What is thermal equilibrium and how do we get there?

arxiv:1507.06479 and more What is thermal equilibrium and how do we get there? Hal Tasaki QMath 13, Oct. 9, 2016, Atlanta 40 C 20 C 30 C 30 C

about the talk Foundation of equilibrium statistical mechanics based on pure quantum mechanical states in macroscopic isolated quantum systems von Neumann 1929 Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zanghi 2010 Macroscopic view point Tasaki 1998 Reimann 2008 Linden, Popescu, Short, Winter 2009 Equilibration from large effective dimension large-deviation theory (Hopefully) the simplest picture for thermalization (approach to thermal equilibrium) we do not use ETH (energy eigenstate thermalization hypothesis)

Why isolated quantum systems? Fashionable answer We can realize isolated quantum systems in ultra cold atoms 7 7 clean system of 10 atoms at 10 K Old-fashioned answer This is still a very fundamental study We wish to learn what isolated systems can do (e.g., whether they can thermalize) After that, we may study the effect played by the environment 2

Settings and main assumptions

The system Isolated quantum system in a large volume V Particle system with constant ρ = N/V Quantum spin systems Hilbert space H tot Hamiltonian Ĥ Energy eigenvalue and the normalized energy eigenstate Ĥ ψ j = E j ψ j ψ j ψ j =1 Suppose that one is interested in a single extensive ˆM [ ˆM,Ĥ] = 0 quantity with in general extension to n quantities ˆM 1,..., ˆM n is easy

Examples Two identical bodies in thermal contact Ĥ = Ĥ1 + Ĥ2 + Ĥint the energy difference ˆM = Ĥ1 Ĥ2 acts on the boundary Ĥ 1 Ĥ 2 General quantum spin chain translationally invariant short-range Hamiltonian and an observable Ĥ = L ĥ i ˆM = L ˆm i i=1 i=1

Microcanonical energy shell u Fix arbitrary and small u, and consider the energy eigeneigenvalues such that u u E j /V u + u relabel j so that this corresponds to j =1,...,D D e σ 0V uv microcanonical average of an observable Ô D Ôu mc := 1 D ψ j Ô ψ j j=1 microcanonical energy shell H sh the space spanned by ψ j with j =1,...,D

A pure state which represents thermal equilibrium extensive quantity of interest ˆM equilibrium value m := lim V ˆM/V u mc projection onto nonequilibrium part (in H tot ) ˆP neq := ˆP ˆM/V m δ fixed const. (precision) DEFINITION: A normalized pure state ϕ H sh (for some V>0 ) represents thermal equilibrium if ϕ ˆP neq ϕ e α V fixed const. ˆM/V ϕ (measurement result) m δ with probability 1 e αv if one measures in such, then From ϕ we get information about thermal equilibrium

Basic assumption which guarantees that the system is healthy ˆP neq projection onto nonequilibrium part THERMODYNAMIC BOUND (TDB): statement in statistical mechanics There is a constant γ > 0, and one has, for any V ˆPneq u mc e γ V simply says large fluctuation is exponentially rare in the MC ensemble (large deviation upper bound) expected to be valid in ANY uniform thermodynamic phase, and can be proven in many cases including the two examples

Two identical bodies in thermal contact Ĥ = Ĥ1 + Ĥ2 + Ĥint the energy difference ˆM = Ĥ1 Ĥ2 Examples ˆP ( Ĥ 1 Ĥ2)/V δ u acts on the boundary General quantum spin chain mc e γ V Ĥ 1 Ĥ 2 thermodynamic bound can be proved easily (except at the triple point) using classical techniques translationally invariant short-range Hamiltonian and an observable ˆP ( ˆM/V ) m δ u mc e γ V Ĥ = L i=1 ĥ i ˆM = L i=1 ˆm i thermodynamic bound follows as a corollary of the general theory of Ogata 2010 7

Typicality of pure states which represent thermal equilibrium

Typicality of thermal equilibrium overwhelming majority of states in the energy shell H sh represent thermal equilibrium (in a certain sense) von Neumann 1929 Llyoid 1988 Sugita 2006 Popescu, Short, Winter 2006 Goldstein, Lebowitz, Tunulkam Zanghi 2006 Reimann 2007 we shall formulate our version (proof is standard and trivial)

Haar measure on H sh a state H sh ϕ = D j=1 α j ψ j with can be regarded as a point on the unit sphere of D j=1 α j 2 =1 C D a natural (basis independent) measure on the uniform measure on the unit sphere H sh is corresponding average ( ):= dα1 dα D δ 1 D j=1 α j 2 ( ) dα1 dα D δ 1 D j=1 α j 2 dα := d(reα) d(imα) From the symmetry α j α k = 1 D δ j,k

