Quantum Ergodicity
Anatoli Polkovnikov Boston University
Thermaliza on: From Glasses To Black Holes., ICTS, Bangalore, June 2013
Simons Foundation
Outline.
Part I. Quantum ergodicity and the eigenstate theramiza on hypothesis (ETH).
Part II. Applica ons of ETH to equilibrium and non- equilibrium thermodynamics. Three different approaches describing isolated systems of par cles (systems with sta onary Hamiltonian). I. Microscopic based on studying long me limit of Hamiltonian dynamics
Works for small few-par cle systems. Can be prohibi vely (exponen ally) expensive in chao c systems. Chao c Regular II. Sta s cal. Start from equilibrium sta s cal descrip on (system is in the most random state sa sfying constraints). Use Hamiltonian (Lindblad) dynamics perturba vely: linear response (Kubo)
III. Mixed: kine c equa ons, Master equa on, Fokker Planck equa on,…: use sta s cal informa on and Hamiltonian dynamics to construct rate equa ons for the probability distribu on: What is the fundamental problem with the first microscopic approach? Imagine Universe consis ng of three par cles (classical or quantum)
Can this universe describe itself? – No. There is no place to store informa on. Increase number of par cles: can simulate three par cle dynamics. Complexity of the total system grows exponen ally, much faster than its ability to simulate itself.
It is fundamentally impossible to get a complete microsopic descrip on of large interac ng systems. 10
as an effective thermal bath? And if this is not possible, noting by (E) the set of eigenstates of H having en- H are there some observable effects on the system dynam- ergies between E and E + δE,wedefineˆρmc(E)= ics? While these questions are definitely connected to 1/ Ψ Ψ ,where is the total num- α (E) N| α α | N quantum ergodicity (Goldstein et al., 2010), a topic with ber∈ ofH states in the micro-canonical shell. Let us now a long history dating back to the early days of quantum ask the most obvious question: given a generic initial mechanics (Deutsch, 1991; Mazur, 1968; von Neumann, condition made out of states in a microcanonical shell, 1929; Pauli and Fierz, 1937; Peres, 1984; Srednicki, 1994; Ψ = c Ψ , is the long time average | 0 α (E) α | α Suzuki, 1971), the past few years have brought a great of the density∈H matrix of the system given by the micro- deal of progress in the context of closed many-body sys- canonical density matrix? The answer to this question tems. The main motivation came from recent experi- for a quantum system is, unlike in the classical case, al- ments on low dimensional cold atomic gases described most always no, as J. von Neumann realized already in in some detail in Sec. IV in this Colloquium (Greiner 1929 (von Neumann, 1929). More precisely, if we assume et al., 2002b; Kinoshita et al., 2006). The experimental the eigenstates of the system not to be degenerate, the availability of essentially closed (on the time scales of ex- time average is periments) stronglyHow do we connect these three approaches? correlated systems together with the awareness of the conceptual importance of these issues Ψ(t) Ψ(t) = c 2 Ψ Ψ =ˆρ , (14) | | | α | | α α | diag in a number of areas (e.g. transport problems, many- α bodyClassical localization,ergodicity integrable: over long periods of me, the me spent by an ensemble of and non-integrable dynam- where Ψ(t) is the time evolved of Ψ0 . This object is ics)par cles in some region of the phase space the same energy (and other conserved have stimulated a lot of interest on quantum ther- known| in the modern literature as the| diagonal ensem- malizationintegrals) is propor onal to the volume of this region. . Below we will give a synthetic view on a ble (Rigol, 2009; Rigol et al., 2008, 2007). Notice now number of recent important developments on this sub- that the most obvious definition of ergodicity, i.e. the ject, starting with the discussions of the general concepts requirement ρ = ρ , implies that c 2=1/ for mc diag | α | N ofIn simple words classical ergodicity and thermalization, andergodicity then moving is delocaliza on in phase space to the every α, a condition that can be satisfied only for a very discussion of many-body systems and integrability. special class of states. Quantum ergodicity in the strict sense above is therefore almost never realizable (Gold- stein et al., 2010; von Neumann, 1929). A. Quantum and classical ergodicity. Our common sense and expectations, which very fre- quently fail miserably in the quantum realm, make us While the idea of ergodicity is well defined in classical nevertheless believe that, in contrast with the arguments mechanics, the concept of quantum ergodicity is some- above, macroscopic many-body systems should behave what less precise and intuitive. Classically, an interacting ergodically almost always, unless some very special con- system of N particles in d dimensions is described by a ditions are met (e.g. integrability). The key to under- point X in a (2 dN)-dimensional phase space. The intu- stand ergodicity in therefore to look at quantum systems itive content of the word ”ergodic”, i.e. the equivalence of in a different way, shifting the focus on observables rather phase space and time averages, can be then formalized by than on the states themselves (Mazur, 1968; von Neu- requiring that if we select an initial condition X having 0 mann, 1929; Peres, 1984). Given a set of macroscopic initial energy H(X )=E,whereH is the Hamiltonian 0 observables M a natural expectation from an ergodic ofChao c ( the system,ergodic then ) Intergrable (non-β ergodic). Number of system would{ be} for every Ψ on a microcanonical shell | 0 1 T conserved quan es = number of (E) δ(X X(t)) lim dt δ(X X(t)) = ρmc(E), H − ≡ T T − →∞ 0 degrees of freedom Ψ(t) Mβ(t) Ψ(t) t + Tr[Mβρˆmc] Mβ mc,(15) (13) | | →→ ∞ ≡ where ρmc(E) is the microcanonical density of the system i.e. that looking at macroscopic observables long after the on the hyper-surface of the phase space of constant en- time evolution started makes the system appear ergodic ergyMost interac ng many-par cle systems are chao c even with E, and X(t) is the phase space trajectory with initial for every initial condition we may choose in (E). One H condition X0. Of course if this condition is satisfied by all needs a certain care in defining the infinite time limit trajectories,regular interac ons (no disorder) then it is also true for every observable. We here, since literally speaking it does not exist in finite immediately see that in order to have ergodicity, the dy- systems because of quantum revivals. A proper way to namics cannot be arbitrary: the trajectories X(t)have understand this limit is to require that Eq. (15) holds in to cover uniformly the energy hyper-surface for almost the long time limit at almost all times. Mathematically every initial condition X0. this means that the mean square difference between the The most obvious quantum generalization of this no- LHS and RHS of Eq. (15) averaged over long times is tion of ergodicity is arduous (von Neumann, 1929). Let vanishingly small for large systems (Reimann, 2008). To us first of all define a quantum microcanonical density avoid dealing with these issues ergodicity can be defined matrix: given a Hamiltonian with eigenstates Ψα of using the time average, i.e. requiring that energies E , a viable definition of the microcanonical| α Ψ(t) M (t) Ψ(t) =Tr[M ρˆ ]= M . (16) ensemble is obtained by coarse graining the spectrum | β | β diag βmc on energy shells of width δE,sufficiently big to con- Notice that if the expectation value of Mβ relaxes to a tain many states but small on macroscopic scales. De- well defined state in the sense described above, this state Famous counter example: Fermi-Pasta Ulam problem. First numerical study of ergodicity (thermaliza on) in an interac ng many-body system.
. . . . 1 2 3 n-2 n-1 n Slow variables
Expecta on: 1. Excite single normal mode 2. Follow dynamics of energies 3. Eventual energy equipar on
Found: 1. Quasiperiodic mo on 2. Energy localiza on in q-space 3. Revivals of ini al state 4. No thermaliza on! Onset of chaos and thermaliza on in classical systems
Integrable system of N degrees of freedom has N integrals of mo on
N constraints in 2N dimensional phase space:
each corresponds to the hyper-surface in phase space
Trajectories are confined to intersec on of all these surfaces forming N-dimensional invariant tori.
KAM theorem: weak perturba on preserves `almost all’ tori under the condi on:
Thermaliza on (chaos) occurs though destruc on of KAM tori Similar scenario for the FPU problem F. M. Izrailev and B. V. Chirikov, Soviet Physics Doklady 11 ,1 (1966) Parametric resonance width matches resonance in 1st order perturba on theory:
Spectral Entropy: (Livi et. al, PRA 31, 2, 1985) Spectral entropy:
where
Effec ve `Reynolds’ number
Strong stochas city threshold: Setup: isolated quantum systems. Arbitrary ini al state. Hamiltonian dynamics.
Easiest interpreta on of density matrix: start from some ensemble of pure states
Look into expecta on value of some observable O at me t.
In order to measure the expecta on value need to perform many measurements unless ψ(t) is an eigenstate of O.
Each measurement corresponds to a new wave func on due to sta s cal fluctua ons. So unless we can prepare iden cal states we can only measure
If Hamiltonian is fluctua ng need to average over the Hamiltonians -> non-unitary evolu on. 10
as an effective thermal bath? And if this is not possible, noting by (E) the set of eigenstates of H having en- H are there some observable effects on the system dynam- ergies between E and E + δE,wedefineˆρmc(E)= ics? While these questions are definitely connected to 1/ Ψ Ψ ,where is the total num- α (E) N| α α | N quantum ergodicity (Goldstein et al., 2010), a topic with ber∈ ofH states in the micro-canonical shell. Let us now a long history dating back to the early days of quantum ask the most obvious question: given a generic initial mechanics (Deutsch, 1991; Mazur, 1968; von Neumann, condition made out of states in a microcanonical shell, 1929; Pauli and Fierz, 1937; Peres, 1984; Srednicki, 1994; Ψ0 = α (E) cα Ψα , is the long time average Suzuki, 1971), the past few years have brought a great | ∈H | 10 of the density matrix of the system given by the micro- deal of progress in the context of closed many-body sys- canonical density matrix? The answer to this question tems. The main motivation came from recent experi- for a quantum system is, unlike in the classical case, al- as an effective thermal bath? And if this is not possible, noting by (E) the set of eigenstates of H having en- ments on low dimensional cold atomic gases described most always no, as J. von Neumann realized already in are there some observable effects on the system dynam- ergies betweenH E and E + δE,wedefineˆρ (E)= in some detail in Sec. IV in this Colloquium (Greinermc 1929 (von Neumann, 1929). More precisely, if we assume ics? While these questions are definitely connected to et al.α , 2002b;(E) 1/ KinoshitaΨα Ψetα al.,where, 2006). Theis the experimental total num- ∈H N| | N the eigenstates of the system not to be degenerate, the quantum ergodicity (Goldstein et al., 2010), a topic with availabilityber of states of essentially in the micro-canonical closed (on the shell. time scales Let us of ex-now time average is a long history dating back to the early days of quantum periments)ask the most strongly obvious correlated question: systems given together a generic with initial the mechanics (Deutsch, 1991; Mazur, 1968; von Neumann, awarenesscondition of made the out conceptual of states importance in a microcanonical of these issues shell, Ψ(t) Ψ(t) = c 2 Ψ Ψ =ˆρ , (14) | | | α | | α α | diag 1929; Pauli and Fierz, 1937; Peres, 1984; Srednicki, 1994; inΨ a0 number= α of( areasE) cα (e.g.Ψα transport, is the longproblems, time average many- α Suzuki, 1971), the past few years have brought a great | ∈H | bodyof the localization, density matrix integrable of the and system non-integrable given by the dynam- micro- where Ψ(t) is the time evolved of Ψ . This object is deal of progress in the context of closed many-body sys- 0 ics)canonical have stimulated density matrix? a lot of The interest answer on toquantum this question ther- known| in the modern literature as the| diagonal ensem- tems. The main motivation came from recent experi- malizationfor a quantum. Below system we is, will unlike give ina synthetic the classical view case, on al- a ble (Rigol, 2009; Rigol et al., 2008, 2007). Notice now ments on low dimensional cold atomic gases described numbermost always of recent no, as important J. von Neumann developments realized on already this sub- in that the most obvious definition of ergodicity, i.e. the in some detail in Sec. IV in this Colloquium (Greiner ject,1929 starting (von Neumann, with the 1929). discussions More of precisely, the general if we concepts assume requirement ρ = ρ , implies that c 2=1/ for et al., 2002b; Kinoshita et al., 2006). The experimental mc diag α ofthe ergodicity eigenstates and of thermalization, the system not and to then be degenerate, moving to the the every α, a condition that can be satisfied| only| for aN very availability of essentially closed (on the time scales of ex- discussiontime average of many-body is systems and integrability. special class of states. Quantum ergodicity in the strict periments) strongly correlated systems together with the sense above is therefore almost never realizable (Gold- awareness of the conceptual importance of these issues Ψ(t) Ψ(t) = c 2 Ψ Ψ =ˆρ , (14) | | | α | | α α | diag stein et al., 2010; von Neumann, 1929). in a number of areas (e.g. transport problems, many- A. Quantum and classicalα ergodicity. Our common sense and expectations, which very fre- body localization, integrable and non-integrable dynam- where Ψ(t) is the time evolved of Ψ0 . This object is quently fail miserably in the quantum realm, make us While| the idea of ergodicity is well| defined in classical ics) have stimulated a lot of interest on quantum ther- known in the modern literature as the diagonal ensem- nevertheless believe that, in contrast with the arguments mechanics, the concept of quantum ergodicity is some- malization. Below we will give a synthetic view on a ble (Rigol, 2009; Rigol et al., 2008, 2007). Notice10 now above, macroscopic many-body systems should behave what less precise and intuitive. Classically, an interacting number of recent important developments on this sub- that the most obvious definition of ergodicity, i.e. the ergodically almost always, unless some very special con- system of N particles in d dimensions is described by a ject, startingas an effective with the thermal discussions bath? And of the if this general is not possible, concepts notingrequirement by (E)ρ the set= ρ of eigenstates, implies of thatH havingc 2 en-=1/ for ditions are met (e.g. integrability). The key to under- point X Hin a (2mcdN)-dimensionaldiag phase space.α The intu- of ergodicityare there and some thermalization, observable effects and on then the system moving dynam- to the ergiesevery betweenα, a conditionE and E that+ δE can,wedefineˆ be satisfiedρmc| ( onlyE)=| for aN very stand ergodicity in therefore to look at quantum systems discussionics? Whileof many-body these questions systems are and definitely integrability. connected to itiveα ( contentE) 1/ ofΨ theα Ψ wordα ,where ”ergodic”,is i.e.the total the equivalence num- of special∈H classN| of states. | QuantumN ergodicity in the strict in a different way, shifting the focus on observables rather quantum ergodicity (Goldstein et al., 2010), a topic with berphase of states space in and the time micro-canonical averages, can shell. be then Let us formalized now by a long history dating back to the early days of quantum asksense the most above obvious is therefore question: almost given never a generic realizable initial (Gold- than on the states themselves (Mazur, 1968; von Neu- requiring that ifErgodicity we select in Quantum Systems an initial condition X0 having mechanics (Deutsch, 1991; Mazur, 1968; von Neumann, conditionstein et made al., 2010; out of von states Neumann, in a microcanonical 1929). shell, mann, 1929; Peres, 1984). Given a set of macroscopic A. Quantum and classical ergodicity. Classical system: me average of the probability distribu on becomes equivalent to initial energy H(X0)=E,whereH is the Hamiltonian 1929; Pauli and Fierz, 1937; Peres, 1984; Srednicki, 1994; Ψ0 Our= commonα (E) cα senseΨα and, is expectations, the long time average which very fre- observables Mβ a natural expectation from an ergodic | of the system,∈H then | Suzuki, 1971), the past few years have brought a great ofthe quently themicrocanonical density fail matrix miserably ensemble. Implies of the insystem thethermaliza on given quantum by of all observables. the realm, micro- make us system would{ be} for every Ψ on a microcanonical shell While the idea of ergodicity is well defined in classical 0 deal of progress in the context of closed many-body sys- canonicalnevertheless density believe matrix? that, The inT answer contrast to this with question the arguments (E) | tems. The main motivation came from recent experi- 1 mechanics, the concept of quantum ergodicity is some- forδ a(X quantumX(t)) systemlim is, unlike indt theδ( classicalX X( case,t)) = al-ρmc(E), H ments on low dimensional cold atomic gases described above,− macroscopic≡ T many-bodyT systems− should behave what less precise and intuitive. Classically, an interacting most always no, as J.→∞ von Neumann0 realized already in Ψ(t) Mβ(t) Ψ(t) t + Tr[Mβρˆmc] Mβ mc,(15) in some detail in Sec. IV in this Colloquium (Greiner 1929ergodically (von Neumann, almost 1929). always, More unless precisely, some if we very assume special(13) con- | | →→ ∞ ≡ system of N particles in d dimensions is described by a Quantum systems (quantum language?) – no relaxa on of the density matrix to the et al., 2002b; Kinoshita et al., 2006). The experimental thewhereditions eigenstatesρ are(E met of) is the the (e.g. system microcanonical integrability). not to be degenerate, density The of key the the to system under- i.e. that looking at macroscopic observables long after the point X in a (2 dN)-dimensional phase space. The intu- microcanonicalmc ensemble (von Neumann, 1929). availability of essentially closed (on the time scales of ex- timeonstand theaverage ergodicity hyper-surface is in therefore of the phase to look space at quantum of constant systems en- time evolution started makes the system appear ergodic itive contentperiments) of the strongly word correlated ”ergodic”, systems i.e. the together equivalence with the of ergyin aE di,ff anderentX way,(t) is shifting the phase2 the space focus trajectory on observables with initialrather for every initial condition we may choose in (E). One phaseawareness space and of time the conceptualaverages, can importance be then of formalized these issues by Ψ(t) Ψ(t) = cα Ψα Ψα =ˆρdiag, (14) H conditionthan| on theX0.| states Of course| themselves if| | this condition (Mazur,| is 1968; satisfied von by Neu- all needs a certain care in defining the infinite time limit requiringin a that number if we of select areas (e.g. an initial transport condition problems,X0 many-having α trajectories,mann, 1929; then Peres, it is 1984). also true Given for every a set observable. of macroscopic We here, since literally speaking it does not exist in finite initialbody energy localization,H(X0)= integrableE,where andH non-integrableis the Hamiltonian dynam- where Ψ(t) is the time evolved of Ψ . This object is Thermaliza onobservables must be built in to the structure of M a natural expectation0 Eigenstates from and revealed through an ergodic systems because of quantum revivals. A proper way to ics) have stimulated a lot of interest on quantum ther- knownimmediately| in the modern seeβ that literature in order as to the| havediagonal ergodicity, ensem- the dy- of the system, then observables (von Neuman{ }, 1929; Mazur, 1968; Sredniki, 1994; Rigol et. al. 2008, Riemann malization. Below we will give a synthetic view on a blenamicssystem(Rigol, cannot would 2009; be Rigol be for arbitrary:et every al., 2008,Ψ the0 2007).on trajectories a microcanonical Notice nowX(t)have shell understand this limit is to require that Eq. (15) holds in T 2008, …) | number of recent important1 developments on this sub- thatto( cover theE) most uniformly obvious definition the energy of ergodicity, hyper-surface i.e. the for almost the long time limit at almost all times. Mathematically δ(X ject,X starting(t)) withlim the discussionsdt δ(X of theX( generalt)) = ρ conceptsmc(E), H 2 − ≡ T T − requirementevery initialρmc condition= ρdiag, impliesX0. that cα =1/ for this means that the mean square difference between the of ergodicity and→∞ thermalization,0 and then moving to the everyΨα(t,) a conditionMβ(t) Ψ that(t) cant be+ satisfiedTr[| M onlyβ|ρˆmc for] aN veryMβ mc,(15) LHS and RHS of Eq. (15) averaged over long times is discussion of many-body systems and integrability. (13) The most| obvious| → quantum→ ∞ generalization≡ of this no- specialtioni.e. that of class ergodicity looking of states. at is Quantum macroscopic arduous ergodicity (von observables Neumann, in the strict long 1929). after Let the vanishingly small for large systems (Reimann, 2008). To where ρmc(E) is the microcanonical density of the system sense above is therefore almost never realizable (Gold- The limit is understood as at long mes at almost all mes. In a way in quantum language avoid dealing with these issues ergodicity can be defined on the hyper-surface of the phase space of constant en- steinustime firstet al.evolution, of 2010; all define von started Neumann, a quantum makes 1929). the microcanonical system appear density ergodic A. Quantum and classical ergodicity. thermaliza on is prebuilt in the system. Time evolu on is just dephasing. ergy E, and X(t) is the phase space trajectory with initial matrix:for every given initial a conditionHamiltonian we with may choose eigenstates in (EΨ).α Oneof using the time average, i.e. requiring that Our common sense and expectations, which very fre-H| condition X . Of course if this condition is satisfied by all quentlyenergiesneeds fail aE certain miserablyα, a viable care in the in definition quantumdefining of realm, the the infinite make microcanonical us time limit While0 the idea of ergodicity is well defined in classical Ψ(t) Mβ(t) Ψ(t) =Tr[Mβρˆdiag]= Mβ mc. (16) neverthelessensemble believe is obtained that, in by contrast coarse with graining the arguments the spectrum trajectories,mechanics, then the it concept is also true of quantum for every ergodicity observable.is some- We here, since literally speaking it does not exist in finite | | above, macroscopic many-body systems should behave immediatelywhat less see precise that and in orderintuitive. to haveClassically, ergodicity, an interacting the dy- onsystems energy because shells of of quantum width δE revivals.,sufficiently A proper big to way con- to Notice that if the expectation value of Mβ relaxes to a ergodically almost always, unless some very special con- system of N particles in d dimensions is described by a tainunderstand many states this limit but small is to require on macroscopic that Eq. (15) scales. holds De- in well defined state in the sense described above, this state namics cannot be arbitrary: the trajectories X(t)have ditions are met (e.g. integrability). The key to under- point X in a (2 dN)-dimensional phase space. The intu- to cover uniformly the energy hyper-surface for almost standthe ergodicity long time in limit therefore at almost to look at all quantum times. systems Mathematically itive content of the word ”ergodic”, i.e. the equivalence of every initial condition X . inthis a diff meanserent way, that shifting the mean the focus square on observables differencerather between the phase space and time averages,0 can be then formalized by thanLHS on and the states RHS themselves of Eq. (15) (Mazur, averaged 1968; over von Neu-long times is Therequiring most obvious that if we quantum select an generalization initial condition ofX thishaving no- 0 mann, 1929; Peres, 1984). Given a set of macroscopic tion ofinitial ergodicity energy isH( arduousX )=E,where (von Neumann,H is the Hamiltonian 1929). Let vanishingly small for large systems (Reimann, 2008). To 0 observables M a natural expectation from an ergodic us firstof theof all system, define then a quantum microcanonical density avoid dealingβ with these issues ergodicity can be defined system would{ be} for every Ψ on a microcanonical shell matrix: given a Hamiltonian with eigenstates Ψ of using the time average,| 0 i.e. requiring that 1 T | α (E) energiesδ(XEα,X a(t viable)) lim definitiondt δ of(X theX( microcanonicalt)) = ρmc(E), H − ≡ T T − Ψ(t) M (t) Ψ(t) =Tr[M ρˆ ]= M . (16) →∞ 0 Ψ(t) Mβ(t) Ψβ(t) t + Tr[Mβρˆmcβ ] diagMβ mc,(15)β mc ensemble is obtained by coarse graining the spectrum(13) | | | →| → ∞ ≡ on energywhere shellsρmc(E) of is the width microcanonicalδE,sufficiently density of big the to system con- i.e.Notice that looking that atif macroscopicthe expectation observables value long of afterMβ relaxes the to a tain manyon the states hyper-surface but small of the on phase macroscopic space of constant scales. en- De- timewell evolution defined started state in makes the sensethe system described appear above, ergodic this state ergy E, and X(t) is the phase space trajectory with initial for every initial condition we may choose in (E). One H condition X0. Of course if this condition is satisfied by all needs a certain care in defining the infinite time limit trajectories, then it is also true for every observable. We here, since literally speaking it does not exist in finite immediately see that in order to have ergodicity, the dy- systems because of quantum revivals. A proper way to namics cannot be arbitrary: the trajectories X(t)have understand this limit is to require that Eq. (15) holds in to cover uniformly the energy hyper-surface for almost the long time limit at almost all times. Mathematically every initial condition X0. this means that the mean square difference between the The most obvious quantum generalization of this no- LHS and RHS of Eq. (15) averaged over long times is tion of ergodicity is arduous (von Neumann, 1929). Let vanishingly small for large systems (Reimann, 2008). To us first of all define a quantum microcanonical density avoid dealing with these issues ergodicity can be defined matrix: given a Hamiltonian with eigenstates Ψα of using the time average, i.e. requiring that energies E , a viable definition of the microcanonical| α Ψ(t) M (t) Ψ(t) =Tr[M ρˆ ]= M . (16) ensemble is obtained by coarse graining the spectrum | β | β diag βmc on energy shells of width δE,sufficiently big to con- Notice that if the expectation value of Mβ relaxes to a tain many states but small on macroscopic scales. De- well defined state in the sense described above, this state In many problems of physical interest, it is necessary to abandon a search for the exact solution, and to turn instead to a statistical approach. Thisinvolvesmentallyreplacingthe answer which we seek, with an ensemble of possibilities, thenadoptingtheattitudethateach member of the ensemble is an equally likely candidate for the true solution. The choice of ensemble then becomes centrally important, and here information theory provides a reliable guiding principle. The principle instructs us to choose the least biased ensemble (the one which minimizes information content), subject to some relevant constraints. A well-known illustration arises in classical statistical mechanics: the least biased distribution in phase space, subject to a fixed normalization and average energy, isthecanonicalensembleofGibbs [1]. Another example appears in random matrix theory: by minimizing the information content of an ensemble of matrices, subject to various simpleconstraints,oneobtainsthe standard random matrix ensembles [2]. The purpose of this paper is to point out that Berry’s conjecture [3] regarding the energy eigenstates of chaotic systems, also emerges naturally from this principle of least bias. by Eq.1), and statesBerry’s that an conjecture eigenstate makesψE at energy two assertionsE will look regarding as if it were the chosen high-lying randomly energy eigenstates ψE of 1 from this ensemble.quantal systems whose classical counterparts are chaotic and ergodic :(1)Sucheigenstates The correlationsappear given to be by random Eq.1 are Gaussian motivated functions by consideringψ(x)onconfigurationspace,(2)withtwo-pointthe Wigner function [4] correspondingcorrelations to the eigenstate givenBerry conjecture, ψ byE , semiclassical limit (M.V. Berry, 1977) by Eq.1), and states that an eigenstate ψE at energy E will look as if it were chosen randomly
