(quantum) chaos theory and statistical physics far from equilibrium

(Quantum) chaos theory and statistical physicsfar from equilibrium:

Introducing the group for Non-equilibrium quantum and statistical physics

Tomaž Prosen

Department of physics, Faculty of mathematics and physics,University of Ljubljana

July, 2011

Tomaž Prosen Non-equilibrium quantum and statistical physics group

People:

dr. Tomaž Prosen, professor, head of the group

dr. Marko Žnudarič, assistant professor, researcher

dr. Martin Horvat, researcher

Bojan Žunkovič, PhD student

Enej Ilievski, PhD student

Simon Jesenko, PhD student

Research themes

We use methods of theoretical and mathematical physics in the intersectionamong the following fields of contemporary physics:

(Hard) condensed matter theory

Non-equilibrium statistical mechanics

Dynamical systems (Nonlinear dynamics, chaos theory)

Quantum information theory

Our group is also a part of the bigger program group (P1-0044) “Condensedmatter theory and statistical physics" shared between Josef Stefan Instituteand the Faculty of Math.& Phys. UL

Topics of main current research interest:

Fundamental:

Non-equilibrium quantum transport in low dimensional interacting systems

Open quantum many-body system – Lindblad master equation approach:Its exact, approximate, and numerical solutions(density-matrix-renormalization group)

Non-equilibrium (quantum) phase transitions

Quantum maps, quantum chaos, random matrix theory:wave-dynamics, wave-chaos, PT-symmetric Hamiltonians

Quantum chaos in many-body systems

Quantum Information Theory and Random Matrix TheoryCHAPTER 5. XY CHAIN FAR FROM EQUILIBRIUM

Figure 5.1: A schematic representation of a quantum spin chain coupled to two reser-voirs with different “temperatures”.

5.1 Lindblad Master equation

The evolution of the System is given in terms of a time-independent generator of in-finitesimal time translations,

(d/dt)! = L̂(!) (5.1)

where the generator L̂ must obey usual requirements when applied to a density opera-tor !, especially to preserve its trace and positivity.

The origin of the generator L̂ is illustrated by considering the Liouville equationfor the density operator R for the Universe, (d/dt)R = i[R, HU] = L̂UR from whichthe state of the System itself may be obtained by tracing over all degrees of freedomin the environment, ! = tr ER and thus !(t) = tr E

!etL̂U R(0)

". From there, one can

derive the generator of infinitesimal time translations assuming that the thermal bathis in a thermal equilibrium. Finally, for sufficiently large times the generator of timetranslations will approach a stationary limit L̂ in (5.1) which we will call the Liouvilleoperator.

In fact, the term Liouville operator applies to unitary evolution of density matrix i.e.Liouville equation in closed quantum systems

(d/dt)! = !i[H, !] " L̂0(!)

which we will denote by L̂0. In open systems the total Liouville operator will consist ofthe unitary evolution L̂0 and the dissipation terms due to the coupling with the envi-ronment L̂diss,

L̂ = L̂0 + L̂diss.

The most convenient way to mimic the coupling to the environment was obtainedby Lindblad (Lin75) who assumed that the System is coupled to the Environment byoperators Lµ and derived the following Master equation describing the time evolutionof the density operator ! by a Liouville operator

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

Operators Lµ are called Lindblad operators and are usually connected to the inflow oroutflow of particles e.g. L = c1, c†

1 speaking of fermions, or L = "±1 in spin chains.As discussed in the original paper (Lin75), equation (5.2) represents “the most generaltime-homogeneous quantum mechanical Markovian master equation” generating com-pletely positive flows.

Applied:

Controlling and rectifying heat flow in quantum/classical lattices

Thermo-electric, thermo-magnetic, or thermo-chemical heat engines, andoptimizing their efficiency from dynamical systems perspective

Fundamental:

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

Applied:

Fundamental:

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

Applied:

Fundamental:

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

Applied:

Fundamental:

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

Applied:

Fundamental:

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

Applied:

Fundamental:

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

Applied:

Fundamental:

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

Applied:

Fundamental:

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

Applied:

A pedestrian path:from one, two, to (infinitely) many dynamical degrees of freedom

Chaotic versus integrable maps: trajectories versus ensembles

(qt+1, pt+1) = F (qt , pt)

Perturbed cat Standard map Suris map Triangle map

Chaos and complexity: transport in Fourier space

Chaotic maps have densities which diffuse exponentially fast in Fourier space.

