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ERGODIC PROPERTIES OF QUANTUM SPIN CHAINS: KICKED TRANSVERSE ISING MODEL

Tomai Prosen

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

The problem of calculation of time-correlation functions and relaxation phenomena in interacting quantum many-body systems or quantized fields is put into the framework of ergodic theory and dynamical systems. In this spirit we discuss spectral analysis of the adjoint propagator in a suitable Hilbert space of quantum observables in Heisenberg picture as an alternative approach to characterize many-body dynamics. We illustrate our approach by working out the time-autocorrelation functions at infinite temperature of a large class of observabIes for a spin s = 1/2 Ising chain in a periodically kicking transverse magnetic field, and show that they decay to their asymptotic values as ex: t3/ 2 General understanding of time-correlation functions and relaxation phenomena of interacting quantum many-body systems is a challenging problem with only very partial existing solutions. Situation is much better if one considers the motion of a one-body or few-body systems where the classical ergodic theory (see e.g. [1]) describes a variety of different types of motion, ranging from quasi-periodic integrable motion characterized by a discrete spectrum of the (Liouville) evolution operator, through KAM quasi-integrability, to ergodicity, mixing and chaos characterized by a continuous evolution spectrum. Mixing, which is equivalent of saying that correlation functions of an arbitrary pair of observables (A,B) decay in time, limt-HXJ ((A(t)B) (A)(B)) = 0, is the necessary dynamical property needed to justify the relaxation to equilibrium (micro/grand/canonical) state and the laws of statistical mechanics, such as the fluctuation-dissipation theorem and transport laws. Mixing in quantum many-body system has been investigated by numerical experiment only quite recently [2]. In another numerical experiment [3], a suggestive possibility of dynamical phase transition from non-ergodic/non-mixing to ergodic/mixing dynamics in a non-integrable periodically kicked chain of interacting spinless fermions (kicked t-V model) has been proposed. General absence of mixing in completely integrable quantum many-body systems (such as those solvable by Bethe ansatz), being the consequence of an infinite sequence of conservation 317 J. Bonca et al. (eds.), Open Problems in Strongly Correlated Electron Systems. 317-322. © 2001 Kluwer Academic Publishers. 318 OPEN PROBLEMS IN STRONGLY CORRELATED ELECTRON SYSTEMS laws, is directly responsible for finite temperature anomalous transport properties of such systems [4, 5]. Quantum mixing property is defined (here) relative with respect to a certain (Hilbert) (sub)space 6 of observables and is most conveniently discussed in Heisenberg picture where the equations of motion for quantum observable A(t) are addressed. Our approach may be interpreted also as a kind ofgeneralization ofLee's recurrence-relations methodof constructing continued-fraction representation of time-correlation functions [6]. After a short outline of the main ideas we will illustrate the method by describing a recent calculation [8] of the time-correlation functions for Ising s = 1/2 spin chain periodically kicked with transverse magnetic field. Quantum dynamics in Heisenberg picture is governed by the Heisenberg equation (d/dt)A(t) = i( adH(t))A(t), where (adH)A:= [H, A1 =HA AH. Our aim is to include description of dynamics of periodically time-dependent (e.g. kicked) quantum systems so we will adopt Floquet formalism and integrate equations of motion over one period of time T (for autonomous systems, T may be arbitrary, e.g. infinitesimally small), defining the so called quantum Floquet map or one-step propagatorU := T exp(-i J; dtH(t) )which induces the adjoint Floquet map U ad acting on the (sub)space 6 of observables A(t +T) = U ad A(t), U A:= utAU. We should require the subspace 6 to be invariat under dynamics, namely A E 6 :::} U ad A E 6. Linear space 6 is turned into the Hilbert space by introducing an innner product of two extensive observables in thermodynamic limit (size L -+ 00) as (AlB) = (AtB), (A):= lim .!.. tr A £--+00 L tr 1 (1) (2) which has a meaning of a canonical average at infinite temperature. 