loss function $\\L(\\theta)$ can be reformed into a form ofdenoised score matching loss of \\cite{song2020denoising,song2020score, song2021maximum}: \n\\bbb \\label{equ:mainloss} \n\\L(\\theta) \n= \\E_{Z \\sim \\Q^\\tg} \n\\left [\n\\frac{1}{2} \\int_0^\\t \\norm{\\sigma(Z_t)^{-1}(s_t^\\theta(Z_t) - b_t(Z_t~|~Z_\\t))}_2^2 \\dt \\right ] + \\mathrm{const}, \n\\eee \nwhich is a score matching term between $s^\\theta$ and $b$.\nThe const term contains the log-likelihood for the initial distribution $\\mu_0$, which is a const in our problem.\nHere $\\theta\\true$ is an global optimum of $\\L(\\theta)$ if \n$$\ns^\\thetat_t(z) = \n\\E_{Z\\sim \\Q^\\tg} [b_t( z | Z_\\t)~|~ Z_t = z].\n$$\nThis means that %\nthe drift term $s_t^\\theta$ should be matched with the conditional expectation of $b_t(z | x)$ with $x= Z_\\t$ conditioned on $Z_t = z$. \n\n\n\n\n\\begin{rem}\nThe SMLD can be viewed as a special case of this framework when we take $\\Q^x$ to be a time-scaled Brownian bridge process: \n\\bbb \\label{equ:ztdfd}\n\\Q^{x, \\mathrm{bb}}: && \n\\d Z_t = \\sigma_t^2\\frac{x-Z_t}{\\beta_\\t - \\beta_t} \\dt + \\sigma_t \\d W_t,~~~~ \nZ_0 \\sim \\normal(x, \\beta_\\t), \n\\eee \nwhere $\\sigma_t \\in [0,+\\infty)$ and $\\beta_t = \\int_0^t \\sigma_s^2 \\d s$. \nThis can be seen by the fact that the time-reversed process $\\rev Z_{t} \\defeq Z_{1-t}$ follows the simple time-scaled Brownian motion $\\d \\rev Z_t = \\sigma_{\\t -t} \\d \\rev W_t$ starting from the data point $\\rev Z_0 = x$, where $\\rev W_t$ is another standard Brownian motion. \nThe Brownian bridge achieves $Z_\\t = x$ because the magnitude of the drift force is increasing to infinite when $t$ is close to time $\\T.$ \\\\ \n\\end{rem}\n\nHowever, \nthe bridge of SMLD above is a relative simple and uninformative process and does not incorporate problem-dependent prior information into the learning procedure. \nThis is also the case of the other standard diffusion-based models \\cite{song2020score}, such as denoising diffusion probabilistic models (DDPM) which can be shown to use a bridge constructed from an Ornstein\u2013Uhlenbeck process. \nWe refer the readers to \\cite{peluchetti2022nondenoising}, \nwhich provides a similar forward time bridge framework \nfor learning diffusion models, \nand it recovers the bridges in SMLD and DDPM as a conditioned stochastic process derived using the $h$-transform technique \\cite{doob1984classical}. However, \nthe $h$-transform method \nis limited to elementary stochastic processes \nthat have an explicit formula of the transition probabilities, and can not incorporate complex physical statistical prior information. \nOur work strikes to construct and use a broader class of more complex \nbridge processes that both reflect problem-dependent prior knowledge and satisfy the endpoint condition $\\Q^x(Z_\\t = x)=1$. \nThis necessitate systematic techniques for constructing a large family of bridges, as we pursuit in Section~\\ref{sec:lyap}.\n \n\\subsection{Designing Informative Prior Bridges} \n\\label{sec:lyap} \nThe key to realizing the general prior-informed learning framework above is to have a general and user-friendly technique to design $\\Q^x$ in \\eqref{equ:Qx} to ensure the bridge condition $\\Q^x(Z_\\t = x) = 1$ while \nleaving the flexibility of incorporating rich prior information. \nTo achieve this, we first develop a general criterion of bridges based on a \\emph{Lyapunov function method} which allows us to identify a very general form of bridge processes; we \nthen propose a particularly simple family of bridges that we use in practice by introducing modification to Brownian bridges. \n\n\\begin{mydef}[\\textbf{Lyapunov Functions}] \\label{def:lyap}\nA function $U_t(z)$ is said to be a Lyapunov function for set $A\\subset \\RR^d$ at time $t=\\t$ if \n$U_\\t (z) \\geq 0$ for $\\forall z\\in \\RR^d$ and $U_\\t(z) = 0$ if and only if $z \\in A$.\n\\end{mydef} \nIntuitively, a diffusion process $\\Q$ is a bridge $A$, i.e., $\\Q(\\Z_\\t \\in A) =1$, if it (at least) approximately follows the gradient flow of a Lyapunov function and the magnitude (or step size) or the gradient flow should increase with a proper magnitude in order to ensure that $\\Z_t \\in A$ at the terminal time $t=\\t$. Therefore, we identify a general form of bridges to $A$ as follows: \n \\bbb \\label{equ:duz}\n \\Q^A:&&\n \\d Z_t =\\left( -\\alpha_t \\dd_z U_t(Z_t) + \\nu_t(Z_t)\\right ) \\dt \n + \\sigma_t(\\X_t) \\dW_t,~~~~~~~t\\in[0,1],~~~ Z_0 \\sim \\mu_0, \n \\eee \n where $\\alpha_t>0$ is the step size of the gradient flow of $U$ and $\\nu$ is an extra perturbation term. \n The step size $\\alpha_t$ should increase to infinity as $t\\to \\T$ sufficiently fast to dominate the effect of the diffusion term $\\sigma_t \\d W_t$ and the perturbation $\\nu_t \\d t$ term to ensure that $U$ is minimized at time $t = \\t$. \n \n \\begin{theoremEnd}[\\isproofhere]{pro}\n \\label{pro:lya}\n Assume $U_t(z) = U(z,t)$ is a Lyapunov function \n of a measurable set $A$ at time $\\t$ and $U(\\cdot, t) \\in C^2(\\RR^d)$ and $U(z, \\cdot) \\in C^1([0,\\T]).$ %\nThen, $\\Q^A$ in \\eqref{equ:duz} is an bridge to $A$, i.e., \n$\\Q^A(Z_\\T \\in A) = 1$, if the following holds: %\n\n 1) $U$ follows an (expected) Polyak-Lojasiewicz condition: $\\E_{\\Q^A}[U_t(Z_t)] -\\norm{\\dd_z U_t(Z_t)}^2]\\leq 0, \\forall t.$\n \n 2) Let $\\beta_t = \\E_{\\Q^A}[\\dd_z U_t(Z_t)\\tt \\nu_t(Z_t)]$, and $\\gamma_t = \\E_{\\Q^A}[\\partial_t U_t(Z_t) + \\frac{1}{2} \\trace(\\dd_z^2 U_t(Z_t) \\sigma^2_t(Z_t))]$, and \n $\\zeta_t = \\exp(\\int_0^t \\alpha_s \\d s)$. Then \n$\\lim_{t\\uparrow 1} \\zeta_t = +\\infty,$ and \n$\\lim_{t\\uparrow 1} \\frac{\\zeta_t}{\\int_0^t \\zeta_s (\\beta_s + \\gamma_s) \\d s} = +\\infty. $\n\\end{theoremEnd} \n\\begin{proofEnd}\nIt is a direct result of Theorem~\\ref{thm:lyap2}. \n\\end{proofEnd}\nBrownian bridge can be viewed as the case when $U_t(z) =\\norm{x-z}^2\/2$ and $\\alpha_t = \\sigma_t^2\/(\\beta_\\t-\\beta_t)$, and $\\nu = 0$. Hence simply introducing an extra drift term into bridge bridge yields that a broad family of bridges to $x$: \n\\bbb \\label{equ:qxbbf}\n\\Q^{x, \\mathrm{bb}, f}: && \n\\d Z_t = \\left ( \\sigma_t f_t(Z_t) + \\sigma_t^2\\frac{x-Z_t}{\\beta_\\t - \\beta_t} \\right) \\dt + \\sigma_t \\d W_t,~~~~ Z_0 \\sim \\mu_0.\n\\eee \n In Appendix~\\ref{thm:perturb} and \\ref{app:cor5},\nwe show that $\\Q^{x, \\mathrm{bb}, f}$ is a bridge to $x$ if $\\E_{\\Q^{x,\\mathrm{bb}}}[\\norm{f_t(Z_t)}^2]<+\\infty$ and $\\sigma_t>0,\\forall t$, which is very mild condition and is satisfied for most practical functions. \nThe intuition is that the Brownian drift $\\sigma^2_t\\frac{x-Z_t}{\\beta_1-\\beta_t}$ is singular and grows to infinite as $t$ approaches $1$. \nHence, introducing an $f$ into the drift would not change of the final bridge condition, unless $f$ is also singular and has a magnitude that dominates the Brownian bridge drift as $t\\to 1$. \n\nTo make the model $\\P^\\theta$ compatible \nwith the physical force $f$, \nwe assume the learnable drift has a form of $s_t^\\theta(z) = \\alpha f_t(z) + \\tilde s_t^\\theta(z)$ where $\\tilde s$ is a neural network (typically a GNN) and $\\alpha$ can be another learnable parameter or a pre-defined parameter. \nPlease refer to algorithm \\ref{alg:learning} and Figure \\ref{fig:algorithm_demo} for descriptions about our practical algorithm.\n\n\n\n\n\n\n \n\\begin{algorithm}[h] \n\\label{alg:learning}\n\\caption{Learning diffusion generative models.} %\n\\begin{algorithmic}\n\\STATE \\textbf{Input}: \nGiven a dataset $\\{x\\datak\\}$, $\\Q^x$ the bridge in \\eqref{equ:qxbbf}, and a problem-dependent prior force $f$,v %\nand a diffusion model $\\P^\\theta$. \n\\STATE \\textbf{Training}: Estimate $\\theta$ by minimizing $\\L(\\theta)$ in \\eqref{equ:mainloss} with stochastic gradient descent and time discretization. \n\\STATE \\textbf{Sampling}: Simulate from $\\P^\\theta$.\n\\end{algorithmic}\n\\end{algorithm} \n\n \n\\begin{figure}[t]\n \\centering\n \\vspace{-5pt}\n \\includegraphics[width=1.\\linewidth]{text\/figures\/pipeline.pdf}\n\n \\caption{\\small{An overview of our training pipeline with molecule generation as an example. Initialized from a given distribution, we pass the data through the network multiple times, and finally get the meaningful output.}}\n \\label{fig:algorithm_demo}\n\n\\end{figure}\n \n\\section{Molecule and 3D Generation with Informative Prior Bridges}\n\nWe apply our method to the molecule \ngeneration as well as point cloud generation. \nInformative physical or statistical priors\nthat reflects the underlying real physical structures \ncan be particularly beneficial for molecule generation as we show in experiments. \n\nIn our problem, each data point $x$ is a collection of\n atoms of different types, more generally marked points, in 3D Euclidean space. \nIn particular, we have $x = [x^r_i, x^h_i]_{i=1}^m$, \nwhere $x^r_i \\in \\RR^3$ is the coordinate of the $i$-th atom, and $x^h_i \\in \\{e_1,\\ldots, e_k\\}$ where each $e_i=[0\\cdots 1\\cdots0]$ is the $i$-th basis vector of $\\RR^k$, which \nindicates the type of the $i$-th atom of $k$ categories.\nTo apply the diffusion generative model, we treat $x^h_i$ as a continuous vector in $\\RR^r$ and round it to the closest basis vector when we want to output a final result or have computations that depend on atom types (e.g., calculating an energy function as we do in sequel). \nSpecifically, for a continuous $x^h_i\\in \\RR^k$, we denote by $\\hat x^h_i = \\ind(x^h_i = \\max(x^h_i))$ the discrete type rounded from it by taking the type with the maximum value. \nTo incorporate priors, we design an energy function $\\eng(x)$ and incorporate $f_t( \\cdot) =-\\dd \\eng( \\cdot)$ into the Brownian bridge \\eqref{equ:qxbbf} to guide the training process. \nWe discuss different choices of $\\eng$ in the following. \n\n\n\n\n\n\n\n\n\\subsection{Prior Bridges for Molecule Generation}\n\nPrevious prior guided molecule or protein 3D structure generation usually depends on pre-defined energy or force \\cite{ luo2021predicting,xu2022geodiff}.