Kx!429

Kx!2867

Kx!308

Kx!1587

Kx!8192

Kx!164

Kx!205

Perturbed cat Standard map Suris map Triangle map

T.P., Complexity and nonseparability of classical Liouvillian dynamics, Phys. Rev. E 83, 031124 (2011).

Open problem: Deterministic diffusion and mixing in non-chaotic maps

Triangle map

qt+1 = qt + pt (mod 2)

pt+1 = pt + α sgnqt+1 + β (mod 2)

on (q, p) ∈ [−1, 1]× [−1, 1]

M. Horvat, M. Degli Esposti, S. Isola, T. Prosen and L.

Bunimovich, Physica D 238, 395 (2009).

Quantum chaos: playing billiards

Integrable dynamics:

Chaos and double-slit experiment

Numerical experiment (Giulio Casati and T.P. Phys. Rev. A 72, 032111 (2005))“Leaking of quantum particles/waves through two slits inside a regular orchaotic billiard.

absorber

screen

absorber

screen

What about changing gears and going to many-body systems?

Universality in spectral statistics of quantum chaotic many body systems

Quantum Chaos Conjecture (Berry 1977, Casati, Guarneri, Vals-Griz 1980,Bohigas, Giannoni, Schmit 1984):

Spectral correlations (and some other statistical properties of spectra andeigenfunctions) of - even very simple - quantum systems, which are chaoticin the classical limit, can be described by universal (no free parameter)ensembles of Gausssian random matrices

0.80.60.40.20

non-integrable1.61.20.80.40

43210s

integrable

Is there a "quantum chaos conjecture" for many body quantum systems whichdo not possess a classical limit?

Case study: Kicked Ising Chain

There were many results reporting random matrix statistics on non-integrablestrongly correlated quantum systems (e.g. Montambaux et al 1993).

There were many results reporting random matrix statistics on non-integrablestrongly correlated quantum systems (e.g. Montambaux et al 1993).Here we describe a recent detailed study (C. Pineda and T.P. PRE 2007) ofquasi-energy level statistics in a non-integrable regime ofKicked Ising spin 1/2 chain (T.P. PTPS 2000, T.P. PRE 2002):

H(t) =L−1Xj=0

(Jσzj σ

zj+1 + (hxσxj + hzσzj )

Xm∈Z

δ(t −m)

UFloquet = T exp„−iZ 1+

0+dt′H(t′)

exp`−i(hxσxj + hzσzj )

`−iJσzj σ

zj+1´

where [σαj , σβk ] = 2iεαβγσγj δjk .

H(t) =L−1Xj=0

(Jσzj σ

Xm∈Z

δ(t −m)

0+dt′H(t′)

`−iJσzj σ

zj+1´

The model is completely integrable in terms of Jordan-Wigner transformation if

hx = 0 (longitudinal field)

hz = 0 (transverse field)

H(t) =L−1Xj=0

(Jσzj σ

Xm∈Z

δ(t −m)

0+dt′H(t′)

`−iJσzj σ

zj+1´

H(t) =L−1Xj=0

(Jσzj σ

Xm∈Z

δ(t −m)

0+dt′H(t′)

`−iJσzj σ

zj+1´

Quasi-energy level statistics

Fix J = 0.7, hx = 0.9, hz = 0.9, s.t. KI is (strongly) non-integrable.

Diagonalize UFloquet|n〉 = exp(−iϕn)|n〉. For each conserved total momentumK quantum number, we find N ∼ 2L/L levels, normalized to mean levelspacing as sn = (N/2π)ϕn.

N(s) = #{sn < s} = Nsmooth(s) + Nfluct(s)

For kicked quantum quantum systems spectra are expected to be statisticallyuniformly dense

Nsmooth(s) = s

For kicked quantum quantum systems spectra are expected to be statisticallyuniformly dense

Nsmooth(s) = s

For classically chaotic systems, statistical distribution of mode fluctuationsNfluct(s) has been predicted to be Gaussian (Aurich, Bäcker, Steiner 1997).

Mode fluctuations

We find perfect agreement with Gaussian mode fluctuations for KI chain.

