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arXiv:1005.3547

Decay of Correlations for Quantum Spin Systems with a Transverse Field: A Dynamic Approach

Alessandra Cipriani, Paolo Dai Pra

Published May 19, 2010• 3 citations
arXivHTMLPDF

Abstract

We consider a wide class of quantum spin systems obtained by adding a transverse field to a classical Hamiltonian. We give explicit high-temperature conditions which guarantee exponential decay of correlations. A stochastic-geometric representation allows to reformulate the model as a classical random field on a space of marked point processes on the circle $[0,β)$, where $β$ is the inverse temperature. We then construct a Markov process having this random field as invariant measure. By the mixing properties of the process, the exponential decay of correlations follows by an adaptation of a general argument.

DOI
10.48550/arxiv.1005.3547

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Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian:

•Jan 1, 1997•8 citations
We consider a wide class of quantum spin systems obtained by adding a transverse field to a classical Hamiltonian. We give explicit high-temperature conditions which guarantee exponential decay of correlations. A stochastic-geometric representation allows to reformulate the model as a classical random field on a space of marked point processes on the circle [0,β)[0,β)[0,β), where βββ is the inverse temperature. We then construct a Markov process having this random field as invariant measure. By the mixing properties of the process, the exponential decay of correlations follows by an adaptation of a general argument.
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 ,
The end-to-end energy-energy correlations of random transverse field quantum Ising spin chains are computed using a generalization of an asymptotically exact real space renormalization group (RG) previously introduced. Away from the critical point, the average energy-energy correlations decay exponentially with a correlation length that is the same as that of the spin-spin correlations. The typical correlations, however, decay exponentially with a characteristic length, proportional to the square root of the primary correlation length. At the quantum critical point, the average correlations decay subexponentially as $\overline{{C}_{L}}\ensuremath{\sim}{e}^{\ensuremath{-}\mathrm{const}\phantom{\rule{0.2em}{0ex}}{L}^{1∕3}}$, whereas the typical correlations decay faster, as $\ensuremath{\sim}{e}^{\ensuremath{-}K\sqrt{L}}$, with KKK a random variable with a universal distribution. The critical energy-energy correlations behave very similarly to the smallest gap, computed previously; this is explained in terms of the RG flow and the excitation structure of the chain. In order to obtain the energy correlations, an extension of the previously used methods was needed; here, this was carried out via RG transformations that involve a sequence of unitary transformations.
Abstract We show that whenever the Gibbs state of a quantum spin system satisfies decay of correlations, then it is stable, in the sense that local perturbations affect the Gibbs state only locally, and it satisfies local indistinguishability, i.e. it exhibits local insensitivity to system size. These implications hold in any dimension, require only locality of the Hamiltonian, and are based on Lieb–Robinson bounds and on a detailed analysis of the locality properties of the quantum belief propagation for Gibbs states. To demonstrate the versatility of our approach, we explicitly apply our results to several physically relevant models in which the decay of correlations is either known to hold or is proved by us. These include Gibbs states of one-dimensional spin chains with polynomially decaying interactions at any temperature, and high-temperature Gibbs states of quantum spin systems with finite-range interactions in any dimension. We also prove exponential decay of correlations above a threshold temperature for Gibbs states of one-dimensional finite spin chains with translation-invariant and exponentially decaying interactions, and then apply our general results.
A new method to investigate the two-point correlation function decay in a finite size random spin system and at finite temperatures is proposed. It is applicable to systems with cyclic boundary conditions and in any local or uniform magnetic field. The efficiency of this method is presented in the case of an isotropic XY spin glass model with Gaussian bond distribution in external field. Using unitary U(1) gauge transformation and operator inequalities we have obtained explicit upper bounds for generalized spin-glass susceptibility κ(T,R) that describes off-diagonal transverse magnetization in two dimensional random XY spin lattices at finite temperatures. It is found that the slowest possible decay for susceptibility in random XY spin glass is of the Kosterlitz–Thouless-type with power-law decay in 2D, and the exponential type in 1D, at low temperatures. These upper bounds rule out the possibility of the corresponding magnetic ordering in 1D and 2D isotropic XY spin glasses at finite temperatures.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian: