Universal dynamics in thermally isolated systems.
AnatoliAnatoli PolkovnikovPolkovnikov,, BostonBoston UniversityUniversity GuyGuy BuninBunin TechnionTechnion LucaLuca DD’’AlessioAlessio BUBU ChristianChristian GogolinGogolin BristolBristol ClaudiaClaudia DeDe GrandiGrandi BUBU VladimirVladimir GritsevGritsev FribourgFribourg
YarivYariv KafriKafri TechnionTechnion LudwigLudwig MatheyMathey NISTNIST
International symposium on Quantum Thermodynamics, Stuttgart, Germany, 09/17/2010 Non-equilibrium. Set of questions to answer.
1. Is there a notion of universality in quantum dynamics? 2. If yes, can we classify perturbations according to their relevance. Is it different from equilibrium? 3. How is complexity of dynamics related to universality? 4. How can we describe relaxation in integrable, non- integrable and weakly non-integrable systems? 5. Can we think about relaxation (dynamics) as an RG process? If yes, what is it? 6. How can we reconcile quantum and classical complexity? 7. … Universality of dynamics
1.Nearly adiabatic dynamics near quantum critical points. Universal scalings from adiabatic fidelity.
2.Diagonal entropy and laws of thermodynamics.
3.Relaxational dynamics as an RG process. Universal dynamics near non-equilibrium KT phase transition.
4.Universal energy distribution in externally driven systems. Quantum phase transitions in a nutshell.
susceptibility
Equilibrium properties of the system are universal near critical points (both classical and quantum).
What can we say about non-equilibrium dynamics?
Sudden quench
Density of single-particle excitations Excess energy =heat
The system is not adiabatic near QCP. Use Landau-Zener criterion
A.P. 2003, Zurek, Dorner, Zoller 2005
analogue of the latent heat Classical Kibble-Zurek mechanism Arbitrary power law quench.
Use the same argument The scalings suggest usability of the adiabatic perturbations theory: small parameter - proximity to the instantaneous ground state.
Non-analytic scalings are usually related to singularities in some susceptibilities. Need to identify the relevant adiabatic susceptibility. Kibble-Zurek scaling through adiabatic pertubation theory.
Expand wavefunction in the instantaneous energy basis
Adiabatic limit: no transitions between levels. Expand in transition amplitudes to instantaneous excited states.
Reduces to the symmetrized version of the ordinary perturbation theory. Perturbative regime
Define adiabatic fidelity: Probability of exciting the system
Where
Two (qualitative) contributions to susceptibility:
UV (high energy) – analytic, cutoff-dependent, IR (low energy) - universal, non-analytic, cutoff-independent.
Smoother protocols suppress high-energy contributions. “Fluctuation-dissipation” relation for the fidelity
Relevant (marginal) operators:
If the scaling dimension of χ is negative then universal contribution to the scaling is the leading one, General scaling results
Main conclusion: quantum Kibble-Zurek scaling and its generalizations are associated with singularities in the adiabatic susceptibilities describing adiabatic fidelity. Non-critical gapless phases (send ν to ∞)
(A.P., V. Gritsev (2008))
(Martin Eckstein, Marcus Kollar, 2008) ThermalizationThermalization inin quantumquantum systemssystems ((EoMEoM areare linearlinear inin timetime –– nono chaoschaos??))
ConsiderConsider timetime averageaverageρ ofof aa certaincertain observableobservable AA inin anan isolatedisolated systemsystem afterafter aa quenchquench.. − ()= = ρ i(Em En )t A t ∑ n,m (t)Am,n ∑ n,m (0)e Am,n n,m n,m
= 1 T = ρ A A(t) dt n,n An,n ∫0 ∑ T n,m
EignestateEignestate thermalizationthermalization hypothesishypothesis ((Srednicki 1994; MM.. RigolRigol,, VV..
DunjkoDunjko && MM.. OlshaniiOlshanii,, NatureNature 452452,, 854854 ,, 20082008.):.): AAn,n~~ constconst ((nn)) soso ρρ therethere isis nono dependencedependence onon nn..
