ERGODIC PROPERTIES OF QUANTUM SPIN CHAINS: KICKED TRANSVERSE ISING MODEL
Tomai Prosen
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 ,
•Jan 1, 2011•0 citations