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+{"text":"\\section*{Supplementary Material: \\\\ \nFloquet-Liouvillian spectrum of decay modes.} \n\nLet us now briefly show how to construct the complete spectrum of decay modes of our quasi-free Floquet-Liouvillian theory. Following Refs.~\\cite{original,3QRedfield,spectral} we define $4n$ Hermitian Majorana maps $\\un{\\hat a}=(\\un{\\hat a}_1,\\un{\\hat a}_2)$, as $\\hat{a}_{1,j} = (\\hat{c}_j +\\hat{c}^\\dagger_j)\/\\sqrt{2}$,  \n$\\hat{a}_{2,j} =  {\\rm i} (\\hat{c}_j -\\hat{c}^\\dagger_j)\/\\sqrt{2}$, where canonical fermionic maps are uniquely defined by their action\n$\\hat{c}_j \\ket{P_{\\un{\\alpha}}} = \\delta_{\\alpha_j,0} \\ket{w_j P_{\\un{\\alpha}}}$,\n$\\hat{c}^\\dagger_j \\ket{P_{\\un{\\alpha}}} = \\delta_{\\alpha_j,1} \\ket{w_j P_{\\un{\\alpha}}}$  over the complete Fock basis $P_{\\un{\\alpha}} = 2^{-n\/2}w_1^{\\alpha_1}\\cdots w_{2n}^{\\alpha_{2n}}$ of the\noperator space. Referring to Sect. 3.5 of \\cite{3QRedfield} we find the Liouvillian propagator \n\\begin{equation}\n\\hat{\\cal U}(t,0) = N(t)\\exp\\left({\\textstyle \\frac{1}{2}}\\un{\\hat a}\\cdot\\left[\\ln\\mm{U}(t)\\right]\\un{\\hat a}\\right),\n\\label{eq:LFmap}\n\\end{equation}\nwhere, after a tedious but direct calculation, one finds \n\\begin{equation}\n\\mm{U}(t) = \\mm{S}^\\dagger \n\\begin{pmatrix}\n\\mm{Q}(t) &  {\\rm i}  \\mm{P}(t) \\cr\n\\mm{0} & \\mm{Q}^{-T}(t) \n\\end{pmatrix} \\mm{S},\\;\\,\n\\mm{S} \\equiv \\frac{1}{\\sqrt{2}}\n\\begin{pmatrix}\n\\mm{1} & - {\\rm i} \\mm{1}\\cr \\mm{1} &  {\\rm i} \\mm{1}\n\\end{pmatrix} ,\n\\end{equation}\nwith $\\mm{P}(t),\\mm{Q}(t)$ given by (\\ref{eq:PQ}) and $N(t)=[\\det\\mm{Q}(t)]^{1\/2}$.\nWrite shortly $\\mm{Q}\\equiv \\mm{Q}(\\tau)$, $\\mm{P}\\equiv\\mm{P}(\\tau)$, $\\mm{U}=\\mm{U}(\\tau)$, and\nassume that $\\mm{Q}$ can be diagonalized  \\cite{note},\n$\\mm{Q} = \\mm{R}\\mm{D}\\mm{R}^{-1}$, with\n$\\mm{D}={\\rm diag}\\{ \\lambda_1,\\ldots,\\lambda_{2n}\\}$.\nThen, $\\mm{U}$ is diagonalized as\n\\begin{equation}\n\\mm{U} = \\mm{V}^{-1}(\\mm{D}\\oplus\\mm{D}^{-1})\\mm{V},\\;\n\\mm{V} = \\begin{pmatrix}\n\\mm{R}^{-1} & \\mm{R}^{-1}\\mm{C}_{\\rm F}\\cr\n\\mm{0} & \\mm{R}^{T}\n\\end{pmatrix} \\mm{S},\n\\end{equation}\nwhere $\\mm{V}^{-1} = \\mm{V}^T \\mm{J}$, $\\mm{J}\\equiv \\sigma^{\\rm x}\\otimes \\mm{1}_{2n}$.\nNow, in exact analogy with the time-independent case, we define normal master mode maps\n$(\\un{\\hat b},\\un{\\hat b}') = \\mm{V}\\un{\\hat a}$, or \n\\begin{equation}\n\\un{\\hat b}=\\mm{R}^{-1}(\\un{\\hat c} + \\mm{C}_{\\rm F} \\un{\\hat c}^\\dagger),\\quad\n\\un{\\hat b}'=\\mm{R}^T \\un{\\hat c}^\\dagger ,\n\\end{equation}\nsatisfying CAR, $\\{\\hat{b}_j,\\hat{b}_k\\}=\\{\\hat{b}'_j,\\hat{b}'_k\\}=0$, $\\{\\hat{b}_j,\\hat{b}'_k\\}=\\delta_{j,k}$\nwhich diagonalize the many-body Liouville-Floquet map\n\\begin{equation}\n\\hat{\\cal U}(\\tau,0) = \\exp\\left(\\sum_{j=1}^{2n} (\\ln \\lambda_j) \\hat{b}'_j \\hat{b}_j\\right).\n\\end{equation}\nAs dynamics should be stable we have $\\forall j, |\\lambda_j| \\le 1$.\nThe complete set of $4^n$ Floquet modes, ${\\hat{\\cal U}}(\\tau,0)\\ket{\\un{\\nu}} = \\Lambda_{\\un{\\nu}}\\ket{\\un{\\nu}}$,  is then\n$\\ket{\\un{\\nu}} = \\prod_{j=1}^{2n} (\\hat{b}'_j)^{\\nu_j} \\ket{\\rho_{\\rm F}}$, $\\nu_j\\in\\{0,1\\}$, with decay rates\n$\\Lambda_{\\un{\\nu}} = \\prod_j \\lambda_j^{\\nu_j}.$ Note that the {\\em largest modulus eigenvalue} $\\lambda_1$ of $\\mm{Q}$ \ndetermines the {\\em spectral gap} of ${\\hat{\\cal U}}(\\tau,0)$, or the relaxation time to $\\rho_{\\rm F}$ as $t^* = \\tau\/(-\\ln|\\lambda_1|)$.\n\nWe note that the covariance flow (\\ref{eq:tdepC}) describes any, possibly non-Gaussian, time-dependent state. Nevertheless, SFS $\\ket{\\rho_{\\rm F}}$ can be thought of as the right vacuum state of a non-Hermitian quadratic periodically time-dependent field theory, determined by $\\hat{b}_j\\ket{\\rho_{\\rm F}}=0$, which implies that $\\ket{\\rho_{\\rm F}}$ {\\em is} a Gaussian state, and all higher-order observables can be expressed in terms of covariances $\\mm{C}_{\\rm F}$ via the Wick theorem.\n\n\n\n\\end{document}\n\n\n \n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction }\n\n\nSince Bekenstein \\cite{Beken} suggested that black holes carry an\nintrinsic entropy proportional to the surface area of the event\nhorizon, and Hawking \\cite{Haw} provided a physical basis for this\nidea by considering quantum effect, there have been various\napproaches to understanding the black hole entropy. One of them is\nthe so-called ``brick-wall model'' introduced by 't Hooft\n\\cite{Hoof}. He has considered a quantum gas of scalar particles\npropagating just outside the event horizon of the Schwarzschild\nblack hole. The entropy obtained just by applying the usual\nstatistical mechanical method to this system turns out to be\ndivergent due to the infinite blue shift of waves at the horizon.\n't Hooft, however, has shown that the leading order term on the\nentropy has the same form as the Bekenstein-Hawking formula for\nthe black hole entropy by introducing a brick-wall cutoff which is\na property of the horizon only and is the order of the Planck\nlength. The appearance of this divergence~\\cite{SUT,DLM} and\nrelationships of this ``statistical-mechanical\" entropy of quantum\nfields near a black hole with its entanglement entropy~\\cite{SE}\nand quantum excitations of the black hole~\\cite{BFZ} have been\nstudied, leading a great deal of interest recently~\\cite{FF}.\n\nThe brick-wall model originally applied to the four dimensional\nSchwarzschild black hole \\cite{Hoof} has been extended to various\nsituations.\nThe application to the case of rotating black holes has also been done\nfor scalar fields in BTZ black holes in three-dimensions~\\cite{KKPS,IS}\nand in Kerr-Newman and other rotating black holes in four-dimensions\n\\cite{LK,HKPS}. In a background spacetime of rotating black holes, it\nis well known that scalar fields have a special class of mode\nsolutions, giving superradiance. It is claimed in Ref.~\\cite{IS} that\nthe statistical-mechanical entropy of a scalar matter is not\nproportional to the ``area''({\\it i.e.}, the circumference in the\nthree-dimensional case) of the horizon of a rotating BTZ black\nhole and that the divergent parts are not necessarily due to the\nexistence of the horizon. Contrary to it, in Ref.~\\cite{KKPS}, the\nleading divergent term on the entropy is proportional to the ``area''\nof the horizon, and it is possible to introduce a universal brick-wall\ncutoff which makes the entropy equivalent to the black hole\nentropy. Moreover, it is claimed in Ref.~\\cite{KKPS} that the contribution from\nsuperradiant modes to entropy is {\\it negative} and its divergence is\nin a {\\it subleading} order compared to that from nonsuperradiant\nmodes. On the other hand, for the case of Kerr black holes in\nRef.~\\cite{HKPS}, the divergence is in the leading order but the\nentropy contribution is still negative.\n\nOne may expect that the leading contribution to the entropy comes from the region\nvery near the horizon as in the case of nonrotating black holes.\nSince the vicinity of a rotating horizon can also be approximated by\nthe Rindler metric, it is seemingly that the essential feature of the leading\ncontribution will be same as that in nonrotating cases.\nOur study in detail shows this naive expectation is indeed true. That is, we point out\nthat previous erroneous results appeared in the literature were\nmainly related to an incorrect quantization of superradiant modes.\nFor the case of rotating BTZ black holes, we have explicitly shown that\nsuperradiant modes also give leading order divergence to the entropy\nas nonsuperradiant ones. Moreover, its entropy contribution is\n{\\it positive} rather than negative found in Refs.~\\cite{KKPS,HKPS}.\nHowever, the total entropy of quantum field can still be identified\nwith the Bekenstein-Hawking entropy by\nintroducing a universal brick-wall cutoff. It also has been shown that\nthe correct quantization of superradiant modes in the calculation of the\n``angular-momentum modified canonical ensemble'' removes various\nunnecessary regulating cutoff numbers as well as ill-defined\nexpressions in the literature.\n\n\n\n\n\n\\vspace{0.5cm}\nLet us consider a quantum gas of scalar particles confined in a box near the\nhorizon of a stationary rotating black hole. The free scalar field satisfies the\nKlein-Gordon equation given by $(\\Box + \\mu^2)\\phi = 0$ with  periodic boundary conditions\n\\begin{equation}\n\\phi (r_++h) = \\phi (L).\n\\label{BC}\n\\end{equation}\nHere, $r_+$, $r_++h$, and $L$ are the radial coordinates of the horizon\nand the inner and outer walls of a ``spherical\" box, respectively.\nSuppose that this boson gas is in a thermal equilibrium state at\ntemperature $\\beta^{-1}$. Due to the existence of \nan ergoregion just outside the event horizon, any thermal\nsystem sitting in this region must rotate with respect to an observer\nat infinity. Accordingly, in order to obtain the appropriate grand\ncanonical ensemble for this rotating thermal system, one should\nintroduce an angular momentum reservoir as well in addition to a heat\nbath\/particle reservoir characterized by temperature $T = \\beta^{-1}$\nand angular speed $\\mbox{$\\Omega$}$ with respect to an observer at infinity\n\\cite{LPW}. All thermodynamic quantities can be derived by the\npartition function $Z(\\beta , \\mbox{$\\Omega$} )$ given by\n\\begin{equation}\nZ(\\beta , \\mbox{$\\Omega$} ) = {\\rm Tr} \\, e^{-\\beta :(\\hat{H}- {\\scriptsize \\mbox{$\\Omega$}}\n  \\hat{J}):},\n\\label{Z0}\n\\end{equation}\nwhere $:\\!\\!\\hat{H}\\!\\!:$ and $:\\!\\!\\hat{J}\\!\\!:$ are the normal ordered\nHamiltonian and angular momentum operators of the quantized field,\nrespectively~\\cite{VA,LPW,FF}. Here, we assume that particle number of \nthe system is indefinite. \n\nAs usual, by using the single-particle spectrum,\none can obtain the free energy $F(\\beta , \\mbox{$\\Omega$} )$ of the system\nin the following form\n\\begin{eqnarray}\n\\beta F &=& - \\ln Z = -\\sum_{\\lambda} \\ln \\sum_{k} [e^{-\\beta (\\varepsilon_{\\lambda}\n  -{\\scriptsize \\Omega} j_{\\lambda})}]^{k}\n\\label{F0}    \\\\\n&=& \\left\\{ \\begin{array}{cc}\n    -\\sum_{\\lambda}\\ln [1+e^{-\\beta (\\varepsilon_{\\lambda}-{\\scriptsize \\Omega} j_{\\lambda})}] &\n    \\mbox{for fermions,}  \\\\\n    \\sum_{\\lambda} \\ln [1-e^{-\\beta (\\varepsilon_{\\lambda}-{\\scriptsize \\Omega} j_{\\lambda})}] &\n    \\mbox{for bosons with $\\varepsilon_{\\lambda}-\\mbox{$\\Omega$} j_{\\lambda} > 0$.}\n\\end{array}\n\\right.\n\\end{eqnarray}\nwhere $\\lambda$ denotes the single-particle states for the free gas in the\nsystem. $\\varepsilon_{\\lambda}= \\langle 1_{\\lambda} |:\\!\\!\\hat{H}\\!\\!:|1_{\\lambda} \\rangle $\nand $j_{\\lambda}= \\langle 1_{\\lambda} |:\\!\\!\\hat{J}\\!\\!:|1_{\\lambda} \\rangle $ are normal ordered\nenergy and angular momentum associated with single-particle states $\\lambda$,\nrespectively. The occupation number $k = 0, 1, 2, \\cdots $ for bosonic\nfields and $k=0, 1$ for fermionic fields.\nNote that, if there exists a bosonic single-particle state with its\nenergy $\\varepsilon_{\\lambda}$ and angular momentum $j_{\\lambda}$ such that\n$\\varepsilon_{\\lambda}-\\mbox{$\\Omega$} j_{\\lambda} < 0$, the expression in Eq.~(\\ref{F0}) becomes\ndivergent and so is ill-defined. In order to compute the free energy\nin Eq.~(\\ref{F0}), one must know all single-particle states and their\ncorresponding values of $\\varepsilon_{\\lambda}$ and $j_{\\lambda}$ for a given system.\n\nAs pointed out in Refs.~\\cite{FT,MDOF}, the quantization of matter fields on\na stationary {\\it rotating} axisymmetric black hole background is somewhat\nunusual due to superradiant modes which occur in the presence of an ergoregion.\nThe mode solutions will be of the form, \n$\\phi (x) \\sim e^{-i{\\scriptsize \\mbox{$\\omega$} }t+im\\varphi }$,\nbecause this background spacetime possesses two Killing vector fields denoted by\n$\\partial_t$ and $\\partial_{\\varphi}$.\nSince the partition function in Eq.~(\\ref{Z0}) is defined with respect to an observer\nat infinity, the vacuum state to be defined by the standard quantization procedure\nshould be natural to that observer at infinity in the far future. Thus, we expand\nthe neutral scalar field in terms of a complete set of mode solutions as follows:\n\\begin{eqnarray}\n\\phi (x) &=& \\sum_m\\int^{\\infty}_{0}d\\mbox{$\\omega$} (b^{\\rm out}_{\\scriptsize\n             \\mbox{$\\omega$} m}u^{\\rm out}_{\\scriptsize \\mbox{$\\omega$} m} + b^{\\dagger\n             {\\rm out}}_{\\scriptsize \\mbox{$\\omega$} m}u^{\\ast {\\rm out}}_{\n             \\scriptsize \\mbox{$\\omega$} m}) + \\sum_m\\int^{\\infty}_{\\scriptsize\n             m\\mbox{$\\Omega$}_{H}}d\\mbox{$\\omega$} (b^{\\rm in}_{\\scriptsize \\mbox{$\\omega$} m}u^{\\rm in}_{\n             \\scriptsize \\mbox{$\\omega$} m} + b^{\\dagger {\\rm in}}_{\\scriptsize\n             \\mbox{$\\omega$} m}u^{\\ast {\\rm in}}_{\\scriptsize \\mbox{$\\omega$} m})   \\\\\n& & + \\sum_m\\int_{0}^{\\scriptsize m\\mbox{$\\Omega$}_{H}}d\\mbox{$\\omega$} (b^{\\rm in}_{\n    \\scriptsize -\\mbox{$\\omega$} -m}u^{\\rm in}_{\\scriptsize -\\mbox{$\\omega$} -m}\n    + b^{\\dagger {\\rm in}}_{\\scriptsize -\\mbox{$\\omega$} -m}\n    u^{\\ast {\\rm in}}_{\\scriptsize -\\mbox{$\\omega$} -m}).\n\\end{eqnarray}\nHere $u^{\\rm out}(x)$ describes unit outgoing flux to the future null infinity\n${\\cal T}^+$ and zero ingoing flux to the horizon ${\\cal H}^+$\nwhile $u^{\\rm in}(x)$ describes unit ingoing flux to ${\\cal H}^+$ and zero\noutgoing flux to ${\\cal T}^+$. These mode solutions are orthonormal\n\\begin{equation}\n\\langle u^{\\rm out}_{\\scriptsize \\mbox{$\\omega$} m}\\, ,\\,\nu^{\\rm out}_{\\scriptsize \\mbox{$\\omega$}'m'} \\rangle\n= \\langle u^{\\rm in}_{\\scriptsize \\mbox{$\\omega$} m}\\, ,\\,\nu^{\\rm in}_{\\scriptsize \\mbox{$\\omega$}'m'} \\rangle\n= \\langle u^{\\rm in}_{\\scriptsize -\\mbox{$\\omega$} -m}\\, ,\\,\nu^{\\rm in}_{\\scriptsize -\\mbox{$\\omega$}' -m'} \\rangle\n= \\delta (\\mbox{$\\omega$} -\\mbox{$\\omega$}')\\delta_{mm'}\n\\end{equation}\nwith respect to the Klein-Gordon inner product\n\\begin{equation}\n\\langle \\phi_1\\, ,\\, \\phi_2 \\rangle\n = \\frac{i}{2}\\int_{t={\\rm const.}} \\phi_1^{\\ast}\\!\n   \\stackrel{\\leftrightarrow }{\\partial_{\\mu}}\\! \\phi_2\\,\\,\n   d\\Sigma^{\\mu}.\n\\end{equation} Note that $u_{\\scriptsize \\mbox{$\\omega$} m}(x) \\sim e^{\\scriptsize -i\\mbox{$\\omega$}\nt+im\\varphi}$ and we suppressed other quantum numbers. Modes with\n$\\tilde{\\mbox{$\\omega$}}=\\mbox{$\\omega$} -\\mbox{$\\Omega$}_H m < 0$ exhibit the so-called\nsuperradiance. Here $\\mbox{$\\Omega$}_H$ is the angular speed of the horizon\nwith respect to an observer at infinity. An observer at infinity\nwould measure positive frequency for all modes $u^{\\rm\nout}_{\\scriptsize \\mbox{$\\omega$} m}$ and $u^{\\rm in}_{\\scriptsize \\mbox{$\\omega$} m}$\nwith $\\tilde{\\mbox{$\\omega$}} >0$, but measure negative frequency for $u^{\\rm\nin}_{\\scriptsize -\\mbox{$\\omega$} -m}$ with $\\tilde{\\mbox{$\\omega$}} <0$. A\nZAMO~\\cite{ZAMO} near the horizon, however, would see positive\nfrequency waves for $u^{\\rm in}_{\\scriptsize -\\mbox{$\\omega$} -m}$ with\n$\\tilde{\\mbox{$\\omega$}} <0$ as well as for $u^{\\rm in}_{\\scriptsize \\mbox{$\\omega$} m}$\nwith $\\tilde{\\mbox{$\\omega$}} >0$. Hence, in the terminology of\nRef.~\\cite{FT}, we adopt the ``distant-observer viewpoint\" for\n$u^{\\rm out}(x)$ and the ``near-horizon viewpoint\" for $u^{\\rm\nin}(x)$. And the conventions are chosen so that they agree with\nviewpoints.\n\nNow the mode solutions for particles confined in the near-horizon box\nwould be constructed by linearly superpose $u^{\\rm in}$ and $u^{\\rm out}$ above\nas follows:\n\\begin{equation}\n\\phi_{\\scriptsize \\mbox{$\\omega$} m}(x) \\sim \\left\\{ \\begin{array}{cc}\n     u^{\\rm out}_{\\scriptsize \\mbox{$\\omega$} m} + \\alpha_{\\scriptsize \\mbox{$\\omega$} m}\n     u^{\\rm in}_{\\scriptsize \\mbox{$\\omega$} m} &\n     \\mbox{for $\\tilde{\\mbox{$\\omega$}} >0$,}  \\\\\n     u^{\\rm out}_{\\scriptsize \\mbox{$\\omega$} m} + \\alpha_{\\scriptsize \\mbox{$\\omega$} m}\n     u^{\\ast \\rm in}_{\\scriptsize -\\mbox{$\\omega$} -m} &\n     \\mbox{for $\\tilde{\\mbox{$\\omega$}} <0$,}\n\\end{array}\n\\right.\n\\end{equation}\nwith appropriate normalization factor. $\\alpha_{\\scriptsize \\mbox{$\\omega$} m}$ is chosen so that\nthe modes satisfy the periodic boundary condition in Eq.~(\\ref{BC}).\nThus, only some discrete (real) values of $\\mbox{$\\omega$}$ will be allowed~\\cite{comp}.