InformationInformation aboutabout steadysteady statestate isis fullyfully containedcontained inin thethe diagonaldiagonal elementselements ofof thethe densitydensity matrixmatrix.. TrueTrue forfor allall thermodynamicthermodynamic observablesobservables:: energyenergy,, pressurepressure,, magnetizationmagnetization,, …….. ((pickpick youryour favoritefavorite).). TheyThey allall areare linearlinear inin ρρ.. NotNot truetrue aboutabout vonvon NeumannNeumann entropyentropy!! = − ρ ρ Sn Tr( ln ) OffOff--diagonaldiagonal elementselements dodo notnot averageaverage toto zerozero..
Von Neumann’s entropy: conserved in time (in thermally isolated systems). Thermodynamics: entropy is conserved only for adiabatic (slow, reversible) processes. Otherwise it increases.
Quantum mechanics: for adiabatic processes there are no transitions ρ = between energy levels: nn (t) const (t) IfIf thesethese twotwo adiabaticadiabatic theoremstheorems areare relatedrelated thenthen thethe entropyentropy should only depend on ρρ . Simple resolution: should only dependρ on nn. Simple resolution: = − ρ TheThe sumsum isis takentaken overover thethe Sd ∑ nn ln nn n instantaneousinstantaneous energyenergy statesstates.. Initial stationary density matrix, closed systems ε S (t) ≥ S (tρ) = S (0) = S (0), d n εn d + ≥ ρ + Boltzmann’s h-theorem Sd1(t) Sd 2 (t) Sd1(0) Sd 2 (0) (thanks to C. Gogolin) ρ Δ ≅ Δ 0 + 0Δ E ∑ n n nρ nn n ε Δ ≅ − Δ 0 + = 1 0Δρ Sd ∑ nn (ln n 1) ∑ n n n T n ⎛ ∂E ⎞ ΔE = λ Δλ + TΔS ⎜ ∂ ⎟ d ⎝ ⎠Sd IfIf λλ standsstands forfor thethe ΔE = −PΔV + TΔS volumevolume thethe wewe findfind d Diagonal entropy and the energy distribution
Entropy is a unique function of energy (in the thermodynamic limit) if the Hamiltonian is local and density matrix is not (exponentially) sparse Sparseness of the density matrix(after quench) is related to inetgarbility
C. Neuenhahn, F. Marquardt , arXiv:1007.5306; G. Biroli, C. Kollath, A. Laeuchli, arXiv:0907.3731
nearly integrable noninterable
Thermalization <-> delocalization of the initial state in the basis of the egienstates (cf. many-body Anderson localization transition, Basko, Aleiner, Altshuller 06) Examples for diagonal entropy.
Diagonal entropy changes suddenly after the quench Hard core bosons in two dimensions Local diagonal entropies after the quench Initial non-equal populations Diagonal entropy and reversibility (periodic modulation of the weak link) Uniqueness of S(E) for different periods of modulation Entropy as a measure of non-adiabaticity
Universal. Not dependent on the choice of observable. Not intensive. Unity unless rate vanishes in the thermodynamic limit.
Universal. Intensive: well defined in thermodynamic limit. Depends on knowing structure of excitations. Ill defined in non-integrable systems.
Universal. Intensive. Measurable. Not easy to separate from adiabatic energy. Not-universal unless end exactly at the critical point.
Universal. Intensive. Sensitive only to crossing QCP (not where we start (end) the quench. (not) Measurable. Time evolution as an RG process. Universal dynamics near non-equilibriumρ KT phase transition. (with L. Mathey, PRA 2010). − ()= = ρ i(Em En )t A t ∑ n,m (t)Am,n ∑ n,m (0)e Am,n n,m n,m
= 1 T ≅ ρ A A(t) dt n,m Am,n ∫0 ∑ T n,m − <π |En Em |t
In quantum mechanics relaxation is like an RG process where high frequency modes are gradually eliminated.
Idea: gradually average over high momentum non-interacting modes and follow time evolution of the remaining low energy modes. Decoupling of two 2D superfluids (with L. Mathey, PRA 2010)
Short times: system relaxes to a steady state with algebraic order.