\nWe introduce our two potential energies. One is formulated inspired by previous works in biology, and the other is an $k$ nearest neighbour statistics directly obtained from the data.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\paragraph{AMBER Inspired Physical Energy.}\nAMBER \\cite{duan2003point} is a family of force fields for molecule simulation. It is designed to provide a computationally efficient tool for modern chemistry-molecular dynamics and free energy calculations. \nIt consists of a number of important forces, including \nthe bond energy, angular energy, torsional energy, \nthe van der Waals energy and \nthe Coulomb energy. \nInspired by AMBER, \nwe propose to incorporate the following energy term into the bridge process: \n\\begin{align}\n \\eng(x) = E_{bond}(x) + E_{angle}(x) + E_{LJ}(x) + E_{Coulomb}(x). \n \\label{eq:our_amber_energy}\n\\end{align}\n$\\bullet$ The bond energy is $ E_{bond}(x) = \\sum_{ij\\in bond(x)} (\\mathrm{Len}(x^r_{ij})- \\ell(\\hat x^h_i, \\hat x^h_j))^2$, where \n$\\mathrm{Len}(x^r_{ij}) = \\norm{x^r_i - x^r_j}$, and \n$bond(x)$ denotes the set of \nbonds from $x$, which is set to be the set of atom pairs with a distance smaller than $1.15$ times the covalent radius; the $\\ell^0(r, c)$ denotes the expected bond length between atom type $r$ and $c$, which we calculate as side information from the training data.\n\n$\\bullet$ \nThe angle energy \nis $ E_{angle}(x) = \\sum_{ijk\\in angle(x)} \n(\\mathrm{Ang}({x^r_{ijk}})- \\omega^0(\\hat x^h_{ijk}))^2$, \nwhere $angle(x)$ \ndenotes the set of \nangles between two neighbour bonds in $bound(x)$, and \n$\\mathrm{Ang}({x^r_{ijk}})$ denotes the angle formed by vector $x^r_i-x^r_j$ and $x^r_k - x^r_j$, and $\\omega^0(\\hat x^h_{ijk})$ is the expected angle between atoms of type $\\hat x^h_i$, $\\hat x^h_j$, $\\hat x^h_k$, which we calculate as side information from the training data.\n\n$\\bullet$ \nThe Lennard-Jones (LJ) energy is defined by $\\eng_{LJ}(x) = \\sum_{i\\neq j} e(\\norm{x^r_i - x^r_j})$ and $e(\\ell) = (\\sigma\/\\ell)^{12} - 2(\\sigma\/\\ell)^6$. \nThe parameter $\\sigma$ is an approximation for average nucleus distance.\n\n$\\bullet$ \nThe nuclei-nuclei repulsion (Coulomb) electromagnetic potential energy is $E_{Coulomb}(x) = \\kappa \\sum_{ij}q(\\hat x^h_i) q(\\hat x^h_j)\/\\norm{x^r_i-x^r_j}$, where $\\kappa$ is Coulomb constant and \n$q(r)$ denotes the point charge of atom of type $r$, which depends on the number of protons.\n\n\n\n\\textbf{Statistical Energy. }\nWhen accurate physic laws are unavailable, \nmolecular geometric statistics, \nsuch as bond lengths, bond angles, and torsional angles, etc, \ncan be directly calculated from the data and shed important insights on the system \\cite{cornell1995second, jorgensen1996development, manaa2002decomposition}. \nWe propose to design a prior energy function in bridges by directly calculate these statistics over the dataset. %\n\nSpecifically, we assume that the lengths and angles of each type of bond follows a Gaussian distribution that we learn from the dataset, and define the energy function as the negative log-likelihood:\n\\begin{align}\nE_{stat}(x) \n= \\sum_{ij\\in knn(x)} \n\\frac{1}{\\hat \\sigma_{\\hat x^h_{ij}}^2}\n\\norm{\\mathrm{Len}(x^r_{ij}) - \\hat \\mu_{\\hat x^h_{ij}}}^2 \n+ \\sum_{ij,jk\\in knn(x)} \n\\frac{1}{\\sigma_{\\hat x^h_{ijk}}^2}\n\\norm{\\mathrm{Ang}(x^r_{ijk}) - \\mu_{\\hat x^h_{ijk}}}^2,\n\\label{eq:our_energy}\n\\end{align}\nwhere $knn(x)$ denotes the K-nearest neighborhood graph constructed based on the distance matrix of $x$; \nfor each pair of atom types $r,c\\in[k]$, \n$\\hat \\mu_{rc}$ and $\\hat \\sigma_{rc}^2$ denotes \nempirical mean and variance of length of $rc$-edges in the dataset; for each triplet $r,c,r'\\in[k]$, \n$\\hat \\mu_{rcr'}$ and $\\hat \\sigma_{rcr'}^2$ is \nthe empirical mean and variance of angle betwen $rc$ and $cr'$ bonds. \n\nIntuitively, depending on the atom type and order of the nearest neighbour, we force the atom distance and angle to mimic the statistics calculated from the data. \nWe thus implicitly capture different kinds of interaction forces. \nCompared with the AMBER energy, \nthe statistical energy \\eqref{eq:our_energy} is simpler and more adaptive to \nthe dataset of interest. \n\n\n\n\\subsection{Prior Bridges for Point Cloud Generation}\nWe design prior forces for \n3D point cloud generation, which is similar to molecule generation except that the points are un-typed so we only have the coordinates $\\{x_i^r\\}$. \nOne important aspect of point cloud generation is to distribute points uniformly on the surface, which is important for producing high-quality meshes and other post-hoc geometry applications and manipulations. %\n\n\\textbf{Riesz Energy.}\nOne idea to make the point distribute uniformly is adding a repulsive force to separate the points apart from each other. We achieve this by minimizing the Riesz energy~\\cite{gotz2003riesz}, \n\\begin{align}\n \\eng_{\\mathrm{Riesz}}(x) = \\frac{1}{2} \\sum_{j \\neq i} ||{x}^r_{i} - {x}^r_{j}||^{-2}.\n \\label{eq:riesz_energy}\n\\end{align}\n\\textbf{KNN Distance Energy.}\nSimilar to molecule design, we directly calculate the average distance between each point and its k nearest neighbour neighbour, and define the following energy: \n\\begin{align}\n \\eng_\\mathrm{knn}(x) = \\sum_i \\left(\\text{knn-dist}_i(x^r) - \\mu_{knn}\\right)^2, \n \\label{eq:knn_energy}\n\\end{align}\nwhere $\\text{knn-dist}_i(x) = \\frac{1}{K} \\sum_{j \\in \\mathcal{N}_K(x_i;x)} ||x^r_i - x^r_j||^2 $ denotes the\naverage distance from $x_i$ to its $K$ nearest neighbors, and $\\mu_{knn}$ is the empirical mean of $\\text{knn-dist}_i(x)$ in the dataset. \nThis would encourage the points to have similar average nearest neighbor distance and yield uniform distribution between points. \nIn common geometric setups, the valence of the point on the surface is 4, which means we set $k = 4$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related works}\n\n\\textbf{Diffuse Bridge Process.}\nDiffusion-based generative models~\\cite{ho2020denoising,liu2022bridge,sohl2015deep,song2020denoising, vargas2021solving} have achieved great successes in various AI generation tasks recently;\nthese methods leverage a time reversion technique and can be viewed as learning variants auto-encoders with diffusion processes as encoders and decoders.\nSchrodinger bridges \\cite{ chen2021likelihood,de2021diffusion,wang2021deep}\n have also been proposed for learning diffusion generative models that guarantee to output desirable outputs in a finite time interval, but %\nthese methods involve iterative proportional fittings and are computationally costly.\nOur framework of learning generative models with diffusion bridges\nis similar to that of \\cite{peluchetti2022nondenoising},\nwhich learn diffusion models as a mixture of forward-time diffusion bridges to avoid the time-reversal technique of \\cite{song2020score}.\nBut our framework is designed to incorporate physical prior into bridges\nand develop a systematic approach for constructing\na broad class of prior-informed bridges.\n\n\n\n\\textbf{3D Molecule Generation.}\nGenerating molecule in 3D space has been gaining increasing interest.\nA line of works ~\\cite{luo2021predicting,mansimov2019molecular,shi2021confgf,simm2019generative,xu2021learning,xu2021end,xu2022geodiff} consider conditional conformal generation, which takes the 2D SMILE structure as conditional input and generate the 3D molecule conformations condition on the input. Another series of works \\cite{NIPS2019_8974, hoogeboom2022equivariant, luo20223d,satorras2021n,wu2022score} focus on directly generating the atom position and type for the molecule unconditionally.\nFor these series of works, improvements usually come from architecture design and loss design.\nFor example, %\nG-Schnet~\\cite{NIPS2019_8974} auto-regressively generates the atom position and type one by one after another; EN-Flow ~\\cite{satorras2021n} and EDM~\\cite{hoogeboom2022equivariant} adopt E(n) equivariant graph neural network (EGNN) \\cite{satorras2021n} to train flow-based model and diffusion model. These methods aim at generating valid and natural molecules in 3D space and outperform previous approaches by a large margin.\nOur work provides a very different approach to\nincorporating the physical information for molecule generation\nby injecting the prior information into the diffusion process,\nrather than neural network architectures.\n\n\\textbf{Point Cloud Generation.}\nA vast literature has been devoted to learning\ndeep generative models for real-world 3D objects in the form of point clouds. %\n\\cite{achlioptas2018learning} first proposed to generate the point cloud by generating a latent code and training a decoder to generate point clouds from the latent code.\nBuild upon this approach,\nmethods have been developed using flow-based generative models \\cite{yang2019pointflow}\nand diffusion-base models \\cite{cai2020learning, luo2021diffusion, luo2022equivariant}.\nHowever, the existing works miss a key important prior information:\nthe points in a point cloud tend to distribute regularly and uniformly.\nIgnoring this information causes poor generation quality.\nBy introducing uniformity-promoting forces\nin diffusion bridges,\nwe obtain a simple and efficient approach to generating\nregular and realistic point clouds.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDans cette partie, on suppose que la repr\u00e9sentation $R(\\alpha,\\beta,\\gamma;l)$ de $W(p,q,r)$ est r\u00e9ductible et qu'il existe une forme bilin\u00e9aire $G$-invariante non nulle. Alors dans ces conditions, on montre que $G'=G\/N(G)$ est isomorphe \u00e0 un groupe di\u00e9dral fini et $N(G)$ est donn\u00e9 explicitement ainsi que l'op\u00e9ration de $G$ sur $N(G)$. The general case will\n\\subsection{Contraintes sur les param\u00eatres}\nLes conditions \u00e9nonc\u00e9es ci-dessus sont \u00e9quivalentes aux deux conditions suivantes:\n\\[\n\\Delta=8-2\\alpha-2\\beta-2\\gamma-\\alpha l-\\beta m=0\\quad \\text{et} \\quad\\alpha l=\\beta m\n\\]\nNous allons voir que cela implique de fortes contraintes sur $p$, $q$, $r$ d'une part et sur $\\alpha$, $\\beta$, $\\gamma$ d'autre part. Pour cela, nous montrons d'abord le r\u00e9sultat suivant:\n\\begin{proposition}\nSoient $a$, $b$ et $c$ trois nombres r\u00e9els. On pose: $\\alpha:=4\\cos^{2}a\\pi$, $\\beta:=4\\cos^{2}b\\pi$ et $\\gamma:=4\\cos^{2}c\\pi$. Alors les deux conditions suivantes sont \u00e9quivalentes:\n\\begin{enumerate}\n \\item (1) $\\exists \\epsilon,\\epsilon' \\in \\{-1,+1\\}$ tels que $c\\equiv \\epsilon a+\\epsilon' b \\mod{\\mathbb{Z}}$;\n \\item (2) $\\alpha\\beta\\gamma=(4-\\alpha-\\beta-\\gamma)^{2}$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n1) Montrons que (1) implique (2).\\\\\nIl existe $\\epsilon_{1}\\in \\{-1,+1\\}$ tel que $\\epsilon_{1}\\cos c\\pi=\\cos a\\pi \\cos b\\pi-\\epsilon\\epsilon'\\sin a\\pi \\sin b\\pi$.\\\\\nNous obtenons:\n\\begin{align*}\n\\cos^{2}c\\pi=\\cos^{2}a\\pi \\cos^{2}b\\pi + \\sin^{2}a\\pi \\sin^{2}b\\pi -2\\epsilon\\epsilon'\\cos a\\pi \\sin a\\pi \\cos b\\pi sin b\\pi\\\\\n=1-\\cos^{2}a\\pi - \\cos^{2}b\\pi +2\\cos^{2}a\\pi cos^{2}b\\pi -2\\epsilon\\epsilon'\\cos a\\pi \\sin a\\pi \\cos b\\pi sin b\\pi\n\\end{align*}\nd'o\u00f9, en multipliant cette \u00e9galit\u00e9 par $8$:\n\\[\n2\\gamma=8-2\\alpha-2\\beta+\\alpha\\beta-4\\epsilon\\epsilon' \\sin2a\\pi \\sin 2b\\pi.\n\\]\nNous en d\u00e9duisons\n\\begin{equation*}\n\\begin{split}\n16\\sin^{2}2a\\pi \\sin^{2}2b\\pi\\\\\n&=(8-2\\alpha-2\\beta-2\\gamma+\\alpha\\beta)^{2}\\\\\n&=16(1-2\\cos^{2}2a\\pi)(1-2cos^{2}2b\\pi)\\\\\n&=16(1-\\cos 2a\\pi)(1+\\cos 2a\\pi)(1-\\cos 2b\\pi)(1+\\cos 2b\/pi)\\\\\n&=16(2-2\\cos^{2}a\\pi)2\\cos^{2}a\\pi (2-2\\cos^{2}b\\pi)2\\cos^{2}b\\pi\\\\\n&=(4-\\alpha)\\alpha(4-\\beta)\\beta\\\\\n&=16\\alpha\\beta-4\\alpha^{2}\\beta-4\\alpha\\beta^{2}+\\alpha^{2}\\beta^{2}\\\\\n&=(8-2\\alpha-2\\beta-2\\gamma)^{2}+\\alpha^{2}\\beta^{2}+2\\alpha\\beta(8-2\\alpha-2\\beta-2\\gamma)\n\\end{split}\n\\end{equation*}\nd'o\u00f9, apr\u00e8s simplifications\n\\[\n4\\alpha\\beta\\gamma=(8-2\\alpha-2\\beta-2\\gamma)^{2}=4(4-\\alpha-\\beta-\\gamma)^{2}\n\\]\nc'est la relation (2)\\\\\n2) Montrons que (2) implique (1).\\\\\nPosons, pour simplifier l'\u00e9criture: $u:=\\cos a\\pi$, $v:=\\cos b\\pi$ et $w:= \\cos c\\pi$ de telle sorte que $\\alpha=4u^{2}$, $\\beta=4v^{2}$ et $\\gamma=4w^{2}$. La relation (2) devient \n\\[\n64u^{2}v^{2}w^{2}=16(1-u^{2}-v^{2}-w^{2})^{2}\n\\]\nd'o\u00f9\n\\[\n(1-u^{2}-v^{2}-w^{2})^{2}-4u^{2}v^{2}w^{2}=0=(1-u^{2}-v^{2}-w^{2}+2uvw)(1-u^{2}-v^{2}-w^{2}-2uvw).\n\\]\nNous brisons maintenant la sym\u00e9trie entre $u$, $v$ et $w$ pour obtenir:\n\\[\n((w-uv)^{2}-(1-u^{2})(1-v^{2}))((w+uv)^{2}-(1-u^{2})(1-v^{2}))=0.\n\\]\nD'apr\u00e8s les valeurs de $u$, $v$ et $w$, nous voyons que $uv=\\cos a\\pi \\cos b\\pi$ et $(1-u^{2})(1-v^{2})=\\sin^{2}a\\pi \\sin^{2}b\\pi$, donc:\n\\[\n0=((w-\\cos a\\pi \\cos b\\pi)^{2}-(\\sin a\\pi \\sin b\\pi)^{2})((w+\\cos a\\pi \\cos b\\pi)^{2}-(\\sin a\\pi \\sin b\\pi)^{2})\n\\]et $0=d_{1}d_{2}d_{3}d_{4}$ o\u00f9:\n\\[\n\\begin{aligned}\nd_{1} &= w-(\\cos a\\pi \\cos b\\pi+\\sin a\\pi \\sin b\\pi) &= w-\\cos(a-b)\\pi\\\\\nd_{2} &= w-(\\cos a\\pi \\cos b\\pi-\\sin a\\pi \\sin b\\pi) &= w-\\cos(a+b)\\pi\\\\\nd_{3} &=w+(\\cos a\\pi \\cos b\\pi+\\sin a\\pi \\sin b\\pi) &= w+\\cos(a-b)\\pi\\\\\nd_{4} &=w+(\\cos a\\pi \\cos b\\pi-\\sin a\\pi \\sin b\\pi) &= w+\\cos(a+b)\\pi\\\\\n\\end{aligned}\n\\]\nComme $\\cos^{2}x-\\cos^{2}y=\\sin (x+y)\\sin(y-x)$, nous obtenons:\n\\[\n\\begin{aligned}\nd_{1}d_{3} &= w^{2}-\\cos^{2}(a-b)\\pi &= \\sin(c+a-b)\\pi \\sin(a-b-c)\\pi\\\\\nd_{2}d_{4} &= w^{2}-\\cos^{2}(a+b)\\pi &= \\sin(c+a+b)\\pi \\sin(a+b-c)\\pi\n\\end{aligned}\n\\]\nd'o\u00f9 finalement:\n\\[\n0=\\sin(c+a-b)\\pi \\sin(a-b-c)\\pi\\sin(c+a+b)\\pi \\sin(a+b-c)\\pi,\n\\]\ncomme $\\sin \\pi x =0$ si et seulement si $x$ est entier, nous voyons qu'il existe $\\epsilon$ et $\\epsilon'$ dans $\\{-1,+1\\}$ tels que $c\\equiv \\epsilon a+\\epsilon' b \\mod{\\mathbb{Z}}$ c'est la condition (1).\n\\end{proof}\n\nPour expliciter les relations entre $p$, $q$, $r$ d'une part et $\\alpha$, $\\beta$, $\\gamma$ d'autre part, nous utilisons le r\u00e9sultat suivant d\u00e9montr\u00e9 dans l'appendice :\nLes deux conditions suivantes $(C)$ et $(D)$ sur le triple d'entiers non nuls $(a_{1},a_{2},a_{3})$ sont \u00e9quivalentes:\n\\begin{align*}\n(C)\n\\begin{cases}\n(C_{1}) & n=ppcm(a_{1},a_{2},a_{3})=ppcm(a_{i},a_{j}) (1\\leqslant i\\neq j \\leqslant 3);\\\\\n(C_{2}) & \\parbox{11 cm}{%\n$\\exists i,j \\in \\mathbb{N}$ tels que $(1\\leqslant i\\neq j \\leqslant 3)$ et $v_{2}(a_{i})=v_{2}(a_{j})=v_{2}(n)$;\nsi $|\\{i,j,k\\}|=3$, $v_{2}(a_{k})$. Alors $L$ poss\u00e8de une repr\u00e9sentation fid\u00e8le irr\u00e9ductible $T$ sur un corps alg\u00e9briquement clos de caract\u00e9ristique $0$ si et seulement si $H$ est un groupe cyclique (et donc $L$ est un groupe di\u00eadral). Dans ce cas $\\deg T=2$.\n\\end{proposition}\n\\begin{proof}\nNous allons utiliser le th\u00e9or\u00e8me suivant de Gasch\\\"utz (\\cite{G}):\"Un groupe fini poss\u00e8de une repr\u00e9sentation irr\u00e9ductible et fid\u00e8le si et seulement si son socle est engendr\u00e9 par une classe de conjugaison\".\\\\\nSoit $T$ une repr\u00e9sentation irr\u00e9ductible de $L$. Comme $H$ est un sous-groupe normal commutatif de $L$, nous savons que $\\deg T\\leqslant 2$. Si $\\deg T=1$, alors $D(L)\\subset \\ker T$ et comme $D(L) \\neq {1}$ car $L$ n'est pas commutatif, nous voyons que $T$ n'est pas fid\u00e8le dans ce cas. Ainsi $\\deg T=2$.\n\nSoit $S$ le socle de $L$. Il est clair que $S \\subset H$. Pour chaque $s$ dans $S$, nous avons $\\sigma s \\sigma^{-1}=s \\,\\text{ou} \\,\\sigma s \\sigma^{-1}=s^{-1}$, donc la classe de conjugaison de $s$ est $\\{s\\}$ ou $\\{s,s^{-1}\\}$. Nous voyons ainsi, gr\u00e2ce au th\u00e9or\u00e8me de Gasch\\\"utz que $L$ poss\u00e8de une repr\u00e9sentation fid\u00e8le et irr\u00e9ductible si et seulement si $S$ est un groupe cyclique. D'apr\u00e8s la structure des groupes commutatifs finis, $S$ est un groupe cyclique si et seulement si $H$ est groupe cyclique. Dans ce cas $L$ est un groupe di\u00e9dral (d'ordre $\\geqslant 6$).\n\\end{proof}\n\\begin{notation}\nOn suppose que le triple d'entiers non nuls $(p,q,r)$ satisfait \u00e0 la condition $(C_{1})$. On pose $n:=ppcm(p,q,r)$, $d:= pgcd(p,q,r)$, $n=pp_{1}=qq_{1}=rr_{1}$ de telle sorte que $pgcd(p_{1},q_{1})=pgcd(q_{1},r_{1})=pgcd(r_{1},p_{1})=1$, $n=p_{1}q_{1}r_{1}d$, $p=q_{1}r_{1}d$, $q=r_{1}p_{1}d$ et $r=p_{1}q_{1}d$.\n\\end{notation}\n\\begin{proposition}\nSoit $G\"$ le quotient de $W(p,q,r)$ (avec comme syst\u00e8me g\u00e9n\u00e9rateur canonique $(t_{1},t_{2},t_{3})$) obtenu en ajoutant la relation $(t_{1}t_{2}t_{3})^{2}=1$.\\\\ Soit $H\":=$. Alors:\\\\\n1) $H\"$ est un sous-groupe commutatif normal de $G\"$.\\\\\n2)Le triple d'entiers $(p,q,r)$ satisfait \u00e0 la condition ($C_{1}$).\\\\\n3) $H\"\\simeq \\mathbb{Z}\/d\\mathbb{Z}\\times \\mathbb{Z}\/n\\mathbb{Z}$.\n\\end{proposition}\n\\begin{proof}\n1) Posons $a:=t_{1}t_{2}$, $b:=t_{2}t_{3}$ et $c:=t_{3}t_{1}$. Il est clair que $abc=1$. De plus $acb=(t_{1}t_{2}t_{3})^{2}=1$, $cb=bc$. On voit de la m\u00eame mani\u00e8re que $ac=ca$ et $ab=ba$: le groupe $H\"$ est commutatif.\\\\\n2) Comme $t_{1}at_{1}^{-1}=a^{-1}$ et $t_{1}ct_{1}^{-1}=c^{-1}$ et comme $G\"=$, nous voyons que $H\"$ est un sous-groupe normal d'indice $2$ de $G\"$ dont tous les \u00e9l\u00e9ments sont invers\u00e9s par $t_{i}$ $(1\\leqslant i \\leqslant 3)$. Il en r\u00e9sulte que l'ordre de $ac$ est $ppcm(p,q)$ donc $r\\,| \\,ppcm(p,q)$. Par sym\u00e9trie, on voit que $p\\,| \\,ppcm(q,r)$ et $q\\,| \\,ppcm(r,p)$: le triple d'entiers $(p,q,r)$ satisfait \u00e0 la condition $(C_{1})$.\\\\\n3) Soit $\\pi$ un nombre premier et soit $\\Pi$ la partie $\\pi$-primaire de $|H\"|$. Nous avons\n\\[\nH\"=\\oplus_{\\pi \\in \\mathcal{P}}\\Pi\n\\]\no\u00f9 $\\mathcal{P}$ d\u00e9signe l'ensemble des nombres premiers positifs.\\\\\nPosons $p=\\pi^{n(p)}p'$, $q=\\pi^{n(q)}q'$ et $r=\\pi^{n(r)}r'$ o\u00f9 $p'$, $q'$ et $r'$ sont premiers \u00e0 $p$. Posons $a':=a^{p'}$, $b':= b^{r'}$ et $c':=c^{q'}$. Alors $a'$ est d'ordre $\\pi^{n(p)}$, $b'$ est d'ordre $\\pi^{n(r)}$ et $c'$ est d'ordre $\\pi^{n(q)}$.\\\\\nNous avons\n\\[\n\\Pi=\n\\]\nNous pouvons supposer que $n(p)=n(r)$ et $n(q)\\leqslant n(p)$ (voir l'appendice ). Nous effectuons des op\u00e9rations \u00e9l\u00e9mentaires sur la matrice des relations de $\\Pi$:\n\\[\n\\begin{pmatrix}\n\\pi^{n(p)} & 0\\\\\n0 & \\pi^{n(p)}\\\\\n\\pi^{n(q)} & \\pi^{n(q)}\n\\end{pmatrix}\n\\to\n\\begin{pmatrix}\n\\pi^{n(p)} & -\\pi^{n(p)}\\\\\n0 & \\pi^{n(p)}\\\\\n\\pi^{n(q)} & 0\n\\end{pmatrix}\n\\to\n\\begin{pmatrix}\n\\pi^{n(p)} & 0\\\\\n0 & \\pi^{n(p)}\\\\\n\\pi^{n(q)} & 0\n\\end{pmatrix}\n\\]\n\\[\n\\to\n\\begin{pmatrix}\n\\pi^{n(q)} & 0\\\\\n0 & \\pi^{n(p)}\\\\\n\\pi^{n(p)} & 0\n\\end{pmatrix}\n\\to\n\\begin{pmatrix}\n\\pi^{n(q)} & 0\\\\\n0 & \\pi^{n(p)}\\\\\n0 & 0\n\\end{pmatrix}\n\\]\nIl en r\u00e9sulte que $\\Pi \\simeq \\mathbb{Z}\/\\pi^{n(q)}\\mathbb{Z} \\times \\mathbb{Z}\/\\pi^{n(p)}\\mathbb{Z}$. Comme $\\pi^{n(q)}$ est la $\\pi$-contribution de $pgcd(p,q,r)$ et $\\pi^{n(p)}$ est la $\\pi$-contribution de $ppcm(p,q,r)$, nous avons le r\u00e9sultat car $H\"$ est le produit de ses sous-groupes de Sylow.\n\\end{proof}\nNous avons maintenant assez d'\u00e9l\u00e9ments pour le r\u00e9sultat suivant:\n\\begin{theorem}\nSoient $W(p,q,r)$ un groupe de Coxeter et $R$ une repr\u00e9sentation de r\u00e9flexion de $W(p,q,r)$. On suppose que le triple d'entiers $(p,q,r)$ satisfait \u00e0 la condition (C). Soit $\\alpha$ une racine de $v_{p}(X)$. On sait (voir l'appendice ) que l'on peut trouver $\\beta$ racine de $v_{q}(X)$ et $\\gamma$ racine de $v_{r}(X)$ de telle sorte que $\\Delta=0$ et $\\alpha l=\\beta m$. Alors $G'$ est un groupe di\u00e9dral d'ordre $2n$, o\u00f9 $n=ppcm(p,q,r)$ et $R$ est r\u00e9ductible.\n\\end{theorem}\n\\begin{proof}\nD'apr\u00e8s la proposition 1 de l'appendice (voir \\cite{Z1}), on sait que le polyn\u00f4me caract\u00e9ristique de $t_{1}=s_{1}s_{2}s_{3}$ est $P_{t_{1}}=(X-1)^{2}(X+1)$, donc $t_{1}$ a comme valeurs propres sur $M'$: $-1$ et $+1$, d'o\u00f9 $(t'_{1})^{2}=id_{M'}$. D'apr\u00e8s la proposition 4, $G'$ est isomorphe \u00e0 un quotient du groupe $G\"$ d\u00e9fini dans cette proposition. Il en r\u00e9sulte que $G'$ est une extension d'un groupe commutatif fini $H$ par un \u00e9l\u00e9ment d'ordre $2$ qui inverse tous les \u00e9l\u00e9ments de $H$. De plus $G'$ op\u00e8re fid\u00e8lement sur $M'$ par d\u00e9finition. Comme $$ $(1\\leqslant i < j \\leqslant 3)$ s'envoie fid\u00e8lement dans $G'$ et comme la repr\u00e9sentation du groupe di\u00e9dral $$ est absolument irr\u00e9ductible sur $M'$, nous voyons que $G'$ est un groupe di\u00e9dral d'ordre $2n$ en appliquant la proposition 3 car toute repr\u00e9sentation irr\u00e9ductible complexe de $D_{n}$ est r\u00e9alisable sur $K_{0}$.\n\\end{proof}\n\\subsection{Le groupe $N(G)$}\nComme $G'$ est un groupe di\u00e9dral d'ordre $2n$ et comme $N$ est un groupe commutatif libre tel que la repr\u00e9sentation de $G'$ sur $N\\otimes_{\\mathbb{Z}}\\mathbb{Q}$ est irr\u00e9ductible, nous allons \u00e9tudier les repr\u00e9sentations rationnelles irr\u00e9ductibles du groupe di\u00e9dral $D_{n}$ d'ordre $2n$ $(n\\geqslant 3)$. En particulier, nous allons voir qu'il n'y en a qu'une seule qui est fid\u00e8le.\n\nNous allons d'abord construire les repr\u00e9sentations irr\u00e9ductibles rationnelles du groupe di\u00e9dral $D_{n}$. D'apr\u00e8s un r\u00e9sultat bien connu (voir \\cite{S}), on a :\\\\\n\"Le nombre des classes de repr\u00e9sentations irr\u00e9ductibles du groupe fini $G$ sur $\\mathbb{Q}$ est \u00e9gal au nombre des classes de conjugaison de sous-groupes cycliques de $G$.\"\n\nNous cherchons donc le nombre de classes de conjugaison de sous-groupes cycliques du groupe $G_{0}$\n\\[\nG_{0}:=\\quad (n\\geqslant 3).\n\\]\nOn pose $C:=$. Il y a d'abord les classes de cardinal $1$ contenant chacune un sous-groupe de $C$: tous les sous-groupes de $C$ sont normaux dans $G_{0}$.\\\\\nSi $n$ est impair, il y a une classe de conjugaison de sous-groupes d'ordre $2$ de $G_{0}$ et si $n$ est pair, il y a deux classes de conjugaison de sous-groupes d'ordre $2$ contenant les involutions non centrales de $G_{0}$.\n\nSoit $\\Phi_{n}(X)$ le n-i\u00e8me polyn\u00f4me cyclotomique . Le polyn\u00f4me $\\Phi_{n}(X)\\in \\mathbb{Z}[X]$, est irr\u00e9ductible et r\u00e9ciproque. On consid\u00e8re le corps $L=\\mathbb{Q}[x]\/(\\Phi_{n}(X))$. C'est un espace vectoriel de dimension $\\varphi(n)=\\deg \\Phi_{n}(X)$ sur le corps $\\mathbb{Q}$. Le groupe de Galois $\\mathcal{G}al(L\/\\mathbb{Q})$ est un groupe commutatif d'ordre $\\varphi(n)$. Soit $\\pi:\\mathbb{Q}[x]\\to L$ la projection canonique. On pose $g:=\\pi(X)$ et l'on fait op\u00e9rer $g$ sur $L$ par multiplication. Comme $n\\geqslant 3$, $\\varphi(n)$ est pair et le groupe de Galois contient un \u00e9l\u00e9ment $s$ d'ordre $2$ qui op\u00e8re comme $g\\mapsto g^{-1}$. Le sous-corps $L_{0}$ des points fixes de $s$ est de degr\u00e9 $\\frac{1}{2}\\varphi(n)$. Il en r\u00e9sulte que $[L,s]$, sous-espace vectoriel de $L$ form\u00e9 des \u00e9l\u00e9ments transform\u00e9s en leurs oppos\u00e9s par $s$, est de dimension $\\frac{1}{2}\\varphi(n)$. Le sous-groupe de $GL(L)$ engendr\u00e9 par $g$ et $s$ est un groupe di\u00e9dral d'ordre $2n$.\n\nNous construisons une base du $\\mathbb{Q}$-espace vectoriel $L$ de la mani\u00e8re suivante: soit $e_{1}\\in [L,s]-\\{0\\}$. Pour $1\\leqslant i \\leqslant\\varphi(n)-1$, on pose $e_{i+1}=g.e_{i}$. Alors $(e_{1},e_{2},\\cdots,e_{\\varphi(n)})$ est une base de $L$ car $\\Phi_{n}(X)$ est un polyn\u00f4me irr\u00e9ductible.\\\\\nSi $\\Phi_{n}(X)=X^{\\varphi(n)}+a_{1}X^{\\varphi(n)-1}+\\cdots+a_{1}X+1$, on a:\n\\[\ng.e_{n}=-e_{1}-a_{1}e_{2}-\\cdots-a_{1}e_{\\varphi(n)}.\n\\]\nDe plus un calcul simple montre que $s(e_{i})=-g^{1-i}.e_{1}$ $(1\\leqslant i \\leqslant \\varphi(n)$).\\\\\nIl est clair que nous avons construit de cette mani\u00e8re une $\\mathbb{Q}$-repr\u00e9sentation irr\u00e9ductible et fid\u00e8le $R_{n}$ du groupe $G_{0}$. Soit maintenant $d\\geqslant 3$ un diviseur de $n$ et soit $C_{d}:=$. Alors $C_{d}\\lhd G_{0}$ et $G_{0}\/C_{d}$ est isomorphe \u00e0 un groupe di\u00e9dral d'ordre $2d$. Nous appliquons ce qui pr\u00e9c\u00e8de au groupe $G_{0}\/C_{d}$ et nous obtenons ainsi une $\\mathbb{Q}$-repr\u00e9sentation irr\u00e9ductible du groupe $G_{0}$, de noyau $C_{d}$ et de degr\u00e9 $\\varphi(d)$.\\\\\n- Si $n$ est impair, alors $C_{1}$ est d'ordre $1$ et nous obtenons la repr\u00e9sentation $R_{1}$ de degr\u00e9 $1$ o\u00f9 $s$ op\u00e8re comme $-1$. Nous avons aussi dans ce cas la repr\u00e9sentation $R_{0}$ o\u00f9 $s$ op\u00e8re comme l'identit\u00e9. D'apr\u00e8s le r\u00e9sultat cit\u00e9 plus haut nous avons obtenu toutes les repr\u00e9sentations rationnelles irr\u00e9ductibles de $G_{0}$.\\\\\n- Si $n$ est pair, alors $C_{2}$ est d'ordre $2$ et nous obtenons la repr\u00e9sentation $R_{2}$ de degr\u00e9 $\\varphi(2)=1$ o\u00f9 $g$ et $s$ op\u00e8rent comme $-1$. Comme dans le cas impair, il y a la repr\u00e9sentation $R_{1}$ de degr\u00e9 $1$ o\u00f9 $s$ op\u00e8re comme $-1$ et $g$ comme l'identit\u00e9; et enfin il y a la repr\u00e9sentation $R_{1}\\otimes R_{2}$ de degr\u00e9 $1$ o\u00f9 $s$ op\u00e8re comme l'identit\u00e9 et $g$ op\u00e8re comme $-1$.\n\nNous pouvons encore remarquer que toutes les repr\u00e9sentations $R_{i}$ sont des $\\mathbb{Z}$-repr\u00e9sentations.\n\nToutes les repr\u00e9sentations absolument simples de $G_{0}$ sont de degr\u00e9 $1$ ou $2$ et peuvent s'\u00e9crire sur le corps $K=\\mathbb{Q}(\\cos \\frac{2\\pi}{n})$. Celles qui sont fid\u00e8les sont celles pour lesquelles le caract\u00e8re $\\chi_{k}$ est tel que $\\chi_{k}(g)=2\\cos \\frac{2k\\pi}{n}$ avec $1\\leqslant k =$ nous voyons que $g \\in G_{2}$.\n\nComme l'ensemble des transform\u00e9s de $e$ par $$ contient une base de $\\Lambda$, nous avons le r\u00e9sultat $G_{2}\\simeq G_{1}$. Il en r\u00e9sulte aussit\u00f4t que $G_{1}$ est isomorphe \u00e0 un quotient du groupe $G$. Comme tous les quotients propres de $G$ sont finis, nous obtenons $G_{1}\\simeq G$.\n\\end{proof}\n\\subsection{Appendice}\nDans cet appendice, nous d\u00e9montrons un r\u00e9sultat d'arithm\u00e9tique (voir \\cite{St}) utilis\u00e9 pour caract\u00e9riser certains triples d'entiers qui donnent les groupes di\u00e9draux.\n\\begin{proposition}\\label{propC1}\nLes deux conditions suivantes $(C)$ et $(D)$ sur le triple d'entiers non nuls $(a_{1},a_{2},a_{3})$ sont \u00e9quivalentes:\n\\begin{align*}\n(C)\n\\begin{cases}\n(C_{1}) & n=ppcm(a_{1},a_{2},a_{3})=ppcm(a_{i},a_{j}) (1\\leqslant i\\neq j \\leqslant 3);\\\\\n(C_{2}) & \\parbox{11 cm}{%\n$\\exists i,j \\in \\mathbb{N}$ tels que $(1\\leqslant i\\neq j \\leqslant 3)$ et $v_{2}(a_{i})=v_{2}(a_{j})=v_{2}(n)$;\nsi $|\\{i,j,k\\}|=3$, $v_{2}(a_{k})0$ et $\\frac{x}{a_{2}}>0$, on a $\\frac{y}{a_{3}}<1$. Ensuite $\\frac{c_{1}}{a_{1}}+\\frac{x}{a_{2}}<2$ donc $-1<\\frac{y}{a_{3}}$. Nous obtenons ainsi $-a_{3}$ un groupe di\u00e9dral d'ordre $2n$ avec $n\\geqslant 3$. On appelle $G$ son sous-groupe cyclique d'ordre $n$. Soient $a_{1},a_{2},a_{3}$ trois entiers $>0$. Les deux conditions suivantes sont \u00e9quivalentes:\n\\begin{itemize}\n \\item (A) Il existe trois involutions non centrales $s_{i}$ $(1\\leqslant i \\leqslant 3)$ de $D$ telles que si $s_{i}s_{j}=g_{k}$ $(1 \\leqslant i \\neq j \\leqslant 3, \\; |\\{i,j,k\\}|=3)$ avec $g_{k}$ d'ordre $a_{k}$, on ait $G= \\;(1 \\leqslant i \\neq j \\leqslant 3)$.\n \\item (B) Le triple d'entiers $(a_{1},a_{2},a_{3})$ satisfait \u00e0 la condition (C).\n\\end{itemize}\n\\end{proposition}\n\\begin{proof}\n1) Montrons que $(A)\\Longrightarrow(B)$. Comme $G=$ $(1 \\leqslant i \\neq j \\leqslant 3)$, n\u00e9cessairement $ppcm(a_{i},a_{j})=n$ $(1 \\leqslant i \\neq j \\leqslant 3)$ donc la condition $(C_{1})$ est satisfaite. Supposons maintenant $n$ pair, $n=2m$. Nous avons vu qu'il existe $i$ et $j$ $(1\\leqslant i \\leqslant 3)$ tels que $v_{2}(a_{i})=v_{2}(a_{j})=v_{2}(n)$. Les conjugu\u00e9s de $s_{3}$ dans $D$ sont les $g^{2l}s_{3}$ $(0\\leqslant l \\leqslant m-1)$ et l'autre classe d'involutions non centrales est l'ensemble des $g^{2l+1}s_{3}$ $(0\\leqslant l \\leqslant m-1)$. Nous avons $s_{1}s_{2}=g_{3}$, $s_{1}s_{3}=g_{2}$, $s_{2}s_{3}=g_{1}$ donc $g_{3}=g_{2}g_{1}^{-1}$. Si nous supposons que $v_{2}(a_{1})=v_{2}(a_{2})=v_{2}(n)$, alors $s_{1}=g^{2\\alpha+1}s_{3}$ et $s_{2}=g^{2\\beta+1}s_{3}$. Il en r\u00e9sulte que $g_{3}=s_{1}s_{2}=g^{2\\alpha-2\\beta}$ et alors $v_{2}(a_{3})-0.85$ the continuum contributions become large\nwhile for $t<-1.15$ the contributions from higher-order terms in the\nOPE become important relative to the leading-order terms.\n\nThe subscript $S_{\\rm V}$ in Eq.~(\\ref{corr}) indicates the\npresence of the external field. Thus,\nthe correlator should be calculated with an additional term\n\\begin{equation}\n\\Delta{\\cal L} \\equiv - S_{\\rm V} [\\overline u (x) u(x)\n-\\overline d(x) d(x)]\\; ,\n\\label{lag}\n\\end{equation}\nadded to the usual QCD Lagrangian, and $-\\Delta{\\cal L}$\nadded to ${\\cal H}_{\\rm QCD}$.\nSince $S_{\\rm V}$ is a scalar constant,\nLorentz covariance and parity allow one to decompose\n$\\Pi(S_{\\rm V},q)$ into two distinct structures\\cite{jin2}\n\\begin{equation}\n\\Pi(S_{\\rm V},q)\\equiv \\Pi^1(S_{\\rm V},q^2)+\n\\Pi^q(S_{\\rm V},q^2)\\rlap{\/}{q}\\ .\n\\end{equation}\nTo obtain QCD sum rules, one needs to construct a phenomenological\nrepresentation for $\\Pi(S_{\\rm V},q)$ and evaluate\n$\\Pi(S_{\\rm V},q)$ using the OPE.\n\n\\subsection{The dispersion relation and phenomenological spectral ansatz}\n\nTo determine the correlator at the hadron level we\nuse the dispersion relation\n\\begin{equation}\n\\Pi^i(S_{\\rm V},q^2)=\\int_0^\\infty{\\rho^i(S_{\\rm V},s)\n\\over s-q^2}ds\\\n\\label{des-rel}\n\\end{equation}\nfor each invariant function $\\{i=1,q\\}$, where\n$\\rho^i(S_{\\rm V},s)={1\\over\\pi}{\\rm Im}\\Pi^i(S_{\\rm V},s)$\nis the spectral density. Here we have omitted polynomial\nsubtractions which will be eliminated by a subsequent Borel\ntransformation.\nWe have also omitted infinitesimal as we are only concerned\nwith large and space like $q^2$ in QCD sum rules.\n\nIn practical applications of QCD sum-rule approach, one usually\nparametrizes the spectral density by a simple pole representing the\nlowest energy baryon state of interest plus a continuum which\nis approximated by a perturbative evaluation of the correlator\nstarting at an effective threshold\\cite{svz1,reinders1,ioffe3}.\nWhen $S_{\\rm V}$\nis present, we add $-\\Delta{\\cal L}$ to ${\\cal H}_{\\rm QCD}$,\nwhich is equivalent to increase $m_u$ and $m_d$ by $S_{\\rm V}$\nand $-S_{\\rm V}$, respectively.\nConsequently at the hadron level, the baryon spectrum will\nbe shifted. Since we are concerned here with the linear response\nto the external source, $S_{\\rm V}$ can be taken to be arbitrarily\nsmall (see below). Thus, there\nis no rearrangement of the spectrum, and we can use a pole\nplus continuum ansatz for the baryon spectral density\n\\begin{equation}\n\\rho^i(S_{\\rm V},s)=\\lambda_{\\rm B}^{*^2}\\phi^i \\delta (s-M_{\\rm B}^{*^2})\n+\\widetilde{\\rho}^i(S_{\\rm V},s)\\theta (s-s_0^{*^i})\\ ,\n\\label{ans}\n\\end{equation}\nwhere $\\phi^{*^i}=\\{M^*_{\\rm B}, 1\\}$ for $\\{i=1,q\\}$, and\n$\\widetilde{\\rho}^i(S_{\\rm V},s)$ is to be evaluated in\nperturbation theory.\nHere $\\lambda^*_{\\rm B}$ is defined by\n$\\langle 0|\\eta_{\\rm B}|{\\rm B}\\rangle_{S_{\\rm V}}\n=\\lambda_{\\rm B}^* v^*_{\\rm B}$\nwith $v^*_{\\rm B}$ the Dirac spinor normalized to\n$\\overline{v}^*_{\\rm B} v^*_{\\rm B}=2M^*_{\\rm B}$, $M^*_{\\rm B}$\nis the mass of the lowest baryon state and $s_0^{*^i}$ is the\ncontinuum threshold in the presence of the external field.\n\nLet us now expand both sides of Eq.~(\\ref{des-rel}) for small $S_{\\rm V}$\n\\begin{equation}\n\\Pi^i_0(q^2)+S_{\\rm V}\\Pi^i_1(q^2)+\\cdots\n=\\int_0^\\infty {\\rho^i_0(s)\\over s-q^2} ds\n+S_{\\rm V}\\int_0^\\infty {\\rho^i_1(s)\\over s-q^2} ds+\\cdots\\ .\n\\label{des-expand}\n\\end{equation}\nSince $S_{\\rm V}$ is arbitrary, one immediately concludes that\n\\begin{eqnarray}\n\\Pi^i_0(q^2)&=&\\int_0^\\infty{\\rho^i_0(s)\\over s-q^2} ds\\ ,\n\\label{des-2}\n\\\\*[7.2pt]\n\\Pi^i_1(q^2)&=&\\int_0^\\infty {\\rho^i_1(s)\\over s-q^2} ds\\ .\n\\label{des-3}\n\\end{eqnarray}\nObviously, Eq.~(\\ref{des-2}) leads to the baryon {\\it mass} sum rules\nin vacuum which have been extensively studied\n\\cite{ioffe3,belyaev3,reinders1,leinweber2}. Here we are interested in\nEq.~(\\ref{des-3}), which corresponds to the linear response of the\ncorrelator to the external source and contains the baryon\nmatrix element under consideration (see below).\n\nExpanding the right-hand side of Eq.~(\\ref{ans}), we find\n\\begin{eqnarray}\n\\rho^i_0(s)&=&\\lambda^2_{\\rm B}\\phi^i_0\\delta(s-M^2_{\\rm B})+\n\\widetilde{\\rho}^i_0(s)\\theta(s-s_0^i)\\ ,\n\\label{ans-0}\n\\\\*[7.2pt]\n\\rho^i_1(s)&=&-2 H_{\\rm B}\\, M_{\\rm B}\\lambda^2_{\\rm B} \\phi_0^i\n\\delta^\\prime (s-M^2_{\\rm B})+\\Delta\\lambda_{\\rm B}^2\\,\\phi_0^i\n\\delta (s-M^2_{\\rm B})\n\\nonumber\n\\\\*[7.2pt]\n& &\n+\\Delta\\phi^i\\, \\lambda^2_{\\rm B}\\delta (s-M^2_{\\rm B})\n-\\Delta s_0^i\\, \\widetilde{\\rho}^i_0(s)\\delta(s-s_0^i)+\n\\widetilde{\\rho}^i_1(s)\\theta (s-s_0^i)\\ ,\n\\label{ans-1}\n\\end{eqnarray}\nwhere we have defined\n\\begin{eqnarray}\nM^*_{\\rm B}&=&M_{\\rm B}+S_{\\rm V} H_{\\rm B}+\\cdots\\ ,\n\\\\*[7.2pt]\n\\lambda^{*^2}_{\\rm B}&=&\\lambda^2_{\\rm B}\n+S_{\\rm V} \\Delta\\lambda^2_{\\rm B}+\\cdots\\ ,\n\\\\*[7.2pt]\ns_0^{*^i}&=&s_0^i+S_{\\rm V} \\Delta s_0^i+\\cdots\\ ,\n\\\\*[7.2pt]\n\\phi^{*^i}&=&\\phi^i_0+S_{\\rm V}\\Delta\\phi^i+\\cdots\\ ,\n\\\\*[7.2pt]\n\\widetilde{\\rho}^{*^i}(s)&=&\\widetilde{\\rho}^i_0(s)+S_{\\rm V}\n\\widetilde{\\rho}^i_1(s)+\\cdots\\ ,\n\\end{eqnarray}\nwhere the first terms are the vacuum spectral parameters\nin the absence of the external field.\nNote that $\\Delta\\phi^1=H_{\\rm B}$ and $\\Delta\\phi^q=0$.\nTreating $S_{\\rm V}$ as a small parameter, one can\n use the Hellman-Feynman theorem\\cite{hellman1,feynman1}\n to show that\n\\begin{equation}\nH_{\\rm B}={\\langle B|\\overline{u}u-\\overline{d}d|B\\rangle\n\\over 2M_{\\rm B}}\\ ,\n\\label{hf-s}\n\\end{equation}\nwhere we have used covariant normalization $\\langle k^\\prime,\nB|k, B\\rangle=(2\\pi)^2 k^0\\delta^{(3)}(\\vec k^\\prime-\\vec k)$.\n\nOne notices that $\\rho^i_1(s)$ has specific new features\nwhich distinguish it from $\\rho^i_0(s)$. The first term\nin Eq.~(\\ref{ans-1}), which is {\\it absent} in $\\rho^i_0(s)$,\ngive rise to a double pole at the baryon mass whose residue\ncontains the matrix element of interest. The second and\nthird terms are single pole terms; the residue at the single\npole contains the information about the\ntransition between the ground state baryon and the excited states.\nIn terms of quantum mechanical perturbation, the double pole term\ncorresponds to the energy shift while the single pole\nterms result from the response of baryon wave function to\nthe external field. The fouth term is due to the response\nof the continuum threshold to the external source and the\nlast term is the continuum contribution. As emphasized in\nthe previous works, the\nsingle pole contributions are not exponentially damped after\nthe Borel transformation relative to the double\nterm and should be retained in a consistent analysis of the sum\nrules.\n\nThe fourth term has been neglected in Ref.~\\cite{jin2}.\nThe contribution\nof this term is suppressed in comparison with the single\npole terms by a factor $e^{-(s_0^i-M^2_{\\rm B})\/M^2}$\n[see Eqs.~(\\ref{sum_p_1}--\\ref{sum_x_q})].\nIf the response of the continuum threshold is small, one can neglect\nthe contribution of the fourth term. However, if\nthe response of the continuum threshold is strong, one needs\nto include the fourth term\nin the calculation. This point has been noticed recently by\nIoffe in Ref.~\\cite{ioffe2}, where a double dispersion\nrelation is considered for the vertex function\n\\begin{equation}\n\\Pi_1(q)=\\int d^4 x e^{iq\\cdot x}\\langle 0|T\n\\eta_{\\rm B}(x)\\left[\\int d^4 z (\\overline{u}(z)u(z)-\\overline{d}(z)d(z))\n\\right]\\overline{\\eta}_{\\rm B}(0)|0\\rangle\\\n\\label{vertex}\n\\end{equation}\nin order to get the appropriate phenomenological representation.\n[This vertex function can be obtained by expanding the\nright-hand side of Eq.~(\\ref{corr}) directly.]\nWe note that our discussion and\nEq.~(\\ref{ans-1}) are consistent with those given\nin Ref.~\\cite{ioffe2}.\nSubstituting Eq.~(\\ref{ans-1}) into Eq.~(\\ref{des-3}),\none obtains the appropriate phenomenological representation.\n\n\n\n\\subsection{QCD representation}\n\n\nThe QCD representation of the correlator is obtained by applying\nthe OPE to the\ntime-ordered product in the correlator. When the external field\nis present, the up and down quark fields satisfy the modified\nequations of motion:\n\\begin{eqnarray}\n(i\\rlap{\\,\/}D-m_u-S_{\\rm V})u(x)&=& 0\\ ,\n\\\\*[7.2pt]\n(i\\rlap{\\,\/}D-m_d+S_{\\rm V})d(x)&=& 0\\ ,\n\\label{eq-mo}\n\\end{eqnarray}\nwhere $\\rlap{\\,\/}D=\\gamma^\\mu(\\partial_\\mu-ig_s{\\cal A}_\\mu)$ is\nthe covariant derivative. (The equation of motion for the\nstrange quark field does not change.)\nIn the framework of the OPE, the external field contributes to\nthe correlator in two ways: It couples directly to\nthe quark fields in the baryon interpolating fields and it also\npolarizes the QCD vacuum. Since the external\n field in the present problem is a Lorentz scalar,\nnon-scalar correlators cannot be induced in the QCD vacuum. However the\nexternal field does modify the condensates already\npresent in the QCD vacuum. To first order in $S_{\\rm V}$,\nthe chiral quark condensates\ncan be written as follows\n\\begin{eqnarray}\n\\langle\\overline{u}u\\rangle_{S_{\\mbox{\\tiny{\\rm\n V}}}}&=&\\langle\\overline{u}u\\rangle_{\\mbox{\\tiny{\\rm 0}}}-\\chi\n S_{\\mbox{\\tiny{\\rm V}}}\\langle\\overline{u}u\\rangle_{\\mbox\n{\\tiny{\\rm 0}}}\\ ,\n\\label{uc}\n\\\\*[7.2pt]\n\\langle\\overline{d}d\\rangle_{S_{\\mbox{\\tiny{\\rm\n V}}}}&=&\\langle\\overline{d}d\\rangle_{\\mbox{\\tiny{\\rm 0}}}+\\chi\nS_{\\mbox{\\tiny{\\rm V}}}\\langle\\overline{d}d\\rangle_{\\mbox{\\tiny\n{\\rm 0}}}\\ ,\n\\label{dc}\n\\\\*[7.2pt]\n\\langle\\overline{s}s\\rangle_{S_{\\mbox{\\tiny{\\rm\n V}}}}&=&\\langle\\overline{s}s\\rangle_{\\mbox{\\tiny{\\rm 0}}}-\\chi_{\\rm s}\nS_{\\mbox{\\tiny{\\rm V}}}\\langle\\overline{s}s\\rangle_{\\mbox{\\tiny\n{\\rm 0}}}\\ ,\n\\label{sc}\n\\end{eqnarray}\nwhere $\\langle \\hat{O}\\rangle_{\\mbox{\\tiny{\\rm 0}}}\\equiv \\langle\n0|\\hat{O}|0\\rangle$.\nThe mixed quark-gluon condensates change in a similar way\n\\begin{eqnarray}\n\\langle g_s\\overline{u}\\sigma\\cdot {\\cal G} u\n\\rangle_{S_{\\mbox{\\tiny{\\rm V}}}}\n&=&\\langle g_s\\overline{u}\\sigma\\cdot {\\cal G} u\\rangle_{\\mbox{\\tiny\n{\\rm 0}}}-\\chi_{\\rm m}\nS_{\\mbox{\\tiny{\\rm V}}}\\langle g_s\\overline{u}\\sigma\\cdot {\\cal G}\nu\\rangle_{\\mbox\n{\\tiny{\\rm 0}}}\\ ,\n\\label{uqc}\n\\\\*[7.2pt]\n\\langle g_s\\overline{d}\\sigma\\cdot {\\cal G} d\n\\rangle_{S_{\\mbox{\\tiny{\\rm V}}}}\n&=&\\langle g_s\\overline{d}\\sigma\\cdot {\\cal G} d\\rangle_{\\mbox{\\tiny\n{\\rm 0}}}+\\chi_{\\rm m}\nS_{\\mbox{\\tiny{\\rm V}}}\\langle g_s\\overline{d}\\sigma\\cdot {\\cal G}\nd\\rangle_{\\mbox\n{\\tiny{\\rm 0}}}\\ ,\n\\label{dqc}\n\\\\*[7.2pt]\n\\langle g_s\\overline{s}\\sigma\\cdot {\\cal G} s\n\\rangle_{S_{\\mbox{\\tiny{\\rm V}}}}\n&=&\\langle g_s\\overline{s}\\sigma\\cdot {\\cal G} s\\rangle_{\\mbox{\\tiny\n{\\rm 0}}}-\\chi_{\\rm ms}\nS_{\\mbox{\\tiny{\\rm V}}}\\langle g_s\\overline{s}\\sigma\\cdot {\\cal G}\ns\\rangle_{\\mbox\n{\\tiny{\\rm 0}}}\\ ,\n\\label{sdc}\n\\end{eqnarray}\nwhere $\\sigma\\cdot {\\cal G}\\equiv\n\\sigma_{\\mu\\nu}{\\cal G}^{\\mu\\nu}$ with ${\\cal G}^{\\mu\\nu}$ the gluon\nfield tensor. One can express $\\chi$, $\\chi_{\\rm s}$, $\\chi_{\\rm m}$,\nand $\\chi_{\\rm ms}$ in terms of correlation functions\n(see Ref.~\\cite{jin2}). Here\nwe have assumed that the response of the up and down quarks is\nthe same, apart from the sign.\nThe Wilson coefficients can be calculated following the methods\noutlined in Ref.~\\cite{jin2}. The results of our calculations for\nthe invariant functions $\\Pi^1_1$ and $\\Pi^q_1$\nare given in Appendix A.\n\n\n\\subsection{Sum rules}\n\nThe QCD sum rules are obtained by equating the QCD representation\nand the phenomenological representation and applying the Borel\ntransformation. The resulting sum rules in the proton case\ncan be expressed as\n\\begin{eqnarray}\n& &{c_1+6c_2\\over 2}M^8 E_2 L^{-8\/9}\n-{c_1+6c_2\\over 2}\\chi a M^6 E_1\n+{3c_2\\over 2}\\chi_{\\rm m}m_0^2 a M^4E_0L^{-14\/27}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{1.0cm}\n+{c_1+3c_2-c_3\\over 3} a^2 M^2\n=\\biggl[2 H_p\\,\\widetilde{\\lambda}_p^2 M_p^2-\n\\Delta\\widetilde{\\lambda}_p^2\\, M_p M^2\n-H_p\\,\\widetilde{\\lambda}_p^2 M^2\\biggr]e^{-M_p^2\/M^2}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{1.5cm}\n+\\left[\n{c_1-6c_2\\over 2}s^1_0 a L^{-4\/9}\n+{3c_2\\over 2}m_0^2 a L^{-26\/27}\\right]\\Delta s_0^1 M^2\ne^{-s^1_0\/M^2}\\; ,\n\\label{sum_p_1}\n\\\\*[14.4pt]\n& &-{4c_1-c_3\\over 4} a M^4 E_0 L^{-4\/9}\n-{c_4+c_5-6c_2\\over 12}m_0^2 a M^2 L^{-26\/27}\n+{2c_1\\over\n3}\\chi a^2 M^2 L^{4\/9}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.4cm}\n-{c_1+2c_2\\over 12}\\chi m_0^2 a^2 L^{-2\/27}\n-{c_1-2c_2\\over 12}\\chi_{\\rm m}m_0^2 a^2L^{-2\/27}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.8cm}=\n\\biggl[2H_p\\,\\widetilde{\\lambda}_p^2 M_p-\n\\Delta\\widetilde{\\lambda}_p^2 M^2\\biggr]e^{-M_p^2\/M^2}\n+{c_3\\over 16}(s_0^q)^2\\Delta s_0^q\nM^2 L^{-8\/9} e^{-s^q_0\/M^2}\\; ,\n\\label{sum_p_q}\n\\end{eqnarray}\nwhere $a\\equiv -4\\pi^2\\langle\\overline{q}q\\rangle_{\\mbox{\\tiny{\\rm 0}}}$,\n$\\widetilde{\\lambda}_p^2\\equiv 32\\pi^4\\lambda_p^2$,\n$\\Delta\\widetilde{\\lambda}_p^2\\equiv 32\\pi^4\\Delta\\lambda_p^2$,\nand $m_0^2\\equiv\n\\langle g_s\\overline{q}\\sigma\\cdot\n{\\cal G} q\\rangle_{\\mbox{\\tiny{\\rm 0}}}\/\n\\langle\\overline{q}q\\rangle_{\\mbox{\\tiny{\\rm 0}}}$.\nHere we have\nignored the isospin breaking in\nthe vacuum condensates (i.e.,\n$\\langle\\overline{u}\\hat{O}u\\rangle_{\\mbox{\\tiny{\\rm 0}}}\n\\simeq\\langle\\overline{d}\\hat{O}d\\rangle_{\\mbox{\\tiny{\\rm 0}}}\n=\\langle\\overline{q}\\hat{O}q\\rangle_{\\mbox{\\tiny{\\rm 0}}}$);\nthe inclusion of the isospin breaking in vacuum condensates\nonly gives small refinements of the results.\nWe have also defined\n\\begin{eqnarray}\n& &E_0\\equiv 1-e^{-s^i_0\/M^2}\\ ,\n\\nonumber\n\\\\*[7.2pt]\n& &E_1\\equiv 1-e^{-s^i_0\/M^2}\\left[{s^i_0\\over\nM^2}+1\\right]\\ ,\n\\nonumber\n\\\\*[7.2pt]\n& &E_2\\equiv 1-e^{-s^i_0\/M^2}\\left[{(s^i_0)^2\\over 2M^4}\n+{s^i_0\\over M^2}+1\\right]\\ ,\n\\label{conform}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n& &c_1=(1-t)^2\\ ,\\hspace{0.8cm}c_2=1-t^2\\ ,\\hspace{0.8cm}c_3=5t^2+2t+5\\ ,\n\\nonumber\n\\\\*[7.2pt]\n& &c_4=t^2+10t+1\\ ,\\hspace{0.8cm}c_5=t^2+4t+7\\ .\n\\label{c-def}\n\\end{eqnarray}\nThe anomalous dimensions\nof the various operators have been taken into\naccount through the factor\n$L\\equiv\\ln(M^2\/\\Lambda_{\\rm QCD}^2)\/\\ln(\\mu^2\/\\Lambda_{\\rm\nQCD}^2)$\\cite{svz1,ioffe3}. We\ntake the renormalization scale $\\mu$ and the QCD scale parameter\n$\\Lambda_{\\rm QCD}$ to be $500\\,\\text{MeV}$\n and $150\\,\\text{MeV}$\\cite{ioffe3}.\n\nThe sum rules in the $\\Sigma^+$ case are given by\n\\begin{eqnarray}\n& &3c_2M^8 E_2L^{-8\/9}\n-3c_2\\chi a M^6 E_1\n+{c_1\\over 2}\\chi_{\\rm s}faM^6 E_1\n+(c_1-2c_3)m_s a M^4 E_0 L^{-8\/9}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n-3c_2m_s f a M^4 E_0 L^{-8\/9}\n+{3c_2\\over 2}\\chi_{\\rm m}m_0^2 a M^4 E_0 L^{-14\/27}\n-{c_2\\over 4}m_s f_s m_0^2 a M^2L^{-38\/27}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n-{2c_1+3c_2-6c_3\\over 12}m_s m_0^2 a M^2 L^{-38\/27}\n+{c_1-2c_3\\over 3} f a^2 M^2\n+{2c_3\\over 3}\\chi m_s a^2 M^2\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n+c_2\\chi m_s f a^2 M^2\n+c_2\\chi_{\\rm s}m_s f a^2 M^2\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{1.0cm}\n=\\biggl[2H_{\\Sigma^+}\n\\,\\widetilde{\\lambda}_{\\Sigma^+}^2 M_{\\Sigma^+}^2\n-\\Delta\\widetilde{\\lambda}_{\\Sigma^+}^2\\, M_{\\Sigma^+} M^2\n-H_{\\Sigma^+}\\,\\widetilde{\\lambda}_{\\Sigma^+}^2 M^2\n\\biggr]e^{- M_{\\Sigma^+}^2\/M^2}\n+\\biggl[{c_1\\over 4}m_s (s^1_0)^2L^{-4\/3}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{1.5cm}\n+{c_1\\over 2} f a s^1_0 L^{-4\/9}\n-3c_2a s^1_0 L^{-4\/9}\n+{3c_2\\over 2} m_0^2 a L^{-26\/27}\\biggr]\\Delta s_0^1\nM^2 e^{-s^1_0\/M^2}\\; ,\n\\label{sum_s_1}\n\\\\*[14.4pt]\n& &3c_2m_sM^6 E_1L^{-4\/3}\n-{2c_1-c_3\\over 2}a M^4 E_0L^{-4\/9}\n+3c_2 f a M^4 E_0 L^{-4\/9}\n-3c_2\\chi m_s a M^4E_0L^{-4\/9}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n-{c_3\\over 4}\\chi_{\\rm s} m_s f a M^4 L^{-4\/9}\n-{c_2\\over 12}m_0^2 a M^2 L^{-26\/27}\n-{5c_2\\over 4}f_s m_0^2 a M^2 L^{-26\/27}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n+{7c_2\\over 4}\\chi_{\\rm m}m_s m_0^2 a M^2 L^{-26\/27}\n-{c_5\\over 12}\\chi_{\\rm ms} m_s f_s m_0^2 a M^2\nL^{-2\/27}\n+{2c_1\\over 3}\\chi a^2 M^2 L^{4\/9}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n-2c_2\\chi f a^2 M^2 L^{4\/9}\n-2c_2\\chi_{\\rm s} f a^2 M^2 L^{4\/9}\n-c_2m_s a^2 L^{-4\/9}\n-{c_3-2c_1\\over 6} m_s f a^2 L^{-4\/9}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n-{c_1\\over 12}\\chi m_0^2 a^2 L^{-2\/27}\n+{5c_2\\over 12}\\chi f_s m_0^2 a^2 L^{-2\/27}\n+{7c_2\\over 12}\\chi_{\\rm s}f m_0^2 a^2 L^{-2\/27}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n-{c_1\\over 12}\\chi_{\\rm m} m_0^2 a^2 L^{-2\/27}\n+{5c_2\\over 12}\\chi_{\\rm ms} f_s m_0^2 a^2 L^{-2\/27}\n+{7c_2\\over 12}\\chi_{\\rm m} f m_0^2 a^2 L^{-2\/27}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{1.0cm}\n=\n\\biggl[2H_{\\Sigma^+}\\,\\widetilde{\\lambda}_{\\Sigma^+}^2 M_{\\Sigma^+}-\n\\Delta\\widetilde{\\lambda}_{\\Sigma^+}^2 M^2\\biggr]e^{-M_{\\Sigma^+}^2\/M^2}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{1.5cm}\n+\\biggl[{c_3\\over 16}(s^q_0)^2-3c_2m_s a\n-{c_3\\over 4}m_s f a\\biggr]\\Delta s_0^q\nM^2 L^{-8\/9} e^{-s^q_0\/M^2}\\; ,\n\\label{sum_s_q}\n\\end{eqnarray}\nwhere $f\\equiv \\langle\\overline{s}s\\rangle_{\\mbox{\\tiny{\\rm 0}}}\n\/\\langle\\overline{q}q\\rangle_{\\mbox{\\tiny{\\rm 0}}}$ and\n$f_s\\equiv\n\\langle g_s\\overline{s}\\sigma\\cdot\n{\\cal G} s\\rangle_{\\mbox{\\tiny{\\rm 0}}}\/\n\\langle g_s\\overline{q}\\sigma\\cdot\n{\\cal G} q\\rangle_{\\mbox{\\tiny{\\rm 0}}}$.