\n$\\phi_{\\scriptsize \\mbox{$\\omega$} m}(x)$ are understood to be cut off everywhere outside the box.\nNote that $\\phi_{\\scriptsize \\mbox{$\\omega$} m}(x) \\sim e^{\\scriptsize -i\\mbox{$\\omega$} t+im\\varphi}$\nfor all $\\tilde{\\mbox{$\\omega$}}$.\n\nThe inner product of these modes becomes\n\\begin{equation}\n\\langle \\phi_{\\scriptsize \\mbox{$\\omega$} m}\\, ,\\, \\phi_{\\scriptsize \\mbox{$\\omega$}'m'} \\rangle\n = \\mbox{$\\delta$}_{\\scriptsize \\mbox{$\\omega$} \\mbox{$\\omega$}'}\\mbox{$\\delta$}_{\\scriptsize mm'}\\!\\! \\int^{L}_{r_++h}\n (\\mbox{$\\omega$} -{\\scriptsize \\Omega_{0}} m)|\\phi_{\\scriptsize \\mbox{$\\omega$} m}|^2 N^{-1}d\\Sigma ,\n\\label{Inner}\n\\end{equation}\nwhere we have used $d\\Sigma^{\\mu}=n^{\\mu}d\\Sigma$ and the unit normal\nto a $t= {\\rm const.}$ surface\n$n^{\\mu}=N^{-1}(\\partial_t+\\mbox{$\\Omega$}_0\\partial_{\\varphi})^{\\mu}$. Here $\\mbox{$\\Omega$}_0(r)$ is the angular\nspeed of ZAMO's~\\cite{ZAMO}. Since $\\mbox{$\\Omega$}_0(r) \\leq \\mbox{$\\Omega$}_H=\\mbox{$\\Omega$}_0(r=r_+)$,\nthe norm of a mode solution with $\\mbox{$\\omega$} > 0$\nis positive if $\\tilde{\\mbox{$\\omega$}}=\\mbox{$\\omega$} -\\mbox{$\\Omega$}_Hm > 0$.\nWhen $\\tilde{\\mbox{$\\omega$}} < 0$, the norm could be either positive or negative\ndepending on the radial behavior of the solution. If the norm of\n$\\phi_{\\scriptsize \\mbox{$\\omega$} m}(x)$ is negative,\nwe can easily see that $\\phi_{\\scriptsize -\\mbox{$\\omega$} -m}(x) \\sim\ne^{\\scriptsize i\\mbox{$\\omega$} t-im\\varphi}$\nhas the positive norm. Let us define a set SR consisting of\nmode solutions $\\phi_{\\scriptsize \\mbox{$\\omega$} m}$ with $\\mbox{$\\omega$} >0$ whose norms are\nnegative. Then, the quantized field inside the box can\nbe expanded in terms of orthonormal mode solutions as\n\\begin{equation}\n\\phi (x) = \\sum_{\\lambda \\not\\in {\\rm SR}}[a_{\\scriptsize \\mbox{$\\omega$} m}\n  \\phi_{\\scriptsize \\mbox{$\\omega$} m}(x) +\n  a^{\\dagger}_{\\scriptsize \\mbox{$\\omega$} m}\\phi^{\\ast}_{\\scriptsize \\mbox{$\\omega$} m}(x)]\n  + \\sum_{\\lambda \\in {\\rm SR}}[a_{\\scriptsize -\\mbox{$\\omega$} -m}\n  \\phi_{\\scriptsize -\\mbox{$\\omega$} -m}(x) + a^{\\dagger}_{\\scriptsize -\\mbox{$\\omega$} -m}\n  \\phi^{\\ast}_{\\scriptsize -\\mbox{$\\omega$} -m}(x)],\n\\label{field}\n\\end{equation}\nwhere the single-particle states are labeled by\n$\\lambda =(\\mbox{$\\omega$} ,m)$. The Hamiltonian operator in the reference frame\nof a distant observer at infinity becomes then \\begin{eqnarray} H &=&\n\\sum_{\\lambda \\not\\in {\\rm SR}} \\mbox{$\\omega$} (a_{\\scriptsize \\mbox{$\\omega$} m}a^{\\dagger}_\n      {\\scriptsize \\mbox{$\\omega$} m} + a^{\\dagger}_{\\scriptsize \\mbox{$\\omega$} m}\n      a_{\\scriptsize \\mbox{$\\omega$} m}) + \\sum_{\\lambda \\in {\\rm SR}}(-\\mbox{$\\omega$} )\n      (a_{\\scriptsize -\\mbox{$\\omega$} -m}a^{\\dagger}_{\\scriptsize -\\mbox{$\\omega$} -m}\n      + a^{\\dagger}_{\\scriptsize -\\mbox{$\\omega$} -m}a_{\\scriptsize -\\mbox{$\\omega$} -m})\n      \\nonumber   \\\\\n&=& \\sum_{\\lambda \\not\\in {\\rm SR}} \\mbox{$\\omega$} (N_{\\scriptsize \\mbox{$\\omega$} m}+\\frac{1}{2})\n    +\\sum_{\\lambda \\in {\\rm SR}}(-\\mbox{$\\omega$} )(N_{\\scriptsize -\\mbox{$\\omega$} -m}+\\frac{1}{2}),\n\\end{eqnarray} where $N_{\\scriptsize \\mbox{$\\omega$} m}=a^{\\dagger}_{\\scriptsize \\mbox{$\\omega$} m}\na_{\\scriptsize \\mbox{$\\omega$} m}$ and $N_{\\scriptsize -\\mbox{$\\omega$} -m}=a^{\\dagger}_\n{\\scriptsize -\\mbox{$\\omega$} -m}a_{\\scriptsize -\\mbox{$\\omega$} -m}$ are number\noperators. Now, by following the standard procedure for defining a\nvacuum state and single-particle states~\\cite{MDOF,FT}, we can\neasily see that $(\\varepsilon_{\\lambda},j_{\\lambda})=(\\mbox{$\\omega$} ,m)$ for single-particle\nstates $\\lambda =(\\mbox{$\\omega$} ,m) \\not\\in {\\rm SR}$ while\n$(\\varepsilon_{\\lambda},j_{\\lambda})=(-\\mbox{$\\omega$} ,-m)$, instead of $(\\mbox{$\\omega$} ,m)$, for\nsingle-particle states $\\lambda = (\\mbox{$\\omega$} ,m) \\in {\\rm SR}$. In\nRefs.~\\cite{KKPS,IS,HKPS}, however, $(\\varepsilon_{\\lambda},j_{\\lambda})=(\\mbox{$\\omega$} ,m)$\nfor $\\lambda \\in {\\rm SR}$ have been used, and our study shows that\nthis error comes from the incorrect quantization of superradiant\nmodes. This important difference is a peculiar feature of the\nquantization of matter fields in the presence of an ergoregion and\nturns out to make our ``angular-momentum modified canonical\nensemble'' in Eq.~(\\ref{Z0}) being well-defined as shall be shown\nbelow in detail. It also makes somewhat unphysical treatment of\nsuperradiant modes and introduction of various cutoff numbers\nunnecessary in the calculation of statistical-mechanical entropy\nin the literature. For example, in Ref.~\\cite{IS}, a cutoff in the\noccupation number $k$ was introduced to avoid the divergent sum\nfor superradiant modes in the log in Eq.~(\\ref{F0}).\n\nIn general, $\\mbox{$\\omega$}$ is discrete due to the finite size of the box,\nbut the gap between adjacent values\ngoes small as the size of the thermal box becomes large.\nIn this continuous limit, one may introduce the density function\ndefined by $g(\\mbox{$\\omega$} ,m)=\\partial n(\\mbox{$\\omega$} ,m)\/\\partial \\mbox{$\\omega$}$ where $n(\\mbox{$\\omega$} ,m)$ is\nthe number of mode solutions whose frequency or energy is below $\\mbox{$\\omega$}$\nfor a given value of angular momentum $m$. Thus, $g(\\mbox{$\\omega$} ,m)d\\mbox{$\\omega$}$\nrepresents the number of single-particle states whose energy lies\nbetween $\\mbox{$\\omega$}$ and $\\mbox{$\\omega$} +d\\mbox{$\\omega$}$, and whose angular momentum is $m$.\nUsing this density function, the free energy in Eq.~(\\ref{F0}) can be\nre-expressed as\n\\begin{equation}\n\\beta F= -\\sum_m \\!\\!\\int \\!d\\mbox{$\\omega$} \\, g(\\mbox{$\\omega$} ,m) \\ln \\sum_{k}\n         [e^{-\\beta (\\varepsilon_{\\lambda} -{\\scriptsize \\Omega} j_{\\lambda})}]^{k}.\n\\label{F1}\n\\end{equation}\n\nThe angular speed $\\mbox{$\\Omega$}$ in Eq.~(\\ref{F1}) is a thermodynamic parameter\ndefined, in principle, by its appearance in the thermodynamic first\nlaw for the reservoir, namely $TdS=dE-\\mbox{$\\Omega$} dJ+ \\cdots $. Since a particle\ncannot move faster than the speed of light, its angular velocity with respect\nto an observer at rest at infinity should be restricted. The possible maximum and minimum\nangular speeds are\n\\begin{equation}\n\\mbox{$\\Omega$}_{\\pm}(r) = \\mbox{$\\Omega$}_0 (r) \\pm \\sqrt{(\\partial_t \\cdot \\partial_{\\varphi}\/\\partial_{\\varphi}\n\\cdot \\partial_{\\varphi})^2 - \\partial_t \\cdot \\partial_t\/\\partial_{\\varphi}\\cdot \\partial_{\\varphi}},\n\\label{Ompm}\n\\end{equation}\nrespectively. We see that, as $r \\rightarrow r_+$, the range of\nangular velocities a particle can take on narrows down({\\it i.e.},\n$\\mbox{$\\Omega$}_{\\pm}(r) \\rightarrow \\mbox{$\\Omega$}_H$), and so the angular speed of particles near\nthe horizon will be $\\mbox{$\\Omega$}_H$. For a rotating body in flat spacetime, one knows\nthat all subsystems must rotate uniformly when the body is\nin a thermal equilibrium state \\cite{LPW}. In fact, it is a part of\nthe thermodynamic zeroth law.\nIn curved spacetimes, the uniform rotation of all\nsubsystems in thermal equilibrium may not be true to hold any more.\nHowever, since we will be finally interested in the quantum gas only in the\nvicinity of the horizon, we shall assume $\\mbox{$\\Omega$} =\\mbox{$\\Omega$}_H$ below.\n\nNow, one can see that the sum in the log in Eq.~(\\ref{F1}) is defined\nwell for states belonging to SR since \n$\\varepsilon_{\\lambda}-\\mbox{$\\Omega$} j_{\\lambda} = -(\\mbox{$\\omega$} -\\mbox{$\\Omega$}_H m) >0$ by the definition of the set SR. \nFor some states with $\\mbox{$\\omega$} -\\mbox{$\\Omega$}_Hm < 0$ \nnot belonging to SR, however, the sum becomes\ndivergent. As will be shown below explicitly, however, the main\ncontribution to the entropy of the system comes from the infinite\npiling up of waves at the horizon. For such localized solutions near\nthe horizon, the signature of norms in Eq.~(\\ref{Inner}) will be\ndetermined by that of $\\mbox{$\\omega$} -\\mbox{$\\Omega$}_Hm$ since $\\mbox{$\\Omega$}_0(r) \\simeq \\mbox{$\\Omega$}_H$ for\nthe range of integration giving dominant contributions. Therefore, we\nassume that all states with $\\mbox{$\\omega$} -\\mbox{$\\Omega$}_Hm < 0$ belong to the set SR.\nThen, the free energy in Eq.~(\\ref{F1}) can be written as\n$F = F_{\\rm NS} +F_{\\rm SR}$.\nHere\n\\begin{eqnarray}\n\\mbox{$\\beta$} F_{\\rm NS} &=& \\sum_{\\lambda \\not\\in {\\rm SR}}\\!\\!\\int d\\mbox{$\\omega$} \\, g(\\mbox{$\\omega$} ,m)\n\\ln [1-e^{-\\beta (\\mbox{$\\omega$} -{\\scriptsize \\Omega_H} m)}],\n\\label{FNS}           \\\\\n\\mbox{$\\beta$} F_{\\rm SR} &=& \\sum_{\\lambda \\in {\\rm SR}}\\!\\!\\int d\\mbox{$\\omega$} \\, g(\\mbox{$\\omega$} ,m)\n\\ln [1-e^{\\beta (\\mbox{$\\omega$} -{\\scriptsize \\Omega_H} m)}].\n\\label{FSR}\n\\end{eqnarray}\n\nIn general cases, it is highly nontrivial to compute $g(\\mbox{$\\omega$} ,m)$\nexactly except for some cases in two-dimensional black\nholes~\\cite{SCZ}. For suitable conditions, however, one can\napproximately obtain $g(\\mbox{$\\omega$} ,m)$ by using the WKB method as in the\nbrick-wall model~\\cite{Hoof}. For simplicity, let us consider a\nscalar field in a rotating BTZ black hole in\n3-dimensions~\\cite{BTZ}. For the case of Kerr black holes in\n4-dimensions, although the essential result is the same as that in\nthe case of BTZ black holes, it requires some modified formulation\nbasically due to the fact that the geometrical property near the\nhorizon changes along the polar angle~\\cite{Kerr}. The metric of a\nrotating BTZ black hole is given by \\begin{equation} ds^2= -N^2dt^2 +\nN^{-2}dr^2 + r^2(d\\varphi - {\\scriptsize \\Omega_{0}} dt)^2, \\end{equation} where \\begin{equation} N^2= r^2\/l^2\n-M +J^2\/4r^2 = (r^2-r^2_+)(r^2-r^2_-)\/r^2l^2,  \\nonumber \\end{equation} and\nthe angular speed of ZAMO's is $\\mbox{$\\Omega$}_0 =J\/2r^2$. Here $r_{\\pm}$\ndenote the outer and inner horizons, respectively. Note that\n$\\mbox{$\\Omega$}_H=\\mbox{$\\Omega$}_0(r=r_+)=r_-\/r_+l $. Then, mode solutions are\n$\\phi_{\\scriptsize \\mbox{$\\omega$} m}(x) = \\phi_{\\scriptsize \\mbox{$\\omega$} m}(r)\ne^{-i{\\scriptsize \\mbox{$\\omega$}}t+im\\varphi }$. Here the radial part\n$\\phi_{\\scriptsize \\mbox{$\\omega$} m}(r)$ satisfies \\begin{equation} rN^2\n\\frac{d}{dr}\\Big[rN^2\\frac{d}{dr}\\phi_{\\scriptsize \\mbox{$\\omega$} m}(r)\\Big]\n+ r^2N^4k^2(r; \\mbox{$\\omega$} ,m)\\phi_{\\scriptsize \\mbox{$\\omega$} m}(r) =0, \\label{REQ}\n\\end{equation} where \\begin{equation} k^2(r;\\mbox{$\\omega$} ,m) = N^{-4}[(\\mbox{$\\omega$} -\\+ m)(\\mbox{$\\omega$} -\\-\nm)-\\mu^2N^2] . \\end{equation} Here $\\mbox{$\\Omega$}_{\\pm}(r) = \\mbox{$\\Omega$}_0 (r) \\pm N\/r$ for a\nrotating BTZ black hole in Eq~(\\ref{Ompm}).\n\nIn the WKB approximation, the discrete value of energy $\\mbox{$\\omega$}$\nin Eq.~(\\ref{REQ}) is related to $n(\\mbox{$\\omega$} ,m)$ as follows\n\\begin{equation}\n\\pi n(\\mbox{$\\omega$} ,m) = \\int^{L}_{r_++h} dr\\, ``k\\textrm{''}(r;\\mbox{$\\omega$} ,m),\n\\label{WKB}\n\\end{equation}\nwhere $``k\\textrm{''}(r;\\mbox{$\\omega$} ,m)$ is set to be zero if $k^2(r;\\mbox{$\\omega$} ,m)$\nbecomes negative for given $(\\mbox{$\\omega$} ,m)$ \\cite{Hoof}.\nSince $``k\\textrm{''}(r;\\mbox{$\\omega$} ,m) \\simeq N^{-2}$\nand $N(r) \\rightarrow 0$ as one approaches to the horizon,\nwe can easily see that the dominant\ncontribution in Eq.~(\\ref{WKB}) comes from the integration in the\nvicinity of the horizon as the inner brick-wall approaches to it(\n{\\it i.e.}, $h \\rightarrow 0$).\n\nNow, Eq.~(\\ref{FNS}) becomes\n\\begin{eqnarray}\n\\beta F_{\\rm NS} &=& \\sum_{\\lambda \\not\\in {\\rm SR}}\\!\\int \\!\\! d\\mbox{$\\omega$} \\,\n  \\frac{\\partial}{\\partial \\mbox{$\\omega$}}\\bigg[ \\frac{1}{\\pi}\\int^L_{r_++h}\\!\\! dr \\,\n  ``k\\textrm{''}(r;\\mbox{$\\omega$} ,m)\\bigg] \\ln [1-e^{-\\beta (\\mbox{$\\omega$} -{\\scriptsize \\Omega_H} m)}]\n  \\nonumber   \\\\\n&=& -\\frac{\\mbox{$\\beta$}}{\\pi} \\int^L_{r_++h}\\!\\! dr \\sum_{m}\\!\\int \\!\\! d\\mbox{$\\omega$}\n  \\, \\frac{k(r;\\mbox{$\\omega$} ,m)}{e^{\\beta (\\mbox{$\\omega$} -{\\scriptsize \\Omega_H} m)}-1}   \\nonumber   \\\\\n& & +\\frac{1}{\\pi} \\int^L_{r_++h}\\!\\! dr \\sum_{m}\\, k(r;\\mbox{$\\omega$} ,m)\n  \\ln [1-e^{-\\beta (\\mbox{$\\omega$} -{\\scriptsize \\Omega_H} m)}] \\Big|^{\\mbox{$\\omega$}_{\\rm max}(m)}_{\\mbox{$\\omega$}_{\\rm\n      min}(m)}\n\\label{F2}\n\\end{eqnarray}\nby using the integration by parts in $\\mbox{$\\omega$}$.\nFor convenience, one can divide $F_{\\rm NS}$ into two parts\n\\begin{equation}\nF_{\\rm NS} = F^{(m>0)}_{\\rm NS} + F^{(m<0)}_{\\rm NS}, \\nonumber\n\\end{equation}\nwhere\n\\begin{equation}\nF^{(m>0)}_{\\rm NS} =  -\\frac{1}{\\pi}\\int^L_{r_++h}\\!\\! dr \\, N^{-2}\n\\int^{\\infty}_0 \\!\\! dm \\!\\!\\int^{\\infty}_{\\+ m}\\!\\! d\\mbox{$\\omega$} \\,\n\\frac{\\sqrt{(\\mbox{$\\omega$} -{\\scriptsize \\mbox{$\\Omega$}_+} m)(\\mbox{$\\omega$} -{\\scriptsize \\mbox{$\\Omega$}_-}\n    m)}}\n{e^{\\beta (\\mbox{$\\omega$} -{\\scriptsize \\Omega_H} m)}-1}\n\\label{NS1+}\n\\end{equation}\nfrom states with positive angular momenta and\n\\begin{eqnarray}\nF^{(m<0)}_{\\rm NS} &=& -\\frac{1}{\\pi}\\bigg( \\int^{r_{\\rm erg}}_{r_++h}\n  dr\\, N^{-2}\\int^{0}_{-\\infty}dm \\int^{\\infty}_{0}d\\mbox{$\\omega$}  \\nonumber \\\\\n& & +\\int^{L}_{r_{\\rm erg}}dr\\, N^{-2}\\int^{0}_{-\\infty}dm\n  \\int^{\\infty}_{\\- m}d\\mbox{$\\omega$} \\bigg)\\, \\frac{\\sqrt{(\\mbox{$\\omega$} -{\\scriptsize \\mbox{$\\Omega$}_+} m)\n  (\\mbox{$\\omega$} -{\\scriptsize \\mbox{$\\Omega$}_-} m)}}{e^{\\beta (\\mbox{$\\omega$} -{\\scriptsize \\Omega_H} m)}-1}\n  \\nonumber  \\\\\n& & -\\frac{1}{\\pi \\mbox{$\\beta$}}\\int^{r_{\\rm erg}}_{r_++h}dr\\, N^{-2}\n  \\int^{0}_{-\\infty}dm\\, \\sqrt{\\+ \\- m^2} \\ln (1-e^{\\beta {\\scriptsize \\Omega_H} m})\n\\label{NS1-}\n\\end{eqnarray}\nfrom states with negative angular momenta. $r_{\\rm erg}=\\sqrt{M}l$ \nis the radius of the outer boundary of the ergoregion where \n$\\mbox{$\\Omega$}_-(r=r_{\\rm erg})=0$. Here we considered a\nmassless scalar field for simplicity.\nSimilarly, from states belonging to SR, Eq.~(\\ref{FSR}) becomes\n\\begin{eqnarray}\nF_{\\rm SR} &=& -\\frac{1}{\\pi}\\int^{r_{\\rm erg}}_{r_++h}dr\\, N^{-2}\n  \\int^{\\infty}_{0}dm \\int^{\\- m}_{0} d\\mbox{$\\omega$} \\, \\frac{\\sqrt{(\\mbox{$\\omega$}\n      -{\\scriptsize \\mbox{$\\Omega$}_+} m)(\\mbox{$\\omega$} -{\\scriptsize \\mbox{$\\Omega$}_-} m)}}\n  {e^{-\\beta (\\mbox{$\\omega$} -{\\scriptsize \\Omega_H} m)}-1}  \\nonumber  \\\\\n& & +\\frac{1}{\\pi \\mbox{$\\beta$}}\\int^{r_{\\rm erg}}_{r_++h}dr\\, N^{-2}\n  \\int^{\\infty}_{0}dm \\sqrt{\\+ \\- m^2}\\, \\ln (1-e^{-\\beta {\\scriptsize \\Omega_H} m}).\n\\label{SR}\n\\end{eqnarray}\nNote that $g=-\\partial n\/\\partial \\mbox{$\\omega$}$ for $\\lambda \\in {\\rm SR}$, and \nthat the boundary term in $F_{\\rm SR}$ exactly cancels that in\n$F^{(m<0)}_{\\rm NS}$.\n\n{}From Eqs.~(\\ref{NS1+}-\\ref{SR}), we can obtain leading order\ndependence on the brick-wall cutoff $h$ for the free energy as\nfollows; \\begin{eqnarray} F^{(m>0)}_{\\rm NS} &=& -\\frac{\\zeta (3)}{\\mbox{$\\beta$}^3}\n\\frac{r^2_{+}l^3}\n  {(r^2_{+}-r^2_{-})^2} \\left[ \\frac{\\sqrt{r^2_+-r^2_-}}{2\\sqrt{2}}\n  \\sqrt{\\frac{r_+}{h}} + \\frac{r_-}{\\pi}\\ln (\\frac{r_+}{h}) +\n  \\vartheta (\\sqrt{h}) \\right] ,    \\nonumber  \\\\\nF^{(m<0)}_{\\rm NS} &=& -\\frac{\\zeta (3)}{\\mbox{$\\beta$}^3} \\frac{r^2_{+}l^3}\n  {(r^2_{+}-r^2_{-})^2} \\left[ \\frac{r^2_+-r^2_-}{2\\pi r_-} \\ln (\\frac{r_+}{h})\n  + \\vartheta (\\sqrt{h}) \\right] ,    \\nonumber   \\\\\nF_{\\rm SR} &=& -\\frac{\\zeta (3)}{\\mbox{$\\beta$}^3} \\frac{r^2_{+}l^3}\n  {(r^2_{+}-r^2_{-})^2} \\Bigg[ \\frac{\\sqrt{r^2_+-r^2_-}}{2\\sqrt{2}}\n  \\sqrt{\\frac{r_+}{h}} - \\frac{r_-}{\\pi}\\ln (\\frac{r_+}{h})\n  - \\frac{r^2_+-r^2_-}{2\\pi r_-} \\ln (\\frac{r_+}{h})  \\nonumber  \\\\\n& & + \\vartheta (\\sqrt{h}) \\Bigg] .