URL:http://link.aps.org/doi/10.1103/PhysRevLett.96.016405
DOI:10.1103/PhysRevLett.96.016405We apply the time-dependent current-density-functional theory to the study of the relaxation of a closed many-electron system evolving from a nonequilibrium initial state. We show that the self-consistent unitary time evolution generated by the exchange-correlation vector potential irreversibly drives the system to equilibrium. We also show that the energy dissipated in the Kohn-Sham system, i.e., the noninteracting system whose particle and current densities coincide with those of the physical system under study, is related to the entropy production in the real system.We acknowledge financial support from NSF Grant No. DMR-0313681 and the kind hospitality of the Scuola Normale Superiore in Pisa, where part of this work was completed

D'Agosta, Roberto

Vignale, Giovanni, 1957-

Physical Review Letters

arXiv

University of Missouri: MOspace

PRL 96, 016405 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending13 JANUARY 2006Relaxation in Time-Dependent Current-Density-Functional TheoryRoberto D’Agosta and Giovanni VignaleDepartment of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA(Received 6 August 2005; published 11 January 2006)0031-9007=We apply the time-dependent current-density-functional theory to the study of the relaxation of a closedmany-electron system evolving from a nonequilibrium initial state. We show that the self-consistentunitary time evolution generated by the exchange-correlation vector potential irreversibly drives thesystem to equilibrium. We also show that the energy dissipated in the Kohn-Sham system, i.e., thenoninteracting system whose particle and current densities coincide with those of the physical systemunder study, is related to the entropy production in the real system.DOI: 10.1103/PhysRevLett.96.016405 PACS numbers: 71.15.Mb, 03.65.YzThe irreversible time evolution of a many-electron sys-tem which is initially prepared in a nonequilibrium state isthe subject of strong research interest [1]. The systemmight be, for example, a large molecule [2] or the electronliquid in a quantum well, and the initial nonequilibriumstate might be created by laser excitation or by the appli-cation and subsequent removal of an electric field [3].From experience, we know that, even if the system isclosed and isolated, i.e., no exchanges of particles andenergy are allowed, the interaction between the particlesdrives the system towards the state of maximum entropy,i.e., the equilibrium state. When this happens, we say thatthe system has relaxed [4]. At each stage of the relaxa-tion process, the system has a certain amount of ‘‘mechani-cal energy’’ which could, in principle, be channeled intowork. One of the tasks of theory is to develop efficienttools—as an alternative to the solution of the many-bodySchro¨dinger equation—to calculate the time evolution ofthe mechanical energy. One would like to do this startingfrom first principles yet without knowing the exact state ofthe many-body system as a function of time.Various formulations of the problem of relaxation of aquantum system have been proposed over the years. Mostof these formulations couple the system of interest to somekind of external environment—the so-called thermal bath[5]. When the information about the microscopic state ofthe thermal bath is discarded, this approach leads to aneffective dynamics of the system that is no longer unitary.The main weakness of this approach lies in the fact thatthe thermal bath and its coupling to the system must bemodeled in an essentially ad hoc manner. In this Letter, wewill describe an alternative approach, based on the time-dependent current-density-functional theory (TDCDFT)[6–10], which allows a first-principle treatment of relaxa-tion in terms of a self-consistent unitary time evolution[11].We assume that the external potential acting on thesystem does not depend on time and that the system isclosed and isolated. We thus consider the time evolution ofa many-electron system described