(Quantum) chaos theory and statistical physics far from equilibrium:

(Quantum) chaos theory and statistical physics far from equilibrium: Introducing the group for Non-equilibrium quantum and statistical physics Department of physics, Faculty of mathematics and physics, University of Ljubljana July, 2011

People: dr., 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 intersection among 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) Condensed matter theory and statistical physics" shared between Josef Stefan Institute and 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 CHAPTER 5. XY Theory CHAIN FAR FROM andequilibrium Random Matrix Theory Applied: Figure 5.1: A schematic representation of a quantum spin chain coupled to two reservoirs with different temperatures. 5.1 Lindblad Master equation Controlling and rectifying heat flow in quantum/classical lattices The evolution of the System is given in terms of a time-independent generator of infinitesimal time translations, (d/dt)ρ = ˆL(ρ) (5.1) Thermo-electric, thermo-magnetic, or thermo-chemical heat engines, and optimizing their efficiency from dynamical systems perspective where the generator ˆL must obey usual requirements when applied to a density operator ρ, especially to preserve its trace and positivity. The origin of the generator ˆL is illustrated by considering the Liouville equation for the density operator R for the Universe, (d/dt)r = i[r, HU] = ˆLUR from which the state of the System itself may be obtained by tracing over all degrees of freedom in the environment, ρ = tr ER and thus ρ(t) = tr E( e t ˆLUR(0) ). From there, one can derive the generator of infinitesimal time translations assuming that the thermal bath is in a thermal equilibrium. Finally, for sufficiently large times the generator of time

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

Chaotic versus integrable maps: trajectories versus ensembles (q t+1, p t+1) = F (q t, p t) Perturbed cat Standard map Suris map Triangle map t 7 t 5 t 3

Chaos and complexity: transport in Fourier space Chaotic maps have densities which diffuse exponentially fast in Fourier space. Perturbed cat Standard map Suris map Triangle map Ky 39 Kx 64 Kx 308 Ky 258 Kx 429 Ky 1720 Ky 794 Ky 154 Kx 1587 Ky 4096 Ky 164 Kx 164 Kx 164 Ky 164 Ky 164 Ky 410 Kx 205 Kx 205 t 7 Ky 410 t 5 Ky 410 t 3 Kx 2867 Kx 8192 Kx 164 Kx 205 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 q t+1 = q t + p t (mod 2) p t+1 = p t + α sgnq t+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 or chaotic billiard. a a l s screen Λ l s screen Λ absorber absorber

a b

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 and eigenfunctions) of - even very simple - quantum systems, which are chaotic in the classical limit, can be described by universal (no free parameter) ensembles of Gausssian random matrices 0.8 0.6 non-integrable 1.6 1.2 integrable p(s) 0.4 0.8 0.2 0.4 0 0 1 2 s 3 4 0 0 1 2 s 3 4 Is there a "quantum chaos conjecture" for many body quantum systems which do not possess a classical limit?

Case study: Kicked Ising Chain There were many results reporting random matrix statistics on non-integrable strongly correlated quantum systems (e.g. Montambaux et al 1993).

Case study: Kicked Ising Chain There were many results reporting random matrix statistics on non-integrable strongly correlated quantum systems (e.g. Montambaux et al 1993). Here we describe a recent detailed study (C. Pineda and T.P. PRE 2007) of quasi-energy level statistics in a non-integrable regime of Kicked Ising spin 1/2 chain (T.P. PTPS 2000, T.P. PRE 2002): ( XL 1 H(t) = Jσj z σj+1 z + (h xσj x + h zσj z ) X ) δ(t m) m Z U Floquet = T exp i j=0 Z 1+ «dt H(t ) = Y j 0+ exp ` i(h xσj x + h zσj z ) exp ` ijσ j z σj+1 z where [σ α j, σ β k ] = 2iε αβγσ γ j δ jk.

Quasi-energy level statistics Fix J = 0.7, h x = 0.9, h z = 0.9, s.t. KI is (strongly) non-integrable. Diagonalize U Floquet n = exp( iϕ n) n. For each conserved total momentum K quantum number, we find N 2 L /L levels, normalized to mean level spacing as s n = (N /2π)ϕ n. N(s) = #{s n < s} = N smooth (s) + N fluct (s) For kicked quantum quantum systems spectra are expected to be statistically uniformly dense N smooth (s) = s

Mode fluctuations We find perfect agreement with Gaussian mode fluctuations for KI chain. n Nfluct 2 0 2 0 6000 12000 dp/dnfluct 0.4 0.3 0.2 0.1 0.0 3 2 1 0 1 2 3 N fluct We plot the mode fluctuation N fluct as a function of the level number n (upper panel) and its normalized distribution (lower panel) for an example of a KI spectrum with L = 18 and K = 6 (N=14599). Gaussian fit: χ 2 = 102.46 and 100 equal size bins.