\n\\label{Free}\n\\end{eqnarray}\nThe entropy of this boson gas which is assumed to be in thermal\nequilibrium with the rotating black hole can be obtained from the free\nenergy by using the thermodynamic relation,\n$S = \\mbox{$\\beta$}^2 \\partial F\/\\partial \\mbox{$\\beta$} |_{\\scriptsize \\mbox{$\\beta$} = \\mbox{$\\beta$}_H}\n=-3\\beta F|_{\\scriptsize \\mbox{$\\beta$} = \\mbox{$\\beta$}_H}$;\n\\begin{eqnarray}\nS_{\\rm NS} &=& \\frac{3\\zeta (3)}{4\\pi^2l}\n  \\left[ \\frac{\\sqrt{r^2_+-r^2_-}}{2\\sqrt{2}}\n  \\sqrt{\\frac{r_+}{h}} + \\frac{r_-}{\\pi}\\ln (\\frac{r_+}{h}) +\n  \\frac{r^2_+-r^2_-}{2\\pi r_-} \\ln (\\frac{r_+}{h}) +\n  \\vartheta (\\sqrt{h}) \\right] ,   \\nonumber  \\\\\nS_{\\rm SR} &=& \\frac{3\\zeta (3)}{4\\pi^2l}\n  \\left[ \\frac{\\sqrt{r^2_+-r^2_-}}{2\\sqrt{2}}\n  \\sqrt{\\frac{r_+}{h}} - \\frac{r_-}{\\pi}\\ln (\\frac{r_+}{h})\n  - \\frac{r^2_+-r^2_-}{2\\pi r_-} \\ln (\\frac{r_+}{h})\n  + \\vartheta (\\sqrt{h}) \\right] ,\n\\label{ENTS}\n\\end{eqnarray}\nwhere the temperature of a rotating BTZ black hole~\\cite{BTZ} is\n\\begin{equation}\n\\mbox{$\\beta$}^{-1}_{H} = (r^2_+ -r^2_-)\/2\\pi r_+l^2.\n\\end{equation}\nNow the total entropy of the system becomes\n\\begin{equation}\nS = \\frac{3\\zeta (3)}{4\\sqrt{2}\\pi^2} \\frac{\\sqrt{r^2_+-r^2_-}}{l}\n    \\sqrt{\\frac{r_+}{h}} + \\vartheta (\\sqrt{h}).\n\\label{ENT}\n\\end{equation}\n\nIn Ref.~\\cite{KKPS}, it is claimed that the contribution from\nsuperradiant modes is a subleading order compared with that from\nnonsuperradiant modes. In our results above, however, we find that \nsuperradiant modes also give a leading order contribution which is in fact \nexactly same as that from nonsuperradiant modes in the leading order of\n$\\sqrt{r_+\/h}$.\nIt should be pointed out that\nthe entropy associated with superradiant modes is {\\it positive} in\nour result whereas it is {\\it negative} in Refs.~\\cite{KKPS,HKPS}.\nIn addition, since the log terms in Eq.~(\\ref{ENTS}) are exactly cancelled,\nour result for the entropy of quantum field smoothly\nreproduces the correct result in the non-rotating limit({\\it i.e.,}\n$J \\rightarrow 0$ or $r_- \\rightarrow 0$) whereas the entropy obtained in\nRefs.~\\cite{KKPS,HKPS} becomes divergent in that limit.\n\nIf we rewrite the entropy in terms of the brick-wall cutoff in\nproper length defined as $\\bar{h} = \\int^{r_++h}_{r_+} \\sqrt{g_{\\rm\nrr}}dr$, Eq.~(\\ref{ENT}) becomes \\begin{equation} S = \\frac{3\\zeta\n(3)\/8\\pi^3}{\\bar{h}} {\\cal C} + \\vartheta (\\bar{h}), \\end{equation} where\n${\\cal C}=2\\pi r_+$ is the circumference of the horizon. Thus, by\nrecovering the dimension and introducing an appropriate brick-wall\ncutoff \\begin{equation} \\bar{h} = \\frac{3\\zeta (3)}{16\\pi^3} l_P \\simeq 7.3\n\\times 10^{-3} l_P \\label{CUT} \\end{equation} which is a universal constant,\none can make the entropy of quantum field finite and being\nequivalent to the Bekenstein-Hawking entropy of a rotating BTZ\nblack hole~\\cite{BTZ} \\begin{equation} S = 4\\pi r_+\/l_P = S_{\\rm BH} \\end{equation} in\nleading order. Here $l_P$ is the Planck length. For a fermionic\nfield, although modes with $\\tilde{\\mbox{$\\omega$}} <0$ do not reveal\nsuperradiance, it turns out that only the overall numerical factor\nin Eq.~(\\ref{ENT}) is different. As mentioned before, the\nextension of our study to the case of Kerr black holes in\nfour-dimensions is straightforward, but requires some\nmodifications mainly due to the polar angle dependence of the near\nhorizon geometry. A calculation in the phase space shows that the\nessential feature of the leading order divergence in the entropy\nof quantum fields is same as that of the present case~\\cite{Kerr}.\n\nOther thermodynamic quantities of quantum field such as the angular\nmomentum and internal energy can also be obtained as follows;\n\\begin{equation}\nJ_{\\rm matter} = -\\frac{\\partial F}{\\partial \\mbox{$\\Omega$}}{\\bigg|}_{\\scriptsize\n  \\mbox{$\\beta$} =\\mbox{$\\beta$}_H, \\mbox{$\\Omega$} =\\mbox{$\\Omega$}_H}\n  = \\frac{3\\zeta (3)\/16\\pi^3}{\\bar{h}}\\, \\frac{2r_+r_-}{l}\n   + \\vartheta (\\ln \\bar{h}).\n\\end{equation}\nHere the derivative with respect to $\\mbox{$\\Omega$}$ has been taken for\nEqs.~(\\ref{NS1+}-\\ref{SR}).\nIf we put the cutoff value in Eq.~(\\ref{CUT}), we have\n\\begin{equation}\nJ_{\\rm matter} = \\frac{2r_+r_-}{l} = J_{\\rm BH}.\n\\end{equation}\nThe internal energy of the system with respect to an observer at\ninfinity is\n\\begin{eqnarray}\nE &=& \\frac{\\partial}{\\partial \\mbox{$\\beta$}}(\\mbox{$\\beta$} F){\\bigg|}_{\\scriptsize\n  \\mbox{$\\beta$} =\\mbox{$\\beta$}_H, \\mbox{$\\Omega$} =\\mbox{$\\Omega$}_H} + \\mbox{$\\Omega$}_HJ_{\\rm matter}  \\nonumber \\\\\n  &=& \\frac{3\\zeta (3)\/16\\pi^3}{\\bar{h}}\\,\n  \\frac{4}{3}\\frac{r^2_++\\frac{1}{2}r^2_-}{l^2} +\\vartheta (\\ln \\bar{h})\n  = \\frac{4}{3}M_{\\rm BH} - \\frac{2}{3} \\frac{r^2_-}{l^2},\n\\end{eqnarray}\nwhere the black hole mass is $M_{\\rm BH}=M=(r^2_++r^2_-)\/l^2$.\nOne can easily see that $J_{\\rm matter} \\rightarrow 0$ and $E\n\\rightarrow \\frac{4}{3}M_{\\rm BH}$ in the limit of non-rotating black\nholes ({\\it e.g.}, $J_{\\rm BH}=J \\rightarrow 0$). Therefore,\nwe find that the entropy and angular momentum of quantum field can be\nidentified with those of the rotating black hole by introducing\na universal brick-wall cutoff although the internal energy is\nnot proportional to the black hole mass.\n\n\n\n\n\n\\vspace{0.5cm}\nWhat kind of relationships could be held among parameters characterizing\na rotating black hole and thermodynamic quantities of\nthe system of quantum fields\nin equilibrium with the black hole? To see this, let us consider\na system whose free energy depends on the temperature such that\n$F(\\mbox{$\\beta$} ,\\mbox{$\\Omega$} ,M,J) = \\mbox{$\\beta$}^{-3}f(\\mbox{$\\Omega$} ,M,J)$~\\cite{footnote}.\nSuppose the entropy of the system is identified with that\nof the black hole after an appropriate regularization,\n\\begin{equation}\nS = \\mbox{$\\beta$}^2 \\left(\\frac{\\partial F}{\\partial \\mbox{$\\beta$}}\\right)_{\\scriptsize \\mbox{$\\Omega$}}\n{\\bigg|}_{\\scriptsize \\mbox{$\\beta$} =\\mbox{$\\beta$}_H, \\mbox{$\\Omega$} =\\mbox{$\\Omega$}_H} =\nS_{\\rm BH} = 4\\pi r_+.\n\\end{equation}\nThe internal energy of the system with respect to a ``corotating''\nobserver is\n\\begin{equation}\nE^{\\prime} = \\left[\\frac{\\partial}{\\partial \\mbox{$\\beta$}}(\\mbox{$\\beta$} F)\\right]_{\\scriptsize \\mbox{$\\Omega$}}\n{\\bigg|}_{\\scriptsize \\mbox{$\\beta$} =\\mbox{$\\beta$}_H, \\mbox{$\\Omega$} =\\mbox{$\\Omega$}_H}\n           = \\frac{2}{3} \\frac{S}{\\mbox{$\\beta$}_H} = \\frac{4}{3} \\frac{r^2_+-r^2_-}{l^2}.\n\\end{equation}\nNow suppose the angular momentum of the system is proportional to that\nof the rotating black hole,\n\\begin{equation}\nJ_{\\rm matter} = -\\left(\\frac{\\partial F}{\\partial \\mbox{$\\Omega$}}\\right)_{\\scriptsize \\mbox{$\\beta$}}\n{\\bigg|}_{\\scriptsize \\mbox{$\\beta$} =\\mbox{$\\beta$}_H, \\mbox{$\\Omega$} =\\mbox{$\\Omega$}_H}\n               = \\mbox{$\\alpha$} J = \\mbox{$\\alpha$} \\frac{2r_+r_-}{l}.\n\\end{equation}\nThe internal energy of the system with respect to an observer at\ninfinity\nis then\n\\begin{equation}\nE = E^{\\prime} + \\mbox{$\\Omega$}_H J_{\\rm matter} = \\frac{4}{3} \\frac{r^2_+ + (3\\mbox{$\\alpha$}\n  \/2-1)r^2_-}{l^2},\n\\end{equation}\nwhich becomes proportional to the mass of the black hole,\n$M=(r^2_++r^2_-)\/l^2$, only if $\\mbox{$\\alpha$} = 4\/3$. Therefore, we expect the\nrelationships are probably\n\\begin{equation}\nJ_{\\rm matter} = \\frac{4}{3} J, \\qquad    E = \\frac{4}{3} M.\n\\label{EJ}\n\\end{equation}\nIf we apply the same argument to the case of Kerr black holes in\n4-dimensions, we obtain\n\\begin{equation}\nJ_{\\rm matter} = \\frac{3}{4} J,  \\qquad  E = \\frac{3}{8} M\n\\end{equation}\nof which the second relationship has been explicitly shown for the\nSchwarzschild black hole in the brick-wall model \nby 't Hooft~\\cite{Hoof}.\n\nWe have not obtained the relationships in Eq.~(\\ref{EJ}) at the\npresent letter. The reason for these discrepancies is not understood\nat the present. It will be very interesting to see how\nthe Pauli-Villars regularization method, which does not require\nthe presence of a brick-wall as shown in Ref.~\\cite{DLM} for the case\nof a charged non-rotating black hole, works for the case of\nrotating black holes.\n\n\n\n\n\\vskip 0.7cm\nAuthors would like to thank, for useful discussions,\nW.T. Kim, Y.J. Park, and H.J. Shin. Especially, GK would like to thank\nJ. Samuel, T. Jacobson, and H.C. Kim for many helpful discussions\nand suggestions. GK was supported by Korea Research Foundation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe role of electron-electron interactions in graphene\n(two-dimensional graphite) is still a debated issue. Whereas most of\nits electronic properties can be understood within a model of\ntwo-dimensional (2D) noninteracting massless Dirac\nfermions,\\cite{CG07} there are some experimental indications\nfor the presence of Coulomb interactions.\\cite{LB08} These\ncorrelations, which may be quantified by the graphene fine-structure\nconstant $\\alpha_G=e^2\/\\hbar \\varepsilon v_{\\rm F}\\simeq 2.2\/\\varepsilon$,\nin terms of the Fermi velocity $v_{\\rm F}$ and the dielectric constant\n$\\varepsilon$, seem, however, to be weak and long-ranged. Therefore,\nstrongly-correlated phases that are expected in the large-$\\alpha_G$\nlimit\\cite{DL09} or for short-range Hubbard\ninteractions\\cite{tosatti,herbut,lee} are unlikely to occur in\nundoped or moderately doped graphene. Theoretically, a perturbative\nFermi-liquid-type treatment of the Coulomb interactions yields a\nlogarithmic divergence of the Fermi velocity,\\cite{GGV94} a\nrenormalization of thermodynamic quantities such as the\ncompressibility\\cite{BM07} as well as to a control of the orbital\nmagnetic susceptibility.\\cite{PPVK10}\n\nThe situation is different if the graphene electrons are exposed to a strong magnetic field that quantizes their kinetic energy into\nnon-equidistant Landau levels (LLs), $\\epsilon_n=(\\lambda\\hbarv_{\\rm F}\/l_B)\\sqrt{2n}$, where $\\lambda=\\pm 1$ is the band index, $l_B=\\sqrt{\\hbar c\/eB}$ is\nthe magnetic length, and $n$ denotes the LL index.\\cite{CG07} The most prominent consequence of this relativistic LL quantization\nand the presence of a zero-energy LL for $n=0$ is a peculiar integer quantum Hall effect (QHE), with an unusual sequence\nof Hall plateaus.\\cite{NF05,ZTSK05} The LLs are highly degenerate, as in the usual 2D electron gas (2DEG) with a parabolic band\ndispersion, where the density of states per LL (and per unit area) is given by the flux density $n_B=1\/2\\pi l_B^2=eB\/h$, which is\nproportional to the perpendicular magnetic field. The filling of the LLs is then characterized by the ratio (filling factor)\n$\\nu=n_{el}\/n_B$ between the 2D electronic density $n_{el}$ and $n_B$.\nA partially filled LL may then be viewed as a strongly-correlated electron\nsystem with a quenched kinetic energy, and its most prominent manifestation is a fractional QHE that has recently been\nobserved in suspended graphene samples.\\cite{DA09,BK09} Prior indications for strong interactions stemmed from\nthe high-field QHE at $\\nu=0$, which indicates a stronger lifting of the four-fold spin-valley degeneracy of graphene than\nwhat one would expect from single-particle effects.\\cite{ZK06,GZ09}\n\nSimilarly to the $B=0$ case, the Coulomb interaction between electrons in completely filled LLs may be viewed as a weak perturbation\nbecause of the energy gap between adjacent LLs. Its role in the dispersion relation and Fermi velocity renormalization of graphene is an open question which has been addressed both theoretically\\cite{IWFB07,BM08,S07} and experimentally, in\nthe framework of transmission spectroscopy.\\cite{SH06,JS07,DG07,HS10,OP10} As compared\nto the 2DEG with a parabolic band dispersion (as in GaAs\/AlGaAs and Si\/SiGe heterostructures), the situation is strikingly different in graphene, where\nthe effect of electron-electron interactions may be probed at zero wave vector. Indeed, in the former,\nLL quantization leads to a set of equidistant LLs separated by the cyclotron frequency.\nKohn's theorem states that in these systems, homogeneous electromagnetic radiation can only couple to the center-of-mass coordinate.\nTherefore internal degrees of freedom associated with the Coulomb interaction cannot be excited by such optical probes.\\cite{K61}\nThen, the dispersion relation of spin-conserving magnetoplasmons at zero wave-vector is equal to the bare cyclotron energy,\nirrespective of existing electronic correlations.\\cite{KH84} A similar consideration holds also for spin wave (SW) modes,\nfor which Larmor's theorem states that the Coulomb interaction does not renormalize the zero-wave-vector dispersion of the\nspin excitons.\\cite{DKW88} However, the dispersion of spin-flip (SF) modes in a 2DEG are shifted from the bare cyclotron resonance\neven at zero wave-vector, due to electron-electron interactions. Therefore, these excitations are the only suitable modes\nto study the many-body effects in a 2DEG by means of optical measurements.\\cite{PW92,KW01,VW06}\n\n\nFurthermore, electron-electron interaction in the regime of the\ninteger QHE yield collective excitations that are different from\nthose in the 2DEG -- instead of inter-LL excitations with a rather\nweak wave-vector dispersion, called magneto-excitons (ME), one finds\nlinear magneto-plasmons that involve superpositions of different LL\ntransitions.\\cite{RFG09,RGF10} Also MEs that may play a role in the\nvicinity of $q=0$ have been studied theoretically in graphene, and\nit has been shown that Coulomb interactions yield a renormalization\nof the transition energy at zero wave vector\\cite{IWFB07,BM08,S10}\nthat indicate that Kohn's theorem does not apply to graphene.\nIn comparison to these works, here we put more emphasis on\nspin-changing modes and on the effect of LL mixing.\n\nA convenient way of assessing the magnetic-field strength is\nin terms of four characteristic length scales: the magnetic length\n$l_B$, the carbon-carbon distance $a$, the Fermi wavelength\n$\\lambda_F$ and the Thomas-Fermi screening length $\\lambda_{TF}\\sim\n\\lambda_F\/\\alpha_G$ where $\\alpha_G\\equiv e^2\/\\varepsilon v_F$\nmeasures the relative strength of Coulomb interactions. In practise,\nthe magnetic length is always much larger than the lattice spacing\n$l_B\\gg a$ because the flux per unit cell is much smaller than the\nflux quantum ($B\\ll 40000$~T). The weak LL mixing approximation --\nwhich we will use when studying the particle-hole excitations --\nrequires $e^2\/(\\varepsilon l_B \\omega_C)\\ll 1$, where $\\omega_C$ is\nthe cyclotron frequency, and corresponds to a strong field such that\n$l_B\\ll \\lambda_{TF}$. As in graphene $\\alpha_G$ is of order 1,\n$\\lambda_{TF}\\sim \\lambda_F$ and the weak LL mixing is also the\nsmall filling factor limit $l_B\\ll \\lambda_F$. In the following, we\nwill consider two limits: either a strong magnetic field (meaning\n$l_B\\ll \\lambda_F$ or typically $B\\gg 20$~T) or a weak magnetic\nfield (meaning $l_B\\gg \\lambda_F$ or typically $B\\ll 20$~T).\n\nIn this paper, we study both spin-conserving ME and spin-dependent\nSF and SW modes in the regime of the integer\nQHE. Following the scheme introduced by Kallin and Halperin\n(KH) for the 2DEG,\\cite{KH84} the Coulomb interaction is treated\nwithin the framework of the time-dependent Hartree-Fock (TDHF) \nand strong-field approximation [for which $e^2\/(\\varepsilon\nl_B\\omega_C)\\ll 1$, which insures that LL mixing is weak],\nthe validity of which is discussed. Indeed, we find that whereas SF\nand SW may be accounted for correctly in graphene within the KH approximation in the limit of a strong magnetic field and\nwhen the Fermi level lies near the $n=0$ LL, its validity in the\ntreatment of spin-conserving ME is questionable even in these\nlimits. This difference between MEs and SF (and SW) excitations\nstems from the depolarization term, accounted for in the\nrandom-phase-approximation (RPA), which is present only in the ME\ndispersion and which yields a strong LL mixing at non-zero values of\nthe wave vector. This LL mixing eventually leads to the formation of\nlinearly dispersing plasmon-type modes that have been obtained\nwithin an RPA treatment of the electronic polarizability in\ngraphene.\\cite{RFG09,RGF10}\n\n\n\nFinally, we consider the opposite limit of graphene in a\nweak magnetic field, and compute the exchange correction to the\nchemical potential. We find that the exchange correction to the\nsingle-particle dispersion presents the same dependence in the two\nlimits, strong and weak magnetic field, proportional to the square\nroot of the ultraviolet cutoff in the Landau levels. On the other\nhand, the exchange correction to the particle-hole dispersion\ndiverges logarithmically with the cutoff.\n\nThe paper is organized as follows. In Sec. \\ref{Sec:ExcMod}, we\nfirst revisit the Kohn's (Sec. \\ref{sec:Kohn}) and Larmor's (Sec.\n\\ref{sec:Larmor}) theorems in the context of graphene and their\nimpact on collective excitations in general. We then study the\nexcitonic modes in graphene in a strong magnetic field, within the\nKH approximation. In Sec. \\ref{Sec:MuEx} we calculate the\nexchange correction to the chemical potential, in the weak magnetic\nfield limit and\/or for highly doped samples. Our main conclusions\nare summarized in Sec. \\ref{Sec:Conc}, and the technical details of\nthe calculations are provided in the appendices.\n\n\n\n\\section{Excitonic modes in graphene in the integer QHE}\\label{Sec:ExcMod}\n\n\\begin{figure}[t]\n  \\centering\n   \\includegraphics[width=0.3\\textwidth]{fig\/Trans-1.pdf}\n\\caption{Sketch of the particle-hole excitations studied in the text. Each Landau level is split in two sub-levels, separated by the Zeeman gap, $\\Delta E_z=g\\mu_BB$. Here, $\\uparrow$ indicates $s_z=+1\/2$ and $\\downarrow$, $s_z=-1\/2$. Label ME stands for magneto-exciton (or magneto-plasmon) where the particle and the hole have the same spin.  SF denotes the spin-flip excitation, in which the electron and the hole do not only reside in different LLs but also have different spin. Finally, SW denotes the spin-wave mode, which is an intra-LL transition where the electron and the hole have a different spin orientation and where we have taken into account a non-zero Zeeman energy.}\n  \\label{Fig:Trans}\n\\end{figure}\n\n\n\n\n\nIn spin-flip modes, an electron is both promoted from one LL to the next one and its spin is reversed. They carry a spin $S_z=s^e_z  - s^h_z=\\pm1$, where $s^{e(h)}_z$ is the $z$-component of the electron (hole) spin. We use the term magneto-exciton (ME, sometimes also called magneto-plasmon in the literature) to denote spin-conserving excitations, where the electron and the hole reside in different LLs and have the same spin. In SW modes, the two particles have the same LL index but opposite spin. SF modes, MEs and SWs are the basic excitations of a quantum Hall system with a finite Zeeman splitting, as sketched in Fig. \\ref{Fig:Trans}.\n\n\\subsection{The fate of Kohn's theorem in graphene}\n\\label{sec:Kohn}\n\nBefore calculating the dispersion relations of the different excitonic modes, we discuss here qualitatively the expectations for\ngraphene with respect to the 2DEG with a parabolic band dispersion. In the latter, Kohn's theorem\\cite{K61} states that electromagnetic\nabsorption, irrespective of the strength of the Coulomb interaction, occurs only at\nthe cyclotron frequency $\\omega_C=eB\/m_b$, in terms of the band mass $m_b$.\nHere and in the remainder of this paper, we consider a system of units with $\\hbar\\equiv c \\equiv 1$. Because this absorption process\nis associated with an inter-LL transition from the last occupied to the first unoccupied level, it means that the lowest-energy\nME must converge to the non-interacting value at zero wave-vector. This statement remains valid also for higher harmonics, such that\nthe ME dispersion has $\\Omega_{ME}(q\\rightarrow 0)\\rightarrow m\\omega_C$, where $m=n_e-n_h$ is the difference between the LL index of the electron ($n_e$) and that of the hole ($n_h$).\n\n\\subsubsection{Kohn's theorem in the 2DEG}\n\nIn order to investigate the fate of Kohn's theorem in graphene, let\nus recall the main steps in the argument for the 2DEG. In the\nlatter, the Hamiltonian of $N$ non-interacting electrons can be\nexpressed in terms of  the gauge-invariant momenta\n$\\mathbf{\\Pi}_j={\\bf p}_j + e{\\bf A}({\\bf r}_j)$, where ${\\bf r}_j$ and ${\\bf p}_j$ are\nthe position and its conjugate (gauge-dependent) momentum,\nrespectively, of the $j$-th electron with charge $-e$ (we choose\n$e>0$ to be the {\\sl positive} elementary charge), \n\\begin{equation}\\label{zeroHam2DEG} H_0=\\frac{1}{2m_b}\\sum_{j=1}^N\n\\mathbf{\\Pi}_j^2. \\end{equation} From the total gauge-invariant momentum\n$\\mathbf{\\Pi}=\\sum_j \\mathbf{\\Pi}_j$, we define the raising and\nlowering operators $\\Pi^{\\pm}=\\Pi_x \\pm i \\Pi_y$, which satisfy the\ncommutation relations $[\\Pi^{\\pm},H_0]=\\mp (1\/m_bl_B^2)\\Pi^{\\pm}$\nwith the Hamiltonian $H_0$. This leads to the equation\n\\begin{equation}\\label{eigen}\nH_0\\left(\\Pi^{\\pm}\\left|\\psi^0\\right\\rangle\\right)=\\left(E^0 \\pm\n\\omega_C\\right)\\left(\\Pi^{\\pm}\\left|\\psi^0\\right\\rangle\\right), \\end{equation}\nwhich means that the application of $\\Pi^{\\pm}$ on an ($N$-particle)\neigenstate $|\\psi^0\\rangle$ (with energy $E^0$) of $H_0$ yields\nanother eigenstate with energy $E^0 \\pm \\omega_C$.\n\nThe first observation is that this equation remains valid also in\nthe presence of electron-electron interactions $V$ that\ncommute with the total momentum $[\\mathbf{\\Pi},V]$=0, such\nas the Coulomb interaction, if one replaces the non-interacting\nstate $|\\psi^0\\rangle$ by an eigenstate $|\\psi\\rangle$ of the full\nHamiltonian $H=H_0+V$, as well as the energy $E^0$ by that,\n$E$, of the state $|\\psi\\rangle$.\n\nSecond, one notices that the electromagnetic light field with\nfrequency $\\omega$ couples to the electronic system via the\nHamiltonian \n\\begin{equation}\\label{LMcoupl} H_{LM}(t) =\n\\frac{e}{2i\\omega} e^{-i\\omega t} {\\bf E}(\\omega)\\cdot \\sum_j {\\bf v}_j +\n\\textrm{ H.c.} , \\end{equation} \nwhere ${\\bf E}(\\omega)$ is the electric component\nof the light field and ${\\bf v}_j$ the velocity operator of the $j$-th\nelectron. In the 2DEG with a parabolic band dispersion, the velocity\noperator is readily expressed in terms of the gauge-invariant total\nmomentum, $\\sum_j {\\bf v}_j=\\mathbf{\\Pi}\/m_b$, such that the\nlight-matter coupling (\\ref{LMcoupl}) is linear in the operators $\\Pi^{\\pm}$.\nAs mentioned above, this induces then a transition from a state $|\\psi\\rangle$ with energy $E$ to a state $\\Pi^{\\pm}|\\psi\\rangle$\nwith energy $E\\pm \\omega_C$, i.e. the only absorption peak for light occurs at the cyclotron frequency $\\omega_C$.\\cite{K61}\n\n\\subsubsection{Difference in graphene}\nAlthough also in graphene the total gauge-invariant momentum\n$\\mathbf{\\Pi}$ commutes with the interaction Hamiltonian $V$\nbut not with $H_0$, one first notices that it may no longer be\nexpressed in terms of the velocity operators of (now\nrelativistic) electrons because of the vanishing band mass. The\nvelocity operator is a $2\\times 2$ matrix ${\\bf v}_j=v_{\\rm F}\n\\boldsymbol{\\sigma}_j=v_{\\rm F} (\\sigma_j^x,\\sigma_j^y)$, in terms of the\nPauli matrices $\\sigma^x$ and $\\sigma^y$, and it is not a conserved\nquantity even in the absence of interactions. The application of the\nvelocity operator on an eigenstate of the non-interacting\nHamiltonian $H_0=\\sum_jv_{\\rm F}\n[{\\bf p}_j+e{\\bf A}({\\bf r}_j)]\\cdot\\boldsymbol{\\sigma}_j$, for which\n$[{\\bf v}_j,H_0]\\neq 0$, yields, even in the absence of a magnetic field, spontaneous inter-band transitions that\nare at the origin of the so-called {\\sl zitterbewegung}.\\cite{K06}\nAs a consequence, the light-matter coupling Hamiltonian\n(\\ref{LMcoupl}), the form of which is also valid for graphene, may\nno longer be expressed in terms of $\\Pi^{\\pm}$. Indeed, the velocity\noperator in Hamiltonian (\\ref{LMcoupl}) yields transitions involving\nLLs with adjacent indices $n$ and $n\\pm 1$, as in the 2DEG, but the\n{\\sl zitterbewegung} translated to the magnetic-field\ncase\\cite{RZ08} furthermore yields inter-band excitations, such that\nthe dipolar selection rules $\\lambda_h,n\\rightarrow \\lambda_e,n\\pm\n1$ are associated with the energies\n\\begin{equation}\\label{CyclRes}\nE_{kin}^{(n,\\lambda_e,\\lambda_h)}=\\frac{v_{\\rm F}}{l_B}\\sqrt{2}\\left(\\lambda_e\\sqrt{n+1}-\\lambda_h\\sqrt{n}\\right),\n\\end{equation}\nwhere one expects absorption peaks. Therefore, already in the non-interacting limit, one expects a plethora of absorption\npeaks, that have indeed been observed experimentally,\\cite{SH06,DG07,JS07,HS10} and not a single cyclotron resonance as in the\ncase of the 2DEG with a parabolic dispersion relation.\n\nFurthermore, because the kinetic Hamiltonian (\\ref{zeroHam2DEG}) becomes $H_0=v_{\\rm F} \\sum_j\\mathbf{\\Pi}_j\\cdot\\boldsymbol{\\sigma}_j$ in\ngraphene, one loses the possibility of writing an equation of the type (\\ref{eigen}) for graphene, neither in\nterms of the total momentum $\\mathbf{\\Pi}$ nor with the help of $\\sum_j{\\bf v}_j$, which as we mentioned is not conserved.\nThere is thus no protection of the energies\n(\\ref{CyclRes}) when interactions are taken into account. Indeed, the latter renormalize the absorption energies, \\cite{IWFB07,BM08,S10}\nas we discuss below, in contrast to the 2DEG, where the absorption energy is protected by Kohn's theorem, and the\nME modes no longer converge to the non-interacting inter-LL transition energies\n\\begin{equation}\\label{TransEn}\nE_{kin}^{(n_e,n_h)}=\\frac{v_{\\rm F}}{l_B}\\sqrt{2}\\left(\\lambda_e\\sqrt{n_e}-\\lambda_h\\sqrt{n_h}\\right)\n\\end{equation}\nin the zero-wave-vector limit.\n\n\n\\subsection{Larmor's theorem applied to graphene}\n\\label{sec:Larmor}\n\nIn addition to ME excitations that do not involve the electronic\nspin, one may investigate spin excitations on rather general\ngrounds. Larmor's theorem states that in the long-wavelength limit,\nthe SW dispersion tends to the (bare) Zeeman splitting,\n$\\Omega_{SW}(q\\rightarrow 0)\\rightarrow g\\mu_BB$.\\cite{DKW88} This\ntheorem may be understood from the symmetries of the Hamiltonian\n$H=H_0 + H_{int} + H_Z$. In the absence of the Zeeman term $H_Z$,\nthe Hamiltonian respects the SU(2) symmetry associated with the\nelectronic spin, i.e. both the total spin operator $\\hat{S}_{tot}^2$\nand any of the components $\\hat{S}_{tot}^{\\mu}$, for $\\mu=x,y,z$,\ncommutes with the Hamiltonian. Since one cannot diagonalize all\ncomponents of the total spin simultaneously, one needs to choose a\nparticular one, and this is naturally the one chosen by the Zeeman\neffect (here $\\hat{S}_{tot}^z$), such that the full Hamiltonian\ncommutes with $\\hat{S}_{tot}^2$ and $\\hat{S}_{tot}^z$. The quantum\nnumbers associated with the spin, $S$ and $S^z$, are therefore good\nquantum numbers for the full interacting $N$-particle Hamiltonian,\nsuch that all possible states have energies $E=E(S,S^z, ...)\n+ g\\mu_B B S^z$, where the dots $...$ represent other quantum\nnumbers that characterize the interacting system. The essence of this expression is that the full interacting\n$N$-particle system may be viewed as a large spin that precesses in \na magnetic field with the fundamental (Larmor) frequency $\\omega_L=g\\mu_B B$.\nWhereas this frequency is affected by the (crystalline) environment via\nthe effective $g$-factor, the latter remains unaltered by the \nelectron-electron interactions.\nApplied to the present problem of collective excitations, this means\nthat the Zeeman term does not represent a further complication to\nthe SU(2) symmetric Hamiltonian $H_0+ H_{int}$, which thus needs to\nbe diagonalized first.\n\nThese rather obvious considerations allow us to understand easily\nLarmor's theorem if one notices that, in the absence of a Zeeman\neffect, the SW mode is just the Goldstone mode of a ferromagnetic\nground state in which all spins are spontaneously polarized. This\nferromagnetic state arises due to exchange-interaction effects when\nnot all subbranches of a particular LL are completely\nfilled.\\cite{MS95} The Goldstone mode is characterized by a\ndispersion relation that vanishes (as $q^2$ for a SW mode\n\\cite{HH69}) in the zero-wave-vector limit, $\\omega_G(q\\rightarrow\n0)\\rightarrow 0$, which means that the different states of the\nground-state manifold (i.e. the different polarizations) are\nconnected by a global rotation of zero energy cost that is precisely\nthe $q=0$ Goldstone mode. In the presence of the Zeeman effect,\nwhich chooses a particular orientation of the total spin, one thus\nobtains a SW mode that tends to the energy $\\Omega_{SW}(q\\rightarrow\n0)\\rightarrow g\\mu_B B S_z$, where $S_z=1$, as stated by Larmor's\ntheorem.\n\nOne notices that, in contrast to the above discussion of Kohn's theorem, the (non-)relativistic character of $H_0$ has never\nplayed a role in the argument, and Larmor's theorem therefore also applies in the case of graphene. Moreover, one is confronted\nin graphene with an additional two-fold valley degeneracy, that may be taken into account by an SU(2) valley isospin. Although\nthe SU(2) valley symmetry is not respected by the interaction Hamiltonian, the symmetry-breaking terms are strongly suppressed\n(by a factor of $a\/l_B\\sim 0.005\\sqrt{B\\text{[T]}}$, in terms of the carbon-carbon distance $a=0.14$ nm) such that the interaction\nHamiltonian is approximately SU(2) valley-symmetric.\\cite{GMD06,G10} Therefore the above arguments apply also to possible valley-ferromagnetic\nstates in graphene, i.e. there are valley-isospin-wave modes that vanish in the $q\\rightarrow 0$ limit and that may become\neventually gapped by a ``valley-Zeeman'' effect $H_{v-Z}$ that, if it may be written in terms of components of the total valley-isospin,\nyields a simple energy offset to the dispersion. In the remainder of the paper, we concentrate on collective excitations that involve\nonly the physical spin.\n\nIn addition to this generalization of Larmor's theorem to the valley isospin, it may also be generalized to the\nSF modes, which involve not only different spin states but also different LLs. The dispersion of the collective SF modes may be fully understood from the\nHamiltonian $H_0 + H_{int}$, whereas the energy $g\\mu_BB S_z$ associated with the Zeeman effect can be simply added at the end\nof the calculation as a global (wave-vector independent) constant. However, as we shall discuss below, the energy of the\nSF modes does not converge to the simple sum of the Zeeman and the transition energies (\\ref{TransEn}) in the zero-wave-vector\nlimit, but they are renormalized by the interaction energy, both in graphene and in the 2DEG.\\cite{PW92}\n\n\n\n\n\n\\subsection{Dispersion relation of the excitonic modes}\\label{Sec:KHappr}\n\nIn graphene, the energies of ME, SW and SF modes can be expressed as:\n\\begin{eqnarray}\n\\Omega_{ME}(q)&=&E_{kin}^{(n_e,n_h)}+\\Delta E^{(n_e,s_z^e;n_h,s_z^h)}(q)\\label{Eq:ME}\\\\\n\\Omega_{SW}(q)&=&g\\mu_BBS_z+\\Delta E^{(n_e,s_z^e;n_h,s_z^h)}(q)\\label{Eq:SW}\\\\\n\\nonumber\n\\Omega_{SF}(q)&=&E_{kin}^{(n_e,n_h)}+g\\mu_BBS_z+\\Delta E^{(n_e,s_z^e;n_h,s_z^h)}(q)\\\\\n\\label{Eq:SF}\n\\end{eqnarray}\nwhere $S_z$ is the $z$-component of the exciton spin, and $E_{kin}^{(n_e,n_h)}$ is the\ntransition energy in the absence of interactions given by Eq.\n(\\ref{TransEn}). The contribution $\\Delta E^{(n_e,s_z^e;n_h,s_z^h)}$\nconsists of three terms (see Appendix \\ref{App:Poles} for details):\na depolarization or exchange term $E_{x}(q)$, which is accounted for\nin the RPA approximation, a direct Coulomb interaction between the\nelectron and hole (vertex corrections) $E_v(q)$, and the difference\nbetween the exchange self-energy of the electron and that of the\nhole, $E_{exch}=\\Sigma_e-\\Sigma_h$. Notice that $E_{x}(q)$ is only\nrelevant for the ME, because only particles with the same spin can\nbe recombined by means of electron-electron interactions.\n\nIt must be kept in mind that, in a 2DEG, the RPA term, which\ndetermines the maximum of the ME dispersion at a wave-vector $q\\sim\n1\/R_C$ in the TDHF approximation, mixes different LLs, with a mixing\namplitude on the order of $e^2\/(\\varepsilon\nl_B\\omega_C)$.\\cite{GV05}  Here $R_C=k_Fl_B^2$ is\nthe cyclotron radius, where the Fermi momentum in terms of the index $N_F$ of the topmost\nfully occupied LL is $k_F=\\sqrt{2N_F+1}\/l_B$ for a 2DEG and $k_F=\\sqrt{2N_F+\\delta_{N_F,0}}\/l_B$ for graphene. This needs to be distinguished from the LL mixing\nat $q=0$, which determines the stability of the LLs in the presence\nof electron-electron interactions and which scales as\n$e^2\/\\varepsilon R_C \\omega_C$. Although the stability of the LLs in\ngraphene is determined by the ratio between the Coulomb energy\n$e^2\/\\varepsilon R_C$ and the LL separation $\\Delta_n=\n(\\sqrt{2}v_{\\rm F}\/l_B)(\\sqrt{N_F+1}-\\sqrt{N_F})\\sim v_{\\rm F}\/R_C$,\\cite{G10}\nwhich happens to be the scale-invariant fine-structure constant\n$\\alpha_G=e^2\/\\varepsilon v_{\\rm F}$, the situation is again different at\nthe maximum of the ME dispersion at $q\\sim 1\/R_C$. The order of\nmagnitude of the $q\\neq 0$ LL mixing in graphene may be obtained by replacing in\n$e^2\/(\\varepsilon l_B\\omega_C)$ the 2DEG cyclotron frequency\n$\\omega_C=eB\/m_b$ by the density-dependent cyclotron frequency\n$\\omega_C(\\mu)=eBv_F^2\/\\mu$, where $\\mu=(v_{\\rm F}\/l_B)\\sqrt{2N_F}$ is the\nchemical potential. As a consequence, the validity of the KH\napproximation fails not only for weak magnetic fields, as in the\nstandard 2DEG, but also at high and intermediate filling factors\nbecause the effective cyclotron frequency in graphene decreases as\nthe number of filled LLs increases, leading to an increase of the LL\nmixing. Therefore, strictly speaking, the results of this section\nwill be valid only in the strong-$B$ limit and for $N_F$  near 0.\nHowever, we will see that the KH approximation can still be\napplied for spin-dependent excitations (SW and SF) slightly away\nfrom half-filling, but not to spin-conserving modes (ME). This is a\nconsequence of the absence of the depolarization term $E_x(q)$,\nwhich is the main source of LL mixing, in SW and SF modes, whereas\nit constitutes the main contribution to the dispersion of ME modes.\n\n\nAfter these general considerations on collective excitations, we now turn to a discussion of the modes at particular integer\nfilling factors, which are described by\n\\begin{equation}\\label{filling}\n\\nu=4N_F-2+2\\left(\\nu_{\\uparrow}^{N_F} + \\nu_{\\downarrow}^{N_F}\\right),\n\\end{equation}\nwhere $N_F$ is the index of the top most fully occupied LL, $0\\leq \\nu_{\\sigma}^n\\leq 1$\nis the filling of the spin-$\\sigma$ branch of the $n$-th LL, and the factor of 2 accounts for the two-fold\nvalley degeneracy of each spin branch.\n\n\n\n\\begin{figure}[t]\n  \\centering\n    \\subfigure[]{\\label{DisperNF0}\\includegraphics[width=0.35\\textwidth]{fig\/DisperNF0.pdf}}\n     \\subfigure[]{\\label{TransNF0}\\includegraphics[width=0.35\\textwidth]{fig\/TransNF0.pdf}}\n \\caption{Dispersions (in units of $e^2\/\\varepsilon l_B$) of the excitonic modes studied for $\\nu=0$, i.e.\n$N_F=0$, $\\nu_{\\downarrow}^0=1$ and $\\nu_{\\uparrow}^0=0$. SW (dotted\ngreen line), ME$_{1,2}$ (dashed blue and red lines, respectively)\nand SF$_{1,2}$ (solid blue and red lines, respectively) are\nrepresented. The thin horizontal line represents the difference in\nkinetic energy between the electron and the hole $E_{kin}^{(1,0)}$.\nWe have used for the Zeeman term an unphysically large value\n$g\\mu_BB=(1\/10)(e^2\/\\varepsilon l_B)$, for illustration reasons. (b)\nSchematic representation of the excitonic modes studied. Notice that\nME$_1$ and ME$_2$ are degenerate in the $N_c\\rightarrow \\infty$\nlimit, as well as the SF$_1$ and SF$_2$.}\n  \\label{NF0}\n\\end{figure}\n\n\n\\subsection{Modes at filling $\\nu=0$}~\n\nAt the charge neutrality point (for a filling factor $\\nu=0$), the\nFermi level is in the $n=0$ LL (i.e. $N_F=0$), with the\nspin-$\\downarrow$ branch completely filled ($\\nu_{\\downarrow}^0=1$)\nand an empty spin-$\\uparrow$  ($\\nu_{\\uparrow}^0=0$). The dispersion\nof the excitonic modes for this situation is shown in Fig.\n\\ref{DisperNF0}.  The transitions corresponding to the different\nexcitations are schematized in Fig. \\ref{TransNF0}. To more easily\ndistinguish between the different modes, we use the notation $\\Delta\nE_{N_F;\\nu^{N_F}_{\\downarrow},\\nu^{N_F}_{\\uparrow}}(q)$. Therefore,\nthe dispersion of the magnetoexciton modes ME$_{1,2}$, Eq.\n(\\ref{Eq:ME}) will correspond to the kinetic particle-hole energy\ndifference plus a renormalization due to electron-electron\ninteractions, $\\Delta E^{ME_{1,2}}_{0;1,0}(q)$, which reads $\\Delta\nE^{(1,-1\/2;0,-1\/2)}(q)=\\Sigma^{ME_1}_{0;1,0}+V^d_{1,0;1,0}(q)+4V^x_{1,0;1,0}(q)$\nfor ME$_1$ and $\\Delta\nE^{(0,+1\/2;1,+1\/2)}(q)=\\Sigma^{ME_2}_{0;1,0}+V^d_{0,-1;0,-1}(q)+4V^x_{0,-1;0,-1}(q)$\nfor ME$_2$, where the expressions for $\\Sigma^{ME_{1,2}}_{0;1,0}$ are\ngiven in Appendix \\ref{App:Self}. Here $V^x(q)$ are matrix elements\nof the Hartree term, in which a particle-hole pair recombines,\nexciting a new particle-hole pair (the usual bubble diagrams). On\nthe other hand, $V^d(q)$ is the Fock term, which accounts for the\ndirect interaction of the excited electron and hole (ladder\ndiagrams). Notice that, due to particle-hole symmetry at this\nfilling,  $\\Delta E^{ME_1}_{0;1,0}(q)=\\Delta E^{ME_2}_{0;1,0}(q)$\nand the two modes are degenerate. The first thing one notices is\nthat the dispersion at $q=0$ is shifted with respect to\n$E_{kin}^{(1,0)}$ [horizontal line in Fig. \\ref{DisperNF0}]. This is\na consequence of the non-applicability of Kohn's theorem in\ngraphene, as discussed in Sec. \\ref{sec:Kohn}, whereas in the 2DEG\nthe theorem is satisfied due to a cancelation between the exchange\nself-energy and the $q=0$ vertex correction, $E_v(q=0)=-E_{exch}$.\nWhereas the behavior of the dispersion at short wavelength is\ndominated by the exchange self-energy and vertex correction terms\n(see Fig. \\ref{NF0decomp}), the peak in the dispersion in the\nlong-wavelength regime is due to the exchange interaction (the RPA\nterm). Furthermore, it is worth  pointing out that this contribution\nrapidly increases as one fills more LLs, as we will see\nbelow. This is a direct consequence of the relativistic quantization\nof the graphene LL spectrum, leading to an important LL mixing \nat higher fillings and, as a consequence, building an\nunusual particle-hole excitation spectrum.\\cite{RFG09}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.35\\textwidth]{fig\/FigDecompNF0.pdf}\n\\caption{(Color online) Decomposition of the ME (full red line) and SF (full blue line) mode for $\\nu=0$ into the interaction-related\ncomponents, exchange self-energy (dotted line, red for ME and blue for SF), vertex correction (dashed-dotted line) and RPA term\n(red dashed line), in units of $e^2\/\\varepsilon l_B$. The kinetic energy, which yields the same constant offset for both modes, is not taken into account in this\ndecomposition.}\n  \\label{NF0decomp}\n\\end{figure}\n\n\nThis RPA contribution is absent, however, in the SF and SW modes\n(see Fig. \\ref{NF0decomp}). As a consequence, the LL mixing for\nthese modes is less important and makes the KH\napproximation, as the one applied here, a justified method\n(especially at strong magnetic fields and for the chemical potential\nat or near the zero energy LL). The results for these modes are also\nshown in Fig. \\ref{DisperNF0}. Electron-electron interactions enter\nin the dispersion of the former through the term $\\Delta\nE^{SF_{1,2}}_{0;1,0}(q)$, which again due to particle-hole symmetry\nleads to degenerate modes with contributions $\\Delta\nE^{(1,+1\/2;0,-1\/2)}(q)=\\Sigma^{SF_1}_{0;1,0}+V^d_{1,0;1,0}(q)$ and\n$\\Delta\nE^{(-1,-1\/2;0,+1\/2)}(q)=\\Sigma^{SF_2}_{0;1,0}+V^d_{0,-1;0,-1}(q)$\nrespectively. As in the 2DEG, the $q\\rightarrow 0$ limit of the\ndispersion of these modes is renormalized from the noninteracting\nvalue, $E_{kin}^{(n_e,n_h)}+g\\mu_BBS_z$. This makes possible the\nstudy of correlation effects by optical measurements.\n\n\n\nOn the other hand, Larmor's theorem still applies in graphene, as one may see from the dispersion of the SW mode. This mode has a $q=0$ dispersion equal to the Zeeman splitting $g\\mu_BBS_z$, and a contribution due to electron-electron interaction $\\Delta E^{(0,+1\/2;0,-1\/2)}(q)=\\Sigma^{SW}_{0;1,0}+V^d_{0,0;0,0}(q)$, which is finite only at non-zero wave-vectors. This implies that, as in the 2DEG, the $g$-factor is not influenced by the Coulomb interaction. The SW disappears if we fill the next LL, for  $\\nu=2$, with $N_F=0$, $\\nu_{\\downarrow}^0=1$, and $\\nu_{\\uparrow}^0=1$. The dispersions of the ME and SF modes (not shown here) are similar to the previous case with the difference that the degeneracy of the latter is lifted, but only by a constant term equal to the double of the Zeeman energy, in agreement with the arguments of Sec. \\ref{sec:Larmor}.\n\n\n\\subsection{Modes at filling $\\nu=4$}~\n\n\n\\begin{figure}[t]\n  \\centering\n    \\subfigure[]{\\label{DisperNF1}\\includegraphics[width=0.35\\textwidth]{fig\/DisperNF1.pdf}}\n     \\subfigure[]{\\label{TransNF1}\\includegraphics[width=0.35\\textwidth]{fig\/TransNF1.pdf}}\n      \\subfigure[]{\\label{SFNF1}\\includegraphics[width=0.35\\textwidth]{fig\/SFNF1.pdf}}\n \\caption{(a) Same as Fig. \\ref{NF0} but for $\\nu=4$, with\n$N_F=1$, $\\nu_{\\downarrow}^1=1$ and $\\nu_{\\uparrow}^1=0$. ME$_{1,2}$ (dashed blue and red lines, respectively) and SF$_{1,2}$ (solid blue and red lines, respectively) are represented. The thin horizontal lines represent the difference in kinetic energy between the electron and the hole  $E_{kin}^{(1,0)}$ and $E_{kin}^{(2,1)}$. (b) Schematic representation of the excitonic modes studied. The degeneracy that occurs at $N_F=0$ is completely lifted at this filling for both, ME and SF modes. For clarity, we show in (c) the SF modes separately.}\n  \\label{NF1}\n\\end{figure}\n\n\nThe relativistic nature of the LLs in graphene is clearly visible if\nwe go beyond $N_F=0$, as shown in Fig. \\ref{TransNF1} for a filling\nfactor of $\\nu=4$, with $N_F=1$, $\\nu^1_{\\downarrow}=1$, and\n$\\nu^1_{\\uparrow}=0$. At this filling, the non-equidistancy of the\nLLs lifts the degeneracy of the two ME modes, as well as the two SF\nmodes. In addition, the exchange contribution to the ME modes, which\nleads to the peak in their dispersion, increases as we decrease the\nseparation between the LLs of the electron and the hole. \nThis yields a strong mixing among the different\nbranches of MEs, as may be seen in  Fig. \\ref{DisperNF1}. In fact,\nthe height of the peak associated with the ME$_1$ [with $n_e=2$ and\n$n_h=1$, as represented in Fig. \\ref{TransNF1}], is larger than that\nof ME$_2$ (with $n_e=1$ and $n_h=0$). This is due to the linear\ndispersion of the spectrum, which enhances the quantum effects as we\ngo to higher filling factors. \nTaking into account that we are showing here only two of the\nspin-conserving excitations possible for this filling (the ones\ninvolving the more adjacent LLs to the chemical potential), one can\nconclude that no {\\it single} MEs will be accessible experimentally\nat finite wave-vectors, but a superposition of them. Therefore, the\nTDHF method in the strong-field approximation is not valid for the\nspin-conserving modes, and the inclusion of a much higher number of\nmodes is necessary to obtain a reliable result. In fact, this\noverlap of different MEs leads to a new set of collective modes, the\nlinear magneto-plasmons, which have been studied\nelsewhere.\\cite{RFG09}\n\nNotice that the above arguments are valid only at non-zero values of the wave-vector, whereas the LL mixing effects\nare less pronounced at $q=0$, which is the relevant ME energy in magneto-optical experiments.\\cite{SH06,DG07,JS07,HS10} However,\nas we have mentioned above, also at $q=0$ the ME energy, which is the inter-LL transition energy measured in spectroscopy, is\nrenormalized due to electron-electron interactions.\n\n\nThe mixing between different contributions is less dramatic for the\nSF modes, as one sees in Fig. \\ref{SFNF1}, where we show a plot with\nonly the SF modes at this filling. One notices how the two modes are\nclearly decoupled, making the use of the KH approximation more\njustified,  because of the absence of the RPA term which is\nresponsible, in the ME case, for the LL mixing at non-zero values of\nthe wave-vector. Although not too clearly, it is appreciable that\nthe number of relative extrema (maxima and minima) in the dispersion\nof SF$_1$ (blue line) is higher than for for SF$_2$ (red line). This\nis directly related to the node structure of the (hole-) LL\nwave-function.\\cite{RGF10} These maxima and minima lead to hot spots\nin the dispersion that may be detected by Raman spectroscopy\ntechniques.\\cite{EW99}\n\n\n\n\n\n\\section{Renormalization of the chemical potential}\\label{Sec:MuEx}\n\nTo gain further insight into the effect of electronic interactions in a\ngraphene flake, we calculate in this section the exchange correction\nto the chemical potential,  from a density-matrix approach. This is\nthe first step toward including electron-electron interaction in the\nsystem. The correction is intrinsically related to the antisymmetry\nof the electronic wave-function, which implies, even in the absence\nof interactions, a certain amount of correlation between the\npositions of two particles with the same spin. Furthermore, its sign\nis always negative, due to the fact that it is the interaction of\neach electron with the positive charge of its exchange hole. \nOne of the effects of Coulomb interaction is a renormalization of\nthe chemical potential $\\mu$, which at zero temperature is the\npartial derivative of the total energy with respect to the number of\nparticles. It contains a contribution from the kinetic energy and also\nfrom interactions. The latter can be written as a mean-field\ncontribution plus correlation: $\\mu=K+\\mu^{ex}+\\mu^c$, where $K$ is\nthe kinetic energy, and $\\mu^{ex}$ and $\\mu^c$ are the exchange and\ncorrelation corrections to the chemical potential, respectively. As usual, the\ndirect (Hartree) mean field contribution does not appear as it is\ncompensated by the positively charged background (or neutralizing\nbackground), see the jellium model. Furthermore, the exchange\ninteraction can lead to a ferromagnetic instability in a dilute\nelectron gas.\\cite{GV05} In graphene, ferromagnetism due to the\nexchange interaction between Dirac fermions has also been\nstudied.\\cite{PGC05}  In a magnetic field, $\\mu^{ex}$ can be\nobtained from the pair correlation function $g(r)$ (see Appendix\n\\ref{App:CorrFunc} for details of the calculation) as\n\\begin{equation}\\label{Eq:muex}\n\\mu^{ex}=\\bar{n}\\int d^2{\\bf r}\\frac{e^2}{\\varepsilon r}[g(r)-1]\n\\end{equation}\nwhere $\\bar{n}=4(1+N_c+N_F)\/(2\\pil_B^2)$ is the electron density for\ngraphene in a magnetic field. $N_F$ is the index of the last\noccupied LL, related to the filling factor by $\\nu=4N_F+2$, and\n$N_c$ is a cutoff chosen such that $(4N_c+2)N_B=2N_{u.c.}$, where\n$N_B={\\cal A}\/2\\pi l_B^2$ is the degeneracy of each LL, ${\\cal A}$\nis the surface of the sample, $N_{u.c.}$ is the number of occupied\nunit cells in the system, the factor 2 is due to spin\ndegeneracy, and $4N_c+2$ is the number of filled sub-levels of the\nvalence band for undoped graphene. $N_c$ is the index of the\nlast LL in the band (a kind of bandwidth) and is roughly given by\n$N_c \\approx N_{u.c.}\/(2N_B)=2\\pi l_B^2\/(3\\sqrt\n3a^2)\\approx40000\/B[T]$ which is always much greater than 1 in\npractise. The fact that $N_c\\gg 1$ is just the statement\nthat, with available magnetic fields, the flux per unit cell is\nalways much smaller than the flux quantum. In this respect, we are\nalways in the weak field limit. An exact solution of Eq.\n(\\ref{Eq:muex}) is possible in the limit $N_c,N_F\\gg 1$, as shown in\nEq. (\\ref{Eq:muexDM}). This correction would eventually involve a renormalization of $N_F$, this is, a shift \nof the chemical potential as compared to the non-interacting case.\n\n\nNotice that, contrary to the strong magnetic field assumption done\nin the previous section, this is the opposite case, namely the weak\nmagnetic field limit.  The strong field limit is actually the\nKH approximation of weak LL mixing.\\cite{KH84} As stated in Sec.\n\\ref{Sec:KHappr}, the criterion for a weak LL mixing is\n$e^2\/(\\varepsilon l_B \\omega_c) \\ll 1$. In graphene, because the\nfine structure constant $\\alpha_G=e^2\/\\varepsilon v_F$ is of order\none, it means that $k_Fl_B \\ll 1$, which means $N_F \\approx 0$ (i.e.\n$N_F \\ll 1$) or in other words $B \\gg 20$T. This is the assumption\nmade in the previous section, whereas in this section we assume the\nopposite limit ($B\\ll 20$T or $N_F\\gg1$). In a standard 2DEG with a\nparabolic band, the weak magnetic field limit implies that the\ntypical Coulomb energy exceeds the cyclotron frequency $\\omega_C$.\nThis allows us to start from the Landau Fermi liquid theory at zero\nmagnetic field.\\cite{AG95} In the case of graphene, this limit is\neven more relevant due to the relativistic quantization of the\nspectrum into non-equidistant LLs, the relative separation of which\ndecreases as the energy increases. Therefore, even in a strong\nmagnetic field, the strength of the Coulomb interaction can be much\nhigher than the separation between the LLs adjacent to the chemical\npotential (the effective cyclotron frequency in graphene) if the\nsystem is sufficiently doped. Further simplification is possible if\n$1\\ll N_F \\ll N_c$. In this limit we obtain (see Appendix\n\\ref{App:CorrFunc}) that the exchange correction to the Fermi\nenergy behaves asymptotically as\n\\begin{equation}\\label{Eq:muexDMAsimp}\n\\mu^{ex}\\simeq -\\frac{e^2}{\\varepsilon l_B} \\frac{16\\sqrt{2}}{3\\pi}\\sqrt{N_c}.