Long times – vortex anti-vortex pairs start to emerge and can unbind. Looks like KT transition in real time. How do we describe it analytically? Need to solve nonlinear Hamiltonian equations of motion:
Idea:θ split θ into low-momentum and high momentum sectors θ = > +θ <
Use perturbation theory in g to treat θ > Time average equation for θ < over fast oscillating modes Rescale coordinates and time and follow the evolution of couplings and energy. Overall formalism is similar to the adiabatic perturbation theory. Result: flow equations for couplings, very similar to usual KT form
Flow parameter l is the real time!
Recover for this problem two scenarios of relevant (normal) and irrelevant (superfluid) vortices with exponentially divergent time scale.
For this problem equilibrium = thermodynamics emerges as a result of the renormalization group process.
RG is a semigroup transformation (no inverse). Lost information in the time averaging of fast modes. Universal energy distribution in thermally isolated driven systems
(with L. D’Alessio, G. Bunin, Y. Kafri)
Conventional heating Nonadiabatic heating Approach:
Fokker-Planck equation (quasi-static process)
Unitarity of the evolution (fluctuation theorems)
Dynamics is highly constrained by the unitarity Results
Two universal regimes
Transition:
If the entropy grows slower than square of time then expect canonical-like distribution. Summary and outlook. • Universal non-adiabatic response of various quantities near quantum critical point. Can be understood through IR divergencies of fidelity susceptibility and its generalizations. • Dependence of dissipation on quench time and shape? • Role of non-integrability (relaxation)? • UV singularities. • Diagonal entropy and heat can be defined microscopically. Laws of thermodynamics follow from many-body delocalization between eigenstates. • Thermalization (time evolution) as an RG process in real time. • Generality of this statement. Role of integrability. • Universal energy fluctuations in thermally isolated driven systems. • Experiments. ClassicalClassical systemssystems..
InsteadInstead ofof energyenergy levelslevels wewe
havehave orbitsorbits.. ρ ↔ nn probabilityprobability toto occupyoccupy anan orbitorbit withwith energyenergy EE..
ρ ∝ exp[i(ε −ε )t] ↔ describesdescribes thethe motionmotion onon nm n m thisthis orbitsorbits.. ε ρ ε ρ ε ClassicalClassical dd--entropyentropy ρ ρ ε δ ε S = − N( ) ( )ln ( )dε ( ) = dΓ ( p,q) ( −ε ( p,q)) d ∫ ∫
TheThe entropyentropy ““knowsknows”” onlyonly aboutabout conservedconserved quantitiesquantities,,
everythingeverything elseelse isis irrelevantirrelevant forfor thermodynamicsthermodynamics!! SSd satisfiessatisfies lawslaws ofof thermodynamicsthermodynamics,, unlikeunlike thethe usuallyusually defineddefined S = ln Γ. PropertiesProperties ofof dd--entropyentropy ((A. Polkovnikov, arXiv:0806.2862. )..
JensenJensen’’ss inequalityinequality:: −ρ ρ + ρ ρ ≤ ρ − ρ ρ = Tr( ln d ln d ) Tr[( d )ln d ] 0 ThereforeTherefore ifif thethe initialinitial densitydensity matrixmatrix isis stationarystationary ((diagonaldiagonal)) thenthen ≥ = = Sd (t) Sn (t) Sn (0) Sd (0)
NowNowρ assumeassume thatthat thethe initialinitial statestate isis thermalthermal equilibriumequilibrium 1 0 = exp[−βε ] n Z n LetLet usus considerconsider anan infinitesimalinfinitesimal changechange ofof thethe systemsystem andand computecompute energyenergy andand entropyentropy changechange.. ThermodynamicThermodynamic adiabaticadiabatic theoremtheorem..
In a cyclic adiabatic process the energy of the system does not change: no work done on the system, no heating, and no entropy is generated .