−D ∗ s s −ip·s/h¯ fromW this(x, p ensemble.) ≡ (2πh¯) ∗ ds ψs x − sψ x +1 e ip·s/,h¯ (2) E ψ x − E ψ x +2 E = 2 dp e δ[E − H(x, p)] (1) ! 2 " #2 " Σ # The correlations given! by" Eq.1! are motivated" # by considering the Wigner function [4] where D is the dimensionality of the system. For high-lying states ψ ,thisWignerfunc- Here, E is theAverage is taken over the energy shall vanishing in the limit but containing many energy of the eigenstate, H(x, p)istheclassicalHamiltoniandescribingE 0 correspondingstates. Equivalently to the eigenstate ψE , → tion, after local smoothing in the x variable, is expected to converge to the microcanonical the system, and Σ ≡ dx dp δ[E − H(x, p)]; ifs the Hamiltonians is time-reversal-invariant, distribution in phase space [3,5–7]:W (x, p) ≡ (2πh¯)−D ds ψ∗ x − ψ x + e−ip·s/h¯ , (2) E $ $ E E then ψ(x)isarealrandomGaussianfunction,otherwise! " 2# " ψ(x2)isacomplexrandomGaussian# 15 1 sm x p x p wherefunction.D is Berry’s theW dimensionalityE conjecture( , ) ≈ δ thus of[E the− uniquelyH system.( , )]. specifies, For high-lying for a given states energyψE,thisWignerfunc-(3)E,anensembleM Σ limit the momentum distribution function is less attention.E In a series of studies of relaxation in fermionic Hubbard models subject to quenches in the tion,of wavefunctions after local smoothingψ(x)(i.e. inM theExisvariable, the Gaussian is expected ensemble to converge with two-point to the microcanonical correlations2 interactions given strength it has been argued that for suffi- By taking the Fourier transformFor non-rela vis c par cles Berry of both sides of Eq.1, andf( thep) nsmoothinglocallyinthe= dp2dp3 ... Ψα(p, p2,...) | | ciently rapid quenches relaxation towards thermal equi- conjecture implies Maxwell- 2 2 distribution in phase space [3,5–7]: p librium occurs through a pre-thermalized phase (Moeckel x-variable rather than averagingBolzmann over distribu on the ensemble ME,itisstraightforwardtoshowthate− 2mkT = 3/2 = fMB(p), (27) and Kehrein, 2008, 2010). Similar two step dynamics oc- 1 (2πmkT ) curs in quenches of coupled superfluids where initial fast sm 1 the correlations givenBy “chaotic by Eq.1 and produce ergodic”, the we desired mean result, that all Eq.3. trajectories, except a set of measure zero, chaotically“light cone” dynamics leads to a pre-thermalized steady WE (x, p) ≈whereδ[ theE − temperatureH(x, p)]. is set by the equipartition law as (3) Σ state, which then slowly decays to the thermal equilib- This follows from E =3/2NkT. Notice that this is expected to happen The assertionand that ergodicallyψE(x)isaGaussianrandomfunctionismosteasilymotivatedby explore the surface of constantα energy in phase space. rium through the vortex-antivortex unbinding (Mathey for every eigenstate of energy close to E , as required by α and Polkovnikov, 2010). In Ref. (Burkov et al., 2007) a By taking the Fourier transform of boththe sides ETH. of Hence Eq.1, thermal and the behaviornsmoothinglocallyinthe will follow for every viewing ψE(x), locally, as a superposition of de Broglie waves with randomphases[3]. very unusual sub-exponential in time decay of correlation initial condition sufficiently narrow in energy. 2 functions was predicted and later observed experimen- 2 For generic many-body systems, such as Hubbard-like When the numberx-variable of theserather waves than becomes averaging infinite, over the the central ensemble limitME theorem,itisstraightforwardtoshowthat tells us that tally (Hofferberth et al., 2007) for relaxational dynamics models and spin chains, the close relation between break- of decoupled 1d bosonic systems. the correlations given by Eq.1 produce theing desired of integrability result, and Eq.3. quantum chaotic behavior is a ψE(x)willlooklikeaGaussianrandomfunction. known fact (Poilblanc et al., 1993). In particular, finite size many-body integrable systems are characterized by We can interpretThe Berry’s assertion conjecture that ψE as(x)isaGaussianrandomfunctionismosteasilymotivatedby making a specificthe Poisson pred spectraliction about statistics the while eigenstate the gradual breaking D. Outlook and open problems: quantum KAM threshold of integrability by a perturbation leads to a crossover to as a many-body delocalization transition ? ψE:oncewecomputeviewing ψEψ(Ex(),x)bysolvingtheSchr¨odingerequation,wecansubjectitto locally, as a superpositionthe Wigner-Dyson of de Broglie statistics. waves The with latter rando is typicallymphases[3]. asso- ciated, in mesoscopic systems or billiards, with diffusive The arguments above clearly pointed to the connection various testsWhen (see e.g. the Ref. number [8]), of and these we will waves observe becomes that,behavior infinite, yes, andψE can the(x)reallybehavesasa be central taken as limit a signature theorem of quantum tells us thatbetween thermalization in strongly correlated systems chaos (Imry, 1997). In many-body disordered systems and in chaotic billiards. This analogy however , rather the emergence of the Wigner-Dyson statistics was ar- than being the end of a quest, opens an entire new kind Gaussian randomψE(x)willlooklikeaGaussianrandomfunction. function with two-point correlations givenbyEq.1.Alternatively,wecan gued to be an indicator of the transition between metallic of questions, which are a current focus of both theoreti- interpret Berry’sWe conjecture can interpret as providing Berry’s us conjecture with the appro as(ergodic) makingpriate and a ensemble insulating specific predof (non-ergodic) wavefunctionsiction about phases the (Mukerjee eigenstatecal and experimental research. In particular, we do know et al., 2006; Oganesyan and Huse, 2007). Inspired by that in a number of models of strongly correlated parti- from which to choose a surrogate for the true eigenstatetheseψ close,ifwecannot(ordonotcare analogies, recent studies gave a boost to our cles eigenstate thermalization is at the root of thermal ψE:oncewecomputeψE(x)bysolvingtheSchr¨odingerequation,wecansubjectittounderstandingE of the crossover from non-ergodic to ther- behavior (Rigol et al., 2008). What is the cause of eigen- mal behavior as integrability is gradually broken and of state thermalization in a generic many-body system, i.e. various tests (see e.g. Ref. [8]), and wethe will origin observe of ergodicity/thermalization that, yes, ψE(x)reallybehavesasa in systems suffi- the analogue of the diffusive eigenstates in phase space 2 ciently far from integrability (Biroli et al., 2010; Kollath of Berry’s conjecture ? And most importantly, while in a This smoothingGaussian is performed random on function a scale which with is two-point large compareet correlationsal.,dwiththelocalcorrelationlength 2007; Manmana givenbyEq.1.Alternatively,wecanet al., 2007; Rigol, 2009; Rigol finite size system the transition from non-ergodic to er- et al., 2008). In particular, a careful study of the asymp- godic behavior takes the form of a crossover, what hap- x of ψ( ), but smallinterpret compared Berry’s with conjecture a classically as relevant providing distancetotics us with of scale density-density the (see appro e.g.priate equations correlators ensemble 3.29 and and momentum of wavefunctions dis- pens in the thermodynamic limit ? Is the transition from tribution function for hard-core bosons in 1d showed that ergodic to non-ergodic behavior still a crossover or it is 3.30 of Ref. [7]). This allows us to replace ensemble averagitheng transition with local from smoothing. non-thermal to thermal behavior in sharp (a quantum KAM threshold )? from which to choose a surrogate for thefinite true size eigenstatesystems takesψE,ifwecannot(ordonotcare the form of a crossover con- At present, research on these questions has just trolled by the strength of the integrability breaking per- started. An interesting idea that has recently emerged is 3 turbation and the system size (Rigol, 2009). Moreover that the study of the transition from integrability to non- there is a universality in state to state fluctuations of integrability in quantum many-body systems is deeply 2This smoothing is performed on a scale whichsimple is observables large compare in thisdwiththelocalcorrelationlength crossover regime (Neuenhahn connected to another important problem at the frontier and Marquardt, 2010), which goes hand-by-hand with of condensed matter physics: the concept of many-body of ψ(x), but small compared with a classicallyan analogous relevant transition distance fromscale Poisson (see e.g. to equations Wigner-Dyson 3.29 andlocalization (Altshuler et al., 1997; Basko et al., 2006), level statistics (Rigol and Santos, 2010; Santos and Rigol, which extends the original work of Anderson on single- 2010a). When integrability is broken by sufficiently particle localization (Anderson, 1958). We note that re- 3.30 of Ref. [7]). This allows us to replace ensemblestrong perturbation averaging ergodic with local behavior smoothing. emerges (Neuen- lated ideas were put forward in studying energy transfer hahn and Marquardt, 2010; Rigol, 2009; Rigol and San- in interacting harmonic systems in the context of large or- tos, 2010),3 which in turn appears to be related to the ganic molecules (Leitner and Wolynes, 1996; Logan and validity of the ETH (Rigol et al., 2008). In this con- Wolynes, 1990). More specifically, it has been noticed text, the anomalous, non-ergodic behavior of integrable that a transition from localized to delocalized states ei- models has been reinterpreted as originating from wide ther in real space (Pal and Huse, 2010) or more gener- fluctuations of the expectation value of natural observ- ally in quasi-particle space (Canovi et al., 2011) is closely ables around the microcanonical average (Biroli et al., connected to a corresponding transition from thermal to 2010). non-thermal behavior in the asymptotics of significant All these statements apply to the asymptotic (or time observables. For weakly perturbed integrable models, averaged) state. So far the relaxation in time, in par- the main characteristic of the observables to display such ticular in the thermodynamic limit, has received much transition appears to be again their locality with respect Thermaliza on through eigenstates (J. Deutsch, 1991, M. Srednicki 1994, M. Rigol et. al. 2008) Main idea: thermaliza on is encoded in eigenstates of the Hamiltonian. Dynamics is just dephasing between the eigenstates.
Prepare a quantum system in some state, characterized by the density matrix ρ0 and let it evolve with the Hamiltonian H.
Look into the long me limit of the expecta on value sta c observable, O
In typical situa ons (both equilibrium and not) energy is extensive and energy fluctua ons are sub-extensive (consequence of locality of the Hamiltonian). If all eigenstates within a subextensive energy shell are similar then
The long me limit of an observable does not depend on the details of the ini al state. M. Rigol, V. Dunjko & M. Olshanii, Nature 452, 854 (2008)
a, Two-dimensional lattice on which five hard-core bosons propagate in time. b, The corresponding relaxation dynamics of the central component n
(kx = 0) of the marginal momentum distribution, compared with the predictions of the three ensembles
c, Full momentum distribution function in the initial state, after relaxation, and in the different ensembles. Fluctua ons of the observables. Off-diagonal matrix elements (M. Srednicki, 1999)
Consider ul mate microcanonical ensemble, a single energy eigenstate
There are exponen ally many terms in this sum ~ exp[S], recall that the entropy is the measure of the density of states (number of states within microscopic energy window)
Make an ansatz (M. Srednicki, 1996)
Expect that fO is a slow func on of E (changing on extensive scales) but fast func on of ω, on inverse relaxa on me scale, σ is a random variable with a unit variance .
The func on fO(ω) should decay faster than exponen ally on intensive energy scales. Let us look into non-equal me correla on func on
Long (Markovian) mes (small frequencies): expect exponen al decay.
Lorentz or Breit-Wigner distribu on
Short (non-Markovian) mes (large frequencies): expect Gaussian decay. Also follows from short me perturba on theory (Zeno effect).
Can recover fluctua on-dissipa on rela ons under the same condi ons (V. Zelevinsky 1995, M. Srednicki 1999, E. Khatami et al 2013) How representa ve is me average? (M. Srednicki, 1999)
Using that
If ETH applies we see that devia ons of the expecta on value from the me average are exponen ally small.
So the system remains expone ally close to the equilibrium state at almost all mes. Quantum chaos and Wigner-Dyson distribu on
Consider a fully chao c random NxN matrix with the Gaussian probability distribu on
One can show that there is a level repulsion so that the probability of the level spacing is:
Wigner-Dyson distribu on for orthogonal ensembles.
Level spacing distribu on in heavy nuclei Level spacing distribu on in Sinai billiard K.H. Böchhoff (Ed.), Nuclear Data for Science and O. Bohigas, M.J. Giannoni, C. Schmit Technology, Reidel, Dordrecht (1983) Phys. Rev. Le ., 52 (1984) A histogram of spacings between bus arrival mes in Cuernavaca (Mexico), in crosses; the solid line is the predic on from random matrix theory. (Source: Krbalek-Seba, J. Phys. A., 2000)
Hydrogen in a strong magne c field Spacing distribu on for a billion zeroes of the Riemann zeta D. Wintgen, H. Friedrich func on, and the corresponding predic on from random matrix Phys. Rev. A, 35 (1987), theory. (Source: Andrew Odlyzko, Contemp. Math. 2001) 3
2. Level spacing distribution TABLE I: Dimensions of subspaces
Bosons The distribution of spacings s of neighboring energy L =18 k =1, 5, 7 k =2, 4, 8 k =3 k =6 levels [2, 7, 8, 10] is the most frequently used observ- 1026 1035 1028 1038 able to study short-range fluctuations in the spectrum. L =21 k =7 other k’s Quantum levels of integrable systems are not prohib- 5538 5537 ited from crossing and the distribution is Poissonian, L =24 odd k’s k =2, 6, 10 k =4 k =8 PP (s)=exp(−s). In non-integrable systems, crossings 30624 30664 30666 30667 are avoided and the level spacing distribution is given by Fermions the Wigner-Dyson distribution, as predicted by random L =18 k =1, 5, 7 k =2, 4, 8 k =3 k =6 matrix theory. The form of the Wigner-Dyson distribu- 1035 1026 1038 1028 tion depends on the symmetry properties of the Hamilto- L =21 k =7 other k’s nian. Ensembles of random matrices with time reversal 5538 5537 invariance, the so-called Gaussian orthogonal ensembles 2 L =24 odd k’s k =2, 6, 10 k =4 k =8 (GOEs), lead to PWD(s)=(πs/2) exp(−πs /4). The 30624 30664 30667 30666 same distribution form is achieved for models (1) and (2) in the chaotic limit, since they are also time reversal in- variant. However, these systems differ from GOEs in the sense that they only have two-body interactions and do the quantum level. Different quantities exist to identify not contain random elements. Contrary to GOEs and to the crossover from the integrable to the non-integrable two-body random ensembles [15], the breaking of sym- regime in quantum systems. We consider both spectral metries here is not caused by randomness, but instead observables associated with the eigenvalues and quanti- by the addition of frustrating next-nearest-neighbor cou- ties used to measure the complexityMany-body quantum systems of the eigenvectors. plings. Notice also that the analysis of level statistics in these systems is meaningful only in a particular symme- try sector; if different subspaces are mixed, level repulsion 2 IntegrableA. Spectral (in a compliated observables way) bosonic(L. Santos and M. may be missedRigol even, 2010) if the system is chaotic [46, 51]. bosonic and fermionic systems studied in Refs. [31] and given by 1 [32]. These systems are clean and have only two-body in-Spectral observables, such as level spacing distribution P t’=0.00 t’=0.02 t’=0.03 t’=0.04 0.5 teractions; the transition to chaos is achieved by increas-and level numberHB = variance are investigated below. They L ing the strength of next-nearest-neighbor (NNN) termsare intrinsic indicators of the integrable-chaos transition. 0 † b 1 b 1 Their analysis are−t basedb b +1 on+ theh.c. unfolded+ V n spectrum− n of each− 1 4 rather than by adding random parameters to the Hamil- " i i % i 2&% i+1 2& i=1 # $ P t’=0.06 t’=0.08 t’=0.12 t’=0.16 tonian. Under certain conditions these systems may alsosymmetry! sector separately. 2 the local density of states is unity. This procedure is the D. Because0.5 the weights |cik| fluctuate, the average over be mapped onto Heisenberg spin-1/2 chains. Several pa- one we! used.† 35 ! b 1 b 1 the ensemble gives number of principal components ∼ − t bi bi+2 + h.c. + V ni − ni+2 − , (1)27,28 pers have analyzed spectral statistics of disordered [38– Given# the unfolded spacings$ of% neighboring2&% levels, the 2&'D/3. 0 42] and clean [43–46] 1D Heisenberg spin-1/2 systems. histogram can now be computed. To compare it with the To study1 the number of principal components for 1. Unfolding procedure t’=0.24 t’=0.48 t’=0.64 Mostly, they were limited to sizes smaller than consid- andtheoretical curves, the distribution needs to be normal- Eqs. (1),P we need to choose a basis.t’=0.32 This choice depends ized, so that its total area is equal to 1. on the0.5 question we want to address. We consider two ered here and, in the case of clean systems, focused on Figure 1 shows the level spacing distribution when the bases, the site- and mean-field basis. The site-basis is The procedureHF = of unfolding consists of locally rescal- properties associated with the energy levels, while here The onset of chaos coincides with the defect is placed on site 1 and on site !L/2".Thefirstcase appropriate0 when analyzing the spatial delocalization of ing the energiesL as follows. The number of levels with 1 eigenvectors are also analyzed. Our goal is to establish onset of correspondsthermaliza on to† an integrable model defined through andf the distribution1 f the1 system. To separate regular from chaotic40 behavior, P t’=0.96 t’=1.28 )L energy lessis a Poisson;than−t orf thei equalfi second+1 + toh.c. case a certain is+ aV chaoticn valuei system,− E is son thegiveni+1 − a more appropriate basis consists of the eigenstates0 of a direct comparison between indicators of chaoticity and 0.5 30 " % 2&% 2& -E by the cumulativethe observables. distribution!i=1 # is spectral Wigner-Dyson. function,$ also known as the the integrable limit of the model, which is known1 as the the results obtained in Refs. [31] and [32] for thermaliza- 27 20 mean-field basis. In our case the integrable(E limit corre- staircase function,! † N(E)= n Θ!(E −f En1), wheref Θ 1is 0 10 tion and the validity of ETH. Our analysis also provides − t f f +2 + h.c. + V n − n − sponds.(2) to024 Eqs. (1) with J024xy &=0,#d &=0,andJ0.01z =0. 0.1 1 the unit step function.i i N(E)hasasmoothparti Ni+2 (E), s s t’,V’ a way to quantify points (i) and (ii) in the previous para- # $! % 2&% sm 2&'We start by writing the Hamiltonian in the site-basis. graph, which can result in the absence of thermalizationwhich is the cumulative mean level density, and a fluctu- Let usFIG. denote 1: these(Color basis online.) vectors Level by |φj # spacing.Intheabsence distribution for hard- ating part Nfl(E), that is, N(E)=Nsm(E)+N†fl(E). of theInterac ng spin-chain with a single impurity † Ising interaction, the diagonalization of the Hamil- in finite systems. Above, L is the size of the chain, bi and bi (fi and fcorei ) bosons averaged over all k’s in Table I, for L =24,and tonian ! leads! to the mean-field basis vectors. They are Unfoldingare the bosonic spectrum (fermionic) corresponds annihilation to mapping and creation the en- oper-t = V .Forcomparisonpurposes,wealsopresentthePoissonD 2 Overall, the crossover from integrability to chaos, given by |ξk# = bkj|φj #. The diagonalization of the ergies {E1,E2,...,ED} ontob{!1, !†2,...!fD},where† !n = and(A. Gubin Wigner-Dyson and L. Santos, 2012) j=1 distributions. Bottom right panel: energy quantified with spectral observables and delocalization ators on site i,andni = bi bi (ni = fi fi) is thecomplete boson matrix, including the Ising interaction, gives Nsm(Enmatrices), so that are pseudo-random the mean level vectors; density that is, of their the! ampli- new differenceL! between firstL excited stateD E1 and ground state E0 measures, mirrors various features of the onset of ther- (fermion) local density operator. Hardcore bosons dothe not eigenstates in thez site-basis, |ψiz# = j=1zaij |φj #.If sequencetudes of energies are random is variables.1. Different20,21†2 All methods the eigenstates are used are inH thez = full spectrumωiSi = timesωSiL,for+ "dSLd =18(circles),(1b) L =21 malization investigated in Refs. [31] and [32], in partic- 2 we use the relation between |φj # and |ξk#, we may also occupyFIG. 1: the (Color same online) site, Level i.e., spacingbi = b distributioni ,sotheoperatorscom- for the (squares),i=1 and L =24(triangles)."i=1 # ! to separatestatistically the smooth similar, part they spread fromL through the fluctuating all basisω vectors one. write the eigenstates! of the total! Hamiltonian in Eqs. (1) ular, the distinct behavior of observables between sys- muteHamiltonian on diff inerent Eqs. (1)sites with but can=15,5spinsup, be taken to anti-commute=0, L−1 with" no. preferencesJ andJ are. therefore ergodic. in the mean-field basis as tems that are close and far from integrability, andStatistics be- d that=05, measurexy =1,and long-rangez =05(arbitraryunits);binsize= correlations are usu- x x y y z z onDespite the same the site. success Thed of NN random (NNN) matrix hoppingd theory and in de- interac-HNN = Jxy Si Si+1 + Si Si+1 + JzSi Si+1 . (1c) ally very0.1. sensitive (a) Defect to on the site adopted=1;(b)defectonsite unfolding! procedure,=7.The! In Figs.D (1)D and (2), we showD P (s) across the transi- tween eigenstates whose energies are close and far from tionscribingdashed strengths lines spectral are the are statistical theoretical respectively properties, curves. t (t it)and cannotV capture(V ). Here, i=1 ∗ while short-range correlations are less vulnerable [50]. tion|ψi# from= ! integrability$ a%ij bkj |ξ tok# = chaoscik& for|ξk# bosons. (8) ' and fermions, the energy of the ground state. We also find that the wethe only details study of real repulsive quantum interactions many-body systems. (V,V ! > The0). fact We take We havek=1 set !j=1equal to 1, L kis=1 the number of sites, contrast between bosons and fermions pointed outHere, in we!that discard random 20% matrices of the are energies completely located filled at with the statisti- edges respectively," " in the case of "L = 24. An average over all =1andt = V = 1 set the energy scale in the remainingSx,y,z = σx,y,z /2 are the spin operators at site i,and of the spectrum,cally independent where elements the fluctuations implies infinite-range are large, inter- and Figureski’s is 2 performed, showsi the number but of we principal emphasize components that the same be- Ref. [32] is translated here into the requirement of larger of the paper.B. Number of principal components σx,y,z are the Pauli matrices. The term H gives the obtain Nactions(E) and by the fitting simultaneous the staircase interaction function of many with parti- a for thehaviori eigenstates is verified in the site-basis also for [(a), each (b)]k and-sector inz the separately. As integrability-breaking terms for the onset of chaos in smThe bosonic (fermionic) Hamiltonian conserves the to-Zeeman splitting of each spin i, as determined by a static polynomialcles. of Real degree systems 15. have few-body (most commonly onlymean-fieldt!,V! basisincreases [(c), (d)] and for symmetriesthe cases where are the broken, defect level repul- fermionic systems. Larger system sizes also facilitate the talWe number now investigate of particles how theN transition(N ) and from is a translational Poisson is placed in-magnetic on site field 1 [(a), in the (c)]z anddirection. on site !L/ All2" sites[(b), are (d)]. assumed two-body) interactions whichb aref usually finite range. A induction of chaos. In addition, we observe that mea- to a Wigner-Dyson distribution affects the structure of Theto level have of delocalizationthe same energy increases splitting significantlyω, except in a the single site variant,better picture being of therefore systems composed with finite-range of independent interactions blocks the eigenstates. In particular, we study how delocalized chaoticd, whose regime. energy However, splitting contraryω + to" randomis caused matrices, by a magnetic sures of the degree of delocalization of eigenstates become eachis provided associated by banded with random a total matrices, momentum whichk were.Inthepar- also d they are in both regimes. the largestfield slightly values are larger restricted than to the the field middle applied of the spec-on the other smooth functions of energy only in the chaotic regime, a ticularstudied caseby Wigner. of k 22=Their 0, parity off-diagonal is also elements conserved, are ran- and at To determine the spreading of the eigenstates in a par- trum,sites. the states This site at the is referrededges being to as more a defect. localized. This behavior that may be used as a signature of chaos. half-filling,dom and statistically particle-hole independent, symmetry but isare present, non-vanishing that is, the ticular basis, we look at their components. Consider propertyA spinis a consequence in the positive of thez direction Gaussian (up)shape is of indicated the by onlyan eigenstate up to a fixed|ψ # distancewritten in from the the basis diagonal. vectors | Thereξ # as aredensity of states of systems1 with two-body interactions. The paper is organized as follows. Section II describes bosonic [fermionic]i model becomes invariantk under the| ↑" or by the vector ; a spin in the negative z direction also ensemblesD of randomL † matricesL that† take into accountThe highest concentration of0 states appears in the middle the model Hamiltonians studied and their symmetries. |ψi# = k=1 cik|ξk#. It will be localized if it has the par- 0 transformation i (bi + bi)[ i (fi + fi)],2 which annihi-of the(down) spectrum, is represented where the bystrong| ↓" mixingor . of An states up spin can on site Section III analyzes the integrable-chaos transition based theticipation restriction of few to basis few vectors, body interactions, that is, if a few so|c thatik| make only the % & 1 lateselementssignificant particles! associated contributions. from( with filled It those will sites beinteractions( and delocalized creates are if them many nonzero; in emptyoccuri has leading energy to widely +ωi/2, distributed and a down eigenstates. spin has energy −ωi/2. on various quantities. After a brief review of the unfold- 2 23–25 % & ones.an|cik example| participate The latter is the with two two-body-random-ensemble similar symmetries values. To will quantify be avoided this cri-(see hereAn byA interesting spin up corresponds difference between to an excitation. the integrable and ing procedure, Sec.III.A focuses on quantities associated selectingreviewsterion, we ink useRefs."=0and the 21,26). sumN of Otherb,f the= square modelsL/3. of For the which probabilities, even describeL, we sys- considerchaoticThe regimes second is the term, fluctuationsHNN, is of known the number as the of XXZ prin- Hamilto- 4 cipal components. For the regular system the number of with the energy levels, such as level spacing distribution ktems|c=1ik| , with(the2,...,L/ sum short-range of2 the−1andforodd probabilities and few-body wouldL interactions, k not=1 be, a2,... good do( notL−1)/nian.2. It describes the couplings between nearest-neighbor D 2 choice, because normalization implies |c | =1), 27 principal(NN) components spins; Jxy showsis the large strength fluctuations. of the In flip-flop con- term and level number variance. Section III.B introduces mea- Theinclude dimension random elements,Dk of each such as symmetry nucleark=1 shellik sector models, studied is and define the number of principal components of eigen- trast,x in thex chaoticy y regime the number of principal com- sures of state delocalization, namely information entropy and the systems of interacting spins which we consider Si Si+1 +Si Si+1,andJz is the strength of the Ising inter- givenstate i inas Table27,28 I. ! ponents approachesz z a smooth function of energy. Chaotic and inverse participation ratio (IPR), showing results for in this article. action Si Si+1. The flip-flop term exchanges the position Exact diagonalization is1 performed to obtain alleigenstates eigen-of neighboring close in upenergy and have down similar spins structuresaccording and to All the matrices weni ≡ have mentioned. can lead to level(7) re-consequently similar values of the number of principal the former in the mean field basis. Results for the inverse values and eigenvectorsD of the4 systems under investiga- pulsion, but differences are observed.|cik| For instance, eigen- participation ratio and discussions about representations ! ! k=1 components.x x y y tion.states When of randomt = V matrices= 0, models are completely (1) and spread (2) are (delo- integrableJxy(Si Si+1 + Si Si+1)| ↑i↓i+1" =(Jxy/2)| ↓i↑i+1", (2) The number of principal! components gives the number are left to the Appendix. Concluding remarks are pre- andcalized)of basis may invectors be any mapped basis, which whereas contribute onto theone to eigenstates each another eigenstate. via of systems the It Jordan- sented in Sec. IV. Wignerwithis small few-body transformationwhen the interactions state is localized [47]. delocalize A and correspondence only large in when themiddle the with theor, equivalently,IV. SYMMETRIES it moves the excitations through the Heisenbergofstate the is spectrum. delocalized. spin-1/226–30 chain also holds, in which case thechain. We have assumed open boundary conditions as InFor this Gaussian paper orthogonal we study ensembles, a one-dimensional the eigenstates system are Theindicated presence by of athe defect sum breaks in HNN symmetrieswhich goes of the from sys-i =1to system may be solved with the Bethe ansatz [48, 49]. L−1. Hence, an excitation in site 1 (or L) can move only ofrandom interacting vectors, spins that is, 1/2. the amplitudes The systemcik are involves indepen- onlytem. In this section we remove the defect and have a II. SYSTEM MODEL nearest-neighbordent random variables. interactions, These states and are in completely some cases, de- alsocloserto look site at 2 (orthe tosymmetries. site L − 1). Closed boundary conditions, next-nearest-neighborlocalized. Complete delocalization interactions. does Depending not mean, how- on the Wewhere refer an to excitation the system in in site the 1 can absence move of also a defectto site L (and ever,III. that SIGNATURES the number of principal OF components QUANTUM is equal CHAOS to (# =0)asdefect-free.Contrarytothecasewhere strength of the couplings, the system may develop chaos, d vice-versa) are mentioned briefly in Sec. IV. We consider both scenarios: hardcore bosons and spin- which is identified by calculating the level spacing distri- The Ising interaction implies that pairs of parallel spins less fermions on a periodic one-dimensional lattice in bution.The concept We also compare of exponential the level divergence, of delocalization which of the is at thehave higher energy than pairs of anti-parallel spins, that the presence of nearest-neighbor (NN) and next-nearest- hearteigenstates of classical in the integrable chaos, has and no chaotic meaning domains. in the It is quan-is, significantly larger in the latter case, where the most de- neighbor (NNN) hopping and interaction. The Hamilto- tum domain. Nevertheless, the correspondence princi- J SzSz | ↑ ↑ " =+(J /4)| ↑ ↑ ", (3) localized states are found in the middle of the spectrum. z i i+1 i i+1 z i i+1 nian for bosons HB and for fermions HF are respectively ple requires that signatures of classical chaos remain in The paper is organized as follows. Section II provides and a detailed description of the Hamiltonian of a spin 1/2 chain. Section III explains how to compute the level J SzSz | ↑ ↓ " = −(J /4)| ↑ ↓ ". (4) spacing distribution and how to quantify the level of de- z i i+1 i i+1 z i i+1 localization of the eigenstates. Section IV shows how For the chain described by Eqs. (1) the total spin in the mixing of symmetries may erase level repulsion even the z direction, Sz = L Sz, is conserved, that is, when the system is chaotic. Final remarks are given in i=1 i [H, Sz] = 0. This condition means that the total number Sec. V. of excitations is fixed; the( Hamiltonian cannot create or annihilate excitations, it can only move them through the II. SPIN-1/2 CHAIN chain. To write the Hamiltonian in matrix form and diagonal- We study a one-dimensional spin 1/2 system (a spin ize it to find its eigenvalues and eigenstates, we need to 1/2 chain) described by the Hamiltonian choose a basis. The natural choice corresponds to arrays of up and down spins in the z direction, as in Eqs. (2), H = Hz + HNN, (1a) (3) and (4). We refer to it as the site basis. In this basis, where Hz and the Ising interaction contribute to the diagonal Quantum ergodicity and localiza on in the Hilbert space Why does ETH work? Why does the single state describe equilibrium?
