2046-4

Summer College on Nonequilibrium Physics from Classical to Quantum Low Dimensional Systems

6 - 24 July 2009

Non-adiabatic dynamics and thermodynamics in isolated out of equilibrium systems

Anatoli Polkovnikov Boston University

Strada Costiera 11, 34151 Trieste, Italy - Tel. +39 040 2240 111; Fax +39 040 224 163 - [email protected], www. Non-adiabatic dynamics and thermodynamics in isolated out of equilibrium systems.

Anatoli Polkovnikov, Boston University

Collaborators:

C. De Grandi, R. Barankov, BU V. Gritsev, Fribourg

Trieste, Italy, July 2009 AFOSR * Basic facts from thermodynamics: energy, heat, work, first and second laws of thermodynamics, fundamental thermodynamic relation. Non-equilibrium work relations (Jarzyisky equality).

* Thermalization in isolated systems; eigenstate thermalization hypothesis. Heat, entropy in driven gapless systems. Consistency of laws of thermodynamics and Hamiltonian dynamics.

* Connection between quantum and thermodynamic adiabatic theorems.

* Universal adiabatic dynamics near critical points

Electric oven heats food

Microwave oven does work on food Usual candidate for entropy:

Thermalization in isolated systems. Ergodic hypothesis In the continuum this systemsystem is equivalentequivalent toto anan integrable KdV equation. The solution splits into non-thermalizing solitons Kruskal and Zabusky (1965 ). Qauntum Newton Craddle. (collisions in 1D interecating Bose gas – Lieb-Liniger model) T. Kinoshita, T. R. Wenger and D. S. Weiss, Nature 440, 900 – 903 (2006)

No thermalization in1D. Fast thermalization in 3D. Quantum analogue of the Fermi-Pasta- Ulam problem. Thermalization in Quantum systems.

Consider the timeρ average of a certain observable A in an isolated system after a quench.

i(Em −En )t A()t = ∑ n,m (t)Am,n =∑ ρn,m (0)e Am,n n,m n,m 1 T A = A(t) dt = ρ A ∫0 ∑ n,n n,n T n,m Eignestate thermalization hypothesis (Srednicki 1994; M. Rigol, V.

Dunjko & M. Olshanii, Nature 452, 854 , 2008.): An,n~ const (n) so there is no dependence on ρρnn.

Necessary assumption: Am,n → 0, | En − Em |<<1/τ 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. Information about equilibrium is fully contained in diagonal elements of the density matrix.matrix.

This is true for all thermodynamic observables: energy, pressure, magnetization, …. (pick your favorite). They all are linear in ρρ.

This is not true about von Neumann entropy!

Sn = −Tr(ρ ln ρ)

Off-diagonal elements do not average to zero.

The usual way around: coarse-grain density matrix (remove by hand fast oscillating off-diagonal elements of ρρ.

Problem: not a unique procedure, explicitly violates time reversibility and Hamiltonian dynamics. ρ ρ Von Neumann entropy: always conserved in time (in isolated systems). More generally it is invariantρ under arbitrary unitary transfomations ρ

+ + ρ − Sn (t) = Tr (t)ln (t) = TrU U lnU U = Tr ln ρ

Thermodynamics: entropy is conserved only for adiabatic (slow, reversible) processes. Otherwise itit increases.increases.

Quantum mechanics: for adiabatic processes there are no

transitions between energy levels: ρnn (t) = const(t)

If these two adiabatic theorems are related then the entropy

should only depend on ρρnn. Thermodynamic adiabatic theorem. 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 E( ) = E()0 + 2 , S( ) = S(0) +α λ2

λ - is the rate of change of external parameter. Adiabatic theorem in quantum mechanics

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. Adiabatic theorem in QM suggests adiabatic theorem in thermodynamics: λ 1. Transitions are unavoidable in large gapless systems. β 2. Phase space availableλ for these transitions decreases with the rate. Hence expect λ E( ) = E(0)+ 2 , S( ) = S(0) +α λ2

Low dimensions: high density of low energy states, breakdown of mean-field approaches in equilibrium

Breakdown of Taylor expansion in low dimensions, especially near singularities (phase transitions). Three regimes of response to the slow linear ramp: A.P. and V.Gritsev, Nature Physics 4, 477 (2008) A. Meanλ fieldfield (analytic)(analytic) – high dimensions: E( ) = E()0 + β λ2 B. Nonλ-analytic – low dimensions E(λ) = E()0 + β | λ | r , r ≤ 2 C. Non-adiabatic – lowβ λ dimensions, bosonic excitations 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 Connection between two adiabatic theorems allows us to define heat (A.P.,ε Phys. Rev. Lett. 101, 220402, 2008 ). λ ρ Considerε an λarbitrary dynamical process and work in the instantaneous energyρ basis (adiabatic basis).ε λ ρ ρ E( t ,t) = ∑ n ( t ) nn (t) = ∑ n ( t ) nn (0) n n λ + ∑ n ( t )[]nn (t) − nn (0) = Ead ( t ) + Q(λt ,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 Isolated systems. Initial stationary state. ρ 0 nm (0) = nδρnm

Unitarity of the evolution: 0 ρ 0 0 nm (t) = n + ∑ pm→n (t)( m − ρn ) m + 2 pm→n = (I −U AU )m,n

∑ 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). What aboutabout entropy? • Entropy should be related to heat (energy), which knows

only about ρρnn. • Entropy does not change in the adiabatic limit, so it should

depend only on ρρnn. • Ergodic hypothesisρ requires that all thermodynamic quantities (includingρ entropy) should depend only on ρρnn. • In thermal equilibrium theρ statistical entropy should • In thermal equilibrium the statisticalρ entropy should coincide with the von Neumann’s entropy:ρ 1 Sn = −Tr( ln ) = −∑ n ln n , n = exp[−βEn ] n Z Simple resolution: diagonal entropy ρ the sum is taken in the Sd = −∑ nn ln ρnn n instantaneous energy basis. ρ Properties ofρ d-entropy (R. Barankov, A. Polkovnikov, arXiv:0806.2862. ). ρ ρ Jensen’s inequality: ρ ρ Tr(− ln + d ln d ) ≤ Tr[( − d )ln ρd ] = 0 Therefore if the initial density matrix is stationary (diagonal) then

Sd (t) ≥ Sn (t) = Sn (0) = Sd (0)

Now assume that the initial state is thermal equilibrium ρ 1 0 = exp[−βε ] n Z n Let us consider an infinitesimal change of the system and compute energy and entropy change. ρ ρ

0 ρ 0 0 nn (t) = n ε+ ∑ pm→n (t)( m − ρn ) mρ yields ε ρ 0 0 Q(t) = ∑ nΔ nn (t) = ∑ n ( m − ρn ) pm→n (t) n n If there is a detailedε balance then ε 1 ρ 0 0 Q(t) = ∑( n − m )( m − ρn ) pm→n (t) 2 n ε Heat is non-negative forε cyclic processes if the initial density ρ 0 0 matrix is passive ( n − m )( m − ρ n ) ≥ 0 . Second law of thermodynamics in Thompson (Kelvin’s form).

The statement is also truetrue witwithout the detailed balance but the proof is more complicated (Thirring, Quantum Mathematical Physics, Springer 1999). ε ρ ε ρ

E ρ 0 0 Δ ≅ ∑ Δ n n + nρΔ nn n

0 1 ε 0 ΔSd ≅ −∑ Δ nn (ln n +1) = ∑ n Δρn n T n Recover the first law of thermodynamics (Fundamental Relation). ⎛ ∂E ⎞ ΔE = ⎜ λ⎟ Δλ +TΔS ∂ d ⎝ ⎠Sd

If λ stands for the volume the we find ΔE = −PΔV +TΔSd Classical systems.

Instead of energy levels we have orbits.

ρnn ↔ probability to occupy an orbit with energy E. ρ ∝ exp[i(ε −ε )t] ↔ describes the motion on nm n m this orbits. ε ρ ε ρ ε Classical d-entropy ρ ρ ε δ S = − N( ) ( )ln ( )dε ( ) = dΓ ( p,q) (ε −ε ( p, q)) d ∫ ∫

The entropy “knows” only about conserved quantities,

everything else is irrelevant for thermodynamics! Sd satisfies laws of thermodynamics, unlike the usually defined S = ln Γ. Classic example: freely expanding gas

Suddenly remove the wall

by Liouville theorem ΔSGibbs = 0 by Liouville theorem ρ 1 (0+) = ρ (0−) double number of occupied states nn 2 nn

result of Hamiltonian dynamics! ΔSd = N ln 2 result of Hamiltonian dynamics! Example

Cartoon BCS model:

+ g + + H = ∑ε k ck↑ck↓ − ∑ck↑c−k↓c− p↓c p↑ k 2N k, p ε =1 k Mapping to spin model (Anderson, 1958) ε k = −1

g H = (S − S ) − (S + S − + S + S − ) 1z 2z 2N 1 2 2 1

In the thermodynamic limit this model has a transition to superconductor (XY-ferromagnet) at g = 1. Change g from g1 to g2.

0.46 τ ∝ N 0.44 Tr ∝ N

0.42 Work with large N.

0.40 Magnetization

0.38

0 50 100 150 200 250 Time Simulations: N=2000

0.46 Full Coarse-grained 0.44

0.42

0.40 Magnetization

0.38

0 1020304050 Cycle

8

6

4

full Heat D-Entropy coarse grained 2 max entropy

0 0 1020304050 Cycle Entropy and reversibility.

δδg = 10-4

δδg = 10-5

Nearly adiabatic dynamics in many-particle systems

Let us assume we are hanging some external parameter in time according to the protocolprotocol

Can we say anything about system response in the limit δ→0? Assume (for now) that we start in the ground state. Need to solve Convenient to work in the adiabatic (co-moving, instantaneous) basis λ  = δ

In the limit →0 only term with n=m (assuming there are no degenarcies) survives. Perturbative analysis, keep only term with m=0 in the sum. Infinite integration limits

where λ* is the complex root of

Finite integration limits: Landau-Zener problem

Change coupling λ in the infinite range.range. ExactExact solution:

Perturbative solution. Eigenstates: Spurious factor π2/9

Now start ti→∞, tf – finite (or vice versa) Gapless systems with quasi-particle excitations

Ramping in generic gapless regime (low energy contribution) dE * dE * E ~ (E*)2 ⇒ λ ~ (E*)2 dt dλ λ E* Uniform system: E( ) = c(λ)k z λ d ln c E* ~  dλ ρ λλ E* n ~ (E)dE ~| λ |d / z Density of quasi-particles (entropy): ex ∫ 0 (d +z)/ z Absorbed energy density (heating): Q ~| λ | Low energyλ contribution:  d / z  (d +z)/ z nex , s ∝| | , Q ∝| λ |

High energy contribution. E | λ |2 pex (E) ∝ 6 E ρ n ~ p (E) (E)dE ∝| λ |2 ex ∫ ex E*

Similarly Q ~| λ |2 λλ High dimensions: high energies dominate dissipation, low- dimensions – low energies dominatedominate dissipation.dissipation. Adiabatic crossing quantum critical points.

Relevant for adiabatic quantum computation, adiabatic preparation ofof correlatedcorrelated states.states.

V=δδt, δδ → 0

How does the number of excitations (entropy, energy) scale with δδ ? Use scaling arguments in the adiabatic perturbation theory dν dν zν +1 A.P. 2003, nex ∝ δ , < 2 νz +1 Zurek, Dorner, Zoller 2005 δ dν n ∝ 2 , ≥ 2 Nontrivial power corresponds ex zν +1 to nonlinear response! dc = 2(z +1/ν ) is analogous to the upper critical dimension. Transverse field Ising model.

zz g →⇒0 σσij → 1

x g →∞ ⇒ σ i → 1

There is a phase transition at g=1.

This problemproblem can be exactly solved using Jordan-Wigner transformation:

xz††† σσiiii=−21,cc =∏ ( cc jjjj −+ 1) ( c c ) ji≤−1 Spectrum: Critical exponents: z=ν=1 ⇒ dνν/(zνν +1)=1/2. Linear response (Fermi Golden Rule):

A. P., 2003 nex ≈ 0.21 δ

Correct result (J. Dziarmaga 2005): nex ≈ 0.11 δ

Interpretation as the Kibble-Zurek mechanism: W. H. Zurek, U. Dorner, Peter Zoller, 2005 Optimal adiabatic passage through a QCP. (R. Barankov and A. Polkovnikov, Phys. Rev. Lett. 101, 076801 (2008) )

Given the total time T, what is the optimal way to cross the phase transition?

Need to slow down near the phase transition:

λλ=(δδt)r, δδ ~ 1/T

optimal power

number of defects at optimal rate. Three regimes of response to the slow linear ramp: A.P. and V.Gritsev, Nature Physics 4, 477 (2008) A. Meanλ field (analytic) – high dimensions: E( ) = E()0 + β λ2 B. Nonλ-analytic – low dimensions E(λ) = E()0 + β | λ | r , r ≤ 2 C. Non-adiabatic – lowβ λ dimensions, bosonic excitations E( ) = E(0)+ |  | r Lη , r ≤ 2,η > 0

In all three situations quantum and thermodynamic adiabatic theorem are smoothly connected.connected. The adiabatic theorem in thermodynamics does follow from the adiabatic theorem in quantum mechanics. Numerical verification (bosons on a lattice).

U (t) = U 0 tanh(δt) Nonintegrable model in all spatial dimensions, expect thermalization.

Use the fact that quantum fluctuations are weak in the SF phase and expand dynamics in the effective Planck’s constant:

= → U / n0 J T=0.02 Heat per site Heat Heat per site Heat

ΔE ∝ T L4/3 δ 1/3 2D, T=0.2 Heat per site Heat Heat per site Heat per

ΔE ∝ T L1/3 δ 1/3 Thermalization at long times (1D).

Correlation Functions 1.0 1.0 Numerical 0.8 Analytical 0.8 t=0 t=3.2/δ 0.6 > t=12.8/δ 0 0.6

a t=28.8/δ

+ j a t=51.2/δ 0.4 < 0.4 t=80/δ

Thermal 0.2

0.2 0.0 0 5 10 15 20 25 0.0 0 20406080 L/π sin(πj/L) Probing quasi-particle statistics in nonlinear dynamical probes. (R. Barankov, C. De Grandi, V. Gritsev, A. Polkovnikov, work in progress.) T

1/3 δ nex ∝ δ LT nex ∝ T>0 T Less adiabatic More adiabatic

T=0 1/3 n ∝ δ 1/ 2 nex ∝ δ ln L ex

0 bosonic-like fermionic-like 1 K transition? massive bosons transition? massive fermions (hard core bosons)