Long-range statistics: spectral form factor Spectral form factor K 2(τ) is for nonzero integer t defined as K 2(t/N ) = 1 tr U t 2 = 1 X e iϕnt 2. N N n

Long-range statistics: spectral form factor Spectral form factor K 2(τ) is for nonzero integer t defined as K 2(t/N ) = 1 tr U t 2 = 1 X e iϕnt 2. N N In non-integrable systems with a chaotic classical lomit, form factor has two regimes: universal described by RMT, non-universal described by short classical periodic orbits. n

Long-range statistics: spectral form factor Note that for kicked systems, Heisenberg integer time τ H = N 1 0.8 K2 0.6 0.02 0.4 0.2 0.02 0.5 1 1.5 0.25 0.5 0.75 1 1.25 1.5 1.75 2 t/τ H We show the behavior of the form factor for L = 18 qubits. We perform averaging over short ranges of time (τ H/25). The results for each of the K-spaces are shown in colors. The average over the different spaces as well as the 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: Average correlator 1 C A (t) = lim tr A(0)A(t), L L2 A(t) L D A = signals quantum ergodicity if D A = 0 T 1 X 1 lim C A (t) T T t=0 = U t AU t Quantum chaos regime in KI chain is compatible with exponential decay of correlations. For integrable, and weakly non-integrable cases, though, we find saturation of temporal correlations D 0.

Decay of time correlatons in KI chain 0.8 (a) D M /L=0.485 Three typical cases of parameters: <M(t)M>/L 0.6 0.4 0.2 0 (a) J = 1, h x = 1.4, h z = 0.0 (completely integrable). (b) J = 1, h x = 1.4, h z = 0.4 (intermediate). (c) J = 1, h x = 1.4, h z = 1.4 ("quantum chaotic"). <M(t)M> /L <M(t)M>/L 0.8 0.6 0.4 0.2 0 0.1 10-2 10-3 (b) (c) <M(t)M>-D M /L 10-1 10-2 10-3 10 20 30 t D M /L=0.293 L=20 L=16 L=12 0.25exp(-t/6) 0 5 10 15 20 25 30 35 40 45 50 t

Loschmidt echo and decay of fidelity Decay of correlations is closely related to fidelity decay F (t) = U t U t δ(t) due to perturbed evolution U δ = U exp( iδa) (Prosen PRE 2002) e.g. in a linear response approximation: F(t) 0.1 10-2 (a)! =0.01! =0.005! =0.0025 10-3 F (t) = 1 δ2 2 tx t,t =1 C(t t ) F(t) 0.1 (b) 10-2! =0.01! =0.005! =0.0025 (a) J = 1, h x = 1.4, h z = 0.0 (completely integrable). (b) J = 1, h x = 1.4, h z = 0.4 (intermediate). (c) J = 1, h x = 1.4, h z = 1.4 ("quantum chaotic"). F(t) 10-3 0.1! =0.02! =0.01 (c) 10-2! =0.04 L=20 10-3 L=16 L=12 10-4 theory 0 50 100 150 200 250 300 350 400 REVIEWED IN: T. Gorin, T. P., T H. Seligman and t 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 Toy Figure models 5.1: Aofschematic interacting representation Heisenbergofspin a quantum 1/2 chains: spin chain coupled to two reservoirs with different temperatures. XY spin chain with transverse magnetic field 5.1 Lindblad Master Xn 1 1 equation + γ H = σj x σj+1 x + 1 γ «nx σ y j The evolution of the System j=1 is2given in terms of2 σy j+1 + hσj z a time-independent j=1 generator of infinitesimal time translations, Anisotropic XXZ spin chain (d/dt)ρ = ˆL(ρ) (5.1) where the generator ˆL must obey usual requirements when applied to a density operator ρ, especially to preserve its Xn 1 trace and positivity. The origin of the generator H = `σx ˆL is j illustrated σj+1 x + σ y j by σy j+1 considering + σz j σj+1 z the Liouville equation for the density operator R for j=1 the Universe, (d/dt)r = i[r, H U ]= ˆL U R from which the state of the System itself may be obtained by tracing over all degrees of freedom ( in the environment, ρ = tr E R and thus ρ(t) = tr E e t ˆL U R(0) ). From there, one can derive the generator of infinitesimal time translations assuming that the thermal bath

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: dρ dt = ˆLρ := i[h, ρ] + X µ 2L µρl µ {L µl µ, ρ} for a general quadratic system of n fermions, or n qubits (spins 1/2) H = 2nX j,k=1 w j H jk w k = w H w L µ = 2nX j=1 l µ,j w j = l µ w where w j, j = 1, 2,..., 2n, are abstract Hermitian Majorana operators Two physical realizations: {w j, w k } = 2δ j,k j, k = 1, 2,..., 2n canonical fermions c m, w 2m 1 = c m + c m, w 2m = i(c m c m), m = 1,..., n. spins 1/2 with canonical Pauli operators σ m, m = 1,..., n, Y w 2m 1 = σj x σ z m w2m = Y σy m m <m m <m σ z m

Quantum phase transition far from equilibrium in XY spin-1/2 chain h critical = 1 γ 2.

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 Lρ NESS = 0

Spin Diffusion in Heisenberg chains

Long-range correlations far from equilibrium

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

ρ NESS = 2 n 1 + Γµ(Z Z ) + Γ 2 µ 2 2 (Z Z ) 2 µ ««2 [Z, Z ] + O(Γ 3 ) Z is a non-hermitian matrix product operator Z = X (s 1,...,s n) {+,,0} n L A s1 A s2 A sn R ny j=1 σ s j j, where σ 0 1 and A 0, A ± is a triple of near-diagonal matrix operators acting on an auxiliary Hilbert space H spanned by { L, R, 1, 2,...}: A 0 = L L + R R + A + = L 1 + c r=1 X cos (rλ) r r, r=1 X sin 2 A = 1 R c 1 X r=1 r +1 2 «λ r r +1, j r k sin 2 +1 λ r +1 r, 2 where λ = arccos R ir and x is the largest integer not larger than x.

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

Improving thermoelectric figure of merit using dynamical systems approach

A simple dynamical model of thermoelectric engine