\n The sum rules in the $\\Xi^0$ case are\n\\begin{eqnarray}\n& &-{c_1\\over 2}M^8 E_2 L^{-8\/9}\n+{c_1\\over 2}\\chi a M^6 E_1\n-3c_2\\chi_{\\rm s}f a M^6 E_1\n-3c_2m_s a M^4 E_0 L^{-8\/9}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n+(c_1-2c_3)m_s f a M^4 E_0 L^{-8\/9}\n+{3c_1\\over 2}\\chi_{\\rm ms}\nf_s m_0^2 a M^4 E_0 L^{-14\/27}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n-{c_2\\over 4}m_s m_0^2 a M^2 L^{-38\/27}\n-{2c_1+3c_2-6c_3\\over 12}m_s f_s m_0^2 a M^2 L^{-38\/27}\n-{c_3\\over 3}f^2 a^2 M^2\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n-c_2f a^2 M^2\n-(c_1-2c_3)\\chi m_s f a^2 M^2\n-(c_1-2c_3)\\chi_{\\rm s}m_s f a^2 M^2\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{1.0cm}\n=\\biggl[2H_{\\Xi^0}\\,\\widetilde{\\lambda}_{\\Xi^0}^2\nM_{\\Xi^0}^2\n-\\Delta\\widetilde{\\lambda}_{\\Xi^0}^2\\, M_{\\Xi^0} M^2\n-H_{\\Xi^0}\\,\\widetilde{\\lambda}_{\\Xi^0}^2 M^2\\biggr]\ne^{-M_{\\Xi^0}^2\/M^2}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{2.0cm}\n+\n\\biggl[-{3c_2\\over 2}m_s (s^1_0)^2 L^{-4\/3}\n+{c_1\\over 2}s^1_0 a L^{-4\/9}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{1.5cm}\n-3c_2s_0 f a L^{-4\/9}\n+{3c_2\\over 2}f_s m_0^2 a L^{-26\/27}\\biggr]\\Delta s_0^1\ne^{-s^1_0\/M^2}\\; ,\n\\label{sum_x_1}\n\\\\*[14.4pt]\n& &3c_2m_s M^6 E_1 L^{-4\/3}\n+{c_3\\over 4} a M^4 E_0 L^{-4\/9}\n+3c_2f a M^4 E_0 L^{-4\/9}\n-3c_2\\chi m_s a M^4 E_0 L^{-4\/9}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n+{2c_1-c_3\\over 2}\\chi_{\\rm s}m_s f a M^4 E_0 L^{-4\/9}\n+{c_5\\over 12}m_0^2 a M^2 L^{-26\/27}\n-{7c_2\\over 4}f_s m_0^2 a M^2 L^{-26\/27}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n+{5c_2\\over 4}\\chi_{\\rm m}m_s m_0^2 a M^2 L^{-14\/27}\n+{c_4\\over 12}\\chi_{\\rm ms}m_s f_s m_0^2 M^2 L^{-26\/27}\n-2c_2\\chi f a^2 M^2 L^{4\/9}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n-2c_2\\chi_{\\rm s} f a^2 M^2 L^{4\/9}\n+{2c_1\\over 3}\\chi_{\\rm s} f^2 a^2 M^2 L^{4\/9}\n-c_2m_s f^2 a^2 L^{-4\/9}\n-{c_3-2c_1\\over 6}m_s f_s a^2 L^{-4\/9}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n+{7c_2\\over 12}\\chi f_s m_0^2 a^2 L^{-2\/27}\n-{c_1\\over 12}\\chi_{\\rm s}f f_s m_0^2 a^2L^{-2\/27}\n+{5c_2\\over 12}\\chi_{\\rm s}f m_0^2 a^2 L^{-2\/27}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{0.5cm}\n+{5c_2\\over 12}\\chi_{\\rm m}f m_0^2 a^2 L^{-2\/27}\n-{c_1\\over 12}\\chi_{\\rm ms}f f_s m_0^2 a^2 L^{-2\/27}\n+{7c_2\\over 12}\\chi_{\\rm ms}f_s m_0^2 a^2 L^{-2\/27}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{1.0cm}=\n\\biggl[2H_{\\Xi^0}\\,\\widetilde{\\lambda}_{\\Xi^0}^2 M_{\\Xi^0}-\n\\Delta\\widetilde{\\lambda}_{\\Xi^0}^2\\biggr]e^{-M_{\\Xi^0}^2\/M^2}\n\\nonumber\\\\*[7.2pt]\n& &\\hspace*{2.0cm}\n+\\biggl[\n{c_3\\over 16}(s^q_0)^2\n-3c_2m_s a\n-{c_3-2c_1\\over 2} m_s f a\\biggr]\\Delta s_0^q\n L^{-8\/9} M^2 e^{-s_0^q\/M^2}\\; .\n\\label{sum_x_q}\n\\end{eqnarray}\n{}.\n\n\n\\section{Sum-rule analysis}\n\\label{anay}\n\nWe now analyze the sum rules derived in the previous section\nand extract the baryon matrix elements of interest. Here\nwe follow Ref.~\\cite{jin2} and use only the\nsum rules Eqs.~(\\ref{sum_p_q}), (\\ref{sum_s_q}), and\n(\\ref{sum_x_q}), which are more stable than the other\nthree sum rules. The pattern that one of the sum rules\n(in each case) works well while the other does not\nhas been seen in various external field\nproblems\\cite{ioffe1,chiu1,chiu2,jin1,jin2}.\nThis may be attributed to the different asymptotic\nbehavior of various sum rules. As emphasized earlier,\nthe phenomenological side of the external field sum rules\ncontains single pole terms arising from the transition\nbetween the ground state and the excited states, whose\ncontribution is {\\it not} suppressed relative to the\ndouble pole term and thus contaminates\nthe double pole contribution. The degree of this\ncontamination may vary from one sum rule to another.\nThe sum rule with smaller single pole contribution\nworks better.\nWe refer the reader to Refs.~\\cite{chiu2,jin1,jin2}\nfor more discussion\nabout the different behavior of various external field\nsum rules.\nIn the analysis to follow, we disregard the sum\nrule Eqs.~(\\ref{sum_p_1}), (\\ref{sum_s_1}), and\n(\\ref{sum_x_1}), and consider only the results from\nthe sum rules Eqs.~(\\ref{sum_p_q}), (\\ref{sum_s_q}), and\n(\\ref{sum_x_q}).\n\n\n\nWe adopt the numerical optimization procedures used in\nRefs.~\\cite{leinweber2,furnstahl1}. The\nsum rules are sampled in the fiducial region of Borel $M^2$, where\nthe contributions from the high-dimensional condensates\nremain small and the continuum contribution is controllable.\nWe choose\n\\begin{eqnarray}\n& &\n0.8\\leq M^2\\leq 1.4\\, {\\mbox{GeV}}^2\\hspace*{2cm}\n{\\mbox{for proton case}}\\ ,\n\\\\*[7.2pt]\n& &\n1.2\\leq M^2\\leq 1.8\\, {\\mbox{GeV}}^2\\hspace*{2cm}\n{\\mbox{for}}\\,\\Sigma^+\\,{\\mbox{and}}\\,\\Xi^0 {\\mbox{case}}\\ ,\n\\end{eqnarray}\nwhich have been identified as the fiducial region for the baryon\nmass sum rules\\cite{ioffe1,ioffe5}. Here we adopt these boundaries as\nthe maximal limits of applicability of the external field sum\nrules. The sum-rule predictions are obtained by\nminimizing the logarithmic measure\n$\\delta (M^2)={\\mbox{ln}}[{\\mbox{maximum}}\\{{\\mbox{LHS,RHS}}\\}\/\n{\\mbox{minimum}}\\{{\\mbox{LHS,RHS}}\\}]$ averaged over $150$ points\nevenly spaced within the fiducial region of $M^2$, where\nLHS and RHS denote the left- and right-hand sides of\nthe sum rules, respectively.\n\nNote that the {\\it vacuum} spectral parameters $\\lambda_{\\rm B}^2$,\n$M_{\\rm B}$ and $s_0^i$, also appear in the external field sum rules\n Eqs.~(\\ref{sum_p_1}--\\ref{sum_p_q}) and\n(\\ref{sum_s_1}--\\ref{sum_x_q}).\nHere we use the experimental values for\nthe baryon masses and extract $\\lambda_{\\rm B}^2$\nand $s_0^i$ from baryon mass sum rules using the same\noptimization procedure as described above.\nWe then extract $H_{\\rm B}$, $\\Delta\\lambda_{\\rm B}^2$, and\n$\\Delta s_0^i$ from the external field sum rules.\n\nFor vacuum condensates, we use $a=0.55\\, {\\mbox{GeV}}^3\\, (m_u\n+m_d\\simeq 11.8{\\mbox{MeV}})$\\cite{ioffe1,ioffe3},\n$m_0^2=0.8\\, {\\mbox{GeV}}^2$\\cite{ioffe1,belyaev3},\nand $f\\simeq f_s=0.8$\\cite{belyaev3,leinweber2}.\nWe take the strange quark mass\n$m_s$ to be $150\\, {\\mbox{MeV}}$\\cite{ioffe5}. The\nparameter $\\chi$ has been estimated in Ref.~\\cite{jin2}.\nThe estimate in chiral perturbation theory gives\n$\\chi\\simeq 2.2\\, {\\mbox{GeV}}^{-1}$. It is also shown that\nto the lowest order in $\\delta m$, $\\chi$ is determined by\n\\begin{equation}\n\\chi\\delta m=-\\gamma+O[(\\delta m)^2]\\ ,\n\\label{chi-est}\n\\end{equation}\nwhere $\\gamma\\equiv \\langle\\overline{d}d\n\\rangle_{\\mbox{\\tiny{\\rm 0}}}\/\\langle\\overline{u}u\n\\rangle_{\\mbox{\\tiny{\\rm 0}}}-1$, and $\\delta m$\nhas been determined by Gasser and Leutwyler,\n$\\delta m \/(m_u + m_d ) = 0.28 \\pm 0.03$\\cite{gasser1}.\nThe value of $\\gamma$ has been estimated previously in various\napproaches\\cite{gasser2,paver1,pascual1,bagan1,dominguez1,%\ndominguez2,narison1,adami2,adami1,eletsky1}\n with results ranging from $-1\\times 10^{-2}$\nto $-2\\times 10^{-3}$, which upon using Eq.~(\\ref{chi-est})\nand a median value for $\\delta m=3.3\\,\\text{MeV}$,\ncorresponds to\n\\begin{equation}\n0.5\\,\\text{GeV}^{-1}\\leq\\chi\\leq 3.0 \\,\\text{GeV}^{-1}\\ .\n\\label{chi-range}\n\\end{equation}\nWe shall consider this range of $\\chi$ values.\nWe follow Ref.~\\cite{jin2} and assume\n$\\chi_{\\rm m}\\simeq \\chi$,\nwhich is equivalent to the assumption that $m_0^2$\nis isospin independent.\n\nThe parameter $\\chi_{\\rm s}$ measures the response of\nthe strange quark condensate to the external field, which\nhas not been estimated previously.\nSince $\\overline{s}s$ is an isospin scalar operator,\n$\\chi_{\\rm s}$ arises from the isospin mixing and\nwe expect $\\chi_{\\rm s}<\\chi$.\nFollowing Ref.~\\cite{jin2},\none may express $\\chi_{\\rm s}$ in terms of a\ncorrelation function\nand estimate it in chiral perturbation theory. It is\neasy to show that $\\chi_{\\rm s}\\langle\\overline{s}s\n\\rangle_{\\mbox{\\tiny{\\rm 0}}}={d\\over d\\delta m}\n\\langle\\overline{s}s\n\\rangle_{\\mbox{\\tiny{\\rm 0}}}$. So, one may determine\n$\\chi_{\\rm s}$ by evaluating ${d\\over d\\delta m}\n\\langle\\overline{s}s\n\\rangle_{\\mbox{\\tiny{\\rm 0}}}$ in effective QCD models.\nHere we shall treat $\\chi_{\\rm s}$ as a free parameter\nand consider the values of $\\chi_{\\rm s}$ in the\nrange of $0\\leq\\chi_{\\rm s}\\leq 3.0\\,\\text{GeV}^{-1}$.\nWe also assume that $\\chi_{\\rm ms}\\simeq \\chi_{\\rm s}$.\n\nWe first analyze the sum rules for Ioffe's\ninterpolating field (i.e., $t=-1$). We start from\nthe proton case. The optimized result for $H_p$\nas function of $\\chi$ is plotted\nin Fig.~\\ref{fig-1}. One can see that\n$H_p$ varies rapidly with $\\chi$. Therefore,\nthe sum-rule prediction for the proton matrix element\n$H_p$ depends strongly on the response of the up and down\nquark condensates to the external source.\n(The sum rules in the proton case\nare independent of $\\chi_{\\rm s}$ and $\\chi_{\\rm sm}$.)\nFor moderate values of $\\chi$ ($1.5\\,\\text{GeV}^{-1}\n\\leq\\chi\\leq 2.0\\,\\text{GeV}^{-1}$), the predictions\nare\n\\begin{equation}\nH_p\\simeq 0.54-0.78\\ .\n\\label{typ-p}\n\\end{equation}\nOn the other hand, for large values of\n$\\chi$ ($2.4\\,\\text{GeV}^{-1}\\leq\\chi\n\\leq 3.0\\,\\text{GeV}^{-1}$), we find $H_p\\simeq\n0.97-1.25$. For small values of $\\chi$\n($\\chi\\leq 1.4\\,\\text{GeV}^{-1}$),\nthe continuum contribution is larger than $50\\%$,\nimplying that the continuum contribution is dominant\nin the Borel region of interest and the prediction\nis not reliable.\n The predictions for $\\Delta\n\\lambda_p^2$ and $\\Delta s_0^q$ also change with $\\chi$\n in the same way as $H_p$.\n\nTo see how well the sum rule works, we plot the LHS, RHS,\nand the individual terms of RHS of Eq.~(\\ref{sum_p_q}) as functions\nof $M^2$ with $\\chi=1.8\\,\\text{GeV}^{-1}$ in Fig.~\\ref{fig-2}\nusing the optimized values for $H_p$, $\\Delta\\lambda_p^2$,\nand $\\Delta s_0^q$. We see that the solid (LHS) and long-dashed (RHS)\ncurves are right on top of each other, showing a very good\noverlap. We also note from Fig.~\\ref{fig-2}\nthat the first term of RHS (curve 1) is larger than\nthe second (curve 2) and third (curve 3) terms. This shows that\nthe double pole contribution is stronger than the single\npole contribution and the predictions are thus stable.