by the Hamiltonian06=96(1)=016405(4)$23.00 01640H^  Xip^2i2m W^ ZdrVrn^r; (1)where n^r is the density operator, Vr is the externalpotential, and W^ describes the electron-electron interaction(here and in the following, @  1). A possible way togenerate the initial nonequilibrium state j ii is by applyingan external potential at t  1 and allowing the system torelax in the presence of this potential. The external poten-tial is then switched off at the initial time t  0, and thesystem evolves henceforth according to the Schro¨dingerequation i@tjti  H^jti. Because of the presence ofthe electron-electron interaction, the system will relax toan equilibrium state of H^. Because the system is closed andisolated during its time evolution, the energy is conservedand equal to its initial value Ei  h ijH^j ii. This impliesthat the final equilibrium state will not be the ground stateof H^ but the most probable state with total energy Ei. Inthis final state, the entropy, as well as the temperature, willhave finite values. Our task is to calculate the time evolu-tion of the mechanical energy between the initial and thefinal states of this evolution.The central tenet of TDCDFT is that the correct timeevolution of the particle and current densities of the many-electron system can be obtained from the time evolution ofa noninteracting reference system, known as the Kohn-Sham (KS) system, described by the state jKSti. Thisstate evolves from an initial state jKS;ii—constructed togive the initial density and energy of the system—accord-ing to the equationi@tjKSti  H^KStjKSti; (2)where the KS Hamiltonian isH^KSt Xi12mp^i  eAxcr^i; t2 Zdrn^rVhxcr; tZdrn^rVr: (3)Vhxcr; t is an effective scalar potential that includes the5-1 © 2006 The American Physical SocietyPRL 96, 016405 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending13 JANUARY 2006Hartree potential h as well as the ground-state exchange-correlation xc potential of static density-functional the-ory (DFT) evaluated at the instantaneous density nr; t.Axcr; t is the xc vector potential, which is a functional ofthe density and of the time-dependent current density jr; tand whose role is to enforce the correct time evolution ofthe current. The time dependence of H^KS arises from itsbeing a functional of n and j. In practice, both Vhxcr; tand Axcr; t need to be approximated, but these approx-imations are nonempirical, in the sense that they are basedon exact physical constraints. The simplest and most popu-lar approximation is the local density approximation(LDA), which expresses the xc potential as a functionalof the density in the following manner:Vhxcr; t Zdr0e2jr r0j nr0; t xcnr; t; (4)where xcn  dxcn=dn and xcn is the xc energydensity of a uniform electron gas [10]. As for Axc, aquasilocal approximation in terms of the current densitywas derived by Vignale and Kohn [12] and can be cast inthe form [13]eddtAxc;kr; t  1nr; t riikxcr; t; (5)where i; k denote Cartesian components and the stresstensor xc is given byikxcr; t  r; tikr  v r; t	 rivk rkvi  23ikr  v; (6)where vr; t  jr; t=nr; t is the velocity, and we have,for simplicity, neglected retardation effects in the relationbetween xc and the velocity. The quantities  and  arethe viscoelastic constants of the homogeneous electronliquid and can be calculated from the linear responsefunctions of the latter [10], evaluated at the instantaneousdensity nr; t. We want to point out that Eqs. (2)–(6)define a self-consistent unitary evolution which is designedto approximate the true density and current of the system. Itis the dependence of the xc vector potential on the velocitythat breaks the time-reversal invariance of the KSHamiltonian and allows for the possibility of irreversiblerelaxation.We now define the key quantity of this Letter, the‘‘Kohn-Sham energy’’Et  hKStjT^Axc  V^jKSti  Ehxcnt hKStjH^KStjKStiZdrVhxcr; tnr; t  Ehxcnt; (7)where T^Axc Pip^i  eAxcr^i; t2=2m is the kinetic en-ergy operator, and Ehxc is the Hartree  xc energy func-tional of the ground-state DFT. Ehxc and Vhxc are related bythe identity