Imagine the most integrable many-body system: collec on of non-interac ng Ising spins. The dynamics is very simple: each spin is conserved.
A typical state with a fixed magne za on is thermal. This is how we derive Gibbs distribu on in sta s cal mechanics. We do not need non-integrability.
More sophis cated language: quantum typicality. If we take a typical many-body state of a big system (Universe) and look into a small subsystem then it will look like the Gibbs state (von Neumann 1929, Popescu et. al. 2006, Goldstein et. al. 2009) Can we create typical states in the lab doing local perturba ons? Let us flip 10 spins in the middle doing the Rabi pulse
By performing a local quench we create a very atypical state, which is not thermal, whether the system is integrable or not. In an integrable system this state will never thermalize. 2 approach. However, contrary to conventional renormal- ization schemes which reduce the size of the effective Hilbert space, the flow equation approach retains the full Hilbert space, which makes it particularly appropriate for nonequilibrium problems (for more details see [19]). Flow equations for the Hubbard model. First we work out the diagonalizing flow equationMore sophis cated example transformation for FIG. 1: The Heisenberg equation of motion for an observable the Hubbard Hamiltonian. The expansion parameter is O is solved by transforming to the B = eigenbasis of the in- the (small)Weak interac on quench in the Hubbard model interaction U and normal-ordering is with (M. Moeckel and S. Kehrein, 2008) 3 teracting Hamiltonian H (forward transformation),∞ where the respect to the zero temperature Fermi-Dirac distribution: time evolution can be computed easily. Time evolution intro- (4) at B = . Integrating back to B = 0 gives the ∞ duces phase shifts, and therefore the form of the observable time evolved creation operator in the original basis, and in the initial basis B = 0 (after a backward transformation) H(B)=it is straightforwardk : tock† evaluateσckσ: the time dependent mo- (2) mentum distribution function with respect to the initial kσ= , changes as a function of time. Fermi gas state↑ ↓ [20]. Nonequilibrium momentum distribution† † function. One + Uppqq(B) :cp cp cq cq : finds the following time-dependent additional↑ ↑ ↓ term↓ to the p pq q 2 distribution nk of the free Fermi gas in O(U ): NEQ NEQ the limit of high dimensions [15], but the calculation also with U (B∆=N 0)( =t)=U.N The(t) flownk of the one-particle(5) en- pStart from U=0. Short me dynamics is pqq k k − 2 (k E)t applies to finite dimensions with the same conclusions up ergies and the generation of highersin normal-ordered− terms approximated by non-interac ng dressed 2 ∞ 2 = 4U dE 2 Jk(E; n) to quantitative details. in the Hamiltonian can− be neglected(k sinceE) we are in- quasi-par cles. −∞ − We study the above real time evolution problem by terested in results in second order in U. The flow of The phase space factor Jk(E; n) resembles the quasipar- the interaction is to leading order given by U (B)= NEQ using the approach introduced in [16]. One solves the ticle collision integral of a quantum Boltzmannp equation:pqq FIG.System relaxes to the atypical state with wrong 2: (a)-(d): Time evolution of N () plotted around 2 defthe Fermi energy for ρF U =0.6. A fast reduction of the dis- Heisenberg equations of motion for the operators that one U exp( B∆ ) with an energy + difference ∆ = p pq q p+q p q ppqq continuity and 1/t-oscillations can be observed. The arrow in Jk(E ; n)= δ δ nknpn− n− n−n−np nq quasi-par cle residue. − p+k p+E p q k p is interested in by performing a unitary transformation + . − (d) indicates the size of the quasiparticle residue in the quasi- p p q q pqp − − steady regime. In (e) the universal curves for ∆Nk = Nk nk to an (approximate) eigenbasis of the interacting Hamil- Next we work out the flow equation transformation forare given for both equilibrium and for the nonequilibrium− For computational convenience we use the limit of infi- tonian. There one can easily work out the time evolution quasi-steady state in the weak-coupling limit. the numberSimpler example: take nite operator dimensions,k specifically(B)= a Gaussiank† (B) densityk (B of), states which We do not need N ↑ 2 C ↑ C ↑ and then transform back to the original basis where the can be obtainedρ()=exp from the(/t transformation∗) /2 /√2πt∗ [15]. of In a the single sequel cre- noninterac ngρ = ρ( = 0)− denotes fermions and the density of states at the Fermi ergodicity, ETH, chaos initial state is specified. In this manner one induces a F ation operatorlevel.k† Results(B). from Under a numerical the evaluation sequence of the of above unitaryor below the Fermi surface one then finds from (5): hit with a local Hammer. C ↑ to understand solution of the Heisenberg equations of motion for an op- transformationsscheme the for operator three time steps changes are presented its form in Fig. to 2. describe ∞ 1 J (E; n) Equilibrium momentum distribution function. Eqs. (4) NEQ 2 sta s cal mechanics. k erator in the original basis but without secular terms, ∆Nk (t )= 4U dE 2 dressing by electron-hole pairs. A truncated ansatz reads: →∞ − 2 (k E) Can not possibly create can also be used to evaluate the equilibrium distribution −∞ − EQU which are usually a major problem in other approxima- function, which will later be important for comparison. =2∆N (7) new Fermi-Dirac k k+p k † In fact,† the asymptotic value hk (B = ) at† the† Fermi tion schemes [17]. Fig. 1 gives a sketch of our approach. k (B)=hk(B)ck + Mp q p(BF)δp +q :cp cq cp : 2 We need them to C ↑ ↑ ∞ ↑ ↓ ↓ since sin in (5) yields a factor 1/2 in the long time limit. distribu on, too much energy is directlyp q relatedp to the quasiparticle residue (Z- Notice that the same general idea was recently also used EQU 2 In the quasi-steady state the momentumunderstand dynamics. distribution factor), Z =[hkF (B = )] [19]. It is easy to solve by Cazalilla to study the behavior of the exactly solvable fine tuning. (4) analytically at the Fermi∞ energy for zero temperature (3)function is therefore that of a zero temperature Fermi one-dimensional Luttinger model subject to a quench [8]. We introducein O the(U 2) zero and one temperature finds for momenta momentumk infinitesimally distri-liquid. However, from (7) one deduces that its Z-factor is smaller than in equilibrium, 1 ZNEQ = 2(1 ZEQU). above or below the Fermi surface def − − Since our model is non-integrable, we implement the bution function of a free Fermi gas nk,definenk− =This factor 2 implies a quasiparticle distribution function ∞ Jk(E; n) in the vicinity of the Fermi surface in the quasi-steady above diagonalizing transformation by the flow equa- ∆N EQU = U 2 dE def (6) 1 n and a phase spacek factor Q [n] =2 n− n−n +state equal to the equilibrium distribution function of k − ppq(k E) p q p tion method [18, 19], which permits a systematic con- − −∞ − the physical electrons, N QP:NEQ = N EQU, as opposed to n n n−. The flow equations for the creation operator k k p q p consistent with a conventional perturbative evaluation. QP:EQU trolled expansion for many equilibrium and nonequi- its equilibrium distribution, Nk = Θ(kF k). are: Short-time correlation buildup. The numerical evalua- Remarkably, Cazalilla’s findings [8] for the interaction− librium quantum many-body problems [19]. One uses tion of the momentum distribution function depicted in quench in the Luttinger model mirror these features: the a continuous sequence of infinitesimal unitary transfor- ∂hk(B) Fig. 2 shows the initial buildup of a correlated2 state from critical exponent describing the asymptotic behavior of k 1 B∆2 kp pq =theU FermiM gas. For( timesB) ∆ 0 kp There are always atypical states. These are the states, which we usually excite in experiments. ETH says that there are no atypical sta onary states (eigenstates of the Hamiltonian) Fluctua ons of observable between eigenstates (M. Rigol et. al. 2008) Nonintegrable Integrable All eigenstates in a non-integrable system are indeed typical. Physical reason: each eigenstate consists of (exponen ally) many eigenstates of other local Hamiltonians, e.g. of a noninterac ng Hamiltonian. FIG. 3: Eigenstate thermalization hypothesis. a,Inournonintegrablesystem,themomentumdistributionn(kx) for two typical eigenstates with energies close to E0 is identical to the microcanonical result, in accordance with the ETH. b,Upperpanel: n(kx =0)eigenstate expectation values as a function of the eigenstate energy resemble a smooth curve. Lower panel: the energy distribution ρ(E) of the three 2 ensembles considered in this work. c,Detailedviewofn(kx =0)(left labels) and |Cα| (right labels) for 20 eigenstates around E0. d,Inthe integrable system, n(kx) for two eigenstates with energies close to E0 and for the microcanonical and diagonal ensembles are very different from each other, i.e., the ETH fails. e,Upperpanel:n(kx =0)eigenstate expectation value considered as a function of theeigenstateenergy gives a thick cloud of points rather than resembling a smooth curve. Lower panel: energy distributions in the integrable system are similar to 2 the nonintegrable ones depicted in b. f,Correlationbetweenn(kx =0)and |Cα| for 20 eigenstates around E0.Itexplainswhyind the microcanonical prediction for n(kx =0)is larger than the diagonal one. Nevertheless, one may still wonder if in this case scenario grable and that enter the generalized Gibbs ensemble, which (i) might hold—if the averages over the diagonal and the was introduced in Ref. [3] as appropriate for formulating sta- microcanonical energy distributions shown in Fig. 3e might tistical mechanics of isolated integrable systems. In the non- agree. Figure 3d shows that this does not happen. This is so integrable case shown in Fig. 3c, n(kx =0)is so narrowly because, as shown in Fig. 3f, the values of n(kx =0)for distributed that it does not matter whether or not it is corre- 2 the most-occupied states in the diagonal ensemble (the largest lated with Cα (we have in fact seen no correlations in the 2 | | values of eigenstate occupation numbers Cα )arealways nonintegrable case). smaller than the microcanonical prediction,| and| those of the least-occupied states, always larger. Hence, the usual thermal The thermalization mechanism outlined thus far explains predictions fail because the correlations between the values why long-time averages converge to their thermal predictions. 2 of n(kx =0)and Cα preclude unbiased sampling of the AstrikingaspectofFig.1b,however,isthatthetimefluc- latter by the former.| These| correlations have their origin in tuations are so small that after relaxation the thermal predic- the nontrivial integrals of motion that make the system inte- tion works well at every instant of time. Looking at Eq. (1), one might think this is so because the contribution of the off- Corollary: each energy eigenstate contains (exponentially) many noninteracting states. Each non-interacting state is projected to many eigenstates (by a local perturbation) C. Neuenhahn, F. Marquardt , 2010; G. Biroli, C. Kollath, A. Laeuchli, 2009 Nearly integrable (non-ergodic) Noninterable (ergodic) Quantum ergodicity is a delocalization of the initial state in the Hilbert space The picture is thus similar to classical. Phase space è Hilbert space. While each eigenstate is enough to thermalize we always populate many! 6 tion measures [11, 12], are not intrinsic indicators of the integrable-chaos transition since they depend on the ba- sis in which the computations are performed. The choice of basis is usually physically motivated. The mean-field basis is the most appropriate representation to separate 6 global from local properties, and therefore capture the transition from regular to chaotic behavior [12]. Here, tion measures [11, 12], are not intrinsic indicators of the this basis corresponds to the eigenstates of the integrable integrable-chaos transition since they depend on the ba- Hamiltonian (t!,V! = 0). Other representations may also sis in which the computations are performed. The choice provide relevant information, such as the site basis, which of basis is usually physically motivated. The mean-field is meaningful in studies of spatial localization, and the basis is the most appropriate representation to separate momentum basis, which can be used to study k-space global from local properties, and therefore capture the localization (see the Appendix for further discussions). transition from regular to chaotic behavior [12]. Here, The degree of complexity of individual eigenvectors this basis corresponds to the eigenstates of the integrable ! ! may be measured, for example, with the information Hamiltonian (t ,V = 0). Other representations may also (Shannon) entropy S or the inverse participation ratio provide relevant information, such as the site basis, which is meaningful in studies of spatial localization, and the (IPR). The latter is also sometimes referred to as number FIG. 6: (Color online.) Shannon entropy in the mean field momentum basis, which can be used to study k-space of principal components. For an eigenstate ψ written in basis vs effective temperature for bosons, L =24,k =2, j localization (see the Appendix for further discussions). D k and t! = V !.ThedashedlinegivestheGOEaveragedvalue the basis vectors φk as ψj = k=1 cj φk,SandIPRare The degree of complexity of individual eigenvectors SGOE ∼ ln(0.48D). respectively given by ! may be measured, for example, with the information (Shannon) entropy S or the inverse participation ratio D k 2 k 2 (IPR). The latter is also sometimes referred to as number FIG. 6: (Color online.) Shannon entropy in the mean field Sj ≡− |c | ln |c | , (5) j j 9 of principal components. For an eigenstate ψj written in basis vs effective temperature for bosons, L =24,k =2, Measures of (de)localiza on in the Hilbert space "k=1 D k and t! = V !.ThedashedlinegivestheGOEaveragedvalue the basis vectors φk as ψj = k=1 cj φk,SandIPRare (L. Santos and M. Rigol, 2010) SGOE ∼ ln(0.48D). be considered when computing T . Our results here show ergyand (Figs. 13 and 14). The IPR values increase sig- respectively given by ! that the differences with considering specific momentum nificantly with t!,V!, but do not reach the GOE result Inverse 1 D sectors are small and decreasing with the system size. IPR = (D +2)/3 [11, 12]. IPR gives essentially the same Shannon k 2 k 2 IPRj ≡ . (6) information as S, although the first showsD larger4 fluctua- Sj ≡− |cj | ln |cj | , (5) Par cipa on |ck| 6 tions. k=1 j entropy "k=1 2. Results in the mean-field basis Ra o ! Thetion measures above quantities [11, 12], are measure not intrinsic the number indicators of basis of the vectors and thatintegrable-chaos contribute transition to each since eigenstate, they depend that on is, the how ba- much Figures 11 and 12 show the mean-field Shannon en- 1 delocalizedsis in which the each computations state is in are the performed. chosen basis. The choice IPRj ≡ . (6) tropy vs energy for all the eigenstates of the k =2sector. D k 4 of basis is usually physically motivated.k The mean-field =1 |cj | Notice that the whole spectrum for the k = 2 sector is For the GOE, the amplitudes cj are independent ran- k presented and not only the energies leading to T ≤ 10 dombasis variablesis the most and appropriate all eigenstates representation are completely to separate delocal- ! The above quantities measure the number of basis vectors as in Figs. 6 and 7. The typical behavior of banded ma- ized.global Complete from local properties, delocalization and therefore does not capture imply, the however, transition from regular to chaotic behavior [12]. Here, that contribute to each eigenstate, that is, how much trices is observed: larger delocalization appearing away that S = ln D. For a GOE, the weights |ck|2 fluctu- from the edges of the spectrum, although not as large as this basis corresponds to the eigenstates of the integrablej delocalized each state is in the chosen basis. ateHamiltonian around ( 1t!/D,V! =and 0). the Other average representations over the may ensemble also is For the GOE, the amplitudes ck are independent ran- the GOE result SGOE = ln(0.48D)+O(1/D), and lower FIG. 7: (Color online.) As in Fig.j 6 for fermions. complexity at the edges [13, 15, 57]. SprovideGOE = relevant ln(0.48 information,D)+O(1/D such) [11, as the 12]. site basis, which dom variables and all eigenstates are completely delocal- is meaningfulFigures 6 and in studies 7 show of the spatial Shannon localization, entropy and in the mean ized. Complete delocalization does not imply, however, fieldmomentum basis basis,S vs which the can effective be used temperature to study k-space for bosons k 2 mf that S = ln D. For a GOE, the weights |cj | fluctu- andlocalization fermions, (see the respectively. Appendix for The further effective discussions). temperature, of theate around effective 1/D temperature,and the average we allow over for the a ensemble direct com- is The degree of complexity of individual eigenvectors Tj of an eigenstate ψj with energy Ej is defined as parisonSGOE of= ln(0 our.48 resultsD)+O here(1/D and) [11, the 12]. results presented in FIG. 7: (Color online.) As in Fig. 6 for fermions. may be measured, for example, with the information Refs.Figures [31] and 6 and [32]. 7 show the Shannon entropy in the mean (Shannon) entropy S or the inverseˆ participation ratio 1 −H/Tj Asfield seen basis in S Figs.mf vs 6 the and eff 7,ective the mixing temperature of basis for vectors, bosons (IPR). The latter isE alsoj = sometimesTr Heˆ referred to, as number (7)FIG. 6: (Color online.) Shannon entropy in the mean field FIG. 13: (Color online.) InverseZ participation ratio in the andand therefore fermions, the respectively. complexity of The the e states,ffective increases temperature, with of the effective temperature, we allow for a direct com- meanof field principal basis vs components. energy for bosons, For an# L eigenstate=24,k $=2,andψj written in basis vs e! ffective! temperature for bosons, L =24,k =2, ! ! ! !T of an eigenstate ψ with energy E is defined as parison of our results here and the results presented in t = V .TheGOEresultIPRGOE ∼DD/3isbeyondthek and t =tV,V.ThedashedlinegivestheGOEaveragedvaluej , but it is only forjTj ! 2 that thej eigenstates of our wherethe basis vectors φk as ψj = k=1 cj φk,SandIPRare Refs. [31] and [32]. chosenrespectively scale. given by SGOE ∼ systemsln(0.48D). approach the GOE result. Similarly, in plots of ! 1 ˆ ˆ S vs energy (see Figs. 11 andˆ 12−H/T inj the Appendix), it is As seen in Figs. 6 and 7, the mixing of basis vectors, −H/Tj mf Ej = Tr He , (7) Z =TrD e . (8) Z and therefore the complexity of the states, increases with only away from the borders# of the spectrum$ that Smf → k 2 k 2 ! ! S ≡− |c #| ln |c | ,$ (5) t ,V , but it is only for Tj ! 2 that the eigenstates of our j j j SGOEwhere; in the borders, the states are more localized and =1 systems approach the GOE result. Similarly, in plots of Above, Hˆ is Hamiltonian"k (1) or (2), Z is the partition therefore less ergodic. This feature is typical of systems ˆ Smf vs energy (see Figs. 11 and 12 in the Appendix), it is with a finite range of interactions,−H/Tj such as models (1) functionand with theNonintegrable Boltzmann constant systems are more delocalized kB =1,andthe Z =Tr e . (8) only away from the borders of the spectrum that S → trace is performed over the full spectrum as in Refs. [31] and (2) and also banded,# embedded$ random matrices, mf SGOE; in the borders, the states are more localized and and [32] (see the Appendix1 for a comparison with effec- and two-body random ensembles [13, 15, 57]. IPRj ≡ D . (6) Above, Hˆ is Hamiltonian (1) or (2), Z is the partition therefore less ergodic. This feature is typical of systems FIG. 11: (Color online.) Shannon entropy in the mean field |ck|4 ! ! tive temperatures obtainedk=1 byj tracing over exclusively The analysis of the structure of the eigenstates hints basis vs energy for bosons, L =24,k =2,andt = V .The function with the Boltzmann constant kB =1,andthe with a finite range of interactions, such as models (1) the sector k = 2). The! figures include results only for on what to expect for the dynamics of the system. In dashed line gives the GOE averaged value SGOE ∼ ln(0.48D). The above quantities measure the number of basis vectors trace is performed over the full spectrum as in Refs. [31] and (2) and also banded, embedded random matrices, Tthatj ≤ contribute10; for high toE eachj , the eigenstate, temperatures that is, eventually how much become theand context [32] (see of the relaxation Appendix dynamics, for a comparison not only with the eff den-ec- and two-body random ensembles [13, 15, 57]. negative.delocalized By each plotting state is inthe the Shannon chosen basis. entropy as a function sitytive of temperatures complex states obtained participating by tracing in the over dynamics exclusively is The analysis of the structure of the eigenstates hints k the sector k = 2). The figures include results only for on what to expect for the dynamics of the system. In For the GOE, the amplitudes cj are independent ran- dom variables and all eigenstates are completely delocal- Tj ≤ 10; for high Ej , the temperatures eventually become the context of relaxation dynamics, not only the den- ized. Complete delocalization does not imply, however, negative. By plotting the Shannon entropy as a function sity of complex states participating in the dynamics is k 2 that S = ln D. For a GOE, the weights |cj | fluctu- ate around 1/D and the average over the ensemble is SGOE = ln(0.48D)+O(1/D) [11, 12]. FIG. 7: (Color online.) As in Fig. 6 for fermions. FIG. 14: (Color online.) Same as in Fig. 13 for fermions. Figures 6 and 7 show the Shannon entropy in the mean field basis Smf vs the effective temperature for bosons and fermions, respectively. The effective temperature, of the effective temperature, we allow for a direct com- Tj of an eigenstate ψj with energy Ej is defined as parison of our results here and the results presented in 3. Results in the k−basis Refs. [31] and [32]. 1 ˆ As seen in Figs. 6 and 7, the mixing of basis vectors, E = Tr Heˆ −H/Tj , (7) Identifying the mean-fieldj Z basis may not always be a and therefore the complexity of the states, increases with # $ ! ! simple task. For example, some 1D models may have t ,V , but it is only for Tj ! 2 that the eigenstates of our morewhere than one integrable point. It may also happen that systems approach the GOE result. Similarly, in plots of one is so far from any integrable pointˆ that there is no rea- S vs energy (see Figs. 11 and 12 in the Appendix), it is −H/Tj mf FIG. 12: (Color online.) Same as in Fig. 11 for fermions. son to believe that suchZ a point=Tr hase any relevance. for the (8) only away from the borders of the spectrum that S → # $ mf chosen system. The latter case may be particularly ap- SGOE; in the borders, the states are more localized and A similar behavior is seen in the plots of the in- plicableAbove, to higher-dimensionalHˆ is Hamiltonian systems (1) or (2), whereZ is integrable the partition therefore less ergodic. This feature is typical of systems verse participation ratio in the mean-field basis vs en- pointsfunction are, in with general, the the Boltzmann noninteracting constant limitkB or=1,andthe other with a finite range of interactions, such as models (1) trace is performed over the full spectrum as in Refs. [31] and (2) and also banded, embedded random matrices, and [32] (see the Appendix for a comparison with effec- and two-body random ensembles [13, 15, 57]. tive temperatures obtained by tracing over exclusively The analysis of the structure of the eigenstates hints the sector k = 2). The figures include results only for on what to expect for the dynamics of the system. In Tj ≤ 10; for high Ej , the temperatures eventually become the context of relaxation dynamics, not only the den- negative. By plotting the Shannon entropy as a function sity of complex states participating in the dynamics is More direct measure of delocaliza on – diagonal entropy (von Neumann’s entropy of the diagonal ensemble) Hard core bosons 3 Bosons Fermions 3 8 1.5 0.4 0.00 0.02 0.3 0.32 0.48 8 1.5 0.4 1 0.00 0.02 0.3 0.32 0.48 1 6 W 0.2 6 W 0.2 s 0.5 0.1 s 0.5 0.1 , S , S d 12 4 0 0 d 4 12 S 0 0 S s -4 -2 0 -4 -20 -10 0 -10 0 s 1 9 -6 -6 -5 -5 1 9 -6 -4 -2 0 -6 -4 -20 -10 -5 0 -10 -5 0 S S /S 0.8 /S 0.8 d d S 2 S 6 6 2 0.8 6 6 0.8 0.00 0.02 0.32 0.48 0.00 0.02 0.6 0.32 0.48 0.6 (a) (b) 3 (a) (b) 3 0.01 0.1 1 0 0.2 0.4 4 0.01 0.1 1 0 0.2 0.4 4 W 0.4 W 0.4 0 0 8 2 8 2 0.2 0.2 0 0 0 0 Nonintegrable systems: delocaliza on in -5 -4 -3 -2 -5 -4 -3 -2 -8 -6 -4 -2 -8 -6 -4 -2 -5 -4 -3 -2 -5 -4 -3 -2 -8 -6 -4 -2 -8 -6 -4 -2 6 6 E E E E E E E E the energy space a er the quench. s s , S 4 d 1 12 FIG. 2: (Color online) Normalized distribution function of energy. s S ! ! , S 4 9 d 12 /S 1 S Bosonic system, L =24, T =3.0 and the values of t = V are FIG. 2: (Color online) Normalized distribution function of energy. d s S ! ! S 9 2 6 indicated. Top panels: quench from tini =0.5,Vini =2.0;bot- /S 0.5 S Bosonic system, L =24, T =3.0 and the values of t = V are d Each eigenstate is thermal (c) (d) 3 2 S 6 0.01 0.1 1 0 0.2 0.4 tom panels: quench from tini =2.0,Vini =0.5.Solidsmoothline: 0.5 indicated. Top panels: quench from tini =0.5,Vini =2.0;bot- − − − 2 2 (c) (d) 3 0.01 0.1 1 0.1 1 best Gaussian fit (√2πa) 1e (E b) /(2a ) for parameters a and b; 0.01 0.1 1 0 0.2 0.4 tom panels: quench from tini =2.0,Vini =0.5.Solidsmoothline: 2 2 2 2 t’=V’ t’=V’ −1 −(E−E ) /(2δE ) −1 −(E−b) /(2a ) dashed line: (√2πδE) e ini . 