\n(Although the second and third terms are sizable\nindividually, their sum is small.)\n\n\nIn Fig.~\\ref{fig-3}, we have displayed the predicted\n$H_{\\Sigma^+}$ as function of $\\chi$ for three\ndifferent values of $\\chi_{\\rm s}$. One notices that\n$H_{\\Sigma^+}$ is largely insensitive to $\\chi_{\\rm s}$,\nbut strongly dependent on $\\chi$ value. For $\\chi$\nvalues in the range of $2.2\\,\\text{GeV}^{-1}\\leq\n\\chi\\leq 3.0\\,\\text{GeV}^{-1}$, we find\n\\begin{equation}\nH_{\\Sigma^+}\\simeq 1.65-2.48\\ .\n\\label{typ-s}\n\\end{equation}\nFor smaller $\\chi$, we obtain smaller values for $H_{\\Sigma^+}$.\nThe predictions for $\\Delta\\lambda_{\\Sigma^+}^2$\nand $\\Delta s_0^q$ change in a similar pattern. The sum rule\nworks very well and the continuum contribution is small for\nall $\\chi$ and $\\chi_{\\rm s}$ values considered here.\n\nThe optimized $H_{\\Xi^0}$ as function of $\\chi_{\\rm s}$\nis shown in Fig.~\\ref{fig-4}. [When $t=-1$, the sum rule\nEq.~(\\ref{sum_x_q}) is independent of $\\chi$\nand $\\chi_{\\rm s}$.] We see that\nthe result is very sensitive to the $\\chi_{\\rm s}$ value.\nThus the prediction for $H_{\\Xi^0}$ has a strong dependence\non the response of the strange quark condensate to the\nexternal field.\nFor moderate $\\chi_{\\rm s}$ ($1.7\\,\\text{GeV}^{-1}\\leq\n2.2\\,\\text{GeV}^{-1}$), we get\n\\begin{equation}\nH_{\\Xi^0}\\simeq 1.57-1.84\\ .\n\\label{typ-x}\n\\end{equation}\nFor larger (smaller) values of $\\chi_{\\rm s}$, we find\nlarger (smaller) values for $H_{\\Xi^0}$.\nAt $\\chi_{\\rm s}=0$, we get\n$H_{\\Xi^0}\\simeq 0.68$. The results for\n$\\Delta\\lambda^2_{\\Xi^0}$ and $\\Delta s_0^q$ increase\n(decrease) as $\\chi_{\\rm s}$ increases (decreases).\n\nAll of the results above use Ioffe's interpolating field\n(i.e., $t=-1$); we now present the results for general\ninterpolating field. In Fig.~\\ref{fig-5}, we have plotted\nthe predicted $H_p$, $H_{\\Sigma^+}$, and $H_{\\Xi^0}$ as\nfunctions of $t$ for $\\chi=2.5\\,\\text{GeV}^{-1}$ and\n$\\chi_{\\rm s}=1.5\\,\\text{GeV}^{-1}$. As $t$ increases,\n$H_p$, $H_{\\Sigma^+}$, and $H_{\\Xi^0}$ all increase; the\nrate of increase is essentially the same for $H_p$\nand $H_{\\Sigma^+}$, but somewhat smaller for $H_{\\Xi^0}$.\nWe note that the {\\it vacuum} spectral parameters $\\lambda^2_{\\rm B}$\nand $s^q_0$ decrease as $t$ increases; this leads to\na large variation of $H_p$, $H_{\\Sigma^+}$, and $H_{\\Xi^0}$\nwith $t$.\n\nThe sensitivity of our results to the assumption of\n$\\chi_{\\rm m}=\\chi$ is displayed in Fig.~\\ref{fig-6},\nwhere $t$ and $\\chi_{\\rm s} (=\\chi_{\\rm ms})$ are\nfixed at $-1$ and $1.5\\,\\text{GeV}^{-1}$, respectively.\nThe three curves are obtained by using $\\chi_{\\rm m}\n=\\chi$, ${1\\over 2}\\chi$, and ${3\\over 2}\\chi$, respectively.\nWe note that $H_p$ and $H_{\\Sigma^+}$\nget larger (smaller) as $\\chi_{\\rm m}$ becomes\nsmaller (larger). The results are more sensitive\nto $\\chi_{\\rm m}$ in the proton case than in the $\\Sigma^+$\ncase. The prediction for $H_p$ changes by about $25\\%$\nwhile the prediction for $H_{\\Sigma^+}$ changes\nby about $15\\%$ when the $\\chi_{\\rm m}$ value\nis changed by $50\\%$. This implies that the terms\nproportional to $\\chi_{\\rm m}$ in the sum rules\ngive rise to sizable contributions. The sensitivity\nof our predictions to the assumption of\n$\\chi_{\\rm sm}=\\chi_{\\rm s}$ is illustrated in\nFig.~\\ref{fig-7}, with $t=-1$ and $\\chi=\\chi_{\\rm m}\n=2.5\\,\\text{GeV}^{-1}$. The three curves correspond\nto $\\chi_{\\rm ms}=\\chi_{\\rm s}$, ${1\\over 2}\\chi_{\\rm s}$,\nand ${3\\over 2}\\chi_{\\rm s}$, respectively. One can see that both\n$H_{\\Sigma^+}$ and $H_{\\Xi^0}$ are insensitive to\nchanges in $\\chi_{\\rm ms}$. This indicates that\nthe terms proportional to $\\chi_{\\rm ms}$ give\nonly small contributions to the sum rules. One\nalso notices that $H_{\\Sigma^+}$ depends only\nweakly on $\\chi_{\\rm s}$. Finally, the effect of\nignoring the response of continuum threshold is\nshown in Fig.~\\ref{fig-8}. The solid (dashed) curve\nis obtained by including (omitting) the third term\non the RHS of Eq.~(\\ref{sum_p_q}). The difference\nbetween the two curves is large for moderate and\nlarge values of $\\chi$. This shows that the response\nof the continuum threshold can be sizable and\nshould be included in the sum rules. Unfortunately,\nthe response\nof the continuum thresholds has been omitted\nin all previous works on external field sum rules.\nThis was first noticed by Ioffe\\cite{ioffe2}.\n\n\n\n\\section{Estimate of baryon isospin mass splittings}\n\\label{isospin}\n\nIn this section we estimate the baryon isospin mass splittings\nusing $\\delta m$ and the baryon matrix elements of\nisovector-scalar current calculated in the previous section.\n\n\nThe observed hadron isospin mass splittings arise from electromagnetic\ninteraction and from the difference between up and down quark masses:\n\\begin{equation}\n\\delta m_h=(\\delta m_h)_{\\rm el}+(\\delta m_h)_{\\rm q}\\ ,\n\\label{dm-sep}\n\\end{equation}\nwhere $(\\delta m_h)_{\\rm el}$ and $(\\delta m_h)_{\\rm q}$ denote the\ncontributions due to electromagnetic interaction and due to\nthe up and down quark mass difference, respectively.\\footnote%\n{This separation is renormalization scale dependent. However,\nthis scale dependence is weak; it is thus meaningful to\nseparate the contribution of quark mass difference\nfrom that due to electromagnetic interaction (see Ref.~\\cite{jin2}).}\nFollowing Ref.~\\cite{jin2}, one can treat $\\delta m$\nas a small parameter and using the Hellman-Feynman\ntheorem~\\cite{hellman1,feynman1} to show that\nthe octet baryon\nisospin mass splittings to first order in $\\delta m$\ncan be expressed as\n\\begin{eqnarray}\n& &M_n-M_p=(M_n-M_p)_{\\rm el}+\\delta m H_p\\ ,\n\\label{mq-np}\n\\\\*[7.2pt]\n& &M_{\\Sigma^-}-M_{\\Sigma^+}=\n(M_{\\Sigma^-}-M_{\\Sigma^+})_{\\rm el}+\\delta m H_{\\Sigma^+}\\ ,\n\\label{mq-sig}\n\\\\*[7.2pt]\n& &M_{\\Xi^-}-M_{\\Xi^0}=(M_{\\Xi^-}-M_{\\Xi^0})_{\\rm el}\n+\\delta m H_{\\Xi^0}\\ .\n\\label{mq-xi}\n\\end{eqnarray}\nNote that $H_n=-H_p$, $H_{\\Sigma^-}=-H_{\\Sigma^+}$,\nand $H_{\\Xi^-}=-H_{\\Xi^0}$ to the lowest order\nin $\\delta m$.\nTherefore, QCD sum rule predictions for $H_p$, $H_{\\Sigma^+}$,\nand $H_{\\Xi^0}$,\nalong with the electromagnetic contributions\\cite{gasser1}\n\\begin{eqnarray}\n& &(M_n-M_p)_{\\rm el}= -0.76\\pm 0.30\\,{\\mbox{MeV}}\\ ,\n\\label{mel-np}\n\\\\*[7.2pt]\n& &(M_{\\Sigma^-}-M_{\\Sigma^+})_{\\rm el}= 0.17\\pm0.3\\,{\\mbox{MeV}}\\ ,\n\\label{mel-sig}\n\\\\*[7.2pt]\n& &(M_{\\Xi^-}-M_{\\Xi^0})_{\\rm el}=0.86\\pm 0.30\\,{\\mbox{MeV}}\\ ,\n\\label{mel-xi}\n\\end{eqnarray}\nwill lead to an estimate of the baryon isospin mass splittings.\nTaking the experimental mass difference\\cite{particle1}, one finds\n\\begin{eqnarray}\n& &(M_n-M_p)_{\\rm q}^{\\rm exp}=2.05\\pm 0.30\\,\\text{MeV}\\ ,\n\\\\*[7.2pt]\n& &(M_{\\Sigma^-}-M_{\\Sigma^+})_{\\rm q}^{\\rm exp}=7.9\\pm 0.33\\,\\text{MeV}\\ ,\n\\\\*[7.2pt]\n& &(M_{\\Xi^-}-M_{\\Xi^0})_{\\rm q}^{\\rm exp}=5.54\\pm 0.67\\,\\text{MeV}\\ .\n\\label{mass-dif-exp}\n\\end{eqnarray}\n\nWe have seen from last section that the uncertainties in our\nknowledge of the response of the quark condensates to the\nexternal field, $\\chi$ and $\\chi_{\\rm s}$, leads to uncertainties\nin the sum-rule determination of the baryon matrix elements\n$H_{\\rm B}$. (There are also uncertainties in $\\delta m$.)\nTherefore, our estimate here are only {\\it qualitative}.\nFor most of the values for $t$, $\\chi$ and $\\chi_{\\rm s}$\nconsidered here, the sum-rule analysis gives\n$0 < H_p < H_{\\Xi^0}\\leq H_{\\Sigma^+}$ (see Figs.~\\ref{fig-5}, \\ref{fig-6}\nand \\ref{fig-7}), which implies\n\\begin{equation}\n0<(M_n-M_p)_{\\rm q}<(M_{\\Xi^-}-M_{\\Xi^0})_{\\rm q}\\leq\n(M_{\\Sigma^-}-M_{\\Sigma^+})_{\\rm q}\\ .\n\\label{quli}\n\\end{equation}\nThis qualitative feature is compatible with the experimental data.\nFor the baryon interpolating fields with $t=-1$\nand moderate $\\chi$ and $\\chi_{\\rm s}$ values\n($1.6\\,\\text{GeV}^{-1}\\leq\\chi\\leq 2.2\\,\\text{GeV}^{-1}$\nand $1.3\\,\\text{GeV}^{-1}\\leq\\chi_{\\rm s}\\leq 1.8\\,\\text{GeV}^{-1}$),\nwe get\n\\begin{eqnarray}\n& &1.95\\,\\text{MeV}\\leq (M_n-M_p)_{\\rm q}\\leq 2.41\\,\\text{MeV}\\ ,\n\\\\*[7.2pt]\n& &4.0\\,\\text{MeV}\\leq (M_{\\Sigma^-}-M_{\\Sigma^+})_{\\rm q}\\leq 6.3\\,\\text{MeV}\\\n,\n\\\\*[7.2pt]\n& &4.5\\,\\text{MeV}\\leq (M_{\\Xi^-}-M_{\\Xi^0})_{\\rm q}\\leq 5.38 \\,\\text{MeV}\\ ,\n\\label{mass-est}\n\\end{eqnarray}\nwhere we have used a median value\n$\\delta m\\simeq 3.3\\,\\text{MeV}$. These results are\ncomparable to the experimental data, though the\nresult in the $\\Sigma$ case is somewhat too small.\nSmaller and larger values of $\\chi$ and $\\chi_{\\rm s}$\nlead to correspondingly smaller and larger values for\nthe baryon isospin mass differences. As $t$ increases\n(decreases), the results increase (decrease).\n\n\\section{Discussion}\n\\label{discussion}\n\nOur primary goal in the present paper has been to extract the\nbaryon matrix element $H_{\\rm B}=\\langle B|\\overline{u}u-\n\\overline{d}d|B\\rangle\/2M_{\\rm B}$ for octet baryons. We observe that\nthe sum-rule predictions for $H_{\\rm B}$ are quite sensitive\nto the response of quark condensates to the external\nisovector-scalar field, which is not well determined.\nThis means that our conclusion about $H_{\\rm B}$ can only be\n{\\it qualitative} at this point. The most concrete conclusion\nwe can draw from this work is that QCD sum rules\npredict positive values for $H_p$, $H_{\\Sigma^+}$,\nand $H_{\\Xi^0}$ and $H_p