Vhxc  Ehxcn=n. In static DFT, Et is just01640the familiar expression from which the ground-state energyis calculated. In TDCDFT, Et is not the true energy of theinteracting many-particle system, but, as we will showbelow, it decreases monotonically with time going fromEi at t  0 to some final value Ef  E0 at t  1, E0being the true ground-state energy of the system.To prove this point, we observe that a straightforwardcalculation shows that (we used H^KS=Axc  j^)dEdtZdrjr; t  dAxcdt: (8)Equation (8) immediately allows us to identify dE=dt asthe work done on the system by the self-consistent xcvector potential. By putting Eq. (5) in Eq. (8), we getdEdt 2ZdrTr! 13 Tr!2 ZdrTr!2; (9)where !ij  rivj rjvi=2. The nonpositivity ofdEt=dt follows from the fact that the viscosity constants and  are positive [10].Next we prove that Et ! Ef  E0 for t! 1. Webegin by observing that, at a fixed time t [14],Et min!nthtjT^Axct V^jtiEhxcntE0Axct;VEhxcntEhxcnt;Axct; (10)where E0A; V is the instantaneous ground-state energy ofthe system, in the presence of static potentials V and A.Ehxcn; A is the Hartree  xc energy functional of thesystem in the presence of a static vector potential Ar.Now it is known [15] that, in LDA and up to second orderin Br  r	 Ar, one has Ehxcn  Ehxcn; A RdrcnrB2r> 0, because cn (equals one-half thedifference between the orbital magnetic susceptibilitiesof the interacting and noninteracting homogeneous elec-tron liquid [15]) is a positive quantity. Assuming that thisinequality is generally true [16], we can then neglect thelast two terms on the right-hand side of Eq. (10) to arriveat Et  E0Axct; V. We now observe that, becausethe electron-electron Coulomb interaction is a positiveoperator, the ground-state energy of an interacting many-particle system is always greater than the ground-stateenergy of the same system where electron-electron inter-action is turned off. We then conclude that E0Axc; V Eni0 Axc; V, where Eni0 is the instantaneous ground-stateenergy of the noninteracting system. On the other hand,the ground-state energy of a noninteracting fermion systemis always greater than the ground-state energy Eni0;b of anoninteracting boson system in the same external poten-tials. We then haveEtEni0 Axc;VEni0;bAxc;VEni0;bAxc0;V; (11)where we have used in the last step the diamagnetic in-equality, which states that the ground-state energy of aspinless boson system in the presence of a vector poten-tial is always greater than the ground-state energy of the5-2PRL 96, 016405 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending13 JANUARY 2006same system without the vector potential [17]. BecauseEni0;bAxc  0; V is a time-independent quantity, we havearrived at our conclusion that Et is bounded. Equa-tions (9) and (11) allow us to conclude that the limitt! 1 of Et is finite and that limt!1dE=dt  0. FromEq. (9) and the requirement that j vanishes at infinity,we deduce that we can have dE=dt  0 if and only ifr  v  0 and r	 v  0, which imply v  0 everywhere(we consider only simply connected geometries). In turn,this implies that the current j is zero everywhere, and,therefore, through the continuity equation, the densityapproaches a stationary limit nfr. This can happen onlyif the asymptotic state is one of the eigenstates of the KSHamiltonian with density nfr. In the following, we willdisregard the unlikely cases when the system gets stuck insome excited state [18], and we will assume nfr  n0r,where n0r is the ground-state density. The time evolutionof Eq. (2) will then drive the KS system to the ground stateof the corresponding KS Hamiltonian.To demonstrate the above ideas in a concrete example,we have computed numerically the time evolution of theelectronic dipole moment dt  R dzznz; t [Fig. 1(a)]and the energy Et  E0 [Fig. 1(b)] in a one-dimensionalquantum well, such that the electrons are confined in the zdirection and yet are free to move in the x-y plane [19]. Thesingle-particle states for this system are of the form nzeikr, where k and r are vectors in the x-y plane.The system is initially prepared in a state that correspondsto the KS ground state in the presence of a uniform electricfield E  0:01 mV=nm in the z direction. This state ishomogeneous in the x-y plane, and this property is main-tained throughout the subsequent time evolution. Thismeans that all the electrons share the same state of motionin the z direction—a time-dependent state that can bewritten as a linear superposition of the lowest-lying sta-tionary states  nz (typically the two lowest ones suffice).The xc vector potential Axc has been approximated in theFIG. 