0.01 0.1 1 0.1 1 best Gaussian fit (√2πa) e for parameters a and b; Always densely occupy eigenstates − − − 2 2 ! ! t’=V’ t’=V’ dashed line: (√2πδE) 1e (E Eini) /(2δE ). FIG. 1: (Color online) Entropies vs t = V .Left:bosons;right: fermions; top: quench from tini =0.5,Vini =2.0;bottom:quench ! ! tegrable (left) and chaotic (right) domains. The sparsity from t =2.0,V =0.5.Filledsymbols:d entropy (1); empty FIG. 1: (Color online) Entropies vs t = V .Left:bosons;right: Either condi on is sufficient for ini ini of the density matrix in the integrable limit is reflected symbols: Ss (2); T =1.5; ! T =2.0; T =3.0.Allpanels: fermions; top: quench from tini =0.5,Vini =2.0;bottom:quench tegrableL. Santos, A. P. and M. (left) and chaotic (right)Rigol (2012) domains. The sparsity ! " by large and well separated peaks, while for the noninte- thermaliza on1/3-filling and L =24;insetsofpanels(a)and(c)show. Sd/Ss for from tini =2.0,Vini =0.5.Filledsymbols:d entropy (1); empty of the density matrix in the integrable limit is reflected L =24,thick(red)line,andL =21,thin(black)lineforT =3.0. grable case W (E) approaches a Gaussian shape similar to symbols: Ss (2); T =1.5; ! T =2.0; T =3.0.Allpanels: 2 2 ! " by large and well separated peaks, while for the noninte- Solid lines in the insets of panels (b) and (d), from bottom to top: (√2πδE)−1e−(E−Eini) /(2δE ),asshownwiththefits.The 1/3-filling and L =24;insetsofpanels(a)and(c)showSd/Ss for microcanonical entropy; canonical entropy Sc for eigenstates with shape of W (E) is determined by the product of the average L =24,thick(red)line,andL =21,thin(black)lineforT =3.0. grable case W (E) approaches a Gaussian shape similar to 2 2 k =0and the same parity as the initial state; Sc for eigenstates with weight of the components of the initial state and the den- Solid lines in the insets of panels (b) and (d), from bottom to top: √ −1 −(E−Eini) /(2δE ) k =0and both parities; and S for all eigenstates with N =8. ( 2πδE) e ,asshownwiththefits.The c sity of states. The latter is Gaussian and the first depends on microcanonical entropy; canonical entropy S for eigenstates with c shape of W (E) is determined by the product of the average the strength of the interactions that lead to chaos, it becomes k =0and the same parity as the initial state; S for eigenstates with c weight of the components of the initial state and the den- Gaussian for large interactions [15]. A plot of ρ vs energy, k =0and both parities; and S for all eigenstates with N =8. nn c sity of states. The latter is Gaussian and the first depends on initial state after the quench. Explicit results for the micro- on the other hand, does not capture so clearly the integrable- canonical entropy with δE determined by the energy uncer- the strength of the interactions that lead to chaos, it becomes chaos transition [12]. tainty are in surprisingly good agreement with those of Sd. Integrable systems. We considera 1DHCB modelwithNN Gaussian for large interactions [15]. A plot of ρnn vs energy, initial state after the quench. Explicit results for the micro- Up to a nonextensive constant, the canonical entropy Sc can hopping and an external potential described by, on the other hand, does not capture so clearly the integrable- also be written in the same form as Sm (2) if we use the canon- canonical entropy with δE determined by the energy uncer- ical width δE2 = ∂ E.ResultsforS are shown for three L−1 L chaos transition [12]. c β c † 2πj † tainty are in surprisingly good agreement with those of Sd. − H = t (b b + H.c.)+A cos b b . (5) Integrable systems. We considera 1DHCB modelwithNN different sets of eigenstates: (i) all the states in the N sec- S − j j+1 " P # j j !j=1 !j=1 Up to a nonextensive constant, the canonical entropy Sc can hopping and an external potential described by, tor, (ii) only the states in the N sector with k =0,(iii)only also be written in the same form as Sm (2) if we use the canon- the states in the N sector with k =0and the same parity as This model is exactly solvable as it maps to spinless noninter- 2 − ical width δE = ∂ E.ResultsforS are shown for three L 1 L the initial state. The latter, as expected, is the closest to Sm c β c † 2πj † acting fermions (see e.g., Ref. [16]). The period P is taken − HS = t (b bj+1 + H.c.)+A cos b bj . (5) (also computed from eigenstates in the same symmetry sec- to be P =5, t =1,andtheamplitudeA takes the values different sets of eigenstates: (i) all the states in the N sec- − j " P # j tor, (ii) only the states in the N sector with k =0,(iii)only !j=1 !j=1 tor as ψini )andSd.Inthethermodynamiclimit,allthree 4, 8, 12, and 16. We study systems with L =20, 25 ...55 sets of| eigenstates" should produce the same leading contribu- the states in the N sector with k =0and the same parity as at 1/5 filling. For the quench, we start with the ground state This model is exactly solvable as it maps to spinless noninter- tion to Sc,butforfinitesystemsitisnecessarytotakeinto of (5) with A =0and evolve the system with a superlattice the initial state. The latter, as expected, is the closest to S m acting fermions (see e.g., Ref. [16]). The period P is taken account discrete symmetries in order to get an accurate ther- (A =0)andvice-versa.Openboundaryconditionsareused (also computed from eigenstates in the same symmetry sec- to be P =5, t =1,andtheamplitudeA takes the values modynamic description of the equilibrium ensemble. in this% case. tor as ψini )andSd.Inthethermodynamiclimit,allthree 4, 8, 12, and 16. We study systems with L =20, 25 ...55 The fact that Sd/Sm 1 in the chaotic limit and that the We first study how the deviation of Sd from Ss,asquanti- | " → sets of eigenstates should produce the same leading contribu- at 1/5 filling. For the quench, we start with the ground state agreement improves with system size provide an important in- fied by Sf /Sd,scaleswithincreasinglatticesizefordifferent dication that S is small and subextensive. Information con- tion to Sc,butforfinitesystemsitisnecessarytotakeinto of (5) with A =0and evolve the system with a superlattice f quenches. As shown in Figs. 3 (a) and (b), Sf /Sd,doesnot tained in the fluctuations of the density matrix becomes neg- account discrete symmetries in order to get an accurate ther- (A =0)andvice-versa.Openboundaryconditionsareused decrease as L increases, rather, we find indications that Sf /Sd ligible in chaotic systems and only the smooth (measurable) saturates to a finite value in the thermodynamic limit. Hence, modynamic description of the equilibrium ensemble. in this% case. part of the energy distribution contributes to the entropy of for these systems Sd is not expected to be equivalent to the The fact that Sd/Sm 1 in the chaotic limit and that the We first study how the deviation of S from S ,asquanti- → d s the system. Also, the close agreement between Sd and Sm in microcanonical entropy. agreement improves with system size provide an important in- fied by Sf /Sd,scaleswithincreasinglatticesizefordifferent the insets of Fig. 1(b) and 1(d) suggests that Sd is indeed the In the lower panels of Fig. 3, we study the scaling of Sd with dication that Sf is small and subextensive. Information con- quenches. As shown in Figs. 3 (a) and (b), Sf /Sd,doesnot proper entropy to characterize isolated quantum systems after increasing system size for the same quenches. A clear linear relaxation. Results for the energy distribution W (E) in Fig. 2 tained in the fluctuations of the density matrix becomes neg- decrease as L increases, rather, we find indications that Sf /Sd behavior is seen, demonstrating that Sd is indeed additive. In ligible in chaotic systems and only the smooth (measurable) saturates to a finite value in the thermodynamic limit. Hence, further support these findings. these panels, we also show the microcanonical (with δE de- Figure 2 shows W (E) for HCBs for quenches in the in- termined as for the interaction quenches [17]) and canonical part of the energy distribution contributes to the entropy of for these systems Sd is not expected to be equivalent to the the system. Also, the close agreement between Sd and Sm in microcanonical entropy. the insets of Fig. 1(b) and 1(d) suggests that Sd is indeed the In the lower panels of Fig. 3, we study the scaling of Sd with proper entropy to characterize isolated quantum systems after increasing system size for the same quenches. A clear linear relaxation. Results for the energy distribution W (E) in Fig. 2 behavior is seen, demonstrating that Sd is indeed additive. In further support these findings. these panels, we also show the microcanonical (with δE de- Figure 2 shows W (E) for HCBs for quenches in the in- termined as for the interaction quenches [17]) and canonical How robust are the eigenstates against small perturba on? Consider two setups preparing the system in the energy eigenstate I A B II A B Setup I: trace out few spins (do not Setup II: suddenly cutoff the link touch them but no access to them) In both cases deal with the same reduced density matrix (right a er the quench in case II) Quench: the reduced density matrix evolves with the Hamiltonian HA. Define two relevant entropies (both are conserved in me a er quench): Diagonal (measure of delocaliza on), entropy Entanglement (von Neumann’s entropy) of me averaged density matrix a er quench A I B II A B B>>A, trace out most of the system. Eigenstate is a typical state so expect that B< Remove one spin. Barely excite the system. Density matrix is almost diagonal (sta onary)? Wrong (in nonintegrable case) Use ETH: Perform quench, deposit non extensive energy δE. Occupy all states in this energy shell. ETH tells us that even cu ng one degree of freedom is enough to recreate equilibrium (microcanonical) density matrix. 3 3 Check: hard-core bosons in 1D 3 (with L. Santos and M. Rigol, 2012) 3 -0.05 -0.1 0.6 0.6 -0.05 -0.1 2 0.6 (a)(a) 0.6 -0.05-0.05(b) (b) -0.1 (a) (a) t’,V’=0.00 0.60.6 (a) 0.60.6 (b)(b) t’,V’=0.00 (b) t’,V’=0.32(b) t’,V’=0.32 L-R R (a) (a) t’,V’=0.00 (b) t’,V’=0.32t’,V’=0.32 -0.12 -0.12 0.4 0.4 0.4 -0.12 -0.1 0.8 0.8 0.40.4 0.40.4 -0.1-0.1 -0.1 for nonintegrable systems, Sd approaches SGC after cutting -0.14 S/(L-R) 0.2 0.2 -0.14 -0.14 S/(L-R) GC S/(L-R) off a few (possibly one) sites, i.e., our hypothesis above is 0.6 0.6 0.20.2 S/(L-R) 0.20.2 0.2 R=L/3 R=L/30.2 GC K/(L-R) -0.15 GC R=L/3 R=L/3 K/(L-R) d R=L/3 R=L/3 K/(L-R) -0.15-0.15 -0.16 R=L/3 R=L/3d K/(L-R) verified. SvN,ontheotherhand,remainsdifferentfromSd 0.4 0.4 0 0 d -0.16 -0.15vN 00 00 vN d -0.16 and SGC until a large fraction of the lattice is traced out [9]. 0.40 0.4 0 S/(L-R) 0.40.4 0.40.4 vN 0.2 0.2 -0.2-0.2 -0.18 -0.2 -0.18 This motivates us to distinguish between the conventional (or (c)(c) 0.40.3 (d)(c)(d) 0.3 (d)0.4 ∼ ∼ R=L/3 0.30.3 0.30.3 -0.2 -0.18 strong) typicality SvN = Sd = SGC and a weaker typicality in 0 0 0.6 0.6 (d) 246 8 10 246 80.20.2 10 0.20.30.2 (c) 0.2 0.30.60.6 (c)(d) t’,V’=0.00 0.6 (d) (c) t’,V’=0.00 t’,V’=0.32 the sense that SvN "= Sd =∼ SGC.Thelatterimpliesthatonly T T t’,V’=0.32t’,V’=0.32 S/(L-R) S/(L-R) the diagonal part of the density matrix of the reduced system S/(L-R) 0.10.1 0.10.1 R=2L/3 0.1 R=2L/3 0.5 0.6 (c) 0.5 t’,V’=0.00 0.6 (d) 0.6 0.6 R=2L/3R=2L/3 0.2 R=2L/3R=2L/3 0.20.50.5 0.5 t’,V’=0.32 in the energy eigenbasis exhibits a thermal structure. 00 000 0 n(k=0) n(k=0) n(k=0) Our results then show that the diagonal entropy satisfies the 0.4 0.4 0.50.5 S/(L-R) 0.50.1 R=2L/3 0.10.40.4 R=2L/3 0.4 0.4 0.5 0.4 0.5 key thermodynamic relation: 0.40.4 (e)(e) 0.250.250.4 (f)(e)(f) 0.25 (f) S/(L-R) 0.2 0.2 0.20.20 0.2 0.30.30 0.3 0.3 0.3 0.30.3 R=1R=1 0.3 R=1R=1 R=1 3 6 9 12 3 366 n(k=0) 699 12 129 12 3 6 9 12 )/(L-R) )/(L-R) )/(L-R) 0.150.150.5 0.15 0.4 0.4 d d ∂S ∂S 1 d R RR S E 0 0 0.20.2 R=2R=2 0.20.10.1 R=2R=2 0.1 R=2 R R = = , (3) 3 6 9 12 3 6 9 12 R=3R=3 R=3(e)R=3 0.25R=3 (f) ∂ES ∂EE T 0.10.1 0.050.050.40.1 (S-S R R (S-S 0.05 (S-S 0.2 Entanglement, diagonal, 00 0.3000 0FIG.FIG. 3: 3: (Color online) (a),(b) KineticFIG. energy 3: (Color per per site site online) and and0.3 (c (c),(d) (a),(b)),(d) Kinetic energy per site and0.3 (c),(d) FIG. 1: (Color online) (a),(b) Entropies per site vs temperature00 for 0.050.05 0.10.1 000 R=10.050.050.05 0.10.10.1 0 0.05R=1 0.1 ! !! 3 6 ! 9! 12 3 6 9 12 where ES and EE are the energies of the subsystems. This fol- 1/L1/L )/(L-R) 1/L1/L nn((kk0.15=0)=0)vs R; L =18; T =4.(a),(c)t = V =0=0(integrable);(integrable); Grand-Canonical entropies d R = L/3.(c),(d)EntropiespersitevsR for a fixed temperature; 0.2 R=2 1/L !! ! R=21/L n(k =0)vs R; L =18; T =4.(a),(c)R t = V =0(integrable); R lows from the fact that Sd coincides with the thermodynamic ! ! ! ! 0.1 ! ! T =4.(a),(c)t = V =0(integrable);Corollary: entropy – energy uncertainty. There is no temperature in a single (b),(d) t =FIG.FIG.V =0 2: 2: (Color. (Color32 online) online) (a)–(d) (a)–(d) Entropies Entropies per perR=3 site site vs vs 11/L/L forforeigenstateTT == . In order (b),(d)(b),(d) tt = V R=3=0.32 (chaotic). (b),(d) t = V =0.32 (chaotic). entropy for S and E simultaneously. In contrast, SvN cannot FIG.0.1 2: (Color online) (a)–(d) Entropies0.05 per site vs 1/L for T = (chaotic). All panels: L =18. (S-S to measure temperature – need to couple to thermometer, e.g. a calibrated two level system. 44andand a a fixed fixed ratio ratioR/LR/L(indicated).(indicated).4 and (e),(f)a (e),(f) fixed Full Full ratio (empty) (empty)R/L symbols:(indicated). symbols: (e),(f) Full (empty) symbols: satisfy this equality, as one can see by considering E # S.In −− −− 0 0 FIG. 3: (Color online) (a),(b) Kinetic energy per site and (c),(d) This always mixes exponen ally many states by ETH. SSGCGC SSdd((SSmcmc SSdd)persitevs)persitevs11/L/L−forfor0RR=1=1,2,and3.(a),(c),(e)−,2,and3.(a),(c),(e)0.05 0.1 0 0.05 0.1 ! ! this case, SvN = SS = SE is proportional to the size of S, !! !! SGC !! Sd (!S! mc Sd)persitevs1/L for CloseRClose=1,2,and3.(a),(c),(e) to the integrable point [Figs. 2(a), 2(c), 2(c), and and 2(e)], 2(e)], tt ==VV =0=0(integrable);(integrable); (b),(d),(f) (b),(d),(f)! tt ==! VV =0=0..3232(chaotic).(chaotic).1/L ! ! 1/L Close to then(k integrable=0)vs R point; L [Figs.=18; 2(a),T =4 2(c),.(a),(c) and 2(e)],t = V =0(integrable); while EE is proportional to the size of E,so∂SE /∂EE → 0. low depend on the regime (integrable vs nonintegrable), but t = V =0(integrable); (b),(d),(f) t = V =0.32 (chaotic). ! ! largelarge fluctuations fluctuations are observed forlarge different fluctuations values values of of areTT observedandand for different values of T and Another of our main goals in this work is to understand the not on specific values of theOtherwise would have contradic ons with fundamental rela on: parameters. !! !! (b),(d) t = V =0.32 (chaotic). FIG. 2: (Color online) (a)–(d) Entropiestt ,V,V ,whichmakesitdifficulttodrawgeneralconclusions.,whichmakesitdifficulttodrawgeneralconclusions. per site vs 1/L for !T != description of few-body observables in the mixed states ob- ˆ t ,V ,whichmakesitdifficulttodrawgeneralconclusions. For our calculations, we select an eigenstate |Ψj& of H (4), 4 and a fixed ratio R/L (indicated).FluctuationsFluctuations (e),(f) are Full indeed (empty) expected symbols: to be larger in in the the integra integrableble tained by the two procedures mentioned before. For that, we with energy E closest to E = E e−Ej /T / e−Ej /T Fluctuations are indeed expected to be larger in the integrable j j j However,However,j in in both both regimes, regimes, all all entropiesapproach entropiesapproach− − each each other otherasas regime than in the chaotic one. In the chaotic regime (and study their expectation values as given by the reduced density corresponding to an effective temperature T .SinceS = SHowever,GC Sd in(S bothmc regimes,Sd)persitevs all entropiesapproach1regime/L for thanR =1 in each the,2,and3.(a),(c),(e) chaotic other as one. In the chaotic regime (and ( the(the fraction fractionvN of of sites sites traced traced out out! increases. increases.! For For the the lattice lattice s sizesizes away! from the! edges of the spectrum)regime all than eigenstates inClose the chaotic of the to the one. integrable In the chaotic point regime [Figs. (and 2(a), 2(c), and 2(e)], matrix, the diagonal ensemble, and the GC ensemble. The Sd =0,thiscanbeseenasthemostdistantchoiceforastate t = V =0(integrable); (b),(d),(f)awayt from= V the=0 edges.32 of(chaotic). the spectrum) all eigenstates of the considered here, we need to tracethe fraction out more of than sites one traced half of out the increases. For the lattice sizes away from the edges of the spectrum) all eigenstates of the results for the first two are similar even if very few sites are with a thermodynamic entropy. We then trace outconsidered a certain here, we need to trace out more than one half of the HamiltonianHamiltonian that are close in energy have (i) (i) a alarge similar similar fluctuations struc struc-- are observed for different values of T and considered here, we need to trace out more than one half of the Hamiltonian that are close in energy have (i) a similar struc- traced out and the system sizes are small. This suggests that number of sites R ≤ 2L/3,ontherightsideofthechain,chainchain for for the the effects effects of of the the off-diagonal off-diagonal elements elements of of the the den den-- ture,ture, as as reflected by the inverse participation ratio ratio! and and! inf infor-or- sity matrix to become irrelevantchain in forS the,leadingthisentropy effects of the off-diagonal elements of the den- ture, as reflectedt ,V by,whichmakesitdifficulttodrawgeneralconclusions. the inverse participation ratio and infor- either tracing out part of the original system or removing the and study the entropies and observables of the reducedsity matrix sys- to become irrelevant in SvNvN,leadingthisentropy mationmation entropy in different bases [10], and (ii) (ii) thermal thermal exp expec-ec- same number of sites and waiting for the reduced system to tem. The reduced density matrix ρˆ describing theto finally remain- approach S and Ssity. matrix to become irrelevant in SvN,leadingthisentropy mation entropyFluctuations in different bases are indeed [10], and expected (ii) thermal to exp beec- larger in the integrable S to finally approach Sdd and SGCGC. tationtation values values of few-body observables [4, 5, 5, 16, 16, 17]. 17]. How- How- relax leads to the same results, up to non-extensive bound- to finally approach S and S . ing system consists of different subspaces each with aThe numbe resultsr presented inHowever, Fig. 1 were obtained in both for regimes,d L =18GC, all entropiesapproachever,ever, this this is not the case each close other totation integrabilityas valuesregime of where where few-body most mostthan observables in the chaotic [4, 5, one. 16, 17]. In How- the chaotic regime (and ary terms. For all practical purposes both procedures are then N ∈ [max(L/3 − R, 0),L/3] of particles. The results presented in Fig. 1 were obtained for L =18, the largest system size that weThe can results study with presented full exact in Fig. di- 1 werequantities obtained fluctuate for L wildly=18 between, ever, eigenstates this is close not inthe en caseergy close to integrability where most equivalent. The agreement with the GC expectation values is Entropies.– The von Neumann, diagonal, and GCthe entropies largest system size thatthe we canfraction study with of sites full exact traced di- outquantities increases. fluctuate For wildly the lattice between s eigenstatesizes away close in from energy the edges of the spectrum) all eigenstates of the [4, 5, 10, 17], and this is affectingquantities our results fluctuate here. wildly between eigenstates close in energy also good and improves with increasing system size. This im- are given by Eq. (1), Eq. (2), and agonalization.agonalization. Figure Figure 2 2 depicts depictsconsideredthe the the largest scaling scaling system here, of of the the sizewe entropie entropie need thatsinsin we to can trace[4, study 5, out 10, with more 17], and full than this exact oneis affecting di- half of our the results here.Hamiltonian that are close in energy have (i) a similar struc- plies that, for few-body observables, only weak typicality is bothboth domains, domains, integrable integrable [(a),(c)] [(a),(c)]agonalization. and and nonintegrable nonintegrable Figure 2 [(b) [(b) depicts,(d)],,(d)], the scalingObservables.–Observables.– of the entropieAn importantsin question[4, 5, 10, we 17], are are left left and to to this address address is affecting our results here. needed to observe thermal behavior in experiments. ES − µNS chain for the effects of the off-diagonalis whether the elements extra information of the carried denObservables.– by- the off-diagonature,An as important reflectedlel- question by the we inverse are left toparticipation address ratio and infor- SGC = ln Ξ + , andand for for(5)TT =4=4.InFigs.2(a)and2(b),when.InFigs.2(a)and2(b),whenboth domains, integrableL/L/33 sitessites [(a),(c)] are are andis whether nonintegrable the extra [(b) information,(d)], carried by the off-diagonalel- System.– We study hard-core bosons in a one-dimensional TGC ements of the reduced density matrix is of relevance to quan- cutcut off, off,SSdandandSSGC approachapproachsityand each each matrixfor other otherT =4 as as toLL.InFigs.2(a)and2(b),whenincreases, becomeincreases, up up irrelevant to to ements in ofSvN the,leadingthisentropyL/ reduced3 sites density are matrixis whether is of relevance themation extra to information quan- entropy in carried different by the bases off-diagona [10],lel- and (ii) thermal expec- lattice with open-boundary conditions described by d GC (µNn−En)/TGC apossiblenon-extensivecorrection.Thistrendisseenforall tities measured experimentally. We focus our analysis on few- respectively. In Eq. (5), Ξ = e apossiblenon-extensivecorrection.Thistrendisseenforis the grand tocut finally off, Sd approachand SGC approachS andall eachS tities other. measured as L increases, experimentally. up to Weements focus our of the analysis reduced on fe densityw- matrix is of relevance to quan- n d GCbody observables. Their expectation values fromtation the reduce valuesd, of few-body observables [4, 5, 16, 17]. How- 1 partition function, TGC is the( GC temperature, µ issystemssystems the chem- we we have have studied studied in in the theapossiblenon-extensivecorrection.Thistrendisseenfor chaotic chaotic regime regime [14], [14], and and opens opens body observables. Their expectationall tities values measured from the experimentally. reduced, We focus our analysis on few- Hˆ = " nˆ1 − (4) ˆ ˆ diagonal, and grand canonical density matricesever, are given this by is not the case close to integrability where most ! 2" ical potential, and ES = Tr[HS ρˆS ] and NS = Trup[upNS aρ aˆS new new] are, question: question: could could cutting cuttingsystemsThe off off results we an an haveinfinitesimal infinitesimal presented studied part part in the ofin of chaotic Fig.diagonal, 1 regime were and [14], obtained grand and canonical opens for L density=18body matrices observables., are given Their by expectation values from the reduced, respectively, the average energy and number of particles in the L−1 the original system lead Sd and SGC to be equal in the thermo- ˆ diagonal, andquantities grand canonical fluctuate density wildly matrices between are given eigenstates by close in energy 1 1 the original system lead Sd andtheupS a largestGC newto be question: equal system in the could size thermo- cutting that we off can an infinitesimal studyOvN with= Tr part[Oˆ fullρˆS of],O exactd = di- ρnnOnn, (6) ˆ†ˆ remaining subsystem S. dynamic limit? In Figs. 2(e) and 2(f), we show the difference OvN = Tr[OρˆS ],Od = ρnnOnn, (6) + −t bi bi+1 + H.c. + V nˆi − nˆi+1 − dynamic limit? In Figs. 2(e)the and original 2(f), we system show the lead differenceS and S to be equal in the thermo- !n $ ! 2"! 2"' Results.– In Fig. 1, we show results for SvN,Sd,andSGC in agonalization. Figured 2 depictsGC the scaling of the entropiesinn [4, 5, 10,ˆ 17], and this is affecting our results here. #i=1 % & between SGC and Sd per site vs system size when tracing out ! OvN = Tr[OρˆS ],Od = ρnnOnn, (6) between SGC and Sd per sitedynamic vs system limit? size when In Figs. tracing 2(e) out and 2(f), we show the difference1 L−2 the integrable [(a),(c)] and chaotic [(b),(d)] domains. Larger 1 (µNn−En)/T Observables.– An important question we are left to address 1 1 one, two, or three sites. Inboth the chaotic domains, regime integrable [Fig. 2(f)], the [(a),(c)] and nonintegrableOGC = Onn [(b)e(µN,(d)],n−En)/T , (7) !n " ˆ†ˆ " fluctuations are seen in Figs. 1(a) and 1(c) as characteristione, two,cof or three sites. In thebetween chaoticS regimeGC and [Fig.Sd per 2(f)], site the vs system size whenOGC tracing= Ξ outOnne , (7) + −t bi bi+2 + H.c. + V nˆi − nˆi+2 − , results are consistent with a vanishing difference in the ther- Ξ !n 1 $ ! 2"! 2"' the integrable regime. Figs. 1(a) and 1(b) show the entropies and for T =4.InFigs.2(a)and2(b),whenL/n 3 sites are is whether the extra(µN informationn−En)/T carried by the off-diagonalel- #i=1 % & results are consistent with aone, vanishing two, ordifference three sites. in the In th theer- chaotic regime [Fig. 2(f)],! the OGC = Onne , (7) vs different eigenstates, which are increasingly awaymodynamic from the limit (even when cutting one site [14]). This find- ˆ Ξ modynamic limit (even whencutresults cutting off, are oneSd consistent siteand [14]).SGC This withapproach find- a vanishing eachrespectively. difference other as Here, inL theincreases,Onn ther-= !ψn| upOˆ|ψ ton" and ements|ψn"’s are of! then the reduced density matrix is of relevance to quan- where, t and t" [V and V "]arenearest-neighbor(NN)and ground state (T increases), when 1/3 of the sitesing are is traced reinforced with the empty symbols, which show the dif- respectively. Here, Onn = !ψn|O|ψn" and |ψn"’s are the ing is reinforced with the emptymodynamic symbols, limitwhich (even show when the dif- cuttingeigenstateseigenstates one site [14]). of the HamiltonianThis find- in the reduced system. system.tities measured experimentally. We focus our analysis on few- next-nearest-neighbor (NNN) hopping [interaction], L is the out. In the chaotic regime, and for all states selected,ference one c betweenan the microcanonicalapossiblenon-extensivecorrection.Thistrendisseenfor entropy (Smc)andSd.In respectively.all Here, Onn = !ψn|Oˆ|ψn" and |ψn"’s are the ference between the microcanonical entropy (Smc)andSd.In Results.– Figure 3 shows results for the kinetic energy chain size, and standard notation has been used [10]. Symme- see that Sd is close to SGC,whileSvN is quite far.this This case, hints finite size effects areing significantly is reinforced reduced, with the lea emptyding symbols,Results.– whichFigure show 3 the shows dif- results for the kinetic energy this case, finite size effectssystems are significantly we have reduced, studied leading in the chaotic regime [14], and openseigenstates ofbody the Hamiltonian observables. in the Their reduced expectation system. values from the reduced, tries, and therefore degeneracies, are avoided by considering athermalstructureinthediagonalpartofthereduceddensito a muchty better agreement betweenference between the two entropies, the microcanonical which entropyL− (SRmc−1)andSd.In L−R−2 to a much better agreementup between a new the twoquestion: entropies, could which cuttingˆ offL− anR− infinitesimal1 ˆ†ˆ part!Results.