\n\\end{equation}\nThis contribution is expected since the energy calculated\nabove includes the interaction energy of the vacuum of negative\nenergy particles. It is interesting to compare this leading behavior\nof the exchange energy, valid for high filling factors, to the\nexchange self-energy obtained in Appendix \\ref{App:Self} valid at\nlow fillings [see e.g. Eqs. (\\ref{Eq:SelfNF0})-(\\ref{Eq:SelfNF1})\nfor $N_F=0$ and 1 respectively]. In the two cases we obtain the same\n$\\sim N_c^{1\/2}$ leading behavior.\\footnote{Notice that this is also\nthe behavior found by Aleiner and Glazman for a 2DEG at high filling\nfactors if we replace $N_c$ by $N_F$. See e. g. Eq. (B28) of Ref.\n\\onlinecite{AG95} where they consider a single parabolic band (no\nneed of ultraviolet cutoff).} Furthermore, our results agree\nwith the exchange contribution calculated for graphene at zero\nmagnetic field, where a $\\Sigma^{ex}\\sim-e^2k_c\/\\varepsilon$\ncontribution was found, $k_c\\sim 1\/a$ being an UV cutoff in\nmomenta.\\cite{HHS07} Taking into account  that $N_c\\sim (l_B\/a)^2$,\nour results for graphene in a magnetic field qualitatively agree\nwith those at $B=0$. Notice that whereas $\\mu^{ex}$ diverges as\n$N_c^{1\/2}$ for the single particle dispersion, the dispersion of a\nparticle-hole pair diverges only logarithmically, because the terms\nproportional to the square root of $N_c$ for each particle cancel\neach other, leading to a behavior $E_{exch}\\propto \\log{N_c}$. This\ndivergence can be reabsorbed into a renormalization of the Fermi\nvelocity,\\cite{GGV94} and its effect for cyclotron resonance\nmeasurements has been studied in detail by Shizuya.\\cite{S10} This\nrenormalization of the chemical potential due to Coulomb interaction\nshould affect the scanning single-electron transistor measurements\nof compressibility in graphene.\\cite{MY07}\n\n\n\n\\section{Summary and conclusions}\\label{Sec:Conc}\n\nIn conclusion, we have studied the SF, SW and\nME (or magneto-plasmon) modes in graphene in the\ninteger QHE regime, in the Kallin-Halperin approximation.\nThe ME dispersion in a 2DEG is not renormalized\nin the long-wavelength limit due to Kohn's\ntheorem for systems with a parabolic band and Galiean invariance. As\na consequence, the correction due to the direct interaction between\nthe electron and the hole is neutralized by their difference in\nexchange self-energy, $E_v(q=0)=-E_{exch}$, leading to a dispersion\nthat tends to $m\\omega_C$ at zero wave-vector.\\cite{PW92} In\ngraphene, Kohn's theorem does not apply and the dispersion of the ME\nmodes is renormalized due to many-body effects even at ${\\bf q}=0$.\n\nOn the other hand, virtual transitions from the vacuum (valence\nband) enhance the depolarization term of the spin-conserving ME\ndispersion, which enters through the RPA contribution and which\nleads to an important LL mixing. We have shown that the mixing is\nhigher as we increase the LL filling and\/or decrease the magnetic\nfield, invalidating the applicability of KH approximation\nfor $\\nu\\ge 2$, which needs to be restricted to the large-field\n$N_F=0$ case.\\cite{IWFB07,BM08} One of our main conclusions is that, for ME modes,\nmethods involving more inter-LL transitions than only one\nneed to be considered in the calculation of the spin-conserving collective\nexcitations. This superposition of several inter-LL transitions is at the\norigin of the strongly-dispersing linear magneto-plasmons, which have been obtained\nwithin an RPA treatment of the electron-electron interactions.\\cite{RFG09,RGF10}\n\nIn contrast to the spin-conserving ME modes, the depolarization term\nis absent in collective excitations where the particle and hole components have \nopposite spin, and the amount of mixing is less\nimportant. Therefore the KH approximation can still be used\nfor these modes in undoped or slightly doped graphene in a strong\nmagnetic field. In a 2DEG, the zero-wave-vector limit of the\nKH correction of SF modes has a finite contribution, because\n$E_v(q=0)=-(1\/2)E_{exch}$ in this case. In graphene, the dispersion\nof these modes is also renormalized at zero wave-vector and leads to\na correction that could be detected in inelastic light scattering\nexperiments, by using the same techniques as for a 2DEG.\\cite{PW92,KW01,VW06,EW99} In contrast to Kohn's theorem, we have shown that Larmor's theorem\napplies to graphene, so that the $q\\rightarrow 0$ limit of the SW\ndispersion is equal to the Zeeman splitting and the $g$-factor is\nindependent of many-body interaction, as in a standard\n2DEG.\\cite{DKW88} In addition, the $g$-factor is also only\nweakly affected by band effects in graphene: the effective\n$g$-factor was measured to be close to its bare value of 2, see Ref.\n\\onlinecite{ZK06}.\n\nFinally, we have calculated the exchange shift of the chemical\npotential in the weak-magnetic-field limit. We have found that, as\nfor strong magnetic fields, the exchange correction to the chemical\npotential diverges with the ultraviolet cutoff as $\\sim N_c^{1\/2}$.\nHowever, when the dispersion of an electron-hole pair is considered,\nthe correction associated with the difference in exchange\nself-energy between the particle and the hole, diverges only\nlogarithmically. This correction leads to a renormalization of\nthe Fermi velocity that seems to explain some recent\nexperimental results.\\cite{JS07,DG07,HS10}\n\n\n\\begin{acknowledgments}\nWe thank M. I. Katsnelson for many useful discussions. This work was\nfunded by ``Triangle de la Physique'' and the EU-India FP-7\ncollaboration under MONAMI.\n\\end{acknowledgments}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFast reliable spin manipulation in quantum dots is one of\nthe challenges in spintronics and semiconductor-based quantum\ninformation. The design of corresponding gates\ncan be based on electric dipole spin resonance where the spin-orbit coupling (SOC)\n\\cite{Rashba1,Nowack,Pioro,Golovach06}\nallows on-chip spin manipulation by external electric field\nas well as electric read-out of spin states.\\cite{Levitov03} Without external driving, SOC effects on localized\nin quantum dots electrons are very weak and lead to long spin relaxation times.\\cite{Khaetskii01,Fabian05}\nThe important questions here are how fast can the gate operate,\nwhat limits the manipulation rate, and how efficient\nis the spin manipulation in terms of the achievable spin configurations.\\cite{GomezLeon11}\nIt seems that a stronger driving field allows for a faster spin manipulation,\nas predicted in a simple Rabi picture of the driven oscillations.\nThis picture is applicable for a single quantum dot with a parabolic confinement,\nwhere the electron displacement from the equilibrium is linear in the applied electric\nfield.\\cite{Jiang06} However, double quantum dots where tunneling plays the crucial role\nfor the orbital dynamics, and the corresponding energy scales are different from a single quantum dot,\nare more promising for observation of new physics and applications in quantum information\ntechnologies.\\cite{Petta05} The tunneling makes the description of the\nSOC puzzling since the electron momentum\nis not a well-defined quantity at under-the-barrier motion, and the tunneling rate\ncan become strongly spin-dependent.\\cite{Amasha,Stano}\nIn addition, the double dots provide a possibility to study free and driven interacting qubits.\\cite{Shitade11,Nowak}\nHere we concentrate on one-dimensional systems attractive for spintronics \\cite{quay2010,Pershin04,Malard11}\nand building quantum dots \\cite{Nadj2010,Nadj-Perge2012,Bringer2011}\nand consider spin manipulation in single-electron double quantum dot\n\\cite{Ulloa06,Sanchez,Wang,Zhu,Wang11,Borhani11} by periodic electric field.\n\nWe show that even for a basic quantum system such a\nsingle electron spin, the efficiency and time scale of the\nmanipulation strongly depend on the electron orbital motion\nand, as result, to an unexpected dependence on the external electric field.\\cite{diamond_science}\nThe nonlinearity of the spin and charge dynamics is expected to lead to\nunusual consequences on the driven spin behavior.\\cite{Rashba11}\nIn a multilevel system Rabi spin oscillations are slowed down if the field is\nsufficiently strong, which challenges efficient spin manipulation.\nWe restrict ourselves to the single electron dynamics to demonstrate\nin the most direct way the nontrivial mutual effect of coordinate and spin motion on the\nRabi oscillations. The slowing of the oscillations down at\nhigh electric fields is a truly unexpected general feature\nof a multi-level system compared to the conventional two-level model\nand thus can occur in a broad variety of structures.\n\nThis paper is organized as follows. In Section II we introduce quantum mechanical description of\nelectron in a double quantum dot with spin-orbit coupling and magnetic field. Section III\npresents the model of driven dynamics. In Section IV we apply the\nstroboscopic Floquet approach for the long-time evolution and obtain the properties of Rabi oscillations under\nvarious conditions. Conclusions of this work are given in Section V.\n\n\n\n\\section{Model, Hamiltonian, and Observables}\n\nThe unperturbed Hamiltonian $H_0={k^{2}}\/{2m}+U(x)$ describes\nelectron in a double quantum dot with the potential (see Fig.\\ref{figure1}) \\cite{Khomitsky}:\n\\begin{equation}\nU(x)=U_{0}\\left[-2\\left(\\frac{x}{d}\\right)^{2}+\\left(\\frac{x}{d}\\right)^{4}\\right],\n\\label{H0}\n\\end{equation}\nwhere $k=-i\\partial\/\\partial x$ is the momentum operator and $\\hbar\\equiv 1$.\nThe minima at $-d$ and $d$ are separated by a barrier of the\nheight $U_{0}$. In the absence of external fields and SOC\nthe ground state is split into the doublet of even ($\\psi_{\\rm g}$) and\nodd ($\\psi_{\\rm u}$) states. The tunneling energy $\\Delta E_{g}\\ll U_{0}$\ndetermines the tunneling time $T_{\\rm tun}=2\\pi\/\\Delta E_{g}$. The Zeeman coupling\nto magnetic field $H_{Z}=\\Delta_{Z}\\sigma_{z}\/2$, where $|\\Delta_{Z}|$ is the Zeeman splitting.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics*[scale=0.3]{for_figure_1.eps}\\\\\n\\includegraphics*[scale=0.45]{figure_1.eps}\n\\caption{(Color online) (a) Qualitative picture of the spin dynamics induced by the interminima tunneling.\n(b,c) Free evolution of coordinate (solid line) and spin components $\\langle\\sigma_x\\rangle$ (dashed\nline), $\\langle\\sigma_y\\rangle$ (dashed-dot line) for\n$B=1.73$ T (b) and $B=6.93$ T (c). The initial state is the even combination of the\nstates corresponding to the tunneling-split doublet with SOC\ntaken into account.}\n\\label{figure1}\n\\end{figure}\n\n\nThe SOC has the form:\n\\begin{equation}\nH_{\\mathrm{so}}=\\left( \\alpha _{D}\\sigma _{x}+\\alpha _{R}\\sigma _{y}\\right)k,\n\\end{equation}\nwhere the bulk-originated Dresselhaus $\\left(\\alpha_{D}\\right) $ and\nstructure-related Rashba $\\left(\\alpha_{R}\\right)$ parameters determine\nthe strength of SOC.\nIn the presence of SOC the spatial parity of the eigenstates is  approximate rather than exact\nbeing the qualitative feature of the\ncoupling linear in the odd $k$-operator, eventually resulting\nin the ability of spin manipulation by electric field.\n\nThe quantities we are interested in are the spin components:\n\\begin{equation}\n\\langle\\sigma_{i}(t)\\rangle=\\int_{-\\infty }^{\\infty }\\sigma _{i}(x,t)dx,\n\\end{equation}\nwhere $\\sigma_{i}(x,t)={\\bm \\psi}^{\\dagger}(x,t)\\sigma_{i} {\\bm \\psi}(x,t)$ is the spin density,\nand expectation value of the coordinate\n\\begin{equation}\n\\langle x(t)\\rangle=\\int_{-\\infty }^{\\infty }x\\rho (x,t)dx,\n\\end{equation}\nwhere $\\rho(x,t)={\\bm \\psi}^{\\dagger}(x,t){\\bm \\psi}(x,t)$, and ${\\bm \\psi}(x,t)$ is the two-component\nelectron wavefunction.\n\nFor numerical studies we diagonalize exactly the time-independent Hamiltonian\n$H_{0}+H_{\\mathrm{so}}+{H}_{Z}$ in the truncated spinor basis\n$\\psi_{n}(x)|\\sigma\\rangle$, where $\\psi_{n}(x)$ are the\neigenfunctions of $H_{0}$ in Eq.(\\ref{H0}), and $|\\sigma\\rangle$\nwhere $\\sigma=\\pm1$ corresponds to the spin parallel (antiparallel)\nto the $z$-axis, find corresponding eigenvalues,\nand obtain the new basis set $\\left|\\psi_{n}\\right\\rangle $.\nWe consider below as an example a GaAs-based structure,\nwhere the effective mass is 0.067 of the free electron mass,\nwith $d=25\\sqrt{2}$ nm and $U_{0}=10$ meV.\nIn the absence of magnetic field the ground state energy is\n$E_{1}=3.938$ meV, and the tunneling splitting $\\Delta E_{g}=0.092$ meV,\ncorresponding to the transition frequency close to 23 GHz.\nTo illustrate the spin dynamics, we consider a moderate external\nmagnetic field with $\\Delta_{Z}=\\Delta E_{g}\/2$\ncorresponding to $B=1.73$ T, and a relatively strong magnetic field with $\\Delta_{Z}=2\\Delta E_{g}$ ($B=6.93$ T)\nwith the Land\\'{e} factor $g=-0.45$. The parameters of the SO coupling are\nassumed to be $\\alpha_{R}=1.0\\cdot 10^{-9}$ eVcm, and $\\alpha _{D}=0.3\\cdot 10^{-9}$ eVcm,\nhowever, our results can be applied to various double quantum dots with different SOC parameters\nand thus have a quite general character.\nIn particular, the change in the interdot barrier shape and geometry would modify only\nquantitatively the system parameters, including the energy levels, spinor\nwavefunctions, and, as a result, the resonant driving frequency. The increase in the interdot\ndistance would decrease the tunneling splitting, making such a system\nmore sensitive to external influence from phonons, fluctuations in the driving field, etc.\n\n\n\\section{Driven dynamics}\n\nTo demonstrate a nontrivial interplay of the tunneling and spin dynamics, we\nbegin with the coordinate and spin evolution of the electron initially\nlocalized near the $-d$ minimum.  Spin evolution of the state\n$\\left(\\left|\\psi_{\\rm g}\\right>+\\left|\\psi_{\\rm u}\\right>\\right)\\left|1\\right>\/\\sqrt{2}$\ncan be described approximately analytically taking into account four spin-split lowest\nlevels and a simpler SOC Hamiltonian $\\alpha_{R}\\sigma_{y}k$ as\n\\begin{equation}\n\\langle\\sigma_{x}(t)\\rangle= \\alpha_{R}\\frac{K\\Delta E_{g}}{A_{+}A_{-}}\\sin (A_{+}t)\\sin(A_{-}t)\n\\end{equation}\nwhere $K=-i\\langle\\psi_{\\rm u}\\left| k\\right|\\psi_{\\rm g}\\rangle$,\nwhich in the $\\Delta E_{g}\\ll U_{0}$ limit can be accurately approximated as\n$K=md\\Delta E_{g}$, and  $A_{\\pm}=\\sqrt{E_{\\pm}^{2}\/4+\\alpha_{R}^{2}K^{2}}$,\nwhere $E_{\\pm}=\\Delta E_{g}\\pm\\Delta_{Z}$.\n{ Numerical results for coordinate and spin are shown in Fig.\\ref{figure1}.\nWith the increase in magnetic field, the effect of SOC decreases, leading\nto smaller amplitudes of precession, as can be seen from comparison of Fig.\\ref{figure1}(b)\nand Fig.\\ref{figure1}(c). In addition, both the initial state and spin precession axis\nchange leading to a different phase shift between the observed spin components.}\n\nNext we consider a periodic perturbation by electric field at $t>0$:\n\\begin{equation}\n\\mathcal{E}(t)=\\mathcal{E}_{0}\\sin (\\widetilde{\\omega}_{Z}t).\n\\end{equation}\nHere $\\widetilde{\\omega}_{Z}$ is the exact, taking into account SOC,\nfrequency of the spin-flip transition. For the chosen set of parameters $\\widetilde{\\omega}_{Z}$\nis very close to $\\Delta_{Z}.$ The field strength is characterized by\nparameter $f$ defined as $e\\mathcal{E}_{0}\\equiv f\\times U_{0}\/2d$,\nwhere $e$ is the fundamental charge. For the chosen system parameters, $f=1$\ncorresponds to the electric field of approximately $1.5\\times 10^{3}$ V\/cm, similar\nto Ref.[\\onlinecite{Nowack}].\nHere we consider different regimes of the strength and see how the change in\nthe shape of the quartic potential produced by the field becomes crucially\nimportant for the spin dynamics in two sets of energy levels produced by\nmagnetic field. We build in the obtained  $|\\psi_{n}\\rangle$ basis the matrix of the Hamiltonian\n$\\widetilde{V}=ex\\mathcal{E}_{0}\\sin(\\widetilde{\\omega}_{Z}t)$ and study the full\ndynamics with the wavefunctions:\n\\begin{equation}\n{\\bm \\psi}(x,t)=\\sum_{n}\\xi_{n}(t)e^{-{i}E_{{n}}t}\\left|\\psi_{{n}}\\right\\rangle.\n\\end{equation}\nThe time dependence of $\\xi_{{n}}(t)$ is given by:\n\\begin{equation}\n\\frac{d\\xi_{{n}}(t)}{dt}={i}e\\mathcal{E}(t)\\sum_{l}\\xi_{l}(t)x_{{ln}}e^{-{i}\\left( E_{{l}}-E_{{n}}\\right)t},\n\\label{maineq}\n\\end{equation}\nwhere $x_{{ln}}\\equiv \\left\\langle \\psi _{{l}}\\right| \\widehat{x}\\left|\\psi_{{n}}\\right\\rangle$.\nThere are two different types of $x_{ln}$: (1) matrix elements of the order of $d$ due to the\ndifferent parity of the wavefunctions in the absence of SOC,\nand (2) those determined by the SOC strength.