0 6000 12000

!3 !2 !1 0 1 2 30.0

Nfluct

We plot the mode fluctuation Nfluct as a function of the level number n (upperpanel) and its normalized distribution (lower panel) for an example of a KIspectrum with L = 18 and K = 6 (N=14599). Gaussian fit: χ2 = 102.46 and100 equal size bins.

Long-range statistics: spectral form factor

Spectral form factor K2(τ) is for nonzero integer t defined as

K2(t/N ) =1N˛̨trUt ˛̨2 =

˛̨̨̨˛X

e−iϕnt

˛̨̨̨˛2

Spectral form factor K2(τ) is for nonzero integer t defined as

K2(t/N ) =1N˛̨trUt ˛̨2 =

˛̨̨̨˛X

e−iϕnt

˛̨̨̨˛2

In non-integrable systems with a chaotic classical lomit, form factor has tworegimes:

universal described by RMT,

non-universal described by short classical periodic orbits.

Note that for kicked systems, Heisenberg integer time τH = N

0.25 0.5 0.75 1 1.25 1.5 1.75 2

0.5 1 1.5

We show the behavior of the form factor for L = 18 qubits. We performaveraging over short ranges of time (τH/25). The results for each of theK -spaces are shown in colors. The average over the different spaces as well asthe theoretical COE(N) curve is plotted as a black and red curve, respectively.

Quantum chaos and non-equilibrium statistical mechanics:Decay of time correlations

Temporal correlation of an extensive traceless observable A:

CA(t) = limL→∞

1L2L trA(0)A(t), A(t) = U−tAUt

Average correlator

DA = limT→∞

T−1Xt=0

signals quantum ergodicity if DA = 0

Quantum chaos regime in KI chain is compatible with exponential decay ofcorrelations. For integrable, and weakly non-integrable cases, though, wefind saturation of temporal correlations D 6= 0.

CA(t) = limL→∞

Average correlator

DA = limT→∞

T−1Xt=0

CA(t) = limL→∞

Average correlator

DA = limT→∞

T−1Xt=0

Decay of time correlatons in KI chain

Three typical cases of parameters:

(a) J = 1, hx = 1.4, hz = 0.0(completely integrable).

(b) J = 1, hx = 1.4, hz = 0.4(intermediate).

(c) J = 1, hx = 1.4, hz = 1.4("quantum chaotic").

0 5 10 15 20 25 30 35 40 45 50

(c) L=20L=16L=12

0.25exp(-t/6)

DM/L=0.293

(a) DM/L=0.485

10 20 30 t

Loschmidt echo and decay of fidelity

Decay of correlations is closely related tofidelity decay F (t) = 〈U−tUt

δ(t)〉 due toperturbed evolution Uδ = U exp(−iδA)(Prosen PRE 2002) e.g. in a linear re-sponse approximation:

F (t) = 1− δ2

tXt′,t′′=1

C(t′ − t′′)

(a) J = 1, hx = 1.4, hz = 0.0(completely integrable).

(b) J = 1, hx = 1.4, hz = 0.4(intermediate).

(c) J = 1, hx = 1.4, hz = 1.4("quantum chaotic").

0 50 100 150 200 250 300 350 400|F(t)|

!’=0.04

!’=0.02 !’=0.01(c)

L=20L=16L=12theory

|F(t)|

!’=0.01!’=0.005

!’=0.0025(b)

|F(t)|

!’=0.01 !’=0.005!’=0.0025(a)

REVIEWED IN: T. Gorin, T. P. , T H. Seligman and

M. Žnidarič: Physics Reports 435, 33-156 (2006)

Exact (analytical and numerical) treatment of large open quantum systems

CHAPTER 5. XY CHAIN FAR FROM EQUILIBRIUM

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

Toy models of interacting Heisenberg spin 1/2 chains:

XY spin chain with transverse magnetic field

H =n−1Xj=1

„1 + γ

2σxj σ

xj+1 +

1− γ2

σyj σyj+1

Anisotropic XXZ spin chain

H =n−1Xj=1

`σxj σ

xj+1 + σyj σ

yj+1 + ∆σzj σ

zj+1´

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

H =n−1Xj=1

„1 + γ

2σxj σ

xj+1 +

1− γ2

σyj σyj+1

H =n−1Xj=1

`σxj σ

xj+1 + σyj σ

yj+1 + ∆σzj σ

zj+1´

(d/dt)! = L̂(!) (5.1)

!etL̂U R(0)

(d/dt)! = !i[H, !] " L̂0(!)