–L of−R−2 ˆFigurediagonal,†ˆ 3 shows and results grand for canonical the kinetic density energy matrices are given by 1/3-filling and by placing an impurity (" "=0)onthefirstsite. matrix in the energy eigenbasis. this case, finite size effects are significantlyK = −t reduced,b†i b leai+1ding+ H.c. −t! bi†bi+2 + H.c. , further improves with L [15]. Hence, one could argue that Kˆ = −t ˆb ˆbi+1 + H.c. −t ˆb ˆbi+2 + H.c. , In what follows, t = V =1sets the energy scale, " =1/5, In Figs. 1(c) and 1(d) we show results for a fixedfurtherT as an improves in- with L [15]. Hence, one could argue that !i=1 " i # !i=1 L"−Ri−1 # L−R−2 " " " " single eigenstates of generictheto many-body a original much better Hamiltonians system agreement lead have betweenSad and theSGC twoto! entropies,i be=1 equal" which in the thermo-# !i=1 " # and t = V .Whent = V =0,themodelisintegrable[11]. creasingly larger fraction of the original system issingle traced eigenstatesout. of generic many-body Hamiltonians have a ˆ ˆ†ˆ (8) O = !Tr[OˆρˆS ]ˆ,O†ˆ = ρ O , (6) thermodynamic entropy [9].further In Figs. improves 2(a) and with 2(b),L one[15]. can Hence, one could argue that K = −t bi bi+1(8)+ H.c.vN −t bi bi+2 +d H.c. , nn nn As the ratio between NNN and NN couplings increases, the One can see again that in the chaotic limit Sd is much closer dynamic limit? In Figs. 2(e) andand the2(f), momentum we show distribution the difference function, thermodynamic entropy [9]. In Figs. 2(a) and 2(b), one can and the momentum distribution function, !i=1 " # !i=1 " !#n system transitions to the chaotic domain [10]. The results be- to SGC than to SvN,evenwhenveryfewsitesaretracedout.also see that for R = L/3single,thevonNeumannentropy(per eigenstates of generic many-body Hamiltonians have a also see that for R = L/3between,thevonNeumannentropy(perSGC and Sd per site vs system size when tracingL−R out (8) site) saturates to a different valuethermodynamic from Sd and entropySGC (per [9]. site), In Figs. 2(a) and 2(b), one1 canL−R 2πk † 1 i L2−R (j−l)ˆ ˆ (µNn−En)/T site) saturates to a different value from Sd and SGC (per site), nˆ(k)= 1 eandi πk the(j− momentuml)b†bl, distribution(9) O function,= O e , (7) as the lattice size increases.one, Asalso shown see two, that in or Figs. for three 2(c)R = and sites.L/ 2(d)3,thevonNeumannentropy(per,it In the chaotic regime [Fig. 2(f)],L− theR ˆj ˆ GC nn as the lattice size increases. As shown in Figs. 2(c) and 2(d),it nˆ(k)=L − R e bjbl, (9) Ξ is only when R>L/2 that the three entropies become com- L − R j,l!=1 L−R n resultssite) saturates are consistent to a different with value a vanishingfrom Sd and differenceSGC (per site), inj,l! the=1 ther- 1 2πk ! † is only when R>L/2 that the three entropies become com- i L−R (j−l)ˆ ˆ parable. However, for our system sizes, this happens only in for an increasingly large fraction of sites tracednˆ( out.k)= The re- e b bl, (9) parable. However, for our systemmodynamicas the sizes, lattice this size limit happens increases. (even only when As in shown cuttingfor in an Figs. increasingly one 2(c) site and large [14]). 2(d) fraction,it This of find- sites traced out. TheL − re-R j the chaotic regime. sults obtained with the three density matrices arerespectively. comparable, j,l! Here,=1 Onn = !ψn|Oˆ|ψn" and |ψn"’s are the the chaotic regime. ingis only is reinforced when R>L/ with2 that the the empty threesults symbols, entropies obtained become with which the com-three show density the dif- matrices are comparable, parable. However, for our system sizes, this happens only in for an increasinglyeigenstates large fraction of the Hamiltonian of sites traced out. in the The reduced re- system. ferencethe chaotic between regime. the microcanonical entropy (Smc)andSsultsd.In obtained withResults.– the threeFigure density 3 matrices shows are results comparab for thele, kinetic energy this case, finite size effects are significantly reduced, leading to a much better agreement between the two entropies, which L−R−1 L−R−2 ˆ ˆ†ˆ ! ˆ†ˆ further improves with L [15]. Hence, one could argue that K = −t bi bi+1 + H.c. −t bi bi+2 + H.c. , !i=1 " # !i=1 " # single eigenstates of generic many-body Hamiltonians have a (8) thermodynamic entropy [9]. In Figs. 2(a) and 2(b), one can and the momentum distribution function, also see that for R = L/3,thevonNeumannentropy(per L−R 2 site) saturates to a different value from Sd and SGC (per site), 1 i πk (j−l) † nˆ(k)= e L−R ˆb ˆb , (9) as the lattice size increases. As shown in Figs. 2(c) and 2(d),it L − R j l is only when R>L/2 that the three entropies become com- j,l!=1 parable. However, for our system sizes, this happens only in for an increasingly large fraction of sites traced out. The re- the chaotic regime. sults obtained with the three density matrices are comparable, thinking. It took some time before the community grew to realize the significance of this work. Neville Mott and David Thouless were probably someofthefirstpeopleto understand the impact of this work and found its connection tophysicalrealizations of metal-insulator transitions. Anderson’s theoretical work atQuantum that time was motivated by exergodicityperiments performed and localiza on in real space in George Feher’s group at the Bell Laboratories [2–4]. They were particularly in- terested in the phenomenonNon-interac ng electrons in a disordered poten al. Anderson localiza on of spin relaxation in phosphorus doped silicon using electron spin resonance techniques. The electronic wavefunction localized on a phos- phorus atom in doped silicon has a Bohr radius of 20 Å. The electron in this state ∼ g – number of conduc ng channels (number of energy levels felt the random environment of Si29 defects in Si28.Therelaxationtimeofthespinswithin the Thouless energy shell, which behave according to the on these donor atoms was of the order of minutes asWigner-Dyson sta s cs opposed to milliseconds which was predicted by theoretical calculations based on Fermi Golden Rule taking into account phonons and spin-spin interactions. Gang of four scaling theory (P.W. Anderson et. al., 1959) Figure 4. According to scaling theory, Anderson localization is a criti- β()g Extended cal phenomenon, at least in three dimensions. The scaling function β(g) describes how—or more precisely, with what exponent—the average dglog β()=g conductance g grows with system size L. For a normal ohmic conductor dLlog in D dimensions, the conductance varies as LD − 2; consequently, β(g)~D − 2 for large g. Thus the beta function is positive for three- dimensional conductors, zero for two- dimensional conductors, and neg- log g ative in one dimension. In the localized regime, g decays exponentially with sample size so that β(g) is negative. In three dimensions, that leads to a critical point at which β vanishes for some special value for g associ- ated with the mobility edge. Lower-dimension systems do not undergo 3D a genuine phase transition because the conductance always decreases 2D with system size. A small 2D conductor, for instance, will look like a Localized 1D Quasi-extended metal in the quasi-extended regime, but all its states are eventually Figure 1.1: The electron on the donor phosphorus impurities are bound in a hydro- localized if the medium is large enough. genic wavefunction with a large Bohr radius. The Si environment is slightly impure due to the presence of Si29.(Figurefrom[5]) A. Lagendijk, B. van Tiggelen, and D. S. Wiersma (2009) Classical waves offer certain advantages for studying lo- Frank Schuurmans and coworkers etched gallium phosphide calization. Unlike electrons, photons don’t interact with each into a porous network. With a mean free path of only 250 nm, other, and wave experiments are easy to control experimen- it is, to date, the strongest scatterer of visible light. In 3D and above there is a transi on between localized and delocalized states. Hi energy tally at room temperature; frequency takes over the role of The scale dependence of diffusion is also studied using 2 electron energy. One drawback of classical waves, however, is time- resolved techniques in which the material is excited by states are typically localized. Localiza on means no that they do not localize at low frequencies, whereergodicity the mean a. pulsed femtosecond source. The time evolution of the op- free path becomes large due to weak, Rayleigh, scattering. tical transmission can be measured down to the one- photon To recognize whether incoming classical waves are local- level. As time increases, so does the sample size explored by ized in a material, one could examine how the transmission the waves. Scale- dependent diffusion may lead to a time- At finite temperature the system is always delocalized (scales with system size. In regular diffusive systems,ergodic the dependent). diffusion constant. As a result, the transmission transmission is dictated by Ohm’s law, in which the signal in- intensity should fall off at a slow, nonexponential rate. In 2006 tensity falls off linearly with thickness. In the regime of An- Maret’s group measured time tails up to 40 ns in titania pow- derson localization, the transmission should decay exponen- ders that had surprisingly large values for the mean free path tially with length. However, one should be careful to exclude (kℓ ≈ 2.5); they found just such a nonexponential time decay absorption effects, which also show up as exponential decay. in transmission.13 A huge advantage of using classical waves is that other properties in addition to conductance—for example, the sta- Microwaves tistical distribution of the intensity, the complex amplitude of At the millimeter wavelengths of microwaves, it’s relatively the waves, and their temporal response—can be measured. easy to shape individual particles, such as metal spheres, that All those properties are expected to be strongly influenced scatter strongly. By randomly placing the spheres in a tubular by localization. In particular, the localized regime is pre- waveguide with transverse dimension on the order of a mean dicted to exhibit large, non-Gaussian fluctuations of the com- free path (typically 5 cm), one can study the statistics of how plex field amplitude and long-range correlations in the inten- the microwave field fluctuates. The quasi-1D geometry of sity at different spots or at different frequencies. the system—essentially a thick wire or multimode fiber—is advantageous because many theoretical predictions become Light relevant, mostly from the DMPK theory. That theory owes its Anything translucent scatters light diffusively. Think, for in- name to its founders—Dorokhov, Mello, Pereyara, and stance, of clouds, fog, white paint, human bones, sea coral, Kumar—and takes arguments from chaos theory to make and white marble. For those and most other naturally disor- precise predictions about the full statistical properties of a dered optical materials, the scattering strength is far from wire’s transmission when its length exceeds the localization that required for 3D Anderson localization. Systems that scat- length. ter more strongly can be synthesized, though. For example, The onset of localization is again governed by the dimen- material can be ground into powder, pores etched into solids, sionless conductance g, which is here essentially equal to the and microspheres suspended in liquids (see figure 5). ratio of the localization length and the sample length. Using For years researchers have worked with titania powder microwaves, Azriel Genack and colleagues have explored a that is used in paints for its scattering properties. Thanks to broad range of g values, including the localized regime g < 1. the powder’s high refractive index (about 2.7) and submicron Indeed, their observations of anomalous time-dependent grain size, mean free paths are on the order of a wavelength. transmission, scale-dependent diffusion, large fluctuations in Experiments reveal clear signs in the breakdown of normal transmission, and long-range correlations of both the inten- diffusion. To observe localization the challenge is to maxi- sity and the conductance of microwaves have led to a rich mize the scattering without introducing absorption. and complete picture of Anderson localization in thick One way is to use light whose frequency is less than the wires.14 Statistics, their work illustrates, can reveal the onset electronic bandgap of a semiconductor so that it cannot be of localization even in the presence of optical absorption. absorbed but whose refractive index is still high. In 1997, two of us (Wiersma and Lagendijk) and coworkers ground gal- Acoustics lium arsenide into a fine powder and observed nearly com- Ultrasound is particularly well-suited for time-dependent lo- plete localization of near- IR light, as deduced from scale- calization studies because of the long times over which energy dependent diffusion that was measured.12 Two years later can be monitored. As early as 1990, using inhomogeneous 2D www.physicstoday.org August 2009 Physics Today 27 Downloaded 14 Jun 2013 to 168.122.66.77. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://www.physicstoday.org/about_us/terms Interac ng were thought to be always ergodic Insulator coupled to phonons. All states are localized Ef Q: !"#$%&'$(")*+$,%"-$./$('/#/#&,%++ 0"*)"1,%$*"#2%$ At finite T can always hop and get the missing energy from the '/(()#2 )#$.'%$+"3%$4"5$"+$('/#/#+$-/ ! phonons. In our language many-body levels are ergodic (sa sfy ETH). A#1: Sure A#2:Pure two-body short range interac ons between electrons. No way (L. Fleishman. P.W. Anderson (1980)) A#3:Altshuller Finite et. al. (1997), temperature MetalBasko-Insulator, Aleiner Transition , Altshuller (2005) (Basko, Aleiner, BA (2006)) Anderson localiza on in N- dimensional hypercube at Drude small interac ons. metal insulator Many-body localiza on with B. Altshuler, the ac va on barrier scaling as V KITP talk the system size. body systems are now relevant to experiments, since such systems can be produced and studied with strongly-interacting ultracold atoms [49]. And they may become rel- evant for certain systems designed for quantum information processing [50, 51]. Also, many-body localization may be underlying some highly nonlinear low-temperature current-voltage characteristics measured in certain thin films [37]. 2.1 The model Many-body localization appears to occur for a wide variety ofparticle,spinorq-bit The model we chose to study has a finite band-width. An infinite temperature models. Anderson’s originallimit of proposal such a system is studied was by considering for states ata hig spinhenergydensitiesi.e. system [1]; the specific simple eigenstates in the middle of the band. We weigh the observables evaluated from these model we study here is alsostates a with spin equal probability model, in order to studynamely their thermal expectation the values.Heisenber A gspin-1/2chainwith practical benifit of working in this limit is the utilization of all the data we acquire from the full diagonalization of the Hamiltonian which is the most computer time- random fields along the z-directionconsuming part of the calculation. [40]: 1D disordered spin chains at infinite temperature There are many distinctions between the localized phase at large random field h>hc and the delocalized phase at(A. Pal, V. h − −0.5 otherwise, we take J =1.ThechainsareoflengthL with periodic−1 boundary con- −1.5 Many-body level sta s cs |] ditions. This is one of the simpler models that shows a many-b(n+1) ody localization Figure 2.1:0.54 The phase diagram as a function of relative interaction strength h/J at −2 ! 8 i GOE T = .Thecriticalpointis(h/J)c 3.5.Forh 0.52 m for h>hc,itismany-bodylocalized. 12 − −2.5 14 8.0 0.5 16 transition. Since we will be studying the system’s behavior by exact(n) diagonalization, ! i 5.0 −3 0.48 3.6 ] ln [|m (n) ! 0.46 −3.5 2.7 working with this one-dimensionalIn the[r ergodic phase model (h FigureStrong numerical indica ons of many-body localiza on as a localiza on in the Hilbert space. No 2.8: The ratio of adjacent energy gaps (defined in the text). The sample size L netizations in adjacent eigenstates (see text). The values of the random field h are is indicatedanaly c theory yet near the transi on. in the legend. In the ergodic phase, the system has GOE level statistics, Ergodicityindicated -> ETH. in the legend. In the ergodic phase (small h)wheretheeigenstatesarether- ature, β =1/T =0,andinthethermodynamiclimit,while in the localized phase the level statistics are Poisson. Lmal these differences,themany-body vanish exponentially in L as L is increased, while they remain large→∞ in the localized phase (large h). which may reverse the direction of the drift and/or reduce thesizeofthefinite-size localization transition at ehffect from= theh irrelevantc = operator.3.5 1.0 does occur in this model. The usual ∼ ± In our figures we show one-standard-deviation error bars. Theerrorbarsare evaluated after a sample-specific average is taken over the different eigenstates and arguments that forbid phase2.5 Spatial transitions correlations at nonzero temperature in one dimension do sites for a particular realization of disorder. Here and in all the data in this work To further explore the finite-size scaling properties of the many-body localization we restrict our attention to the many-body eigenstates that are in the middle one- transition in our model, we next look at spin correlations on length scales of order not apply here, since theythe rely length L of on our samples. equilibrium One of the simplest correlation statistical functions within a mechanics,third of the energy-ordered which list is of exactlystates for their sample. Thus we look only at many-body eigenstate n of the Hamiltonian of sample α is | ! high energy states and avoid states that represent low temperature. In this energy what is failing at the localization transition. We also presentrange, the indications difference in energy density that between this adjacent states n and (n +1)is of Czz (i, j)= n SˆzSˆz n n Sˆz n n Sˆz n . (2.13) nα " | i j | !α −" | i | !α" | j | !α order √L2−L and thus exponentially small in L as L is increased. If the eigenstates 4833 are thermal then adjacent eigenstates represent temperatures that differ only by this ˆz exponentially small amount, so the expectation value of Si should be the same in 39 What happens when the systems are not ergodic. Chaotic system: rapid Integrable system: relax to (exponential) relaxation to constraint equilibrium: microcanonical ensemble Quantum language: in both cases relax to the diagonal ensemble Integrable systems: generalized Gibbs ensemble (Jaynes 1957, Rigol 2007, J. Cardy, F. Essler, P. Calabrese, J.-S. Caux, E. Yuzbashyan …) F. Essler, talk at KITP, 2012 5. Generalized Gibbs Ensemble Let IProve for a par cular (transverse field m be local (in space) integralsIsing) model of motion [Im, In]=[Im, H(h)]=0 Works both for equal and non-equal me correla on func ons. in ourNeed only integrals of mo on, which “fit” to the subsystem case If we can not measure In – have too many fi ng parameters. In = In(j, j +1,...,j+ n) j What if integrability is slightly broken? j ... j+ln Tuesday, 28 August 12 302 3 Stationary Spectra of Weak Wave Turbulence More familiar example of GGE: Kolmogorov turbulence so-called Zakharov-Kraichnan transformations. They factorize the collision integral. As a result one can (i) prove directly that Kolmogorov spectra reduce the collision integral toImages from Wikipedia zero and (ii) find that the Rayleigh-Jeans and Kolmogorov distributions are the only universal stationary power solutions of kinetic equation. 3.1.1 Dimensional Estimations and Self-similarity Analysis This section deals with universal flux distributions corresponding to con- stant fluxes of integrals of motion in the k-space. In this subsection we shall A. N. Kolmogorov show that for scale-invariant media, these solutions may be obtained from dimensional analysis (see also [3.1,2]). For complete self-similarity we shall first discuss the possible form of universal flux distributions n(k) and the corresponding energy spectra E(k)=(2k)d−1ω(k)n(k). We shall recall how to find the form of the spec- Pump energy at long wavelength. Dissipate at short wavelength. Non-equilibrium steady state trum E(k) for the turbulence of an incompressible fluid: in this case here is Scaling solu on of the onlyNavier one relevant Stokes equa ons parameter, the density ρ; and E(k) may be expressed via ρ,kand the energy flux P .This energy can be thought of as the Comparing the dimensions, we obtain ∂v + (v∇)v = −∇p + ν △v mode dependent temperature. A ∇ − v = 0 E(k) P 2/3k 5/3ρ1/3 par cular type of GGE. (3.1.1) ! Zakharov, L’vov, Fal’kovichwhich: derived this solu on from the kine c equa ons is the famous Kolmogorov-Obukhov “5/3 law” [3.3,4]. As we have seen in Sect. 1.1, in the case of wave turbulence there are always two relevant parameters. We can choose the medium density to be the first one. In contrast to eddies, waves have frequencies, which may be chosen as the second parameter. The frequency enables us to arrange dimensionless parameter Pk5−d ξ = , ρω3(k) so E(k) may be determined from dimensional analysis up to an approxima- tion of the unknown dimensionless function f(ξ): 2 d−6 5−d 3 E(k)=ρωkk f Pk /ρωk . (3.1.2) In particular, if we demand! that ω"(k) be eliminated from (3.1.2), we obtain f(ξ) ξ2/3, and (3.1.2) coincides with (3.1.1). In the case of weak wave tur- bulence∝ the connection between P (k) and n(k) follows from the stationary kinetic equation: − dP (k)/dk = (2k)d 1πω(k)I(k)(3.1.3) − which holds in the limit ξ 1. $ For the three-wave kinetic equation (2.1.12) I(k) n2(k) and n(k) P 1/2, and for the four-wave one, n(k) P 1/3. These∝ expressions may be∝ unified into one: ∝ 2. The Probability Distribution According to the program outlined in the Introduction, first of all we need to study all possible integrals of motion for the system (1.1). Let us see how one can construct † those integrals. We shall start by assigning the variables ap and ap the initial values 0 † †0 ap(t =0)=ap, ap(t =0)=ap .Thenbysolvingtheequationsofmotiononecanfind a(t)anda†(t)asfunctionsofa0 and a†0 and time. By inverting those functions, one can find a0 and a†0 as functions of time, a(t)anda†(t). But by definition a0 and a†0 do not depend on time.δ! Soappears they as are we the have integrals to use (5.12) of motion. and (5.13) If we in giving want to sense construct to the expressions something we obtained out of them whichusing can the play rules the discussed role of the above density while of the waves, convergence we should of the considerintegrals the in linear (2.11) allows us combination not to worry about the boundary terms. †0 0 The expression here,F = whenfp combinedap ap with the kinetic equation (2.11),(2.1) clearly gives the correction to the frequency!p in the equation (2.11) † where fp are some arbitrary coefficients. After being expressed in terms of a(t), a (t)and λpqpq 1. Introduction ωp → ωp + dq (5.15) t it becomes a valid integral of motion. Its explicit time depe! ndencefq can be eliminated by passingThe to theory the limit of wavet →∞ turbulence. studies the stationary states of the statistical classical While (5.14) is just the first term of the expansion due to (5.15) we can easily prove that In order to find the explicitVictor form forGurarieF we need (1994): Scaling solu on from as the GGE to solve the equations of motion. The (not quantum) system(5.15) can consisting be obtained of waves up to with any ordera small if interaone sumsction. up (on the thediagra theoryms with of wave all possible Hamiltonian (1.1) allows us to solve them perturbatively andwecanobtainF in terms of turbulence see ref.tadpole [1] and graphs references on the external therein). lines. Its Hamilt However,onian more can complex be writtendiagrams, down like in the the second aseriesinpowersofWeakly interac ng (weakly λ.Wecanevenavoidsolvingtheequationsofmotionifweusethnonintegrable) Bose gas e following form one from the fig. 2, can lead to the corrections which cannot be interpreted that easily. following procedure. One should look for F in terms of a power series We would like to† conclude this section with† saying† that the technique described here H = ωpapap + λp1p2p3p4 ap1 ap2 ap3 ap4 (1.1) 4 is more general† !p than the standardp1p!2†p3p†4 field theory approach of the ϕ theory. In fact, if we F = fpapap + Λp1p2p3p4 ap1 ap2 ap3 ap4 + take!p the limit of!fp → ωp we shall discover that the prefactor computed according to our It is just a collectionLook for a sta onary probability distribu on of waves with the energy spectrum ωp and the four wave(2.2) inter- rules goes into 1 and the whole expression† † turns† into the standard ϕ4 theory diagram. + Ωp p p p p p a a a ap ap ap + .... action λp p p p with the evident properties1 2 3 λ4p5 p6 ppp1 p=2 λp3p p4p p5 =6 λp p p p .Letusnote 1 2 3 4This should of course! be expected1 as2 in3 this4 limit2 1F3 →4 H.SoourFeynmanrulesshould3 4 1 2 that this Hamiltonian conserves the total wave number Here Λ and Ω arebe some treated still as theunknown “turbulent” functions. generalization We impose of the the standar conditiondfieldtheoryrules.Webelieve on F that it is an integral of motion,manyFind F beautiful or {HF(1), F results} =0,(2) perturba vely are{ and hidden} being in this the new. Problem: perturba on theory is very Poisson technique. brackets. It allows us to N = a†a . (1.2) find those functions to get p p singular because of small denominators !p 6. Epsilon Expansionfp1 + fp2 − fp3 − fp4 The simplest stationaryΛp1p2p3p4 state= of this system is a thermodynaλp1p2p3p4 mic equilibrium(2 and.3) its ωp + ωp − ωp − ωp − i# The kinetic equation1 (2.11)2 is3 in general4 very difficult to solve (that is, to find such f probability density is given by the well-known Gibbs distribution exp(− H+µN ). The basic p Possible nonsingular solu ons: T that it is satisfied). However(λp7p1p5 itp6 hasΛp2p long3p4p been7 − Λ realizedp7p1p5p6 thatλp2p3 ifbothp4p7 ) λ and ω are homogeneous property of thermodynamicΩp1p2p3p4p5p6 =4 equilibrium is that the detailed balance principle. is satisfied.(2.4) functions of momenta,ωp1 then+ ω wep2 + canωp solve3 − ω thatp4 − equationωp5 − Thermal equilibrium with chemical poten als. ωp exactl6 − 2iy.