\nIn the weak SOC limit $\\left|\\Delta_{Z}-\\Delta E_{g}\\right|\\gg\\alpha_{R}|K|$,\nthe SO-determined matrix element of coordinate in the lowest spin-split doublet can be evaluated as\n\n\\begin{equation}\nx_{\\rm so}=2dK\\alpha_{R}\\frac{\\Delta_{Z}}{\\Delta_{Z}^{2}-\\left(\\Delta E_{g}\\right)^{2}}.\n\\end{equation}\n\nIn our calculations we assume that the initial state is the ground state of the\nfull Hamiltonian, that is $\\xi_{1}(0)=1$ and $\\xi_{n>1}(0)=0$.\nThe entire driven motion of the system can be approximately\ncharacterized as a superposition of two types of\ntransitions: resonant ``spin-flip'' transitions with the matrix\nelement of coordinate determined by the SOC and off-resonant ``spin-conserving''\ntransitions with a larger matrix element of coordinate.\\cite{leakage}\nBoth types are crucially important\nfor the understanding of the spin dynamics. { With the estimate $K\\approx m\\Delta E_{g}d$,\nin both cases considered by us ($\\Delta_{Z}=E_{g}\/2$ and $\\Delta_{Z}=2E_{g}$),\nwe obtain $d\\approx 10x_{\\rm so}$. As a result, the off-resonant\ntransitions are not weak compared to the required once.} Throughout the calculation we neglect orbital\nand spin relaxation processes assuming that the driving force is sufficiently strong\nto prevent the decoherence on the time scale of the spin spin-flip transition.\nIt is known that the periodic field forms a well-established driven dynamics even in the presence\nof damping as long as the level structure is not deeply disturbed by the broadening. For our parameters\nit means that one can expect the observation of the predicted results in the currently available semiconductor\nstructures at temperatures moderately below 1 K.\\cite{Nadj2010}\n\n{ We begin with presentation of the short-time dynamics of coordinate\n$\\langle x \\rangle \/d$ and spin $\\langle \\sigma_{x} \\rangle$ for four initial periods of the driving field\n(Fig.\\ref{figure2}).\nThese resuts were obtained by the explicit numerical integration of Eq.(\\ref{maineq}) with a\ntime step on the order of $10^{-4}T_{Z}$.}\nThe other component $\\langle\\sigma_{z}\\rangle$ changes much\nslower and will be treated later on a long timescale, which is the primary topic of our interest.\nIt can be seen in Fig.\\ref{figure2} that the fast oscillations are accompanying mainly\nthe local variations of observables, especially of the spin. { Considerable changes such as\nRabi oscillations of spin can be achieved only after many periods\nof the driving field. We will focus on this slow dynamics below.}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics*[scale=0.45]{for_figure_2.eps}\\\\\n\\includegraphics*[scale=0.45]{figure_2.eps}\n\\caption{(Color online) (a) Scheme of the levels and spin components involved in the Zeeman resonance, $B=1.73$ T (left)\nand $B=6.93$ T (right), (b) short-term time-dependent $\\langle x\\rangle\/d$ and $\\langle\\sigma_{x}\\rangle$, as marked\nnear the plots, induced by external field with $f=0.1$, $B$=1.73 T, { $T_{Z}$=88 ps;} (c) same as in (b) for $B$=6.93 T,\n{ $T_{Z}$=22 ps.}}\n\\label{figure2}\n\\end{figure}\n\n\n\\section{Floquet stroboscopic approach}\n\nTo { consider the long-term time dependence of the periodically driven system we apply the\nFloquet approach \\cite{Shirley,Platero,Kohler05,Jiang06,Wu} in the stroboscopic form.}\nHere we remind the reader main features of this approach developed in Ref.[\\onlinecite{Demikhovskii}].\nAs the first step, the one-period propagator matrix $\\mathbf{U}_{ln}(T_{Z})$ is obtained by a\nhigh-precision numerical integration of the system (\\ref{maineq}) at one period of the\ndriving $T_{Z}=2\\pi\/\\widetilde{\\omega}_{Z}$\nin the basis of all unperturbed states. For numerically accurate $\\mathbf{U}_{ln}(T_{Z})$,\nwe obtain its eigenvalues $E_{Q}$ which are the quasienergies of the driven system, and the corresponding orthogonal\neigenvectors $A^{Q}_{l}$. As a result,\nthe one-period propagator $\\mathbf{U}_{ln}(T_{Z})$ can be presented as:\n\\begin{equation}\n\\mathbf{U}_{ln}(T_{Z})=\\sum_{Q} A^{Q}_{l} \\left( A^{Q}_{n} \\right)^{*} e^{-i E_{Q}T_{Z}}.\n\\label{uoneper}\n\\end{equation}\nIts $N$-th power obtained by taking into account the orthogonality of the eigenvectors $A^{Q}_{l}$\ngives the stroboscopic propagator $\\mathbf{U}_{ln}(NT_{Z})$ for $N$ periods as\n\\begin{equation}\n\\mathbf{U}_{ln}(NT_{Z})=\\sum_{Q} A^{Q}_{l} \\left( A^{Q}_{n} \\right)^{*} e^{-i E_{Q}NT_{Z}}.\n\\label{umanyper}\n\\end{equation}\nFor any integer $N$ the system state is\ngiven by $\\left|\\Psi(NT_{Z})\\right\\rangle=\\mathbf{U}_{ln}(NT_{Z})\\left|\\Psi(0)\\right\\rangle $.\n{ The similarity of Eq.(\\ref{uoneper}) for a single-period propagator and Eq.(\\ref{umanyper})\nfor any $N\\ge1$ is a highly nontrivial fact demonstrating that\n${U}_{ln}(NT_{Z})={U}_{ln}^{N}(T_{Z})$ can be simply expressed by the right-hand-side\nin Eq.(\\ref{umanyper}).}\nThe stroboscopic approach allows us to study very\naccurately the long-time evolution since the $N$-period propagator (\\ref{umanyper}) is constructed\nexplicitly in a finite algebraic form. Although this propagator describes the dynamics\nexactly, it allows to watch only the stroboscopic evolution rather than the entire\ncontinuous one. However, if we are interested in slowly evolving phenomena such as\nchaos development \\cite{Demikhovskii} and Rabi oscillations which occur here on many periods\nof the driving field, the stroboscopic approach is fully justified and highly efficient.\n{The experiment \\cite{Nowack} uses stroboscopic approach with the intervals\non the order of 100 ns to measure the slow dynamics of the driven electron spin.}\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics*[scale=0.45]{figure_3.eps}\n\\caption{(Color online) Stroboscopic time dependence of $\\langle x\\rangle\/d$ for field $f=0.35$ in a given time window:\n(a)  $B=1.73$ T, (b) $B=6.93$ T. {Solid lines serve only as a guide for the eye.}}\n\\label{figure3}\n\\end{figure}\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics*[scale=0.45]{figure_4.eps}\n\\caption{(Color online) Stroboscopic time dependence of $\\langle\\sigma_{z}(NT_{Z})\\rangle$ for different external\ndriving fields $f$ marked near the plots for two different magnetic fields: (a)  $B=1.73$ T, vertical dashed line marks\nthe operational definition of the spin-flip period, (b) same as in (a) for $B=6.93$ T.}\n\\label{figure4}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics*[scale=0.45]{figure_5.eps}\n\\caption{(Color online) Nonlinear dependence of the Rabi spin-flip frequency on the electric field amplitude in\nmultilevel system for two different magnetic fields (a) $B=1.73$ T, the characteristic spin-flip rate of the order\nof 50 MHz, (b) $B=6.93$ T, the characteristic spin-flip rate of the order of 400 MHz.}\n\\label{figure5}\n\\end{figure}\n\n{ The results of calculations of electron displacement at discrete times $NT_{Z}$ are presented in Fig.\\ref{figure3}.\nAs one can see in Fig.\\ref{figure3}, the time dependence of the\ndisplacement becomes strongly nonperiodic with the typical values being considerably less than $d$.\nThat is in a strong electric field the electron probability density redistribution between the dots\nis not complete. The motion can be qualitatively analyzed in the pseudospin model\nof the charge dynamics,\\cite{Rashba11} where the tunneling splitting is described by the $\\sigma_{z}$ matrix,\nand the driving field is coupled to the $\\sigma_{x}-$matrix. The decrease in the electron displacement\nwith the increase in the electric field can be viewed as a suppressed spin precession in a strong periodic field\nor as a coherent destruction of tunneling.\\cite{Platero}\nThis nonperiodic behavior and decrease in the displacement eventually result in a less efficient spin driving.}\nIt should be mentioned that the fast oscillations in Fig.\\ref{figure2} which are in general absent\nin Fig.\\ref{figure3} reflect the difference between the continuous time scale in the former Figure and\nthe stroboscopic Floquet times $NT_{Z}$ in the latter one. { Figure \\ref{figure3} clearly illustrates\nthe role of the spin in the orbital dynamics: the curves in Fig.\\ref{figure3}(a) and \\ref{figure3}(b)\nare very different. Tracking of the system at stroboscopic times $NT_{Z}$ may not allow seeing the complete\nfast orbital dynamics, thus masking some details. As a result,\nthere is no simple way to describe this stroboscopic picture directly in terms\nof the Hamiltonian parameters.}\n\nThe slow long-term spin dynamics is presented in Fig.\\ref{figure4}. Here the \"unit of time\" $T_{Z}$ is short enough and the time dependence of\n$\\langle\\sigma_{z}\\rangle$ is accurately described by the stroboscopic approach.\nSince the spin dynamics is not strictly periodical and full spin flips do not always appear\nin this system, we use the operational definition of the ``spin-flip'' time $T_{sf}$:\nspin flip occurs when spin component shows a broad minimum albeit accompanied by fast oscillations\n(see in Fig.\\ref{figure4}(a)). The fast dynamics in the spin-flip doublet shown in Fig.\\ref{figure4}\nbecomes slow with the field increase as a result of a weaker\neffective coupling of the states with different parity. The resulting spin\nbehavior, arising solely due to the SOC, is shown in Fig.\\ref{figure4}.\nThe Rabi frequency for the spin-flip is smaller for some higher\nvalues of $f$ (which we vary through Fig.\\ref{figure3}-Fig.\\ref{figure4}) than for some weaker values of $f$\nin contrast to what can be expected for the weak fields\nemployed, e.g. in the experiments \\cite{Nowack}, being a manifestation of the generally\nnonmonotonous behavior of the Rabi frequency on the electric field amplitude.\nIn addition, in contrast to the simple Rabi oscillations, the\nflips become incomplete, with $\\left<\\sigma_{z}(t)\\right>=-1$ never reached. These two\nqualitative effects are the results of the enhanced electron tunneling\nbetween the potential minima: the spin precession in the driven interminima motion\nestablishes corresponding spin dynamics and prevents the electric field to flip\nthe spin efficiently. This effect makes a qualitative difference to the model\nof Ref.\\cite{Nowack}, where electron is assumed to be always located in the orbital ground state near\nthe minimum of the potential formed by the parabolic confinement and weak external electric field.\n\nTo present a broader outlook onto the dependence of\nthe spin flip rate on the driving field, we plot in Fig.\\ref{figure5} the spin flip rate\nfor $B=1.73$ T and $B=6.93$ T. In contrast to the linear dependence\nfor a conventional two-level Rabi resonance formula, one can see a strongly\ndifferent much more complicated non-monotonous\ndependence in a multi-level structure, especially at high fields. The regime in Fig.\\ref{figure5}(a)\nshows more irregularities since all four lowest\nlevels are equidistant (Fig.\\ref{figure2}(a))\nand involved in the resonance while in Fig.\\ref{figure5}(b) more regular dependence is observed,\nreflecting a simpler nature of the resonances here.\n\nWe would like to mention here that the observed slowing down\nof spin dynamics can be seen on a more general ground,\nnot restricted to the exact form of Eq.(\\ref{maineq}), as\nthe Zeno effect of freezing evolution of a measured quantity.\\cite{Streed,Echanobe,Sokolovski10}\nIndeed, the operator $-i\\sigma_i\\partial\/\\partial x$ makes the orbital dynamics spin-dependent,\nand, as a result, performs the measurement of the $\\sigma_i$ component \\cite{Sokolovski11,Allahverdyan}\nin the sense of von Neumann procedure. This can be seen in the evolution\nof a two-component wave function:\\cite{Sokolovski11}\n\\begin{eqnarray}\n&&e^{-\\alpha t\\sigma_z{\\partial}\/{\\partial x}}\\phi(x)\n\\left(\\zeta_{1}\\left|1\\right>+\\zeta_{-1}\\left|-1\\right>\\right)\n= \\nonumber \\\\\n&&\\phi(x-\\alpha t)\\zeta_{1}\\left|1\\right>+\\phi(x+\\alpha t)\\zeta_{-1}\\left|-1\\right>,\n\\end{eqnarray}\nwhere we took $i=z$ as an example,\n$\\zeta_{1}$ and $\\zeta_{-1}$ correspond to $\\pm1$ eigenvalues of $\\sigma_{z}$, respectively,\nand $\\alpha$ is the coupling constant. The SOC thus The SOC thus entangles the orbital and spin motion, destroys\nthe coherent superposition of spin-up and spin-down states, and performs the von Neumann-like spin measurement\nby mapping spin state on the electron position.\nThis von Neumann measurement, is, however, different from the experimental measurement procedure applied,\ne.g. in Ref.[\\onlinecite{Nowack}].  The spin-orbit coupling coupling drives the coherent superposition of different\nspin components and at the same time, by constant strong measurement, destroys it leading to a slow spin dynamics.\n\n\n\n\\section{Conclusions}\n\nWe have considered the interplay between the tunneling and\nspin-orbit coupling in a driven by an external electric\nfield one-dimensional single-electron double quantum dot.\nIn the regime of the electric dipole spin resonance, where the electric\nfield frequency exactly matches the Zeeman transition, the complex interplay of these\nmechanisms results in two unexpected effects. The first effect is the\nnonmonotonous change in the Rabi spin oscillations\nfrequency with the electric field amplitude. The Rabi oscillations  become much slower\nthan expected for a two-level system. The second effect is the incomplete Rabi spin flips.\n{ This behavior results from the fact that the interminima motion establishes a competing\nspin dynamics, leading to the physics somewhat similar to the Zeno effect, preventing a fast\nchange in a measured quantity.}\nThese results indicating the slowdown and nonlinearity of the spin resonance in multilevel systems can be useful\nfor pointing out certain fundamental challenges for the future experimental and spintronics device applications\nof  phenomena based on spins in double quantum dots.\n\n\n\\section{Acknowledgements}  D.V.K. is supported by the RNP Program of Ministry of Education and Science RF,\nand by the RFBR (Grants No. 11-02-00960a, 11-02-97039\/Regional).  This work of EYS was supported by the MCINN of Spain\nGrant FIS2009-12773-C02-01, by \"Grupos Consolidados UPV\/EHU del Gobierno Vasco\" Grant IT-472-10,\nand by the UPV\/EHU under program UFI 11\/55.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOur aim in this note is to show how basic results about the survival probability\nof branching processes can be used to give an essentially best possible result\nabout the emergence of the giant component in $G_{n,p}$, the random graph with\nvertex set $[n]=\\{1,2,\\ldots,n\\}$ in which each edge is present independently\nwith probability $p$.\nIn 1959, Erd\\H os and R\\'enyi~\\cite{ER_evol} showed that if we take\n$p=p(n)=c\/n$ where $c$ is constant, then there is a `phase transition' at $c=1$.\nWe write $L_1(G)$ for the maximal number of vertices in a\ncomponent of a graph $G$. Also, as usual, we say that an event holds \\emph{with high\nprobability} or \\emph{whp} if its probability tends to $1$ as $n\\to\\infty$.\nErd\\H os and R\\'enyi showed that, whp, if $c<1$ then $L_1(G_{n,c\/n})$ is of\nlogarithmic order, if $c=1$ it is of order $n^{2\/3}$, while if $c>1$ then there\nis a unique `giant' component containing $\\Theta(n)$ vertices, while the second\nlargest component is much smaller.\n\nIn 1984, Bollob\\'as~\\cite{BB_evol} noticed that this is only the starting point,\nand an interesting question remains: what does the component structure of\n$G_{n,p}$ look like for\n$p=(1+\\eps)\/n$, where $\\eps=\\eps(n)\\to 0$? He and {\\L}uczak~\\cite{Luczak_near}\nshowed that if $\\eps=O(n^{-1\/3})$ then $G_{n,p}$ behaves in a qualitatively similar way\nto $G_{n,1\/n}$; this range of $p$ is now called the \\emph{scaling window}\nor \\emph{critical window} of the\nphase transition. The range $\\eps^3n\\to\\infty$ is the \\emph{supercritical}\nregime, characterized by the fact that there is whp a unique `giant' component\nthat is much larger than the second largest component. The range $\\eps^3n\\to -\\infty$\nis the \\emph{subcritical} regime.\n\nIn this paper we are interested in the size of the giant component\nas it emerges. Thus we consider the (weakly) supercritical regime\nwhere $p=p(n) = (1+\\eps)\/n$, with $\\eps=\\eps(n)$ satisfying\n\\begin{equation}\\label{a1}\n \\eps\\to 0 \\hbox{\\qquad and \\qquad} \\eps^3n\\to\\infty\\hbox{\\qquad as }n\\to\\infty.\n\\end{equation}\nOur aim here is to use branching processes\nto give a very simple new proof of the following\nresult, originally due to Bollob\\'as~\\cite{BB_evol}\n(with a mild extra assumption) and {\\L}uczak~\\cite{Luczak_near}.\n\\begin{theorem}\\label{th1}\nUnder the assumption \\eqref{a1} we have\n\\[\n L_1(G_{n,p}) = (2+\\op(1))\\eps n.