1 We note that the adjoint Floquet map U ad is a unitary operator in 6 w.r.t. inner product (1), or equivalently, that ad H is selfadjoint. Thus a pair (U ad , 6) defines a unitary evolution to which we apply standard terminology and results of spectral theory of dynamical systems (see e.g. part III of ref. [1]). To any observable A E 6 we assign spectral measure dI-'A(fJ), where the spectral parameter lies on a unit circle () E 8 1 == [0, 21f), which is directly related to the time-autocorrelation function of A (t) (AtA(mT)) = (AI[U 1 A) = / dI-'A(fJ)e • So, ergodic properties of the unitary evolution are dual to its spectral properties: IThis is the simplest choice which makes sense also for time-dependent systems. For autonomous systems one may work with more general finite-temperature canonical inner product (see e.g. [6]). Ergodic Properties ofQuantum Spin Chains: Kicked Transverse Ising Model 319 • Quantum dynamics is ergodic in 6 iff 1 is (at most) simple (nondegenerate) eigenvalue of Uad (with eigenvector 1 if 1 E 6). Then limT-too ~ JOT dt {(A(t)IB) (All)(IIB)} = O. • Quantum dynamics is mixing in 6 iflimt-too(A(t)IB) = (All)(IIB) for arbitrary A, B E 6. Then U ad has an absolutely continuous spectrum, except possibly for a simple eigenvalue 1. • If 1 is a multiply degenerate eigenvalue of U ad then the corresponding eigenvectors Qn are the conservation laws of 6, U ad Qn = Qn Let Qn be a complete set of orthonormalized (QnIQm) = dnm eigenvectors with eigenvalue 1. The averaged time-autocorrelation function is the spectral weight at point 1 1 jT DA:= lim -T dt(A(t)IA) = L. I(QnIA )1 • T-too 2 -T n (3) For example, if A = J is the particle/spin current D J is proportional to the charge/spin stiffness at infinite temperature and D J > 0 indicates ballistic transport. Even when the set of known conservation laws is incomplete, the above relation gives rigorous lower bound on DA [7, 5]. Next we propose a simple but nontrivial application of the above picture. We consider aHilbert space 6 ofobservables over infinite spin 1/2 chains described by Pauli operators a}, s E {x, y, z}, j E Z, which is spanned by two infinite sequences of selfadjoint observables Un and Vn,nEZ, namely n ~ 1, n =0, n <-1 ,

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The problem of calculation of time-correlation functions and relaxation phenomena in interacting quantum many-body systems or quantized fields is put into the framework of ergodic theory and dynamical systems. In this spirit we discuss spectral analysis of the adjoint propagator in a suitable Hilbert space of quantum observables in Heisenberg picture as an alternative approach to characterize many-body dynamics. We illustrate our approach by working out the time-autocorrelation functions at infinite temperature of a large class of observabIes for a spin s = 1/2 Ising chain in a periodically kicking transverse magnetic field, and show that they decay to their asymptotic values as ex: t3/ 2 General understanding of time-correlation functions and relaxation phenomena of interacting quantum many-body systems is a challenging problem with only very partial existing solutions. Situation is much better if one considers the motion of a one-body or few-body systems where the classical ergodic theory (see e.g. [1]) describes a variety of different types of motion, ranging from quasi-periodic integrable motion characterized by a discrete spectrum of the (Liouville) evolution operator, through KAM quasi-integrability, to ergodicity, mixing and chaos characterized by a continuous evolution spectrum. Mixing, which is equivalent of saying that correlation functions of an arbitrary pair of observables (A,B) decay in time, limt-HXJ ((A(t)B) (A)(B)) = 0, is the necessary dynamical property needed to justify the relaxation to equilibrium (micro/grand/canonical) state and the laws of statistical mechanics, such as the fluctuation-dissipation theorem and transport laws. Mixing in quantum many-body system has been investigated by numerical experiment only quite recently [2]. In another numerical experiment [3], a suggestive possibility of dynamical phase transition from non-ergodic/non-mixing to ergodic/mixing dynamics in a non-integrable periodically kicked chain of interacting spinless fermions (kicked t-V model) has been proposed. General absence of mixing in completely integrable quantum many-body systems (such as those solvable by Bethe ansatz), being the consequence of an infinite sequence of conservation 317 J. Bonca et al. (eds.), Open Problems in Strongly Correlated Electron Systems. 317-322. © 2001 Kluwer Academic Publishers. 