# Namely, following ref. [1] !p7 There are as manywe choose waves going from the wave-number p1 to p2 as there are ones going We can in principle find recursively all the terms in the series(2.2),oneafteranother. back. However thereWith addi onal power-law constraint – addi onal Kolmogorov solu on are other stationary states where that principle is not satisfied, or in α β Afewwordsmustbesaidaboutω = p , λ #’s= whichλ (p p appearp p ) 4 inU( thep% , p% denominators., p% , p% )δ(p% + p% Technically,− p% − %p )(6.1) other words, while thep total numberp"1p"2p"3p" of4 waves0 1 coming2 3 4 to the1 2 giv3en4 wave1 number2 3 p is4 zero, when we compute Poisson brackets no #’s appear. But we must introduce them to avoid there is a flux of waves through the system. Which state will be chosen by the system whereWeak interac ons select possible GGE! U is a function depending only on the ratio of lengths of the momenta and the depends on the external conditions. If it interacts5 with a heat bath satisfying the detailed angles between them and λ0 is a small constant. The parameter α is called the energy balance principle,spectrum it will dimension, soon settle while intoβ theis the thermodynam interaction dimension.ic equilibrium. Then the If, onkinetic the equation other can hand, the heatbe bath solved is so with special the aid that of the it injects so-called waves Zakharov to the transformat system ations, one to wave give number and removes them at a different one, then theγ system2 will neces2sarily chooseα one of those fp = p , γ = β + d or γ = β + d − (6.2) extra states. A simple example of the latter case is3 the waves on3 the surface3 of water which are injected by, for example, the ship, at wave lengths20 of the order of ship length, and are dissipated at much smaller lengths by viscosity. The theory of turbulence concerns itself with studying the “flux” states as the thermo- dynamic equilibrium has already been studied in great details by Statistical Physics. The standard hydrodynamics turbulence is also the example of thefluxstatessinceherewe have a very complicated motion of liquid with a stationary probability distribution which is characterized by the flux of energy or other conserved quantities through various scales of vertex motion. 2 Summary of part I • Ergodic quantum systems sa sfy ETH: each eigenstate of the Hamiltonian (each sta onary state) is equivalent to the micro-canonical ensemble. • Ergodicity can be understood as a process of delocaliza on in the eigenstate basis. Delocalized states are always ergodic (irrespec ve of integrability). • Direct analogy between (many-body) Anderson localiza on and ergodicity. • Many-body states are very fragile: ny perturba on mixes exponen ally many eigenstates. Ensembles are stable. • Integrable systems relax to asympto c states which can be described by the GGE (generalized Gibbs ensembles) composed of local integrals of mo on. • Weak interac on can select possible classes of stable GGE state, which ul mately thermalize (prethermaliza on scenario). But s ll many open problems remain. Part II. Applica ons of ETH to thermodynamics Image taken from Thermodynamics (unlike sta s cal physics) typically deals with non-equilibrium processes, which at each stage can be approximated as approximate local equilibrium Setups considered I. II. 1. Prepare system A in a sta onary 1. Prepare systems A and B in state (diagonal ensemble) sta onary states (0) 0 ρnm = ρnδ nm 2. Apply some me-dependent 2. Connect them by a weak coupling perturba on (quench) (quench) for a period of me . 3. Let the system relax to a new 3. Disconnect and let the systems steady state relax to the steady states. (Markov process). Fundamental thermodynamic rela on for open and closed systems Start from a sta onary state. Consider some dynamical process Assume ini al Gibbs Distribu on Combine together Recover fundamental rela on with the only assump on of Gibbs distribu on What if we do not have the Gibbs distribu on? Imagine we are umping some energy into an isolated system. Does fundamental rela on s ll apply? Entropy is a unique function of energy (in the thermodynamic limit) if the Hamiltonian is local and density matrix is not (exponentially) sparse. I.e. if the system is not localized in the Hilbert space Gaussian approximation for W(E), applies even for small systems: In delocalized regime, which is always the case if ETH applies Recover fundamental rela on + sub-extensive correc ons from ETH Integrable systems: sparse distribu ons Gives extensive contribu on comparable to Sm Can we use GGE density of states? Integrable Hamiltonian (with L. Santos and M. Rigol) Filling 1/5, period P=5, quench A Solid line Sd Dashed lines Sm, SGGE GGE is not constraining enough Isolated systems. Unitary dynamics. Ini al sta onary state. 0 ρnm (0) = ρnδ nm Unitarity of the evolu on: Transi on rates pm->n are non-nega ve numbers sa sfying sum rule p (t) p (t) 1 ∑ n→m = ∑ m→n = n n In general there is no detailed balance even for cyclic processes (but within the Fremi-Golden rule or for symmetric protocols there is). The only stable distribu on under these transforma ons is the infinite temperature maximum entropy state Second law of thermodynamics p (t) p (t) 1 ∑ n→m = ∑ m→n = n n Start from a sta onary state with monotonically decreasing probability (e.g. Gibbs distribu on). Energy can only increase or stay constant, see picture Thirring, Quantum Mathema cal Physics, 1999, A. E. Allahverdyan, Th. M. Nieuwenhuizen, (2002). Likewise (diagonal) entropy sa sfies the second law: For any ini al sta onary state Also immediately follows from the sum rule. Sd (t) ≥ Sn (t) = Sn (0) = Sd (0), S (t) S (t) S (0) S (0) Follows from Araki-Lieb subbadi vity d1 + d 2 ≥ d1 + d 2 (thanks to C. Gogolin) Fluctua on theorems (Bochkov, Kuzovlev, Jarzynski, Crooks) Ini al sta onary state + me reversability: Microscopic probabili es are the same Eigenstate thermaliza on hypothesis: microscopic probabili es are smooth (independent on m,n) Bochkov, Kuzivlev, 1979, Crooks 1998 Probability to do work w If we assume the Gibbs distribu on Crooks equality, C. Crooks, 1998, Jarzynski equality, 1997 Jarzynski equality heavily relies on having Gibbs distribu on, no equivalence of ensembles. Probe large devia ons However, if interested in cumulants can use arbitrary ensembles. Assume for simplicity a cyclic process Zf=Zi. Now can integrate with an arbitrary distribu on ρ(E). Require that high cumulants are small, narrow distribu on Second law of thermodynamics. Einstein-like dri diffusion rela ons. Fokker-Planck (diffusion) equa on: Exercise: expand to the second order in w. Beat subtle es It is sufficient to know only the hea ng rate to find the energy distribu on! Simple way to derive the dri diffusion rela on For unitary dynamics with arbitrary me dependent Hamiltonian the a ractor is Plug back to the FP equa on The last term is usually suppressed in large systems. It Solu on can be also recovered from the Crooks rela om Example (par cle in a deforming cavity C. Jarzynski , 1992) Exercise: compute Equivalent to the Lorenz gas solvable by kinetic equations: L. D’Alessio and P. Krapivsky. Example: universal energy fluctua ons for driven thermally isolated systems (microvawe hea ng) (G. Bunin, L. D’Alessio, Y. Kafri, A. P., 2011) Conven onal hea ng Nonadiaba c (microwave) hea ng Universal non-Gibbs distribu on Dynamical phase transi ons as a func on of the hea ng protocol (to the superheated regime). Can prepare arbitrarily narrow distribu ons. Open systems (A can be a single spin or macroscopic) Time reversal symmetry implies Crooks equality: Sum over eigenstates of B. Use ETH for the system B. Detailed balance follows from the Crooks equality for the two systems and ETH for the system B. Open systems (G. Bunin and Y. Kafri, 2011) Time reversal symmetry implies Crooks equality: Hence Jarzynski type rela on for an open system (require narrow distribu ons) Heat flows from hot to cold because of ETH 5 [2] R. Schmitz and E.D.G. Cohen, J. Stat. Phys. 39,285 Here a, a† and b, b† denote creation and annihilation op- (1985) erators of photons with momentum k and polarization α [3] J. R. Dorfman, T. R. Kirkpatrick and J. V. Sengers, in each of the two cavities and k = |k|. We impose reflect- Annu. Rev. Phys. Chem. 45,213-239(1994) ing boundary conditions at the surface z = 0 separating [4] H. Spohn, J. Phys. A: Math. Gen. 16,4275(1983) [5] A. L. Garcia, M. M. Mansour, G. C. Lie and E. Cementi, the two cavities, so that kz in the sums is restricted to J. Stat. Phys. 47 209 (1987) positive values. The interaction term [6] C. Giardin`a, J. Kurchan and F. Redig, J. Math. Phys. 48 033301 (2007) 2 † [7] B. Schmittmann and R. K. P. Zia., “Phase Transitions Hint = λ!c dx dy a b + h.c. (17) " (x,y,0),α (x,y,0),α and Critical Phenomena vol 17,” ed C. Domb and J. α!=1 Lebowitz, Academic Press, London (1995). 5 [8] B. M. Law, R. W. Gammon, and J. V. Sengers, Phys. effectively converts outgoing a photons from the first cav- Rev. Lett. 60 1554 (1988) † † ity into b photons in the second cavity. The prefactor λ [2] R. Schmitz and E.D.G. Cohen, J. Stat. Phys. 39,285 Here a, a and b, b[9]denote W. G. van creation der Wiel, and et al., annihilation Rev. Mod. Phys. op-75 1(2002) (1985) erators of photons[10] with S. Ciliberto, momentum et al.,k arXiv:1301.4311and polarization (2013) α plays the role of the transmission through the barrier sep- arating the two systems and we assume that the barrier [3] J. R. Dorfman, T. R. Kirkpatrick and J. V. Sengers, [11] T. Fujisawa, et al.,k Science 312 1634 (2006). K¨ung, B., in each of the two cavitieset al., and Physicalk = Review| |. We X 2 impose011001 (2012) reflect- does not radiate, i.e. it is effectively at zero temperature. Annu. Rev. Phys. Chem. 45,213-239(1994) ing boundary conditions at the surface z = 0 separating [4] H. Spohn, J. Phys. A: Math. Gen. 16,4275(1983) [12] C. Jarzynski and D. K. W´ojcik, Phys. Rev. Lett. 92, Usual blackbody radiation corresponds to λ =1. 230602 (2004) [5] A. L. Garcia, M. M. Mansour, G. C. Lie and E. Cementi, the two cavities, so that kz in the sums is restricted to To calculate the energy transfer between the two sys- positive values. The[13] G. interaction Bunin and term Y. Kafri, J. Phys. A: Math. Theor. 46 J. Stat. Phys. 47 209 (1987) 095002 (2013) tems we will use Fermi Golden rule treating Hint as [6] C. Giardin`a, J. Kurchan and F. Redig, J. Math. Phys. 48 [14] I. Tikhonenkov, A. Vardi, J.R. Anglin, and D. Cohen, a perturbation. Then, the matrix element squared for 033301 (2007) 2 Phys. Rev. Lett. 110,050401(2013). a photon of a given polarization with momentum k = [15] A. Imparato,† V. Lecomte, and F. van Wijland, Phys. Rev. (k ,k ,k ) in box a to transfer to box b with momentum [7] B. Schmittmann and R. K. P. Zia., “Phase Transitions Hint = 5λ!c dx dy a b + h.c. (17) x y z 80,011131(2009)(x,y,0),α (x,y,0),α ! ! and Critical Phenomena vol 17,” ed C. Domb and J. " k =(kx,ky,kz) is α!=1[16] L. Bertini, D. Gabrielli and J. L. Lebowitz, J. Stat. Phys. Lebowitz, Academic Press,† London† (1995). 121 843 (2005) [2] R. Schmitz and E.D.G. Cohen, J. Stat. Phys. 39,285 Here a, a and b, b denote creation and annihilation op- 2!2 2 [8] B. M. Law, R. W. Gammon, and J. V. Sengers, Phys. [17] C. Kipnis, C. Marchioro and E. Presutti, J. Stat. Phys. |λ| c (a) (b) (1985) erators of photons with momentum k and polarizationeffectivelyα converts outgoing a photons from the first cav- n (1 + n ! )(18) Rev. Lett. 60 1554 (1988) L2 k k [3] J. R. Dorfman, T. R. Kirkpatrick and J. V. Sengers, in each of the two cavities and k = |k|. We imposeity into reflect-b photons in27 the65 second (1982) cavity. The prefactor λ Annu. Rev. Phys. Chem. 45,213-239(1994)[9] W. G. van der Wiel, et al., Rev. Mod. Phys. 75 1(2002) [18] B. Derrida, M.R. Evans, V. Hakim and V. Pasquier, J. ing boundary conditions at the surface z =plays 0 separating the role of the transmissionPhys. A 26,1493-1517(1993) through the barrier sep- (a,b) −β !ck [4] H. Spohn, J. Phys. A: Math. Gen. 16[10],4275(1983) S. Ciliberto, et al., arXiv:1301.4311 (2013) where nk =1/(e a,b − 1) are the Bose occupation the two cavities, so that k in the sums is restricted to [19] C. W. Gardiner, Handbook of stochastic methods for [5] A. L. Garcia, M. M. Mansour, G. C. Lie[11] and T. E. Fujisawa, Cementi, et al., Science 312 1634 (2006).z K¨ung, B., arating the two systems and we assume that the barrier factors for the two cavities, and L is the linear size of physics, chemistry, and the natural sciences, Springer J. Stat. Phys. 47 209 (1987) et al., Physical Reviewpositive X values.2 011001 The (2012) interaction term does not radiate, i.e. it is effectively at zero temperature. (1994) the cavities along the contact surface. Each such process [6] C. Giardin`a, J. Kurchan and F. Redig,[12] J. Math. C. Jarzynski Phys. 48 and D. K. W´ojcik, Phys. Rev. Lett. 92, Usual blackbody[20] radiation L. Bertini, corresponds A. De Sole, D. to Gabrielli,λ =1. G. Jona-Lasinio, and leads to an energy transfer !ck. Then, using the Fermi 033301 (2007) 230602 (2004) 2 golden rule and summing over initial and final states, we ! † To calculate the energyC. Landim, transfer Phys. betweenRev. Lett. 87 the040601 two sys- (2001) [7] B. Schmittmann and R. K. P. Zia., “Phase[13] G. Transitions Bunin and Y. Kafri,Hint = J.λ Phys.c A:dx Math. dy a Theor.b46 + h.c. (17) [21] J. Tailleur, J. Kurchan, and V. Lecomte, J. Phys. A: " (x,y,0),α (x,y,0),α find the energy transfer rate from box a to b to be and Critical Phenomena vol 17,” ed C.095002 Domb and (2013) J. α!=1 tems we will use FermiMath. Theor.Golden41 rule505001 treating (2008) Hint as Lebowitz, Academic Press, London (1995).[14] I. Tikhonenkov, A. Vardi, J.R. Anglin, and D. Cohen, a perturbation.[22] Then, G. Bunin, the matrix Y. Kafri, element and D. squared Podolsky, forEPL 99 20002 [8] B. M. Law, R. W. Gammon, and J. V. Sengers, Phys. (2012) 2! 2 2 2 Phys. Rev. Lett. 110effectively,050401(2013). converts outgoing a photons froma the photon first cav- of a given polarization with momentum k = |λ| c L 3 ! Wa→b = d k dk Rev. Lett. 60 1554 (1988) [23] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and (2π)3 " " z [15] A. Imparato, V. Lecomte,ity into b andphotons F. van in Wijland, the second Phys. cavity. Rev. The(k prefactorx,ky,kz)λ in box a toC. transferLandim, J. to Stat. box Mech.b with L11001 momentum (2010) α!=1 [9] W. G. van der Wiel, et al., Rev. Mod. Phys. 75 1(2002) ! ! 80,011131(2009)plays the role of the transmission through thek barrier=(k sep-,k ,k ) is[24] G. Bunin, Y. Kafri and D. Podolsky, J. Stat. Mech. (a) (b) ! [10] S. Ciliberto, et al., arXiv:1301.4311 (2013) x y z !ck nk (1 + nk! )δ(!ck − !ck ). [11] T. Fujisawa, et al., Science 312 1634[16] (2006). L. Bertini, K¨ung, B., D. Gabrielliarating and the J. two L. Lebowitz, systems and J. Stat. we assume Phys. that the barrier L10001 (2012) 121 843 (2005) [25] T. Bodineau, B. Derrida, V. Lecomte and F. van Wijland,6 et al., Physical Review X 2 011001 (2012) does not radiate, i.e. it is effectively at zero temperature. 2!2 2 ! [17] C. Kipnis, C. Marchioro and E. Presutti, J. Stat. Phys. |λ| J.c Stat.(a Phys) 133(b1013) (2008) Taking into account that kz > 0andkz > 0, and using [12] C. Jarzynski and D. K. W´ojcik, Phys. Rev. Lett. 92, Usual blackbody radiation corresponds to λ =1. ! ! kz ! [26]2 L. Bertini,nk (1 A. + Denk Sole,)(18) D. Gabrielli, G. Jona-Lasinio, C. the fact that δ(!ck − !ck )= δ(k − k ), we obtain 230602 (2004) 27 65 (1982) can compute the energy fluctuations in a similar fashion: L Appendix B: Derivation of Eq. (15) !ck z z To calculate the energy transfer between the two sys- Landim, arXiv:0705.2996 (2007) [18] B. Derrida, M.R. Evans, V. Hakim and V. Pasquier, J. [13] G. Bunin and Y. Kafri, J. Phys. A: Math. Theor. 46 [27] C. Enaud and B. Derrida, J. Stat. Phys 114 537 (2004) tems we will use2 2 Fermi2 2 Golden rule treating Hint as 2! 2 2 ∞ 095002 (2013) Phys. A 26,1493-1517(1993) ! (a,b) −β !ck |λ| c L 3 (a) (b) |λ| c L 3 ! ! 2 where! n! ! =1/Here(e wea,b derive− 1) the are bound the given Bose in occupation Eq. (15). We as- W = dk k n (1 + n ). (19) a perturbation.Bab = Then, thed matrixk dk elementz ( ck) δ( squaredck − ckk for) a→b 2 k k [14] I. Tikhonenkov, A. Vardi, J.R. Anglin,[19] and C. D.W. Cohen, Gardiner, Handbook of(2 stochasticπ)3 " methods" for sume that ∂ A > 0and∂ A < 0, meaning that 4π "0 α!=1 factors for the two cavities,Ei ij and L is theEj ij linear size of Phys. Rev. Lett. 110,050401(2013). physics, chemistry,a photon and the of natural a given sciences, polarization Springer with momentum k = (a) (b) (b) (a) the average currents grow when the difference between [15] A. Imparato, V. Lecomte, and F. van Wijland, Phys. Rev. (k ,k ,k ) in box× an to(1 transfer + n ! )+ ton box(1b +withn ! the) momentum cavities along the contact surface. Each such process (1994) x y z k k k k Ei and Ej grows.Appendix The matrix A: CoupledR is then blackbodies given by This reduces to the usual result for blackbody radiation 80,011131(2009) k! ! # leads$ to an energy transfer !ck. Then, using the Fermi 5 [20] L. Bertini, A. De Sole,=( D.kx|,kλ Gabrielli,|2y!,k2cz3)L is2 G. Jona-Lasinio, and when βb = ∞ and λ =1. [16] L. Bertini, D. Gabrielli and J. L. Lebowitz, J. Stat. Phys. = dk k4 n(a)(1 + n(b))+n(b)(1 + n(a)) C. Landim, Phys. Rev. Lett. 872 040601 (2001)k k k goldenk rule and summing over initial and final states, we The net energy transfer rate from a to b is then 121 843 (2005) (2π) " # $ Let’s consider† − two†γ1 − coupledγ2 cavitiesγ3 at different temper- [21] J. Tailleur, J. Kurchan,[2] R. and Schmitz V. Lecomte, and2!2 E.D.G.2 J. Cohen, Phys. A:J. Stat. Phys. 39,285 Here a, a Rand= b, b denote creation and, annihilation op- |λ| c (a) (b) find the energy transfer rate from* γ box2 a−toγ3 b−toγ4 + be [17] C. Kipnis, C. Marchioro and E. Presutti, J. Stat. Phys. (1985) ! eratorsatures. of Each photons cavity with has momentum a thermalk gasand of polarization photons, andα Math. Theor. 41 505001Numerically (2008)Example: two coupled black bodies at different temperatures we find2 thatnk this(1 + expressionnk )(18) is well approxi- 2 4 4 27 65 (1982) [3] J. R. Dorfman,L T. R. Kirkpatrick and J. V. Sengers, Aab = Wa→b − Wb→a = |λ| σ(Ta − Tb )(20) mated by inthe each photons of the cantwo leakcavities from and onek = cavity|k|. We to the impose other reflect- at the [18] B. Derrida, M.R. Evans, V. Hakim and[22] V. G. Pasquier, Bunin, J. Y. Kafri, andAnnu. D. Podolsky, Rev. Phys.Chem. EPL 9945(with G. Bunin, Y. ,213-239(1994)20002 Kafri, V. Leconte, D. Podolsky) 2whereinginterface2 boundary2γ1 =2 − between∂E1 conditionsA01, theγ2 = two.∂ atE1 theTheA12, surface Hamiltonianγ3 = −z∂E=2 A 012 is separating,and (2012) ! 2 2 Phys. A 26,1493-1517(1993) [4] H.(a,b Spohn,) J.− Phys.β !ck A: Math. Gen. 16,4275(1983)2 |λ| c L 3 ! π L a,b 45 γ4 = ∂E2 A23 are all positive numbers. As |Tr (R)| = 2 3 where nk =1/(2e 5−/2 1) are the5/2 Bose5/ occupation2Wa→b = the two cavities,d k so thatdk kz in the sums is restricted to where σ = 60c ! is the Stefan constant. Note that since [23] L. Bertini, A. De Sole,[5] D. A.B Gabrielli,ab L.=8 Garcia,|λ| σ G. M.( Jona-Lasinio,T M.aTb Mansour,) + G. aTnd C. Lie− T and E. Cementi,(21) 3 z [19] C. W. Gardiner, Handbook of stochastic methods for 4 a b (2|γπ1 )+ γ2 + γ3 "+ γ4| then"† max γ ≡ max {γ†i} ≤ |Tr (R)|. 4 factors for the two% cavities, andπ L is the linear( size of positiveH = values.!cka Thek interactionak + ! termckbk bk + H . (16) the energy density of a blackbody is proportional to T physics, chemistry, and the natural sciences,C. Landim, Springer J. Stat. Mech.J. L11001 Stat. Phys. (2010)47 209 (1987) & ' α!=1 ,α ,α ,α ,α int T The correlationk is given by k the black body radiation results in a linear A model. We [24] G. Bunin, Y. Kafrithe cavities and[6] C. D.A Giardin`a, along Podolsky, the J. Kurchan contact J. Stat. and surface. F. Mech. Redig, Each J. Math. such process Phys. 48 !,α (a) (b) !,α ! (1994) which satisfies Eq. (4) when Ta ≈ Tb.T ApplyingB our for- !ck nk (1 + nk! )δ(!ck − !ck ). L10001 (2012) leads to033301 an energy (2007) transfer !ck. Then, using the Fermi 2 [20] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and malism in a similar fashion to that described in the dis- B (γ + γ )γ +† B γ (γ + γ ) [7] B. Schmittmann and R. K. P. Zia., “Phase Transitions Hint = λ!c23 1 dx2 dy3 a 01 2 b 3 4 + h.c. (17) C. Landim, Phys. Rev. Lett. 87 040601[25] (2001) T. Bodineau, B. Derrida,golden rule V. Lecomte and summing and F. over van Wijland,initial and final states, we C12 = (x,y,0),α (x,y,0),α cussionand of Critical the Fermi Phenomena gas, we find vol that 17,” the ed correlationsC. Domb and are J. " 2Tr(R!)det(R) 6 [21] J. Tailleur, J. Kurchan, and V. Lecomte, J. Phys. A: Taking into account that kz α!>=10andk > 0, and using J. Stat. Phys 133find1013 thealways (2008)Lebowitz, energy positive. transfer Academic rate Press, from London box a (1995).to b to be z Math. Theor. 41 505001 (2008) ! Recoloring operator, ! ! B12 k(γz2γ4 + γ1γ3 !+2γ1γ4) [26] L. Bertini, A. De Sole,[8] D.As B. Gabrielli, a M. side Law, remark R. G. W. Jona-Lasinio,we Gammon, note that and the C. same J. V.the result Sengers, fact for Phys. thatthe δ( ck − ck−)= !ck δ(kz − kz), we obtain [22] G. Bunin, Y. Kafri, and D. Podolsky, EPL 99 20002arXiv:0705.2996 can computeeffectively the converts energy2Tr( fluctuations outgoingR)det(aRphotons) in a similar from the fashion: first cav- Appendix B: Derivation of Eq. (15) Landim, blackRev.(2007) body Lett. radiation60 15542 can(1988) be obtained using a different λ – transmission amplitude (2012) |λ|2!c2L2 ity into b photons in the second cavity. The prefactor λ [27] C. Enaud and B. Derrida,[9]type W. J.of G. perturbation Stat. van derPhys Wiel,114 et537 al.,3 Rev. (2004) Mod.! Phys. 75 1(2002) 2! 2 2 ∞ [23] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and Wa→b = d k dkz |λplays| c theL role of2 the transmission(a) ( throughb) the barrier sep- [10] S. Ciliberto,(2π)3 et al., arXiv:1301.4311" " (2013) W = so that|λ|2!c2L2 dk k3n (1 + n ). (19) C. Landim, J. Stat. Mech. L11001 (2010) ! 2 α!=1 a→b arating2 the two systems3k and we! !k assume2 that! the! barrier! Here we derive the bound given in Eq. (15). We as- [11] T. Fujisawa,λ c et al., Science† 312 1634 (2006). K¨ung, B.,Bab = 4π "0 d k dkz ( ck) δ( ck − ck ) H = dx dy a(a) ∂(b) + h.c. 3 iπ/2 [24] G. Bunin, Y. Kafri and D. Podolsky, J. Stat. Mech. intet al., Physical Review! X(x,y,2 0110010),α z ( (2012)x,y,z!),α ! ! Different gauge b->b does(2 notπ) radiate," i.e. ite is" eff – flux operatorectively2 at zero temperature. sume that ∂Ei Aij > 0and∂Ej Aij < 0, meaning that i " ck nk (1 + nk! )δ( ck)z−=0 ck ). 9max(α!=1 Bij )(maxγ) L10001 (2012) [12] C. Jarzynskiα!=1 and D. K. W´ojcik, Phys. Rev.) Lett. 92, Usual|C12 blackbody| ≤ radiation corresponds to λ =1. the average currents grow when the difference between This) reduces(22) to the usual(a) result2 |Tr (R( forb)| det blackbody (R(b) radiation(a) [25] T. Bodineau, B. Derrida, V. Lecomte and F. van Wijland,Appendix A: Coupled230602 (2004) blackbodies × nk (1 + nk! )+nk (1 + nk! ) 6 ! To calculate the energy transfer between the two sys- Ei and Ej grows. The matrix R is then given by TakingAt intoλ = account 1 this perturbation that k > 0and is nothingk > but0, the and energy using |Tr (R)| −1 J. Stat. Phys 133 1013 (2008) [13] G. Bunin and Y. Kafri,z J. Phys. A:z Math.when Theor.βb =46∞ and λ =1.# ≤ 9 'J( max |β − β $ | ! kz ! 2 tems2 3 2 we will use Fermi Goldeni rulei+1 treating H as the factUse Fermi golden rule (exercise) fluxcan095002 that operator computeδ( (2013)!ck of the− photons.! energyck )= fluctuationsThe easiestδ(k − in wayk a), similar to we check obtain fashion: that |λ| ! c L Appendix B:det( Derivationa ()R) i=0,1(,b2) of Eq.(b (15)) (a) int [26] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. !ck z z The net energy transfer rate4 from a to b is& then ' arXiv:0705.2996 [14]this I. perturbation Tikhonenkov, gives A. Vardi, the same J.R. result Anglin, as Eq. and (17) D. Cohen, is to= a perturbation.2 dk k Then,nk (1 the + n matrixk )+n elementk (1 + n squaredk ) for Landim, (2007) Let’s consider two coupled cavities at different temper- (2π) " −γ1 − γ2 γ3 discretize the Hamiltonian2 along the z-direction a photon of a given# polarization with momentum$ k = R [27] C. Enaud and B. Derrida, J. Stat. Phys 114 537 (2004) Phys. Rev.2 Lett.! 2 2110,050401(2013).∞ = , atures. Each cavity has a thermal|λ2| gas! c2L of2 photons,3 and! ! 2 ! ! ! whereHere in we the derive second the equalitybound given we used in Eq. Eq. (15). (4), We that6 as- at * γ2 −γ3 − γ4 + [15] A.B Imparato,ab =|λ| c V.L Lecomte,d andk 3 F.dk(a van)z ( Wijland,ck) (δb() ck Phys.− ck Rev.) (kx,ky,kz) in box a to2 transfer4 to4 box b with momentum W = (2π2)3 dk k n (1 + n ). A(19)abNumerically= Wa→b − weW findb→a that= |λ| thisσ(T expression− T )(20) is well approxi- the photons can leak from onea→b cavity to2 the other" at" k the k thesume! steady that ∂ stateEi Aij! '>J(0and= A01∂aEj=AijA12 0and1 ∂Ej2Aπij <3 0, meaning that( k gaugek transformationα!=1 b → be . This gauge transfor- |C12|R≤=18 'J( τ max &|βi − βi+1,| '. !,α The[18] net! B.,α Derrida, energy transfer M.R. Evans, rate V. from Hakima to andb isthe V. then Pasquier, black body J.the average radiation currents results grow* inγ when2i a=0 linear,1, the2−γ3 di−Affγerence4model.+ between We The correlation is given by Let’s consider two coupled cavities at different temper- mation obviously(a) does(b) not aff(bect) H (whilea) H reduces & ' NumericallyPhys.× A 26n ,1493-1517(1993) we(1 +findn that! )+ thisn (1 expression +0n ! ) is wellint approxi- (a,b) −βa,b!ck k k k k whichEi and(approximate expression) satisfieswhereEj grows.nk Eq. The=1 (4) matrix/ when(e RTis≈− then1)T . given are Applying the by Bose our occupation for- atures. Each cavity has a thermal gas of photons, and [19]tomated Eq. C. W. (17) by Gardiner,# in the continuum Handbook limit, of stochastic using$ 1/d methods→ δ(z). for a b 2 2 3 2 2 4 4 |λ| ! c L malismwherefactors in aγ1 similar= for−∂ theE1 fashionA two01, γ2 cavities,= to∂E that1 A12 and, describedγ3L= is−∂ theE2 A in12 linear the,and dis- size of the photons can leak from one cavity to the other at the =HenceAphysics,ab we= W recover chemistry,a→dkb k−4 theWn(b equivalencea→ and)(1a += then|λ(b|) natural)+σ of(Tn thea(b)−(1 sciences, twoT +bn)(20) choices(a)) Springer for B23(γ1 + γ2)γ3 + B01γ2(γ3 + γ4) 2 k k 45 k k 2 γ4the= cavities∂E2 A23 are along all the positive contact numbers. surface. As Each|Tr (R such)| = process C = A-B rela on is sa sfied when otherwise high third order the(2(1994) perturbation.π) " 2 # 5/2 5/2 5/2 $ cussion of the Fermi− gas,γ1 −cumulant weγ2 findγ3 that the correlations are 12 interface between the two. The Hamiltonian is Bab =8|λ| σ (TaTb) + 4 Ta − Tb (21) R = , R . 2Tr(R)det(R) 2 2 π |γleads1 + γ2 to+ γ an3 + energyγ4| then transfer max γ ≡!maxck.{ Then,γi} ≤ |Tr using ( )| the Fermi [20] L. Bertini,π L A. De% Sole, D. Gabrielli,& G. Jona-Lasinio,' ( always and positive. * γ2 −γ3 − γ4 + 2 3 The correlation is given by whereNumericallyσC.