L̂ = L̂0 + L̂diss.

L̂(!) = L̂0(!) + !µ

![Lµ!, L†

µ] + [Lµ, !L†µ]

". (5.2)

H =n−1Xj=1

„1 + γ

2σxj σ

xj+1 +

1− γ2

σyj σyj+1

H =n−1Xj=1

`σxj σ

xj+1 + σyj σ

yj+1 + ∆σzj σ

zj+1´

Open Many-Body Quantum Systems, method I:Quantization in the Liouville-Fock space of density operators

Analytical solution for quasi-free fermionic systems

Consider a general solution of the Lindblad equation:

= L̂ρ := −i[H, ρ] +Xµ

“2LµρL†µ − {L†µLµ, ρ}

”for a general quadratic system of n fermions, or n qubits (spins 1/2)

H =2nX

wjHjkwk = w ·Hw Lµ =2nXj=1

lµ,jwj = lµ · w

where wj , j = 1, 2, . . . , 2n, are abstract Hermitian Majorana operators

{wj ,wk} = 2δj,k j , k = 1, 2, . . . , 2n

= L̂ρ := −i[H, ρ] +Xµ

H =2nX

lµ,jwj = lµ · w

{wj ,wk} = 2δj,k j , k = 1, 2, . . . , 2n

Two physical realizations:

canonical fermions cm, w2m−1 = cm + c†m,w2m = i(cm − c†m),m = 1, . . . , n.

spins 1/2 with canonical Pauli operators ~σm, m = 1, . . . , n,

w2m−1 = σxjY

m′<m

σzm′ w2m = σymY

m′<m

σzm′

= L̂ρ := −i[H, ρ] +Xµ

H =2nX

lµ,jwj = lµ · w

{wj ,wk} = 2δj,k j , k = 1, 2, . . . , 2n

w2m−1 = σxjY

m′<m

σzm′ w2m = σymY

m′<m

σzm′

= L̂ρ := −i[H, ρ] +Xµ

H =2nX

lµ,jwj = lµ · w

{wj ,wk} = 2δj,k j , k = 1, 2, . . . , 2n

w2m−1 = σxjY

m′<m

σzm′ w2m = σymY

m′<m

σzm′

Quantum phase transition far from equilibrium in XY spin-1/2 chain

hcritical = 1− γ2.

Open Many-Body Quantum Systems, method II:time-dependent density-matrix-renormalization group in operator-space

Non-equlibrium steady state as a fixed point of Liouville equation LρNESS = 0

Spin Diffusion in Heisenberg chains

Long-range correlations far from equilibrium

Solving a long standing problem: Proof of ballistic spin transport ineasy-plane (|∆| < 1) anisotropic Heisenberg spin-1/2 chain

ρNESS = 2−n„1 + Γµ(Z − Z †) + Γ2

„µ2

2(Z − Z †)2 − µ

2[Z ,Z †]

««+O(Γ3)

Z is a non-hermitian matrix product operator

(s1,...,sn)∈{+,−,0}n〈L|As1As2 · · ·Asn |R〉

σsjj ,

where σ0 ≡ 1 and A0,A± is a triple of near-diagonal matrix operators actingon an auxiliary Hilbert space H spanned by {|L〉, |R〉, |1〉, |2〉, . . .}:

A0 = |L〉〈L|+ |R〉〈R|+∞Xr=1

cos (rλ) |r〉〈r |,

A+ = |L〉〈1|+ c∞Xr=1

sin„2—r+12

«|r〉〈r+1|,

A− = |1〉〈R| − c−1∞Xr=1

sin““

k+1”λ”|r+1〉〈r |,

where λ = arccos∆ ∈ < ∪ i< and bxc is the largest integer not larger than x .

Towards application:Improving efficiency of thermo-electric (thermo-chemical) heat-to-powerconversion

Improving thermoelectric figure of merit using dynamical systems’ approach

A simple dynamical model of thermoelectric engine

(quantum) chaos theory and statistical physics far from equilibrium

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