318 OPEN PROBLEMS IN STRONGLY CORRELATED ELECTRON SYSTEMS laws, is directly responsible for finite temperature anomalous transport properties of such systems [4, 5]. Quantum mixing property is defined (here) relative with respect to a certain (Hilbert) (sub)space 6 of observables and is most conveniently discussed in Heisenberg picture where the equations of motion for quantum observable A(t) are addressed. Our approach may be interpreted also as a kind ofgeneralization ofLee's recurrence-relations methodof constructing continued-fraction representation of time-correlation functions [6]. After a short outline of the main ideas we will illustrate the method by describing a recent calculation [8] of the time-correlation functions for Ising s = 1/2 spin chain periodically kicked with transverse magnetic field. Quantum dynamics in Heisenberg picture is governed by the Heisenberg equation (d/dt)A(t) = i( adH(t))A(t), where (adH)A:= [H, A1 =HA AH. Our aim is to include description of dynamics of periodically time-dependent (e.g. kicked) quantum systems so we will adopt Floquet formalism and integrate equations of motion over one period of time T (for autonomous systems, T may be arbitrary, e.g. infinitesimally small), defining the so called quantum Floquet map or one-step propagatorU := T exp(-i J; dtH(t) )which induces the adjoint Floquet map U ad acting on the (sub)space 6 of observables A(t +T) = U ad A(t), U A:= utAU. We should require the subspace 6 to be invariat under dynamics, namely A E 6 :::} U ad A E 6. Linear space 6 is turned into the Hilbert space by introducing an innner product of two extensive observables in thermodynamic limit (size L -+ 00) as (AlB) = (AtB), (A):= lim .!.. tr A £--+00 L tr 1 (1) (2) which has a meaning of a canonical average at infinite temperature. 1 We note that the adjoint Floquet map U ad is a unitary operator in 6 w.r.t. inner product (1), or equivalently, that ad H is selfadjoint. Thus a pair (U ad , 6) defines a unitary evolution to which we apply standard terminology and results of spectral theory of dynamical systems (see e.g. part III of ref. [1]). To any observable A E 6 we assign spectral measure dI-'A(fJ), where the spectral parameter lies on a unit circle () E 8 1 == [0, 21f), which is directly related to the time-autocorrelation function of A (t) (AtA(mT)) = (AI[U 1 A) = / dI-'A(fJ)e • So, ergodic properties of the unitary evolution are dual to its spectral properties: IThis is the simplest choice which makes sense also for time-dependent systems. For autonomous systems one may work with more general finite-temperature canonical inner product (see e.g. [6]). Ergodic Properties ofQuantum Spin Chains: Kicked Transverse Ising Model 319 • Quantum dynamics is ergodic in 6 iff 1 is (at most) simple (nondegenerate) eigenvalue of Uad (with eigenvector 1 if 1 E 6). Then limT-too ~ JOT dt {(A(t)IB) (All)(IIB)} = O. • Quantum dynamics is mixing in 6 iflimt-too(A(t)IB) = (All)(IIB) for arbitrary A, B E 6. Then U ad has an absolutely continuous spectrum, except possibly for a simple eigenvalue 1. • If 1 is a multiply degenerate eigenvalue of U ad then the corresponding eigenvectors Qn are the conservation laws of 6, U ad Qn = Qn Let Qn be a complete set of orthonormalized (QnIQm) = dnm eigenvectors with eigenvalue 1. The averaged time-autocorrelation function is the spectral weight at point 1 1 jT DA:= lim -T dt(A(t)IA) = L. I(QnIA )1 • T-too 2 -T n (3) For example, if A = J is the particle/spin current D J is proportional to the charge/spin stiffness at infinite temperature and D J > 0 indicates ballistic transport. Even when the set of known conservation laws is incomplete, the above relation gives rigorous lower bound on DA [7, 5]. Next we propose a simple but nontrivial application of the above picture. We consider aHilbert space 6 ofobservables over infinite spin 1/2 chains described by Pauli operators a}, s E {x, y, z}, j E Z, which is spanned by two infinite sequences of selfadjoint observables Un and Vn,nEZ, namely n ~ 1, n =0, n <-1 ,
Spectral analysis of the adjoint propagator in a suitable Hilbert space (and Lie algebra) of quantum observables in Heisenberg picture is discussed as an alternative approach to characterize infinite temperature dynamics of non-linear quantum many-body systems or quantum fields, and to provide a bridge between ergodic properties of such systems and the results of classical ergodic theory. We begin by reviewing some recent analytic and numerical results along this lines. In some cases the Heisenberg dynamics inside the subalgebra of the relevant quantum observables can be mapped explicitly into the (conceptually much simpler) Schrödinger dynamics of a single one-(or few)-dimensional quantum particle. The main body of the paper is concerned with an application of the proposed method in order to work out explicitly the general spectral measures and the time correlation functions in a quantum Ising spin 1/2 chain in a periodically kicking transversal magnetic field, including the results for the simpler autonomous case of a static magnetic field in the appropriate limit. The main result, being a consequence of a purely continuous non-trivial part of the spectrum, is that the general time-correlation functions decay to their saturation values as t-3/2.