= Landim,60c we! findis Phys. the that Rev.Stefan this Lett. expression constant.87 040601 is well Note (2001) approxi- that since golden rule and summing over initial and final states, we B12 (γ2γ4 + γ1γ3 +2γ1γ4) 4As a side remark we note that the same result for the ! † ! † matedwhich by satisfies Eq. (4) when Ta ≈ Tb. Applying our for- − H = ckak ak,α + ckbk bk,α + Hint. (16) the[21] energy J. Tailleur, density J. of Kurchan, a blackbody and V. is proportional Lecomte, J. Phys. to T A: find the energy transfer rate from box a to b to be 2Tr(R)det(R) ,α ,α malism in a similar fashion to that described in the dis-blackwhere bodyγ1 = radiation−∂E1 A01, γ can2 = ∂ beE1 A obtained12, γ3 = − using∂E2 A12 a,and different !k,α !k,α the blackMath. body Theor. radiation41 505001 results (2008) in a linear A model. We B23(γ1 + γ2)γ3 + B01γ2(γ3 + γ4) 45 2 γ4 = ∂E2 A23Care= all positive numbers. As |Tr (R)| = [22]B cussion G.=8 Bunin,|λ of|2σ the Y.(T Fermi Kafri,T )5/ gas,2 + and we D. findT Podolsky,5/ that2 − T the5/2 correlations EPL (21)99 20002 aretype of perturbation12 2Tr(R)det(R) ab a b 4 a b |γ + γ + γ + γ | then max γ 2≡ max {γ } ≤ |Tr (R)|. always(2012) positive.% π & ' ( 1 2 3 4 |λ|2!c2L2 i so that The correlation isB given12 (γ2γ by4 + γ1γ3 +23 γ1γ4) ! [23] L.As Bertini, a side A. remark De Sole, we noteD. Gabrielli, that the G. same Jona-Lasinio, result for the and Wa→b2 =− d k dkz λ!c (2π2Tr()3 R)det(" R) " whichblackC. satisfies Landim, body Eq. radiation J. (4) Stat. when Mech. canTa be≈ L11001T obtainedb. Applying (2010) using our adi for-fferentH = dx dy a†α!=1 ∂ b + h.c. malism in a similar fashion to that described in the dis- int (x,y,0),α z (x,y,z),α 2 [24]type G. Bunin, of perturbation Y. Kafri and D. Podolsky, J. Stat. Mech. i B"23(γ1 + γ2)γ3 +! B01(γa2)(γ3 + γ(4b)) )z!=0 ! ! 9max(Bij )(maxγ) C12 α!==1 ck nk (1 + nk! )δ( ck − ck ). |C | ≤ cussionL10001 of the (2012) Fermi gas, we find that the correlations are so that 2Tr(R)det(R) ) 12 2 ) (22) 2 |Tr (R)| det (R) always[25] T. positive. Bodineau,λ!c B. Derrida, V. Lecomte and F. van Wijland, † B12 (γ2γ4 + γ1γ3 +2γ1γ4) ! As aH sideint = remark we notedx dy that a(x,y, the0), sameα∂zb(x,y,z result),α for the+ h.c.At λ =Taking 1 this− into perturbation account that isk nothing>2 0and butk > the0, energy and using |Tr (R)| −1 J. Stat. Physi 133" 1013 (2008) z=0 9max(Bij )(maxγz ) z α!=1 ) 2Tr(R)det(!R) kz ! ≤ 9 'J( max |βi − βi+1| black[26] L. body Bertini, radiation A. De can Sole, be D. obtained Gabrielli, using G. Jona-Lasinio,a di)fferent flux C. operatorthe fact|C12 of| that≤ photons.δ(!ck − The!ck )= easiestδ way(k − tok check), we obtain that i=0,1,2 ) (22) 2 |Tr (R)| det (R) !ck z z det (R) & ' type ofLandim, perturbationarXiv:0705.2996 (2007) this perturbation gives the same result as Eq. (17) is to At λ = 1 this perturbation is nothing but the energyso that |Tr (R)| −1 [27] C. Enaud and B. Derrida, J. Stat. Phys 114 537 (2004)discretize the Hamiltonian≤ 9 'J( 2! 2 along2 max∞ the|βzi-direction− βi+1| flux operator! 2 of photons. The easiest way to check that |λ|detc (RL) i=0,1,2 3 (a) (b) λ c † W = dk& k n (1 + 'n ). (19) where in the second equality we used Eq. (4), that at H this= perturbationdx dy gives a the same∂ b result as Eq.+ (17)h.c. is to a→b 2 k k int (x,y,0),α z (x,y,z),α 4π "20 discretizei the" Hamiltonian along the z-direction)z=0 9max(2 Bij )(maxγ) the steady state 'J( = A01 = A12 = A23,andthat α!=1 |C12|!≤c ) where in the2 | secondTr (R)| equalitydet (R†) we used Eq. (4), that at 2 2 −1 ) (22) Hint = λ dx dy a b(x,y,d),α − h.c. , (max γ) ≤ [Tr(R)] .As|Tr(R)| / det (R)=λ1 + 2 theThis steady reduces state to'J the( = usualA01(x,y,= result0)A,α12 for= blackbodyA23,andthat radiation At λ = 1 thisAppendix perturbation!c A: Coupled is nothing blackbodies but the energy id "|Tr (R)| # $ −1 H = λ dx dy a† b − h.c. , (max γ)2 ≤α!=1[Tr(R)]2.As|Tr(R)| / det (−R1)=λ−1 + λ2 ≤ 2τ, one obtains Eq. (15) flux operatorint of photons. The easiest(x,y,0) way,α ( tox,y,d check),α that when≤βb9='J∞( and λ =1.max |βi − βi+1| 1 id " # $ −1 det (R) i=0,1,2 & ' (23) α!=1 λ2 ≤ 2τ, one obtains Eq. (15) thisLet’s perturbation consider gives two thecoupled same cavities result as at Eq. diff (17)erent is temper- to (23)where d isThe the net lattice energy spacing transfer andrate from anda thento b makeis then the discretize the Hamiltonian along the z-direction −1 atures.where Eachd is cavity the lattice has spacing a thermal and gas and of then photons, make thegauge andwhere transformation in the second equalityb → be we−iπ used/2. ThisEq. (4), gauge that transfor- at |C12| ≤ 18 'J( τ max |βi − βi+1| . −iπ/2 |C | ≤ 18 'J( τ max |β − β 2 |−14 . 4 i=0,1,2 gauge transformation2 b → be . This gauge transfor-the steady stateA12ab '=J(W=a→Ab − W=b→Aa =i =|λA|i+1σ(,andthatT − T )(20) & ' the photons!c can leak from one cavity to the other atmation the obviously does not01i=0 aff,1ect,2 &12H0 while23 a'Hintb reduces mation obviously does not† affect H0 while Hint reduces 2 2 −1 interfaceHint = λ between thedx dy two.a(x,y, The0),α Hamiltonianb(x,y,d),α − h.c. is , (max γ) ≤ [Tr(R)] .As|Tr(R)| / det (R)=λ1 + id " to Eq.−1 (17) in the2 continuum2 limit, using 1/d → δ(z). to Eq. (17)α!=1 in the continuum# limit, using 1/d$ → δ(z).λ ≤ 2τ, one obtainsπ L Eq. (15) 2 where σ = 2 3 is the Stefan constant. Note that since Hence we recover the equivalence of the two choices(23) Hence for we recover60 thec ! equivalence of the two choices for ! † ! † the energy density of a blackbody is proportional to T 4 wheretheHd =is perturbation. the latticeckak,αa spacingk,α + andckb andk,α thenbk,α make+ Hint the. (16)the perturbation. −1 gauge transformation!k,α b → be−i!kπ,/α2. This gauge transfor- the|C black12| ≤ 18 body'J( radiationτ max |β resultsi − βi+1 in| a linear. A model. We i=0,1,2 & ' mation obviously does not affect H0 while Hint reduces to Eq. (17) in the continuum limit, using 1/d → δ(z). Hence we recover the equivalence of the two choices for the perturbation. (Non-equilibrium) Onsager rela ons. Two or more conserved quan es. in progress with L. D’Alessio, G. Bunin, Y. Kafri, also P. Gaspard and D. Andrieux (2011) Time reversibility and ETH imply the Crooks rela on Two independent Jarzynski rela ons: Onsager rela ons (cumulant expansion) This is not a gradient expansion. E.g. temperatures can but need not be close! Two currents and one current fluctua on set the other fluctua on and the cross- correla on. Possible applica ons from spintronics to black hole radia on. 7 Fermi Acceleration The so-called Fermi acceleration – the acceleration of a particle through col- lision with an oscillating wall – is one of the most famous model systems for understanding nonlinear Hamiltonian dynamics. The problem was introduced by FermiEnergy localiza on transi on in periodically driven systems [1] in connection with studies of the acceleration mechanism of cos- mic particles through fluctuating(with L. magneticD’Alessio fields. Similar) mechanisms have been studied for accelerating cosmic rockets by planetary or stellar gravita- tional fields. One of the most interesting aspects of such models is the deter- minationInstead of a single quench consider a periodic sequence of pulses: of criteria for stochastic (statistical) behavior, despite the strictly deterministic dynamics. g1 Here,g we study the simplest example of a Fermi acceleration model, which was originally studied by Ulam [2] : a point mass moving between a fixed and oscillating wall (see Fig. 7.1). Since then, this modelWhat is the long me limit in system has been investigated by many authors, e.g., Zaslavskii and Chirikovthis system? [3], Brahic [4], g0Lichtenberg, Lieberman, and their coworkers [5]–[8], and it is certainly one of T1 T2 the first examples of the variety of kicked systems which appear in numerous studies of nonlinear dynamics. time 7.3 Computer Experiments 149 Small energies: chaos Fermi-Ulam problem (prototype of 0.5 the Fermi accelera on problem). Cubic Sawtooth and diffusion. Large fixed wall energies – periodic 6 0 1 mo on. Energy stays L M = localized within the L v 16a s chao c region. ? 1 Sine Triangle ? 62a 0 1 ? Stochas c mo on – Fig. 7.14. Poincar´ephasespacesec- infinite accelera on. Fig. 7.15. Poincar´ephasespacesec- oscillating wall tion for a harmonic wall oscillation with tion for a harmonic wall oscillation with (G. M. Zaslavskii and B. V. Chirikov, 1964 M =20.Iterationsofseveralselected M =40.Iterationsofseveralselected M. A. Fig.Liberman 7.1. and J. ModelLichenberg for Fermi 1972) acceleration. Fig. 7.2.trajectories.Different wall oscillations.trajectories. For the absolute barrier, one finds the simple approximate expression [6] u 2.8√M, (7.17) b ≈ which is supported by the double logarithmic plot in Fig. 7.13 (taken from Ref. [5] ). These data agree, of course, with the values ub =12.9 and 27.5 obtained numerically in the above experiments. Equation (7.17), however, describes only the overall features of the growth of the stochastic sea with increasing M and does not account for the finer details. When we compare the Poincar´ephase space sections for M =10 (Fig. 7.6) and M = 100 (Fig. 7.5), we observe the highest period-one fixed point at u = M/m,withm =1inthefirstcaseandm =4inthesecond. Therefore, the three fixed points with m =1,2,and3disappearfromthe chaotic sea when M is increased from 10 to 100. Here, we study this mechanism in some more detail. Figures 7.14 and 7.15 show Poincar´esectionsforM =20and40.Locating the fixed points in the centers of the large stability islands embedded in the chaotic sea at u =10or13.3, respectively, we readily identify them as m =2 or m = 3 fixed points. The maximum values of the velocity accessible for acceleration from small velocities is given by ub =12.5 and 15.8, respectively. The fixed points with lower m are located above these ub values and a KAM curve separates these higher fixed points from the lower ones. It should also be noted that the value of ub =12.5 for M =20issmaller than that that of ub =12.9 for M = 10, despite the overall increase predicted in (7.17) and shown in Fig. 7.13. This justifies a more detailed numerical study Kicked rotor (realiza on of standard Chirikov map) Transi on from regular (localized) to chao c (delocalized) mo on as K increases. Chirikov, 1971 K=0.5 K=Kg=0.971635 K=5 (images taken from scholarpedia.org) Delocaliza on transi on at Kc≅1.2 (B. Chirikov (1979)). Quantum systems: (dynamical) localiza on due to interference even in the chao c regime (F. Izrailev, B. Chirikov, … 1979). What about periodically driven ergodic systems in thermodynamic limit? The equation of motion of the Kapitza pendulum reads a θ¨ = ω2 + γ2 cos (γt) sin θ (1) − 0 l where θ is the angle measured from the downward position (see Fig. 1), g ω0 = l is the frequency of small oscillations and a, γ are the amplitude and frequency of the driving of the point of suspension: yc = a cos (γt). This − dynamical system has an extremely rich behavior containing both regions of chaotic and regular motion (see Ref. [14] and references therein). For our purposes we consider the limit of small driving amplitude a/l 1 and New non-equilibrium phases and phase transi ons describe how the dynamical behavior qualitatively changes as a function of Example: Kapitza pendulum (emerged from par cle accelerators, 1951) the driving frequency. For a small amplitude drive the lower equilibrium at θ = 0 remains stable γ 2 unless particular parametric resonance conditions with n = integer, ω0 ≈ n are met [22]. As we increase the frequency of the external drive from γ 2ω ≈ 0 we observe qualitatively different regimes. First the motion in phase space is completely chaotic and both the lower and upper equilibrium (θ =0, π) are unstable, then the lower equilibrium becomes stable while the upper a γ √ equilibrium remains unstable, finally when l ω > 2 both the upper and 0 lower equilibrium are stable. The surprising phenomenon that the upper position becomes stable (and the pendulum performs oscillations around The equation of motion of the Kapitza pendulumthis inverted reads position) is known in the literature as dynamical stabilization and was first explained by Kapitza. He showed that for small amplitude and high frequency driving the dynamic of the driven pendulum can be ¨ 2 a 2 θFigure= 1:ω Schematic0 + γ representationcos (γt) ofaccurately thesin Kapitzaθ described pendulum, i.e. by a a rigidtime-independent pendulum(1) effective Hamiltonian, moreover with− vertically oscillatingl point of suspension, and its phase portraits. (a) The Kapitza pendulum. (b) Non-driven regime: thethe pendulum effective performs potential small oscillations energy around develops the a local minimum at θ = π when γ Stable inverted equilibrium for stable lower equilibrium which are representeda > by√ the2 explainingred line in the phase the oscillations portrait. (c) around the inverted position. where θ is the angle measuredDynamical stabilization from regime:the downward the penduluml ω0 performs position small oscillations (see Fig. around 1), the g Stability: experimentally proven by Kapitsa using stable upper equilibrium which are representedUsually by the thered line dynamical in the phase stabilization portrait. In is obtained by splitting the degrees of ω0 = is the frequency of small oscillations and a, γ are the amplitude and l Singer sewing machine and by Arnold using razor. the phase portraits the green lines correspondfreedom to rotations, into fast the blackand linesslow to modes, oscillations, eliminating the fast modes, and obtaining the blue lines are the separatrices and the points represent the region of chaotic motion. frequency of the driving (For of interpretation the point of the of references suspension: tothe colour eff inectivey thisc = figure potential legend,a cos the ( forγ readert). the is This referred slow modes [22]. This procedure has a limi- − dynamical system hasTheore cally proven by Arnold using KAM anto the extremely web version of this rich article.) behaviortation containing that it can not both be easily regions extended to either interacting systems or to of chaotic and regulartheorem motion (see Ref. [14] andthe quantum references domain. therein). It is also For unclear whether averaging over fast degrees with the recent experimental findings on a AC-driven electron glass [19]. In our purposes we consider the limit of small drivingof freedom amplitude will leada/l to the Hamiltonian1 and equation of motion in each order of this experiment, the energy absorbedthe expansion.by the electrons Here from we the show AC-driving, that the dynamical stabilization phenomenon describe how the dynamicalis related behavior to the variation qualitatively in the conductance changes whichas can a function be directly mea- of sured and it is convincingly showncan that be at understood high frequency through (short period) the Magnus the expansion of the quantum evolution the driving frequency. electron glass does not absorb energy.operator Moreover, in powers it is shown of the that inverse the criti- frequency (a relate perturbative analysis in For a small amplitudecal drive frequency the is set lower by the equilibrium electron-phononpowers at of interactionsθ the= inverse 0 remains and frequency it is much stable lower was studied in [20, 21]) . The advantage of unless particular parametricthan the resonance maximum rate conditions of energythis exchange methodγ which2 iswith is that set byn allows electron-electron= integer us to analyze, behavior of the periodically driven interactions. Finally, we will show a strongω0 ≈ evidencen for this transition using are met [22]. As we increase the frequency of theinteracting external systems. drive from γ 2ω examples of classical and quantum interacting spin systems. ≈ 0 we observe qualitatively different regimes. First the motion in phase space 2. The Kapitza pendulum is completely chaotic and both the lower and upper equilibrium (θ =0, π) 5 are unstable, then the lowerBefore equilibrium addressing the many-particle becomes stable problem while we will the discuss upper a much equilibrium remains unstable,simpler example finally of a when periodicallya γ driven> √ system,2 both the so the called upper Kapitza and pen- dulum [13] and show how thel Magnusω0 expansion can be used to derive the lower equilibrium are stable.effective potential. The surprising The Kapitza pendulum phenomenon is a classical that rigid the pendulum upper with position becomes stablea vertically (and the oscillating pendulum point of suspension performs (see oscillations Fig. 1). around this inverted position) is known in the literature as dynamical stabilization and was first explained by Kapitza. He showed4 that for small amplitude and high frequency driving the dynamic of the driven pendulum can be accurately described by a time-independent effective Hamiltonian, moreover the effective potential energy develops a local minimum at θ = π when a γ > √2 explaining the oscillations around the inverted position. l ω0 Usually the dynamical stabilization is obtained by splitting the degrees of freedom into fast and slow modes, eliminating the fast modes, and obtaining the effective potential for the slow modes [22]. This procedure has a limi- tation that it can not be easily extended to either interacting systems or to the quantum domain. It is also unclear whether averaging over fast degrees of freedom will lead to the Hamiltonian equation of motion in each order of the expansion. Here we show that the dynamical stabilization phenomenon can be understood through the Magnus expansion of the quantum evolution operator in powers of the inverse frequency (a relate perturbative analysis in powers of the inverse frequency was studied in [20, 21]) . The advantage of this method is that allows us to analyze behavior of the periodically driven interacting systems. 5 Further experimental progress Light induced Superconduc vity in a Exciton-Polariton condensates in driven- Stripe-ordered Cuprate dissipia ve system D. Faus , et all, Science 331, 189 (2011) J. Kasprzak et. al, Nature, 443, 409 (2006) Interes ng unpublished results from R. Averi group in VO2 driven by THz pump. Image taken from A. Cavalleri web page Periodic drive: wave func on (density matrix) a er n-periods 7 Time evolu on is like a single quench to the Floquet Hamiltonian FIG. 2: Two equivalent description of the driving protocol: (left) sequence of sudden quenches between H0 and H1 and (right) single quench from H0 to the effective Floquet Hamiltonian Heff and back to H0. external magnetic field and the Hamiltonian H1 to be interacting and ergodic: Magnus expansion: (7) H0 = BxHBx,H1 = JzHz + Jz Hz + J H + J H where, we have defined the shorthand notations:: x z z x x y y HBx = n sn,Hz = n snsn+1 ,H = n snsn+1 + snsn+1 z z x x y y Hz = n snsn+2 ,H= i snsn+2 + snsn+2 Let us point that this system is invariant under space translation and π rotation around the • − Each term in the expansion is extensive and local (like in high temperature expansion) x x y y z z x axis (s s , sn sn, s s ). For numerical calculations we choose the following − n → n →− n →−n • Higher order terms are suppressed by the period T but become more and more non-local. 1 1 1 parameters: Bx =1,Jz = J = 2 ,Jz = 40 ,J = 4 . We checked that our results are not tied − − • Compe on between suppression of higher order term and their non-locality – similar to to any particular choice of couplings. many-body localiza on. As pointed out earlier, we can expect two qualitatively different regimes depending on the period of the driving. At long periods the system has enough time to relax to the stationary state • The expansion is well defined classically if we change between the pulsescommutators and thus is expected to to the Poisson brackets. constantly absorb energy until it reaches the infinite temperature. This situation is similar to what happens for driving with random periods26.Onthe contrary if the period is very short we can expect that the Floquet Hamiltonian converges to the time averaged Hamiltonian. Since the whole time evolution can be viewed as a single quench to the Floquet Hamiltonian (right panel in Fig. 2) we expect that the energy will be localized even in the infinite time limit as long as the Floquet Hamiltonian is well defined and local. Noticing that the commutator of two local extensive operators is local and extensive we see from Eq. (5) that the Floquet Hamiltonian is local an extensive in each order of ME. Thus the question of whether the energy of the system is localized in the infinite time limit or reaches the maximum possible dynamical stabilization condition, first obtained by Kapitza. The quantum time-dependent Hamiltonian for the Kapitza pendulum is: 1 Hˆ (t)= pˆ2 + f(t) cos θˆ (A.1) 2m θ dynamical stabilization condition, first obtained by Kapitza. The quantum 2 a 2 ˆ where f(t)= m ωtime-dependent+ γ cos (γt) Hamiltonianand θ, pˆθ are for quantum the Kapitza operators pendulum with is: − 0 l ˆ dynamical stabilization condition, first obtained by Kapitza. The quantum canonical commutation relations θ;ˆpθ = i. The explicit form of the first Theory progress: no general framework yet but a good progress time-dependent1 Hamiltonian for the Kapitza pendulum is: Hˆ (t)= pˆ2 + f(t) cos θˆ (A.1) threedynamical terms stabilization in the ME condition, are (see first the obtained review by article Kapitza. [24]):2m Theθ quantum Magnus (short-period) expansion for the ˆ Kapitza1 2 pendulum ˆ time-dependent Hamiltonian for the Kapitza pendulum is: H(t)= pˆθ + f(t) cos θ (A.1) L. D’Alessio(1) and A.P. 2013 (original explana on, 2 a 2 Kapitza 1951) 2m where f(t)=ˆ m 1ω +ˆ γ cos (γt) and θˆ, pˆθ are quantum operators with Heff = T 0 H(lt1) 1 2− 2 a 2 ˆ Hˆ ((2)t)= pˆ + f(t) coswhereθˆ f(t)= ˆm ω0 +(A.1)γ cos (γt) and θ, pˆθ are quantum operators with canonicalHˆ = commutation1θ Hˆ ( relationst ); Hˆ (t )−θ;ˆpθ = il. The explicit form of the first eff 2m2T (i) 1 2 ˆ canonical commutation relations θ;ˆpθ = i. The explicit form of the first (3) three terms in the ME are (see the review article [24]): ˆ 1 2 a 2ˆ ˆ ˆ ˆ ˆ ˆ whereH f(=t)= m2 ω + γHcos(t1 ();γt)Hand(t2);θˆ,Hpˆ(tarethree3) quantum terms+ H in(t operators3 the); MEH( aret2 with); (seeH(t the1) review article [24]): eff 6T−(i) 0 l θ ˆ (1) 1 canonical commutation relations θ;ˆpθ = i . The explicitˆ form of theˆ first (1) (A.2) Heff = H(t1) ˆ 1 ˆ T Heff = T H(t1) wherethree terms the time in the integration ME are (see domains the review are article ordered,(2) [24]): i.e.1 0 19 19 Specific model: classical or quantum spin chain J Start in the ground state of the noninterac ng system. Follow the noninterac ng energy. time T1 T0 Analy cally tractable limit: Classical limit: T1 0 commutators -> Poisson brackets. → Singularity (phase transi on?) at hT0 = π 9 Quantum spin chain: energy in the infinite me limit Two different regimes. Is it a crossover or a transi on? FIG. 3: (Color online) Excess energy of the quantum spin chain in the long time limit: Q = ψ(t) H0 ψ(t) t Egs,whereEgs is the ground state energy of the Hamiltonian H0, as a function | | →∞ − of the pulse times T0 and T1 in units of /Bx. Blue (orange) regions correspond to small (large) excess 1 energy. The data is obtained by the exact diagonalization of a spin- 2 chain with N = 15 spins. of /Bx) no matter how small T1 or the coupling constants in H1 (the Js) are. The location of the singularity is also manifestly independent of the system size. As it is seen from the numerical simulations (Fig. 3), the singularity in the effective Hamiltonian is also manifested in the excess energy of the system in the infinite time limit. We point out that, in the limit of small T1, our system directly extends the kicked rotor model to the many-spin domain. Indeed most of the time the spins precess around the magnetic field Bx getting periodically short kicks by the interacting Hamiltonian H1. Thus in this limit the many-body localization transition directly generalizes the well known kicked rotor localization transition11,12. Away from the T axis, nested commutators of order T n for n>1 need to be included in 0 − 1 the effective Hamiltonian. These commutators become difficult to compute analytically at high n. Like in the high temperature expansion in statistical physics they involve multiple spin interactions and become non-local in space. Therefore we have to rely on numerics. In Fig. 4 we analyze the long time limit of the excess energy along the generic direction T = T 2 for 2 T 3(pink 1 0 − ≤ 0 ≤ arrow in Fig. 3). In Fig. 5 we show the error between the exact asymptotic value of the normalized excess energy (see Fig. 4) and the corresponding excess energies obtained by truncating the ME. From the left panel we see that, as expected, for short periods the ME becomes asymptotically exact. From the Quantum spin chain (comparison with Magnus expansion) 4 III. BEHAVIOR OF THE DIAGONAL ENTROPY ACROSS THE LOCALIZATION Energy 4TRANSITION FOR THEEntropy INTERACTING QUANTUM SPIN CHAIN 1.2 1 L=10 N=10 1 L=11 N=11 L=12 0.8 N=12 L=13 N=13 0.8 L=14 N=14 L=15 N=15 L=16 0.6 N=16 0.6 L=17 N=17 Heff(1) Heff(1) 0.4 Heff(3) 0.4 Heff(3) Heff(5) Heff(5) Heff(∞+5) Heff(∞+5) 0.2 0.2 0 0 2 2.2 2.4 2.6 2.8 3 0 0.2 0.4 0.6 0.8 1 FIG. 2: (Color online) Long time absorbed energy, ψ(t = T0 T0 ) H0 ψ(t = ) , as a function of the parameters, T0 and ∞ | | ∞ T1, of the driving protocol. Blue (orange) regions corresponds to little (large) absorbed energy. The data are obtained by FIG. 3: (Color online) Normalized asymptotic valueFIG. 