1. (a) Plot of the dipole moment dt for the electrons inthe quantum well as described in the text. (b) Plot of the Kohn-Sham energy Et  E0. (c) Plot of St=2Ei  E0pasgiven by Eqs. (12) and (15). A fit of Et using the data in (b)gives   0:0034 a:u:01640form of Eq. (5) [20]. We see from Fig. 1 that the conditiondE=dt  0 is verified at all times. Moreover, the energyEt is well approximated by the analytic formEt  E0  Ei  E0et; (12)where the parameter  can be estimated from the slope ofthe curve in Fig. 1.We now wish to argue that Et  E0 is the maximumwork that can be extracted from the system at a given timeand that its time derivative is related to the rate of entropyproduction [21]. To do this, let us take a closer look to thequantum well (see Fig. 2). In the 3D system, we separatetwo different sets of degrees of freedom: the motion in the zdirection confined in the quantum well and the free motionin the x-y plane. These two sets of degrees of freedom canbe studied as two coupled ‘‘subsystems’’ (see Fig. 2). Thecoupling is induced by the electron-electron Coulombinteraction: No empirical modeling is needed. As the mac-roscopic state of motion in the z direction evolves withtime, energy is transferred from this motion into low-lyingexcitations of the two-dimensional electron gas (2DEG) inthe x-y plane. These excitations are, in the simplest case,double electron-hole pairs with zero total momentum.We start from an initial state in which the 2DEG is in theground state but all the electrons occupy an excited state ofmotion in the z direction. At the end of the relaxation, allthe electrons are in lowest energy state in the z direction,but the 2DEG is in an excited state. In the process, we havetransferred energy from a macroscopic state to the manydegrees of freedom of the 2DEG. The total energy isconserved and given by Ei. On the other hand, theinfinite-time limit of Et is the ground-state energy ofthe system E0, as one can see by noting that the finalKohn-Sham Hamiltonian, according to Eq. (3), coincidesFIG. 2 (color online). The motion in the z direction is coupledthrough the Coulomb interaction to a 2DEG in the x-y plane. Theenergy is lost to the motion in the z direction at a rate given bydE=dt.5-3PRL 96, 016405 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending13 JANUARY 2006with the Kohn-Sham Hamiltonian of static DFT which, bydefinition, yields the correct ground-state energy. Since theelectrons are finally at rest in the z direction, the differenceEi  E0 must be entirely ascribed to the excitation of the2DEG; i.e., Ei  E0 is the total heatQ, transferred from themacroscopic motion in the z direction to the many degreesof freedom of the 2DEG. Denoting by Ti and Tf the initialand final temperatures of the 2DEG, respectively, we haveQ Z TfTidTCVT; (13)where CVT is the heat capacity of the 2DEG. At lowtemperature, we have CVT  T, with  AmkB=3, where A is the area occupied by the gas inthe x-y plane and m is the particle mass renormalized byelectron-electron interactions. Because in our model cal-culation Ti  0, we get for the final temperature Tf 2=Ei  E0p , and, by recalling that CV=T  dS=dT,we get the increase in entropy S  Tf. The fact that theenergy Ei  E0 is converted into heat that increases thetemperature and entropy of the 2DEG allows us to identifyEi  E0 as the maximum energy that could, in principle, beextracted from the motion in the z direction to do work.We can also use Eq. (13) to define a suitable ‘‘tempera-ture’’ for