We study a generic but simple nonintegrable quantum many-body system of locally interacting particles, namely, a kicked-parameter (t,V) model of spinless fermions on a one-dimensional lattice (equivalent to a kicked Heisenberg XX-Z chain of 1/2 spins). The statistical properties of the dynamics (quantum ergodicity and quantum mixing) and the nature of quantum transport in the thermodynamic limit are considered as the kick parameters (which control the degree of nonintegrability) are varied. We find and demonstrate ballistic transport and nonergodic, nonmixing dynamics (implying infinite conductivity at all temperatures) in the integrable regime of zero or very small kick parameters, and more generally and importantly, also in the nonintegrable regime of intermediate values of kicked parameters, whereas only for sufficiently large kick parameters do we recover quantum ergodicity and mixing implying normal (diffusive) transport. We propose an order parameter (charge stiffness D) which controls the phase transition from nonmixing and nonergodic dynamics (ordered phase, D>0) to mixing and ergodic dynamics (disordered phase, D=0) in the thermodynamic limit. Furthermore, we find exponential decay of time correlation functions in the regime of mixing dynamics. The results are obtained consistently within three different numerical and analytical approaches: (i) time evolution of a finite system and direct computation of time correlation functions, (ii) full diagonalization of finite systems and statistical analysis of stationary data, and (iii) algebraic construction of quantum invariants of motion of an infinite system, in particular the time-averaged observables.
Using an efficient one and two qubit gate simulator operating on graphical processing units, we investigate ergodic properties of a quantum Ising spin model on a two-dimensional lattice, which is periodically driven by a δ -pulsed transverse magnetic field. We consider three different dynamical properties: (i) level density, (ii) level spacing distribution of the Floquet quasienergy spectrum, and (iii) time-averaged autocorrelation function of magnetization components. Varying the parameters of the model, we found transitions between ordered (non-ergodic) and quantum chaotic (ergodic) phases, but the transitions between flat and non-flat spectral density do not correspond to transitions between ergodic and non-ergodic local observables. Even more surprisingly, we found good agreement of level spacing distribution with the Wigner surmise of random matrix theory for almost all values of parameters except where the model is essentially non-interacting, even in regions where local observables are not ergodic or where spectral density is non-flat. These findings question the versatility of the interpretation of level spacing distribution in many-body systems and stress the importance of the concept of locality.
A dynamical Lie algebraic method for the construction of quantum invariants of motion in non-integrable many-body systems of infinite size is proposed and applied to a simple but generic toy model, namely an infinite kicked t - V chain of interacting spinless fermions. The transition from an integrable via quasi-integrable ( intermediate ) to a quantum ergodic (quantum mixing) regime in parameter space is investigated. A dynamical phase transition between an ergodic and intermediate (neither ergodic nor completely integrable) regime in thermodynamic limit is proposed. The existence or non-existence of local conservation laws corresponds to the intermediate or ergodic regime, respectively. The computation of time-correlation functions of typical observables by means of local conservation laws is found to be fully consistent with direct calculations on finite systems.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.
We analytically study the time evolution of the expectation values of observables in periodically kicked many-body quantum systems. Starting from an initial state, we compute both the transient and the long-time properties of the observables. Our derivation explains the criteria and the mechanism that lead to the infinite-temperature statistical average of observables at long times, irrespective of the initial state. When the criteria are violated, the observables oscillate with time. These oscillations are subharmonic and robust to small perturbations, suggesting the emergence of a discrete time crystal phase. We demonstrate these features explicitly in periodically kicked nonintegrable spin chains. For a spin chain with two kicks per cycle, we show that the kicked chain can exhibit an ergodic or a discrete-time crystal phase for the same kicking strengths, depending on the initial state preparation. We complement our time-evolution study of observables with the spectral form factor of these kicked models.