1: of (Color the online) (Color online) Asymptotic value of the normalized diagonal entropy, Sd/Smax where H0 (t= ) Egs energy, 2 ∞ − where Egs and Emax are the lowest exact diagonalization (in the k = 0 momentum sector) of a (Emax Egs) Smax is the logarithm of the total number of the microstates. The exact results for different system sizes are 1 − quantum spin- chain with N =15spins. and highest eigenvalues of H0, along the line pink line of Fig. 2 compared with the predictions obtained by truncating the Magnus Expansion to different orders: Heff (k) 2. TheClear evidence for the phase transi on as a func on of the driving period. exact results for different system sizes are compared contains terms of order T mT n with m + n k and H ( + 5) denotes the non-perturbative result in T with the predictions obtained by truncating the Magnus Ex- 0 1 ≤ eff ∞ 0 m n pansion to different orders (see main text). (Eq. (8) in the main text) together with all the other terms T0 T1 with m + n 5. The results from the can be computed to first order in T1 using the Hausdorff- Very similar behavior in a classical chain. ≤ Baker-Campbell formula (HBC) [14]. For the present Magnus Expansion for different system sizes are identical within the image resolution. problem, due to the simple form of the H0, it is pos- Exact-Heff(3) sible to resum this series and obtain the following non- 1 In this Section we show our numerical results for the behavior of the diagonal entropy across the 1 0.9 T0=2.4 perturbative expression in T0 (see Appendix): L=15 0.8 T0=2.5 k k localizationT0=2.6 transition. The diagonal entropy [6] is defined as Sd(t)= k p0(t) log p0(t)where -1 T0=2.7 10 0.7 T0=2.8 − 0.6 pk(t) is the occupation probability at time t of the k th eigenstate of H . The diagonal entropy T1 λ 2 0 0 -2 0.5 − Heff = Hav 1 λ cot + λ cot(λ) M+ (T1 ) 10 Exact-Heff(1) −2T − 2 O Exact-Heff(3) 0.4 serves as a measure of the occupation in the Hilbert space. As in Fig. 4 in the main text we study Exact-Heff(5) 0.3 (9) -3 10 Exact-Heff(∞+5) 0.2 the behavior along the generic line T1 = T0 2 for 2 T0 3. In particular, in Fig. 1 we show the 1 0.1 − ≤ ≤ where Hav T (H0T0 + H1T1) is the time-averaged ≡ -4 0 asymptotic value the diagonal entropy for different system sizes and we compare it to the values Hamiltonian, M is a simple operator that couples 10 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 10 12 14 16 18 20 T0 L nearest-neighbor and next-nearest-neighbor spins and obtained by truncating the ME to different orders. The asymptotic values have been computed B T λ x 0 .Theeffective Hamiltonian above is singu- by projecting the initial state to the eigenstates of H and then back to H .Thisprocedureis ≡ FIG. 4: (Color online) Error for fixed system size, L =15,and eff 0 lar for λ = n π with n = integer which corresponds to different approximations of the ME (left panel) and error for equivalent to the assumption of infinite time averaging with respect to Heff and the asymptotic T0 = n π (we have set = 1 and Bx = 1). Since the different system sizes and the same approximation, Heff (3), time-evolution for the time-dependent problem can be of the ME (right panel). The horizontal dottedvalues lines are obtained a correspond to the prediction of the diagonal ensemble of Heff . Fig. 1 shows that guide for the eye. obtained by a single quench from H0 to Heff (see Fig. the energy increase observed in Fig. 4 of the main text is indeed caused by a delocalization of 1) we expect that any singularity in Heff will show up as a singularity in the dynamical behavior of the sys- tem. This is clearly shown in Fig. 2. We stress that larger cluster are generated at each order of perturba- the location of the singularity, T ,isdeterminedbythe tion theory). Thus away from the T axis we rely on c 0 − single particle energy scale (Bx) and it is independent on the numerics. In Fig. 3 we study the behaviour along the system size. Before moving forward we note that for the line-cut T = T 2 for 2 T 3(pinklinein 1 0 − ≤ 0 ≤ T1 0, in each cycle of the driving the spins precess Fig. 2). In Fig. 3 the exact results for different system independently→ around the x axis (under the action of sizes, L = 10,...,17, are shown together with the predic- − H0) and are “δ-kicked” by switching on the spin interac- tions obtained by truncating the ME at different orders: tions (H ) for an infinitesimal time. It seems plausible H (k) contains terms of order T mT n with m + n k. 1 eff 0 1 ≤ that the transition we have just described is the exten- In particular Heff (1) is the time averaged Hamiltonian sion to coupled-systems of the well known kicked rotor and Heff ( +5) contains the not-perturbative result Eq. localization transition [15]. 9 together∞ with all the terms T mT n with m + n 5. In 0 1 ≤ Away from the T0 axis, nested commutators of order Fig. 4 we study the error between the exact asymptotic n − T1 for n>1needtobeincludedintheeffective Hamil- value and the one obtained by truncating the ME. In tonian. These commutators becomes difficult to compute particular, in the left panel it is shown the error for fixed very quickly and the effective Hamiltonian becomes non- system size, L = 15, and different approximations of the local (this is similar to what happens in the HTE where ME. We note that by adding more terms in the ME the Temporal simula ons of a quantum spin chain. Exact Time Evolution, NS=16 T0=T1=0.6 1.2 T0=T1=1.0 T0=T1=1.2 T0=T1=1.4 T0=T1=1.6 1 0.8 7.3 Computer Experiments 149 0.6 0.4 Excess Energy divided by HalfBandwidth 0.2 0 0 200 400 600 800 1000 # number of cycles Fig. 7.14. Poincar´ephasespacesec- Fig. 7.15. Poincar´ephasespacesec- tion for a harmonic wall oscillation with tion for a harmonic wall oscillation with M =20.Iterationsofseveralselected M =40.Iterationsofseveralselected trajectories. trajectories. For the absolute barrier, one finds the simple approximate expression [6] u 2.8√M, (7.17) b ≈ which is supported by the double logarithmic plot in Fig. 7.13 (taken from Ref. [5] ). These data agree, of course, with the values ub =12.9 and 27.5 obtained numerically in the above experiments. Equation (7.17), however, describes only the overall features of the growth of the stochastic sea with increasing M and does not account for the finer details. When we compare the Poincar´ephase space sections for M =10 (Fig. 7.6) and M = 100 (Fig. 7.5), we observe the highest period-one fixed point at u = M/m,withm =1inthefirstcaseandm =4inthesecond. Therefore, the three fixed points with m =1,2,and3disappearfromthe chaotic sea when M is increased from 10 to 100. Here, we study this mechanism in some more detail. Figures 7.14 and 7.15 show Poincar´esectionsforM =20and40.Locating the fixed points in the centers of the large stability islands embedded in the chaotic sea at u =10or13.3, respectively, we readily identify them as m =2 or m = 3 fixed points. The maximum values of the velocity accessible for acceleration from small velocities is given by ub =12.5 and 15.8, respectively. The fixed points with lower m are located above these ub values and a KAM curve separates these higher fixed points from the lower ones. It should also be noted that the value of ub =12.5 for M =20issmaller than that that of ub =12.9 for M = 10, despite the overall increase predicted in (7.17) and shown in Fig. 7.13. This justifies a more detailed numerical study Entropy (log of number if occupied states) Exact Time Evolution, NS=16 T0=T1=0.6 1.2 T0=T1=1.0 T0=T1=1.2 T0=T1=1.4 T0=T1=1.6 1 0.8 0.6 0.4 Diagonal Entropy divided Smax=Log(N) 0.2 0 0 100 200 300 400 500 600 700 800 900 1000 # number of cycles Poten al implica ons for driven dissipa ve systems Applica on to non-equilibrium heat engines. (with P. Mehta, 2013) Standard heat engine: two reservoirs hot an cold More common heat engines: only one reservoir like atmosphere. 4 A B Energy Isothermal Thermalization QH Equilibrate with bath Adiabatic Adiabatic Tq Perform Work Ergodic and non-ergodic single reservoir enginesIsothermal 4 QL A B C D 0.8 Energy Energy Isothermal 0.7 η 0.6 c Q Thermalization Thermalization η H 0.5 ηmt η3 Equilibrate Equilibrate 0.4 with bath with bath η5/3 Adiabatic Adiabatic 0.3 Tq Tq 0.2 Efficiency Perform Perform 0.1 0 Work Work 0 0.5 1 1.5 2 2.5 3 Isothermal τ QL C D FIG. 2: Comparison of Carnot engines and single-heat bath engines (A) Carnot engines function by using two heat reservoirs, Ergodic engine a hot reservoir thatNon-ergodic serves as a source engine of energy and a cold reservoir that serves as an entropy sink. (B) In the ergodic regime, energy is injected into the engine. The gas within the engine quickly equilibrates with itself. The gas then performs 0.8 mechanical work and then relaxes to back to its initial state. (C). In the non-ergodic regime, the system thermalizes on time Energy 0.7 scales much slower than time scales on which work is performed. (D).(blue)Maximumefficiency as a function of excess energy η 0.6 c (ratio of injected energy to initial energy), τ, for Carnot engine, ηc, (red) true thermodynamic bound, ηmt,(magenta)actual Thermalization η η efficiency of a non-ergodic engine which acts as an effective one-dimensional gas, η (see the text), and (green) actual efficiency What0.5 does secondmt law tell us about maximum efficiency? 3 η3 of three-dimensional ideal gas Lenoir engine, η . Equilibrate 0.4 5/3 with bath η5/3 Tq 0.3 0.2 external parameter λ from λ to λ . In this case, Efficiency 1 2 0.1 Perform (2) (1) (2) Work 0 T0 Sr(q p ) Sr(p p ) 0 0.5 1 1.5 2 2.5 3 ηmne = || − || , (8) τ ∆Q (1) (2) where p and p stand for equilibrium Gibbs distributions corresponding to the couplings λ1 and λ2 at the beginning FIG. 2: Comparison of Carnot engines and single-heat bath engines (A) Carnot engines function by using twoand heat the reservoirs, end of the process I respectively. Since the second term is negative, changing the external parameter during a hot reservoir that serves as a source of energy and a cold reservoir that serves as an entropy sink. (B)theIn first the stage ergodic can only reduce the engine efficiency, though this may be desirable for other practical reasons unrelated regime, energy is injected into the engine. The gas within the engine quickly equilibrates with itself. The gasto thermodynamics.then performs mechanical work and then relaxes to back to its initial state. (C). In the non-ergodic regime, the system thermalizes on time scales much slower than time scales on which work is performed. (D).(blue)Maximumefficiency as a function of excess energy A. Efficiency of Ergodic Engines (ratio of injected energy to initial energy), τ, for Carnot engine, ηc, (red) true thermodynamic bound, ηmt,(magenta)actual efficiency of a non-ergodic engine which acts as an effective one-dimensional gas, η3 (see the text), and (green) actual efficiency of three-dimensional ideal gas Lenoir engine, η5/3. An important special case of our bound is the limit where the the relaxation of particles within the engine is fast compared to the time scale on which the engine preforms work (see Figure 2). This is the normal situation in mechanical engines based on compressing gases and liquids. In this case, after the injection of energy the particles 1 external parameter λ from λ1 to λ2. In this case, in the engine quickly thermalize and can be described by a gas at an effective temperature T (E) (dS/dE)− that depends on the energy of the gas. It is shown in Sec. V, that in this case, (7) reduces to ≡ (2) (1) (2) T0 Sr(q p ) Sr(p p ) ηmne = || − || , (8) I E+∆Q ∆Q T0∆S 1 T0 ηmt =1 = dE 1 . (9) − ∆Q ∆Q − T (E) (1) (2) E where p and p stand for equilibrium Gibbs distributions corresponding to the couplings λ1 and λ2 at the beginning and the end of the process I respectively. Since the second term is negative, changing the external parameter during the first stage can only reduce the engine efficiency, though this may be desirable for other practical reasons unrelated to thermodynamics. A. Efficiency of Ergodic Engines An important special case of our bound is the limit where the the relaxation of particles within the engine is fast compared to the time scale on which the engine preforms work (see Figure 2). This is the normal situation in mechanical engines based on compressing gases and liquids. In this case, after the injection of energy the particles 1 in the engine quickly thermalize and can be described by a gas at an effective temperature T (E) (dS/dE)− that depends on the energy of the gas. It is shown in Sec. V, that in this case, (7) reduces to ≡ I E+∆Q T0∆S 1 T0 η =1 = dE 1 . (9) mt − ∆Q ∆Q − T (E ) E Application to heat engines (with P. Mehta). Consider cyclic process. Small perturbation: 3 Large perturbations. Can write inequalities Equilibrium (1) Nonequilibrium Equilibrium (2) (1) (2) (2) λ I λ II λ Energy Relaxation injection p(1)ε(1) q ε(2) p(2)ε(2) n , n T n, n TTn , n FIG. 1: Generalized nonequilibrium quenches. A system parameterized by λ is coupled to an external bath at temperature T . (1) (1) Initially, λ = λ and the system is in equilibrium and is described by the Boltzmann distribution pn . I, energy is suddenly injectedThe into second the system while inequality changing λ from impliesλ(1) to λ(2) .that The system the is free now described energy by possibly can nonthermal only distribution, qn. During stage II of the process, the system relaxes and equilibrates with the external bath after which it is described by a (2) (2) Boltzmanndecrease distribution duringpn with λ the= λ .relaxation; the first inequality is less known. The importance of relative entropy for describing relaxation of nonequilibrium distributions has been discussed in previous for different setups both in quantum and classical systems10,12–15. Taken together, (4), (5), and (6) constitute the nonequilibrium identities that will be exploited next to calculate bounds for the efficiency of engines that operate with a single heat bath. II. MAXIMUM EFFICIENCY OF ENGINES Figure 2 summarizes the single-reservoir engines analyzed in this work and compares them with Carnot engines (1). The engine is initially in equilibrium with the environment (bath) at a temperature T0 and the system is described by the equilibrium probability distribution peq. In the first stage, excess energy, ∆Q, is suddenly deposited into the system. This can be a pulse electromagnetic wave, burst of gasoline, current discharge etc. In second stage, the engine converts the excess energy into work and reaches mechanical equilibrium with the bath . Finally, the system relaxes back to the initial equilibrium state. Of course splitting the cycle into three stages is rather schematic but it is convenient for the analysis of the work of the engine. Such an engine will only work if the relaxation time of the system and environment is slow compared to the time required to perform the work. Otherwise the energy will be simply dissipated to the environment and no work will be done (see discussion in Ref. [16]). I The initial injection of energy, ∆Q results in the corresponding entropy increase ∆S = S(q) S(peq) of the system, where S is the diagonal entropy and q describes the system immediately after the addition− of energy. Because by assumption the environment is not affected during this initial stage, the total entropy change of the system and environment is also just ∆SI . By the end of the cycle, the entropy of the system returns to its initial value. Thus, from the second law of thermodynamics, the increase in entropy of the environment must be greater than equal to ∆SI . This implies that the minimal amount of heat that must be dissipated into the environment during the cycle is I I T0∆S . An engine will work optimally if no extra entropy beyond ∆S is produced during the system-bath relaxation since then all of the remaining energy injected into the system is converted to work. Thus, the maximal work that can be performed by the engine during a cycle is W = ∆Q T ∆SI . For a cyclic process such as the one considered m − 0 here, substituting (5) into the expression for Wm implies that the maximum efficiency of a nonequilibrium engine, ηnme, is given by W T ∆SI T S (q p) η = m =1 O = 0 r || . (7) mne ∆Q − ∆Q ∆Q Equation (7) is the main result of this paper. It relates the maximum efficiency of an engine to the relative entropy of the intermediate nonequilibrium distribution and the equilibrium distribution. We next consider various limits and applications of this result. We point that Eq. (5) also allows us to extend the maximum efficiency bound to a more general class of engines, like Otto engines, where during the first stage of the cycle one simultaneously changes the Extension of the fundamental relation to arbitrary processes.3 Relative entropy (Kullback-Leibler Divergence). Equilibrium (1) Nonequilibrium Equilibrium (2) (1) (2) (2) λ I λ II λ Energy Relaxation injection p(1)ε(1) q ε(2) p(2)ε(2) n , n T n, n TTn , n FIG. 1:Define Generalized nonequilibriummicroscopic quenches. Aheat system parameterized by λ is coupled to an external bath at temperature T . (1) (1) Initially,andλ = λ adiabaticand the system is work: in equilibrium and is described by the Boltzmann distribution pn . I, energy is suddenly injected into the system while changing λ from λ(1) to λ(2). The system is now described by possibly nonthermal distribution, qn. During stage II of the process, the system relaxes and equilibrates with the external bath after which it is described by a (2) (2) BoltzmannMain distributionresultsp n(alsowith λDeffner,= λ . Lutz, 2010). Proofs are straightforward calculus. The importance of relative entropy for describing relaxation of nonequilibrium distributions has been discussed in previous for different setups both in quantum and classical systems10,12–15. Taken together, (4), (5), and (6) constitute the nonequilibrium identities that will be exploited next to calculate bounds for the efficiency of engines that operate with a single heat bath. II. MAXIMUM EFFICIENCY OF ENGINES All inequalities follow from non-negativity of the relative entropy. Figure 2 summarizes the single-reservoir engines analyzed in this work and compares them with Carnot engines (1). The engine is initially in equilibrium with the environment (bath) at a temperature T0 and the system is described by the equilibrium probability distribution peq. In the first stage, excess energy, ∆Q, is suddenly deposited into the system. This can be a pulse electromagnetic wave, burst of gasoline, current discharge etc. In second stage, the engine converts the excess energy into work and reaches mechanical equilibrium with the bath . Finally, the system relaxes back to the initial equilibrium state. Of course splitting the cycle into three stages is rather schematic but it is convenient for the analysis of the work of the engine. Such an engine will only work if the relaxation time of the system and environment is slow compared to the time required to perform the work. Otherwise the energy will be simply dissipated to the environment and no work will be done (see discussion in Ref. [16]). I The initial injection of energy, ∆Q results in the corresponding entropy increase ∆S = S(q) S(peq) of the system, where S is the diagonal entropy and q describes the system immediately after the addition− of energy. Because by assumption the environment is not affected during this initial stage, the total entropy change of the system and environment is also just ∆SI . By the end of the cycle, the entropy of the system returns to its initial value. Thus, from the second law of thermodynamics, the increase in entropy of the environment must be greater than equal to ∆SI . This implies that the minimal amount of heat that must be dissipated into the environment during the cycle is I I T0∆S . An engine will work optimally if no extra entropy beyond ∆S is produced during the system-bath relaxation since then all of the remaining energy injected into the system is converted to work. Thus, the maximal work that can be performed by the engine during a cycle is W = ∆Q T ∆SI . For a cyclic process such as the one considered m − 0 here, substituting (5) into the expression for Wm implies that the maximum efficiency of a nonequilibrium engine, ηnme, is given by W T ∆SI T S (q p) η = m =1 O = 0 r || . (7) mne ∆Q − ∆Q ∆Q Equation (7) is the main result of this paper. It relates the maximum efficiency of an engine to the relative entropy of the intermediate nonequilibrium distribution and the equilibrium distribution. We next consider various limits and applications of this result. We point that Eq. (5) also allows us to extend the maximum efficiency bound to a more general class of engines, like Otto engines, where during the first stage of the cycle one simultaneously changes the Ergodic engines. Assume that the energy is first deposited without change of 3 external couplings. Equilibrium (1) Nonequilibrium Equilibrium (2) (1) (2) (2) λ I λ II λ Energy Relaxation injection p(1)ε(1) q ε(2) p(2)ε(2) n , n T n, n TTn , n FIG. 1: Generalized nonequilibrium quenches. A system parameterized by λ is coupled to an external bath at temperature T . (1) (1) Initially, λ = λ and the system is in equilibrium and is described by the Boltzmann distribution pn . I, energy is suddenly injected into the system while changing λ from λ(1) to λ(2). The system is nowMaximum described by possibly efficiency nonthermal distribution,is given qn. During stage II of the process, the system relaxes and equilibrates with the external bath after which it is described by a (2) (2) Boltzmann distribution pn with λ = λ . by the relative entropy The importance of relative entropy for describing relaxation of nonequilibrium distributions has been discussed in previous for different setups both in quantum and classical systems10,12–15. Taken together, (4), (5), and (6) constitute the nonequilibrium identities that will be exploited next to calculate bounds for the efficiency of engines that operate with a single heat bath. II. MAXIMUM EFFICIENCY OF ENGINES Figure 2 summarizes the single-reservoir engines analyzed in this work and compares them with Carnot engines (1). The engine is initially in equilibrium with the environment (bath) at a temperature T0 and the system is described by the equilibrium probability distribution peq. In the first stage, excess energy, ∆Q, is suddenly deposited into the system. This can be a pulse electromagnetic wave, burst of gasoline, current discharge etc. In second stage, the engine converts the excess energy into work and reaches mechanical equilibrium with the bath . Finally, the system relaxes back to the initial equilibrium state. Of course splitting the cycle into three stages is rather schematic but it is convenient for the analysis of the work of the engine. Such an engine will only work if the relaxation time of the system and environment is slow compared to the time required to perform the work. Otherwise the energy will be simply dissipated to the environment and no work will be done (see discussion in Ref. [16]). I The initial injection of energy, ∆Q results in the corresponding entropy increase ∆S = S(q) S(peq) of the system, where S is the diagonal entropy and q describes the system immediately after the addition− of energy. Because by assumption the environment is not affected during this initial stage, the total entropy change of the system and environment is also just ∆SI . By the end of the cycle, the entropy of the system returns to its initial value. Thus, from the second law of thermodynamics, the increase in entropy of the environment must be greater than equal to ∆SI . This implies that the minimal amount of heat that must be dissipated into the environment during the cycle is I I T0∆S . An engine will work optimally if no extra entropy beyond ∆S is produced during the system-bath relaxation since then all of the remaining energy injected into the system is converted to work. Thus, the maximal work that can be performed by the engine during a cycle is W = ∆Q T ∆SI . For a cyclic process such as the one considered m − 0 here, substituting (5) into the expression for Wm implies that the maximum efficiency of a nonequilibrium engine, ηnme, is given by W T ∆SI T S (q p) η = m =1 O = 0 r || . (7) mne ∆Q − ∆Q ∆Q Equation (7) is the main result of this paper. It relates the maximum efficiency of an engine to the relative entropy of the intermediate nonequilibrium distribution and the equilibrium distribution. We next consider various limits and applications of this result. We point that Eq. (5) also allows us to extend the maximum efficiency bound to a more general class of engines, like Otto engines, where during the first stage of the cycle one simultaneously changes the Magnetic gas engine. B I) Flip spins with probability 0 Ergodic engine: allow to thermalize with kinetic degrees of freedom and then push the piston. Non-ergodic engine: for inverted spin population (R>1/2) first perform the work on B by rotating around x-axis and then use the residual energy to push the piston. Ergodic engine: allow to thermalize with B kinetic degrees of freedom and then push the piston. Non-ergodic engine:8 for inverted spin B. Magnetic Gas Engine population (R>1/2) use macroscopic It is possible exceed the thermodynamic efficiency ηmt by considering more complicatedmagnetic engines withenergy an additional to extract work magnetic degree of freedom. Then as we show below one can create a non-ergodic engine with efficiency higher than the thermodynamic bound and which can be arbitrarily close to 100%. Assume that we have a gas composed of N atoms which have an additional magnetic degree of freedom like a spin. For simplicity we assume that the spin is equal to 1/2, i.e. there are two magnetic states per each atom. As will be clear from the discussion, this assumption is not needed for the main conclusion and the calculations can easily be generalized to the case where we consider electric dipole moments or some other discrete or continuous internal degree of freedom instead of the spin. 1.0 ηmne 0.8 ηmt η 0.6 y η5/3 ηmgne Can beat maximum ienc 0.4 equilibrium efficiency ηmge ffic E by using non-ergodic 0.2 setup. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Possible applications in R small systems? FIG. 3: Efficiency of magnetic engine with initial temperature equal to 10% of the Zeeman energy: T0 =0.1hz as a function of the spin flipping rate (see Sec. III). (black) ηmne, maximum non-ergodic efficiency, (red) ηmt, maximum ergodic efficiency, (pink) ηmgne,efficiency of magnetic gas engine in non-ergodic regime, (green) η5/3,efficiency of ergodic ideal gas engine,(blue) ηmge,efficiency of ergodic magnetic gas engine. The Hamiltonian of the system is then mv2 = j h σz, (25) H0 2 − z j j z where σj are the Pauli matrices. To simplify notations we absorbed the Bohr magneton and the g factor into the magnetic field. The first term in the Hamiltonian is just the usual kinetic energy and the second term is due to the interaction of the spin degrees of freedom with an external field in the z-direction. Initially the system is in equilibrium at a temperature T and a fixed magnetic field hz. Now let us assume that via some external pulse we pump energy to the atoms by flipping their spins with some probability. This can be done by a resonant laser pulse or by e.g. a Landau-Zener process where we adiabatically turn on a large magnetic field in x-direction then suddenly switch its sign and slowly decrease it back to zero. Ideally this process creates a perfectly inverse population of atoms (i.e. number of spin up and spin down particles is exchanged) but in practice there will be always some imperfections. In general unitary process the new occupation numbers can be obtained from a single parameter describing the flipping rate: R [0, 1], with ∈ q = p (1 R)+p R, q = p (1 R)+p R (26) ↑ ↑ − ↓ ↓ ↓ − ↑