nonequilibrium states. We observe that Eq. (13)can be cast in the formR10 dtdEdt  Tt dSdt  0, whichsimply states the global conservation of energy. This sug-gests that we define the nonequilibrium temperature of thesystem in such a way that the equationdEdt Tt dSdt(14)is satisfied at all t. The possibility of this definition stemsfrom the fact that we can always follow a nonequilibriumthermodynamical transformation through a series of smallquasiequilibrium transformations. Equation (14) is, in fact,a differential equation for the temperature that can beeasily solved to give Tt  2=Ei  Etp [we haveused again CVT  T for a 2DEG] and, finally [seeFig. 1(c)],St 2Ei  Etq: (15)In conclusion, we have shown that the time-dependentcurrent-density-functional theory naturally opens the pos-sibility of a first-principle description of relaxation inmany-body systems. To show this, we have identified aquantity, the Kohn-Sham energy, that defines a ‘‘timearrow’’ in the evolution of the system by monotonicallydecreasing with time. We are also able to relate the energydissipated to the increase in entropy and temperature. Thisthermodynamical approach allows us to identify the Kohn-Sham energy with the maximum amount of work that canbe extracted from the system.We thank Carsten Ullrich for assistance with the numeri-cal calculations and for many useful discussions and01640Janmin Tao for a critical reading of the manuscript. Weacknowledge financial support from NSF Grant No. DMR-0313681 and the kind hospitality of the Scuola NormaleSuperiore in Pisa, where part of this work was completed.5-4[1] U. Weiss, Quantum Dissipative Systems (World Scientific,Singapore, 1999), 2nd ed.[2] Introduction to Nanoscale Science and Technology, editedby M. Di Ventra, S. Evoy, and J. R. Heflin (Springer,Berlin, 2004).[3] Atoms in Intense Laser Fields, edited by M. Gavrila(Academic, Boston, 1992).[4] We assume that the system is large enough for thePoincare´ recurrence time to be practically infinite.[5] R. P. Feynman and F. L. Vernon, Ann. Phys. (N.Y.) 24, 118(1963).[6] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864(1964).[7] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133(1965).[8] E. K. U. Gross, J. F. Dobson, and M. Petersilka, in DensityFunctional Theory, edited by R. F. Nalewajski (Springer,Berlin, 1996), p. 81.[9] R. van Leeuwen, Int. J. Mod. Phys. B 15, 1969 (2001).[10] G. F. Giuliani and G. Vignale, Quantum Theory of theElectron Liquid (Cambridge University, Cambridge,England, 2005).[11] For a different approach to dissipative systems in theframework of time-dependent density-functional theory,see K. Burke, R. Car, and R. Gebauer, Phys. Rev. Lett. 94,146803 (2005).[12] G. Vignale and W. Kohn, in Electronic Density FunctionalTheory: Recent Progress and New Directions, editedby J. F. Dobson, G. Vignale, and M. P. Das (Plenum,New York, 1996), p. 199.[13] G. Vignale, C. A. Ullrich, and S. Conti, Phys. Rev. Lett.79, 4878 (1997).[14] We have E0A;VminfhjT^A V^jiEhxcn;AgminfhjT^A V^Wjig.[15] G. Vignale, M. Rasolt, and D. J. W. Geldart, Phys. Rev. B37, 2502 (1988).[16] In a homogeneous electron liquid, the presence of amagnetic field tends to localize the electrons, thus deep-ening the xc hole and, therefore, decreasing the xc energy.If the inhomogeneity of the system is not so strong, weexpect this to be more generally true.[17] B. Simon, Phys. Rev. Lett. 36, 1083 (1976).[18] For the system to get stuck in an excited state, the initialwave function should have certain symmetries that do notbelong to the ground state. In experiments, it is difficult toprepare a many-particle system in such a state.[19] H. O. Wijewardane and C. A. Ullrich, Phys. Rev. Lett. 95,086401 (2005).[20] A more accurate study of this system, including thefrequency-dependence of the viscoelastic coefficients,has been recently presented in Ref. [19].[21] L. D. Landau and E. M. Lifshitz, Statistical Physics(Pergamon, New York, 1980).

Relaxation in Time-Dependent Current-Density-Functional Theory

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