\n\\]\n\\end{theorem}\nHere $\\op(1)$ denotes a quantity that tends to 0 in probability:\nthe statement is that for any fixed $\\delta>0$, $L_1(G_{n,p})$\nis in the range $(2\\pm\\delta)\\eps n$ with probability tending\nto $1$ as $n\\to\\infty$.\n\nSince the original papers~\\cite{BB_evol,Luczak_near} (which in fact gave\na more precise bound than that above), many different\nproofs of many forms of Theorem~\\ref{th1} have been given. For example, Nachmias\nand Peres~\\cite{NP_giant} used martingale methods to reprove the result as stated\nhere. Pittel and Wormald~\\cite{PWio} used counting methods to prove an even more\nprecise result; a simpler martingale proof of (part of) their result is given\nin~\\cite{BR_walk}.  A proof of Theorem~\\ref{th1} combining tree counting and\nbranching process arguments appears in~\\cite{rg_bp}. More recently, Krivelevich\nand Sudakov~\\cite{KS} gave a very simple proof of a variant of Theorem~\\ref{th1}\nwhich is even weaker than the original Erd\\H{o}s--R\\'enyi result:\n$\\eps>0$ is taken to be constant, and the size of the giant component is determined\nonly up to a constant factor.\n\n\n\\section{Branching process preliminaries}\n\nLet us start by recalling some basic concepts and results.\nThe \\emph{Galton--Watson} branching process with offspring\ndistribution $Z$ is the random rooted tree constructed as follows:\nstart with a single root vertex in generation 0. Each vertex in\ngeneration $t$ has a random number of children in generation $t+1$,\nwith distribution $Z$. The numbers of children are independent of each\nother and of the history. It is well known and easy to check that\nif $\\E[Z]>1$, then the process \\emph{survives} (is infinite) with\nprobability $\\rho$ the unique solution in $(0,1]$ to $1-\\rho=f_Z(1-\\rho)$,\nwhere $f_Z$ is the probability generating function of $Z$.\nWhen $\\E[Z]<1$, the expectation of the total\nnumber of vertices in the branching process is\n\\begin{equation}\\label{totsize}\n 1+\\E[Z]+\\E[Z]^2+\\cdots= \\frac{1}{1-\\E[Z]},\n\\end{equation}\nand in particular the survival probability is $0$.\n\nLet us write $\\bp{n,p}$\nfor the {\\em binomial branching process} with parameters $n$ and $p$,\ni.e., for the branching process as above with offspring distribution ${\\rm Bi}(n,p)$.\nSince the generating function of ${\\rm Bi}(n,p)$ satisfies\n\\[\n f(x)=\\sum_{k=0}^n \\binom{n}{k} p^k (1-p)^{n-k} x^k=\\big(1-p(1-x)\\big)^n,\n\\]\nwhen $np>1$ the survival probability\n$\\rho=\\rho_{n,p}$ satisfies\n\\[\n 1-\\rho=(1-p \\rho)^n.\n\\]\nFrom this it is easy to check that if $\\eps=np-1 \\to 0$ with $\\eps >0$ then\n\\begin{equation}\\label{2eps}\n\\rho \\sim 2\\eps.\n\\end{equation}\n\nConditioning on a suitable branching process dying out (i.e., having finite total size)\none obtains another branching process, called the \\emph{dual branching process}.\nIn the binomial case, one way to see\nthis is to think of $\\bp{n,p}$ as a random subgraph of the infinite $n$-ary rooted tree $\\bp{n,1}$\nobtained\nby including each edge independently with probability $p$, and retaining only the component\nof the root. For a vertex of $\\bp{n,1}$\nin generation 1 there are three possibilities: it may (i) be \\emph{absent}, i.e.,\nnot joined to the root, (ii) \\emph{survive}, i.e., be joined to the root\nand have infinitely many descendents, or (iii) \\emph{die out}. The probabilities\nof these events are $1-p$, $p\\rho$ and $p(1-\\rho)$, respectively.\nLet $\\cD$ denote the event that the process $\\bp{n,p}$ dies out, i.e., the total population\nis finite. Since $\\cD$ happens if and only if every vertex of $\\bp{n,1}$ in generation 1 is absent\nor dies out, the conditional distribution of $\\bp{n,p}$ given $\\cD$\nis the unconditional distribution of $\\bp{n,\\pi}$, with $\\pi= p(1-\\rho) \/ (1-p\\rho)$.\nThus the dual of $\\bp{n,p}$ is $\\bp{n,\\pi}$.\n\nNote that when $np-1=\\eps\\to 0$, then\n\\[\n 1-n\\pi = \\frac{1-p\\rho-np+np\\rho}{1-p\\rho} \\sim np\\rho - (np-1) -p\\rho \\sim \\eps.\n\\]\nHence the mean number of offspring in the dual process $\\bp{n,\\pi}$ is $1-(1+o(1))\\eps$,\nand from \\eqref{totsize} its expected size total is $(1+o(1))\\eps^{-1}$.\nWriting $\\cS=\\cD^\\cc$ for the event that $\\bp{n,p}$ \\emph{survives} (is infinite),\nand $|\\bp{n,p}|$ for its total size (number of vertices),\nit follows that for any integer $L=L(n)$ we have\n\\begin{eqnarray}\n \\Pr(|\\bp{n,p}|\\ge L) &=& \\Pr(\\cS) + \\Pr(\\cD)\\Pr(|\\bp{n,\\pi}|\\ge L) \\nonumber \\\\\n &\\le& \\Pr(\\cS) + \\Pr(|\\bp{n,\\pi}|\\ge L) \\nonumber \\\\\n &\\le& (1+o(1))(2\\eps + 1\/(\\eps L)), \\label{Tup}\n\\end{eqnarray}\nwith the second inequality following from Markov's inequality.\n\nWe shall use one further property of $\\bp{n,p}$, which can be proved in a number\nof simple ways. Suppose, as above, that $\\eps=np-1\\to 0$, and let $M=M(n)$ satisfy\n$\\eps M\\to\\infty$. Let $w(\\cT)$ denote the \\emph{width} of a rooted tree $\\cT$, i.e.,\nthe maximum (supremum) of the sizes of the generations. Then\n\\begin{equation}\\label{unwide}\n \\Pr\\bigl( \\{w(\\bp{n,p})\\ge M \\}\\cap \\cD\\bigr) = o(\\eps).\n\\end{equation}\nTo see this, consider testing whether the event $\\cW_M=\\{w(\\bp{n,p})\\ge M\\}$\nholds by constructing $\\bp{n,p}$\ngeneration by generation, stopping at the first (if any) of size at least $M$.\nIf such a generation exists then (since the descendents of each vertex in this generation\nform independent copies of $\\bp{n,p}$), the conditional probability\nthat the process dies out is at most $(1-\\rho)^M\\le e^{-\\rho M}\\to 0$. Hence\n\\begin{equation}\\label{wdie}\n \\Pr(\\cD\\mid \\cW_M)=o(1).\n\\end{equation}\nThus\n\\[\n \\Pr(\\cW_M) \\sim \\Pr(\\cS\\cap \\cW_M) \\le \\Pr(\\cS) \\sim 2\\eps,\n\\]\nwhich with \\eqref{wdie} gives \\eqref{unwide}.\n\n\\section{Application to $G_{n,p}$}\n\nThe binomial branching process is intimately connected to the component exploration\nprocess in $G_{n,p}$. Given a vertex $v$ of $G_{n,p}$, let $C_v$ denote the component of\n$G_{n,p}$ containing $v$, and let $\\cT_v$ be the random tree obtained by\nexploring this component by breadth-first search. In other words, starting with $v$, find\nall its neighbours, $v_1, \\dots , v_{\\ell}$, say, next find all the neighbours of $v_1$\ndifferent from the vertices found so far, then the new neighbours of $v_2$, and so on,\nending the second stage with the new neighbours of $v_{\\ell}$. The third stage consists of\nfinding all the new neighbours of the vertices found in the second stage, and so\non. Eventually we build a tree $\\cT_v$, which is a spanning tree of $C_v$.\n\nNote that our notation suppresses the fact that the distributions of $\\cT_v$ and of $C_v$ \ndepend on $n$ and $p$.\nIn the next lemma, as usual, $|H|$ denotes the total number of vertices in a graph $H$.\n\n\\begin{lemma}\\label{couple}\n{\\rm (i)} For any $n$ and $p$, the random rooted trees $\\cT_v$ and\n$\\bp{n,p}$ may be coupled so that $\\cT_v\\subset \\bp{n,p}$.\n\n{\\rm (ii)} For any $n$, $k$ and $p$ there is a coupling of the integer-valued\nrandom variables $|C_v|$ and $|\\bp{n-k,p}|$ so that either\n$|C_v|\\ge |\\bp{n-k,p}|$ or both are at least $k$.\n\\end{lemma}\n\n\\begin{proof}\nFor the first statement we simply generate $\\cT_v$ and $\\bp{n,p}$\ntogether, always adding fictitious vertices to the vertex set of $G_{n,p}$\nfor the branching process to take from, so that in each step a vertex\nhas $n$ potential new neighbours (some fictitious) each of which\nit is joined to with probability $p$.\nAll the descendants of the fictitious vertices are themselves fictitious.\n\nTo prove (ii) we slightly modify the exploration, to couple\na tree $\\cT_v'$ contained within $C_v$ with $\\bp{n-k,p}$ such that one of two\nalternatives holds: either $\\cT_v' \\supset \\bp{n-k,p}$, or else both\n$\\cT_v'$ and $\\bp{n-k,p}$ have at least $k$ vertices. Indeed, construct\n$\\cT_v'$ exactly as $\\cT_v$, except that at each step at the start\nof which we have not yet reached more than $k$ vertices,\nwe test for edges from the current vertex to exactly $n-k$ potential new\nneighbours. Since $|C_v|\\ge |\\cT_v'|$, this coupling gives the result.\n\\end{proof}\n\n\nFrom now on we take $p=p(n)=(1+\\eps)\/n$, where $\\eps=\\eps(n)$ satisfies \\eqref{a1}.\nWe start by using the two couplings described above to give bounds on the expected number of vertices\nin large components. In both lemmas, $N_{[L,n]}$ denotes\nthe number of vertices of $G_{n,p}$ in components with between $L$ and $n$ vertices (inclusive);\n$\\Pr_{n,p}$ and $\\E_{n,p}$ denote the probability measure and expectation associated\nto $G_{n,p}$.\n\\begin{lemma}\\label{elb}\nSuppose that $L=L(n)=o(\\eps n)$.\nThen\n$\\Pr_{n,p}(|C_v|\\ge L)\\ge (2+o(1))\\eps.$\nEquivalently, $\\E_{n,p} (N_{[L,n]})\\ge (2+o(1))\\eps n$.\n\\end{lemma}\n\\begin{proof}\nTaking $k=L$ in Lemma~\\ref{couple}(ii),\n\\begin{eqnarray*}\n\\Pr_{n,p} (|C_v|\\ge L) &\\ge& \\Pr(|\\bp{n-L,p}|\\ge L)\\\\\n                 &\\ge& \\Pr (\\bp{n-L,p} \\ \\text{survives})\\sim 2\\big( (n-L)p-1\\big) \\sim 2 \\eps,\n\\end{eqnarray*}\nwhere the approximation steps follow from \\eqref{2eps} and the assumption on $L$.\n\\end{proof}\n\n\\begin{lemma}\\label{eub}\nSuppose that $L=L(n)$ satisfies $\\eps^2L\\to\\infty$.\nThen\n$\\E_{n,p} (N_{[L,n]})\\le (2+o(1))\\eps n$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{couple}(i) and \\eqref{Tup},\n\\[\n \\Pr_{n,p}(|C_v|\\ge L) \\le \\Pr(|\\bp{n,p}|\\ge L) \\le (1+o(1))(2\\eps +1\/(\\eps L))\\sim 2\\eps.\n\\]\n\\end{proof}\n\nTogether these lemmas show that the expected number of vertices in components of size at\nleast $n^{2\/3}$, say, is asymptotically $2\\eps n$. Two tasks remain: to establish\nconcentration, and to show that most vertices in large components are in a single\ngiant component. For the first task, one can simply count tree components.\n(This is a little messy, but theoretically trivial.\nThe difficulties in\nthe original papers~\\cite{BB_evol,Luczak_near} stemmed from the fact that non-tree\ncomponents had to be counted as well. What is surprising is that here it suffices to count\ntree components.) Indeed, applying the first\nand second moment methods to the number $N$ of vertices in tree components\nof size at most $n^{2\/3}\/\\omega$, where $\\omega=\\omega(n)\\to\\infty$ sufficiently\nslowly, shows that this number is within $\\op(\\epsilon n)$\nof $(1-\\rho)n$, reproving Lemma~\\ref{eub} and (together with Lemma~\\ref{elb})\ngiving the required concentration.\nSee~\\cite{rg_bp} for a version of this argument with a (best possible) $\\Op(\\sqrt{n\/\\eps})$\nerror term.\nSince the calculations, though requiring no ideas, are\nsomewhat lengthy, we take a different approach here.\n\n\n\\begin{lemma}\\label{llarge}\nSuppose that $L=L(n)$ satisfies $\\eps^2L\\to\\infty$ and $L=o(\\eps n)$. Then\n\\[\n N_{[L,n]}(G_{n,p}) = (2+\\op(1))\\eps n.\n\\]\n\\end{lemma}\n\\begin{proof}\nLet $N$ be the number of vertices of $G_{n,p}$ in components\nof size at least $L$. From Lemmas~\\ref{elb} and~\\ref{eub} the expectation $\\E[N]$ of $N$\nsatisfies $\\E[N]\\sim 2\\eps n$,\nso it suffices to show that\n\\begin{equation}\\label{aim}\n \\E[N^2] \\le (4+o(1))\\eps^2n^2.\n\\end{equation}\n\nFix a vertex $v$ of $G_{n,p}$.\nLet us reveal a tree $\\cT_v'$ spanning a subset $C_v'$ of $C_v$ by exploring using\nbreadth-first search as before,\nexcept that we stop the exploration if at any point (i) we have reached $L$ vertices\nin total, or (ii) there are $\\eps L$ vertices that have been reached (found as a new\nneighbour of an earlier vertex) but not yet explored (tested for new neighbours).\nNote that condition (ii) may happen partway through revealing a generation of $\\cT_v'$,\nand indeed partway through revealing the new neighbours of a vertex. We call a vertex\nreached but not (fully) explored a \\emph{boundary vertex}, and note\nthat there are at most $\\eps L+1\\le 2\\eps L$ boundary vertices. Let $\\cA$ be the event\nthat we stop for reasons (i) or (ii), rather than because we have revealed\nthe whole component:\n\\[\n \\cA=\\{\\hbox{ the exploration stops due to (i) or (ii) holding }\\}.\n\\]\nNote that if $|C_v|\\ge L$, then $\\cA$ holds.\n\nAs before, we may couple $\\cT_v'$ with $\\bp{n,p}$ so that $\\cT_v'\\subset \\bp{n,p}$.\nSince the boundary vertices correspond to a set of vertices of $\\bp{n,p}$ contained\nin two consecutive generations, if $\\cA$ holds, then either $|\\bp{n,p}|\\ge L$\nor $w(\\bp{n,p})\\ge \\eps L\/2$. From \\eqref{Tup} and \\eqref{unwide} it follows that\n$\\Pr(\\cA)\\le (2+o(1))\\eps$.\n\nSince all vertices are equivalent and $|C_v|\\ge L$ implies that $\\cA$ holds, we have\n\\begin{equation}\\label{eN2}\n \\E[N^2] =n\\E[1_{|C_v|\\ge L}N] \\le n \\E[1_{\\cA} N] = n\\Pr(\\cA) \\E[N\\mid \\cA] \\le (2+o(1))\\eps n \\E[N\\mid \\cA].\n\\end{equation}\nSuppose that $\\cA$ does hold. Given any vertex $w\\notin C_v'$,\nwe explore from $w$ as usual, but within $G'=G_{n,p}\\setminus V(C_v')$, coupling the resulting\ntree $\\cT_w'$ with $\\bp{n,p}$ so that $\\cT_w'\\subset \\bp{n,p}$. Let $C_w'$ be the component\nof $w$ in $G'$, so $C_w'$ is spanned by $\\cT_w'$.\nLet $\\cS$ be the event that (this final copy of) $\\bp{n,p}$ is infinite, and let $\\cD=\\cS^\\cc$.\nNote that $C_w'\\subset C_w$, and that the two are equal unless there is an edge from $C_w'$\nto some boundary vertex. Since there are at most $2\\eps L$ boundary vertices,\nthis last event has conditional probability at most $2\\eps L |C_w'| p \\le 3\\eps L|C_w'|\/n$, say.\nSince $|C_w'|\\le |\\bp{n,p}|$, it follows that\n\\begin{eqnarray*}\n \\Pr(|C_w|\\ge L\\mid \\cA) &\\le& \\Pr(\\cS) + \\Pr(\\cD)\\Pr(|C_w'|\\ge L \\mid \\cD) +3\\Pr(\\cD)\\eps L n^{-1} \\E[|C_w'| \\mid \\cD] \\\\\n  &\\le& \\Pr(\\cS) + \\Pr(|\\bp{n,p}|\\ge L \\mid \\cD) +3\\eps L n^{-1} \\E[|\\bp{n,p}| \\mid \\cD] \\\\\n  &\\le& \\Pr(\\cS) + (L^{-1} + 3\\eps L n^{-1}) \\E[|\\bp{n,p}|\\mid \\cD],\n\\end{eqnarray*}\nby Markov's inequality.\nSince the final expectation above is $\\sim \\eps^{-1}$ and\nour assumptions give that both $L^{-1}$ and $3\\eps Ln^{-1}$ are $o(\\eps^2)$, we see that\n$ \\Pr(|C_w|\\ge L\\mid \\cA) \\le (2+o(1))\\eps$. Hence, recalling that\nthere are at most $L$ vertices in $C_v'$,\n\\[\n \\E[N\\mid \\cA] \\le L+(n-L)\\Pr(|C_w|\\ge L\\mid \\cA) \\le L+(2+o(1))\\eps n \\sim 2\\eps n.\n\\]\nCombined with \\eqref{eN2} this gives \\eqref{aim}.\n\\end{proof}\n\nTo complete the proof of our main result,\nit remains only to show that almost all vertices in large components are in a single\ngiant component. For this we use a simple form of the classical sprinkling argument\nof Erd\\H os and R\\'enyi~\\cite{ER_evol}.\n\n\\begin{proof}[Proof of Theorem~\\ref{th1}]\nIt will be convenient to write $\\eps=\\omega n^{-1\/3}$, with $\\omega=\\omega(n)\\to\\infty$ and\n$\\omega=o(n^{1\/3})$. Also, let $\\omega'\\to\\infty$ \\emph{slowly},\nsay with $\\omega' =o(\\log \\log \\omega)$.\n\nSet $L=\\eps n\/\\omega'$.\nBy Lemma~\\ref{llarge} there are in total at most $(2+\\op(1))\\eps n$ vertices\nin components of size larger than $L$, which gives the upper bound on $L_1$.\n\nFor the lower bound, set $p_1=n^{-4\/3}$, and define $p_0$ by $p_0+p_1-p_0p_1=p$, so that if first we choose\nthe edges with probability $p_0$ and then (we sprinkle some more) with probability $p_1$ then the random\ngraph we get is exactly $G_{n,p}$. Since $np_0-1=(1+o(1))\\eps$, for any $\\delta>0$\nLemma~\\ref{llarge} shows that\nwith probability $1-o(1)$ the graph $G_{n,p_0}$ has at least $(2-\\delta)\\eps n$ vertices\nin components of size at least $L$.\n\n\nLet $U_1, \\dots , U_{\\ell}$ be the vertex sets of the components of $G_{n,p_0}$ of size at least $L$.\nThe probability that no edge sprinkled with probability $p_1$ joins $U_1$ to $U_j$ is\n\\[\n(1-p_1)^{|U_1| |U_j|} \\le e^{-p_1L^2} = \\exp \\big(-n^{-4\/3} \\omega^2 n^{4\/3}\/(\\omega')^2\\big),\n\\]\nso the expected number of vertices of $U$ not contained in the component of $G_{n,p}$ containing\n$U_1$ is at most\n\\[\n \\sum_{j=2}^{\\ell} \\exp\\big(-(\\omega\/\\omega')^2\\big) |U_j|=o(|U|).\n\\]\nConsequently, with probability $1-o(1)$ all but at most $\\delta|U|$ vertices of $U$ are contained\nwithin a single component of $G_{n,p}$, in which case $L_1(G_{n,p})\\ge (1-\\delta)(2-\\delta)\\eps n$.\nSince $\\delta>0$ was arbitrary, it follows that $L_1(G_{n,p})\\ge (2-\\op(1))\\eps n$,\ncompleting the proof.\n\\end{proof}\n\nTo conclude,\nlet us remark that although Theorem~\\ref{th1} is a key result about the phase\ntransition, as discussed in the introduction it is far from the final word\non the topic.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}