Average over H sh and mc-average operator Ô normalized state ϕ = D j=1 α j ψ j quantum mechanical expectation value D ϕ Ô ϕ = average over H sh j,k=1 α j α k ψ j Ô ψ k α j α k = 1 D δ j,k ϕ Ô ϕ = j,k α j α k ψ j Ô ψ k D = 1 D ψ j Ô ψ j = Ôu mc j=1

Average over H sh and mc-average operator Ô normalized state ϕ = D j=1 α j ψ j quantum mechanical expectation value D ϕ Ô ϕ = α j α k ψ j Ô ψ k average over H sh j,k=1 α j α k = 1 D δ j,k α j α k ψ j Ô ψ k ϕ Ô ϕ = j,k average over infinitely many states in the shell = 1 D average over D energy eigenstates D ψ j Ô ψ j = Ôu mc j=1 Another way of looking at the microcanonical average

Typicality of thermal equilibrium Assume Thermodynamic bound (TDB) ϕ ˆP neq ϕ = ˆPneq u mc e γ V provable for many models Markov inequality THEOREM: Choose a normalized ϕ H sh randomly according to the uniform measure on the unit sphere. one has Then with probability 1 e (γ α)v ϕ ˆP neq ϕ e α V the pure state ϕ represents thermal equilibrium Almost all pure states thermal equilibrium!! ϕ H sh represent

12 What is thermal equilibrium? Thermal equilibrium is a typical property shared by the majority of (pure) states in the energy shell Almost all pure states (with respect to the Haar measure) ϕ H sh represent thermal equilibrium the microcanonical energy shellh sh nonequilibrium nonequilibrium nonequilibrium thermal equilibrium

What is thermal equilibrium? Thermal equilibrium is a typical property shared by the majority of (pure) states in the energy shell Almost all pure states (with respect to the Haar measure) ϕ H sh represent thermal equilibrium the microcanonical energy shellh sh nonequilibrium nonequilibrium nonequilibrium thermal equilibrium How do we get there? 12

Thermalization or the approach to thermal equilibrium

Question initial state ϕ(0) H sh unitary time-evolution ϕ(t) = e iĥt ϕ(0) Does ϕ(t) approach thermal equilibrium? numerical mathematical Jensen, Shanker 1985 Satio, Takesue, Miyashita 1996 Rigol, Dunjko, Olshanni 2008 many many recent works von Neumann 1929 Tasaki 1998 Reimann 2008 Linden, Popescu, Short, Winter 2009 Goldstein, Lebowitz, Mastrodonato, Tumulka, Zanghi 2010 many recent works We shall formulate possibly the simplest version which is directly related to macroscopic physics

Derivation (easy) initial state H sh ϕ(0) = D j=1 c j ψ j time-evolution ϕ(t) = e iĥt ϕ(0) = D j=1 c j e ie jt ψ j expectation value of the projection on nonequilibrium c j c k e i(e j E k )t ψ j ˆP neq ψ k ϕ(t) ˆP neq ϕ(t) = j,k long-time average lim τ 1 τ τ 0 dtϕ(t) ˆP neq ϕ(t) = j oscillates (assume no degeneracy) diagonal ensemble c j 2 ψ j ˆP neq ψ j c j 4 j ψ j ˆP neq ψ j 2 c j 4 Tr Hsh [ ˆP neq ] j j

lim τ 1 τ τ 0 dtϕ(t) ˆP neq ϕ(t) j c j 4 Tr Hsh [ ˆP neq ] effective dimension of ϕ(0) with respect to Ĥ D eff := D c j 4 1 Reimann 2008, Linden, Popescu, Short, Winter 2009 the effective number of energy eigenstates contributing to the expansion thermodynamic bound lim τ 1 τ τ 0 ϕ(0) = D j=1 c j ψ j 1 dtϕ(t) ˆP neq ϕ(t) D j=1 1 D eff D D Tr H sh [ ˆP neq ]= ˆP neq u mc e γv D eff 1/2 e γv/2 exponentially small if D eff is large enough

no degeneracy Thermalization j = j E j = E j provable for many models thermodynamic bound (TDB) effective dimension is large enough D D eff := j=1 c j 4 1 e ηv D with small η > 0 coefficients in the expansion are mildly distributed ˆPneq u mc e γ V ϕ(0) = D j=1 c j ψ j essential assumption! THEOREM: For any initial state ϕ(0) satisfying the above condition, ϕ(t) represents thermal equilibrium t for most in the long run

for most t in the long run THEOREM: For any initial state ϕ(0) satisfying the above condition, ϕ(t) represents thermal equilibrium t for most in the long run there exist a (large) constant τ and a subset B [0, τ] with B /τ e νv such that for any t [0, τ]\b ϕ(t) ˆP neq ϕ(t) e α V B t 0 τ

How do we get there? A pure state evolving under the unitary time evolution thermalizes, i.e., represents thermal equilibrium for most t in the long run The main assumption is that the effective dimension of the initial state is large enough (exactly the same conclusion for mixed initial states) the microcanonical energy shellh sh nonequilibrium nonequilibrium nonequilibrium 18 thermal equilibrium

On the effective dimension of easily preparable initial states

Conjecture CONJECTURE: For most realistic Hamiltonian of a macroscopic system, most easily preparable initial state (with energy density u ) has an effective dimension D eff not much smaller than the dimension thermalizes. D of the energy shell, and hence we still don t know how to characterize easily preparable states but product states, Gibbs states of a different Hamiltonian, FCS=MPS,... are easily preparable

A theoretical support for large D eff Let the initial state ˆρ(0) be a sufficiently disordered state, e.g., a product state or a Gibbs state (of a different Hamiltonian) at high enough temperatures Quantum Central Limit Theorem : In such a state, the probability distribution of the energy (eigenvalues ˆρ(0) of Ĥ ) converges to the Gaussian distribution as p(e) V there are almost D energy eigenstates! E uv We expectd eff D D eff := D j=1 ψ j ˆρ(0) ψ j 2 1 O( V )

Integrable vs non-integrable Hamiltonians If the Hamiltonian Ĥ is integrable, one has D eff D for most sufficiently disordered initial states (there is equilibration, but not to thermal equilibrium) conserved extensive quantities Â 1, Â2,...,Ân Rigol 2016 Â i /V ā i in the initial state (QCT or Q Large-Deviation) then D eff exp[v σ(ā 1,...,ā n )] exp[v max σ(a 1,...,a n )] D a 1,...,a n the energy distribution converges to Gaussian, but is extremely sparse similar situation is expected for many-body localization If Ĥ is non-integrable, the distribution is dense, and one has D eff D (at least numerically, for some models)

Conjecture (refined) CONJECTURE: For a non-random non-integrable Hamiltonian of a macroscopic system, most easily preparable initial state (with energy density u ) has an effective dimension D eff not much smaller than the dimension hence thermalizes. D of the energy shell, and D eff := D ψ j ˆρ(0) ψ j 2 1 j=1 D eff := D j=1 c j 4 1 Ĥ ψ j = E j ψ j ϕ(0) = D j=1 c j ψ j Is this true?? (Numerical works are not yet conclusive) It must be extremely difficult to justify this rigorously (we only have very artificial and simple examples)

Toy model for two identical bodies in contact Ĥ 1 Ĥ 2 Ĥ = Ĥ1 + Ĥ2 + Ĥint Example Ĥ 1 ξ k = k ξ k Ĥ 2 η = η Ĥ int ξ k η = ε τ,τ =±1 ξ k+τ η +τ ψ j = k, c j;k, ξ k η where c j;k, are dominant and comparable for k, with E j ( k + ) ε We can show D eff D for ˆρ(0) = ˆρ Gibbs β 1 ˆρ Gibbs β 2 an easily preparable initial state We can shown that the system thermalizes! β 1 β 2 β eq β eq

What is thermal equilibrium? Typical property of states in the energy shell Most pure states represent thermal equilibrium How do we get there? Initial states with large effective dimension thermalizes only by unitary time evolution Easily preparable initial states are conjectured to have large effective dimension D eff := D ψ j ˆρ(0) ψ j 2 1 Remaining issues Verify (partially) the conjecture about the effective dimension of easily preparable states Time scale of thermalization j=1

related issues: 1 Energy Eigenstate Thermalization Hypothesis (ETH) ψ j ˆP neq ψ j e κv for any j =1,...,D each energy eigenstate ψ j equilibrium represents thermal von Neumann 1929, Deutsch 1991, Srednicki 1994, and many more ϕ(0) H sh THEOREM: For ANY initial state, ϕ(t) represents thermal equilibrium for most t in the long run The result is strong but probably it is not necessary to cover ANY initial states It is extremely difficult to verify the assumption in nontrivial quantum many body systems

related issues: 2 Time scale of thermalization: first step von Neumann s random Hamiltonian Ĥ = ÛĤ0Û Ĥ 0 fixed Hamiltonian Û random unitary on H sh THEOREM: With probability close to 1, for ANY initial ϕ(0) H sh τ τ 1 τ 0 dtϕ(t) ˆP neq ϕ(t) β/τ state and any, we have { } Goldstein, Hara, Tasaki 2015 Out-of-Time-Ordered (OTO) correlator in the MC ensemble W (t)vw(t)v = φ(t) 4 WVWV +2 { φ(2t){φ( t)} { { 2 + φ( 2t){φ(t)} 2 2 φ(t) 4} WV 2 φ(t) =D 1 D φ(t) 2 = {1+(t/β) 2 } j=1 eie jt 1 The only time scale is the Boltzmann time h/(k B T ) Only limited aspect of time-dependence in many-body quantum systems is captured in this approach Reimann 2016