λ β λ General expectation: λ
2 α 2 E( &) = E()0 + & , S( &) = S(0) + λ&
λ& -- isis thethe raterate ofof changechange ofof externalexternal parameterparameter.. AdiabaticAdiabatic theoremtheorem inin quantumquantum mechanicsmechanics
Landau Zener process:
In the limit δδ→0 transitions between different energy levels are suppressed.
This, for example, implies reversibility (no work done) in a cyclic process. AdiabaticAdiabatic theoremtheorem inin QMQM suggestssuggests adiabaticadiabatic theoremtheorem inin thermodynamicsthermodynamics:: λ 1. Transitions are unavoidable in large gapless systems. 2. Phase space availableβ forλ these transitions decreases with the rate. Hence expect λ
2 α 2 E( &) = E()0 + & , S( &) = S(0) + λ&
Low dimensions: high density of low energy states, breakdown of mean-field approaches in equilibrium
BreakdownBreakdown ofof theTaylortheTaylor expansionexpansion inin lowlow dimensionsdimensions,, especiallyespecially nearnear singularitiessingularities ((phasephase transitionstransitions).). ThreeThree regimesregimes ofof responseresponse toto thethe slowslow rampramp:: A.P. and V.Gritsev, Nature Physics 4, 477 (2008) A.A. MeanMeanλ fieldfield ((analyticanalytic)) –– highhigh dimensionsdimensions::
E( &) = E()0 + β λ&2
λ B.B. NonNon--analyticanalytic –– lowlow dimensionsdimensions λ β E( &) = E()0 + | λ& | r , r ≤ 2
β C.C. NonNon--adiabaticadiabatic –– lowlowλ dimensionsdimensions,, bosonicbosonic excitationsexcitations
E( &) = E()0 + | & | r Lη , r ≤ 2,η > 0
In all three situations quantum and thermodynamic adiabatic theorem are smoothly connected. The adiabatic theorem in thermodynamics does follow from the adiabatic theorem in quantum mechanics. λ ConnectionConnection betweenbetweenε twotwo adiabaticadiabatic theoremstheorems allowsallows usus toto definedefine heatheatλ ((A.Pρ., Phys. Rev. Lett. 101, 220402, 2008 )).. Consider εan arbitraryλ dynamical process and work in the instantaneous energy basis (adiabaticρ basis). ε λ ρ ρ = = E( t ,t) ∑ n ( t ) nn (t) ∑ n ( t ) nn (0) n n λ + []− = + λ ∑ n ( t ) nn (t) nn (0) Ead ( t ) Q( t ,t) n
• Adiabatic energy is the function of the state. • Heat is the function of the process. • Heat vanishes in the adiabatic limit. Now this is not the postulate, this is a consequence of the Hamiltonian dynamics! ρρ IsolatedIsolated systemssystems.. InitialInitial stationarystationary statestate.. = ρ 0δρ nm (0) n nm Unitarity of the evolution: ρ = 0 + 0 − ρ 0 nm (t) n ∑ pm→n (t)( m n ) m = − + 2 pm,n (I U AU )m,n
Transition probabilities pm−>n are non-negative numbers satisfying
= ∑ pm→n (t) ∑ pn→m (t) mn
In general there is no detailed balance even for cyclic processes (but within the Fremi-Golden rule there is). ρ ρ
ρ = 0 + 0 − ρ 0 nn (t) n ε ∑ pm→n (t)( m n ) mρ yields ε ρ = Δ = 0 − ρ 0 Q(t) ∑ n nn (t) ∑ n ( m n ) pm→n (t) n n If there is a detailed εbalance then ε ρ = 1 − 0 − ρ 0 Q(t) ∑( n m )( m n ) pm→n (t) 2 n Heat is non-negativeε for cyclic processes if the initial density matrix is passive ε .Second law of thermodynamicsρ in Thompson (Kelvin’s form). − 0 − ρ 0 ≥ ( n m )( m n ) 0
The statement is also true without the detailed balance (Thirring, Quantum Mathematical Physics, Springer 1999, A. E. Allahverdyan, Th. M. Nieuwenhuizen, Physica A 305, 542 (2002).).