Revisiting the Mazur bound and the Suzuki equality

Abstract

Among the few known rigorous results for time-dependent equilibrium correlations, important for understanding transport properties, are the Mazur bound and the Suzuki equality. The Mazur inequality gives a lower bound, on the long-time average of the time-dependent auto-correlation function of observables, in terms of equilibrium correlation functions involving conserved quantities. On the other hand, Suzuki proposes an exact equality for quantum systems. In this paper, we discuss the relation between the two results and in particular, look for the analogue of the Suzuki result for classical systems. This requires us to examine as to what constitutes a complete set of conserved quantities required to saturate the Mazur bound. We present analytic arguments as well as illustrative numerical results from a number of different systems. Our examples include systems with few degrees of freedom as well as many-particle integrable models, both free and interacting.

Introduction

Time dependent equilibrium auto-correlation functions of physical observables play an important role in understanding dynamical properties of a system. In particular they tell us about the ergodicity of a given Hamiltonian system. In a seminal paper [1], Mazur discussed the long time average of such auto-correlation functions in the context of ergodicity. A rigorous lower bound was obtained for this quantity. In the classical description, we consider systems with phase space degrees of freedom $(x,p)=\{x_i,p_i\},i=1,2,\dots ,N$ and $A=A(x,p)$ is some function of the phase space variables and describes some physical observable. The systems dynamics is described by a Hamiltonian $H(x,p)$ and let us assume that apart from $H$ there are $R-1$ other conserved quantities, not necessarily independent ones. We denote the set of conserved quantities as ${\mathbf{I}}_R=(I_1,I_2,\dots ,I_R),$ with $I_1=H-〈H〉,$ where $〈\dots 〉$ denotes a thermal average over the Gibbs distribution $\rho =e^{-\beta H}/Z$ with $Z=\int dxdp{e}^{-\beta H}$. Without loss of generality, one can assume that $〈I_k〉=0$ for all $k$. We define the correlation matrix, $C,$ with elements $C_{ij}=〈I_i{I}_j〉,$ and consider the quantity

$$D_A:=\sum _{k,\ell }〈I_{k}A〉{\left[C^{-1}\right]}_{k\ell }〈I_{\ell }A〉,$$

which we will refer to as the Mazur bound. Let us also define the long time average of the temporal auto-correlation function of the observable $A$:

$$C_A:=\underset{\tau \to \infty }{\lim }\frac{1}{\tau }{\int }_0^{\tau }dt〈A\left(t\right)A\left(0\right)〉,$$

where the average is again over the equilibrium Gibbs distribution. Mazur proved that

$$C_A\ge D_A.$$

In [1] it was shown that this result could provide insights on the ergodicity of the variable or its absence. We briefly discuss the notion of ergodicity as indicated in the behavior of the correlation function. Let $\overline{A}\left(E\right)=\int dqdpA(q,p)\delta (E-H)/\int dqdp\delta (E-H)$ denote the microcanonical average of the observable. Then ergodicity implies $C_A=〈{\overline{A}}^2\left(E\right)〉$ and in the thermodynamic limit this leads to the equality,

$$C_A=\frac{{〈A\Delta H〉}^2}{〈{\left(\Delta H\right)}^2〉},$$

where $\Delta H=H-〈H〉$. Allowing for the presence of extra conservation laws, the notion of sub-ergodicity was discussed in [2], which in modern terms relates to the idea of generalized Gibbs ensembles.

The Mazur bound can be proven for both classical and quantum systems as already noted in the original paper. For the case of quantum systems, an exact Mazur-type equality was derived by Suzuki [3], which hold when one includes a “sufficient” number of constants of motion. For example, in a quantum system with a Hilbert space of finite dimension $\mathcal{D},$ a trivial complete set of constants of motion are the energy projection operators ${\widehat{P}}_n=|n〉〈n|$ where $|n〉,$ with $n=1,2,\dots ,\mathcal{D},$ denotes the energy eigenstates. Then (see later) it is easy to show that one obtains the equality in the Mazur relation in Eq. (3). Two natural questions that one could ask are: (i) instead of the projection operators, is it possible to obtain the equality with a smaller number of “local” constants of motion and (ii) does this result have a classical analogue. One of the aims of the present work is to discuss these questions and provide illustrative examples that clarify some subtle issues.

An important application of the Mazur relation has been in the context of transport properties of integrable systems [4], [6], [7], [8], [9], [10]. The auto-correlation functions involving currents corresponding to conserved quantities are related to transport coefficients via the Green-Kubo formulas. In particular, the asymptotic long time saturation value of the auto-correlation, $C_A,$ gives the so-called Drude weight which is the strength of the zero-frequency component of the conductivity and implies ballistic transport. It was pointed out in [4] that the Mazur bound can be used to prove the presence of a finite Drude weight for integrable systems. This was used to prove ballistic transport in one-dimensional systems such as the quantum spin-$1/2$ XXZ chain [5] and the Toda lattice [6]. For classical integrable systems with $N$ degrees of freedom, the set of exactly $N$ independent constants of motion, which we denote by $\left\{Q_i\right\},$ is in some sense special and in fact their existence defines integrability (e.g one can construct action-angle variables). One might expect that this set should be sufficient to saturate the Mazur bound. However, the numerical study in [11] found that one needs to include bilinear combinations of the conserved quantities, of the form $Q_j{Q}_k,$ in order to approach the equality. The study was restricted to systems of sizes $N=4,6,8$ and an important question is whether the contribution of the bilinear terms vanishes in the thermodynamic limit. On the other hand, the situation is even more complicated in the quantum case since the notion of quantum integrability is not so well defined and a basic question is on the choice of the set of constants $\mathbf{I}$ required to saturate the Mazur bound. These aspects will also be discussed in this paper.

The plan of the paper is as follows: In Section 2 we outline the proofs of the Mazur inequality and the Suzuki equality. In Section 3 we describe a procedure which leads to the classical analogue of the Suzuki equality. As illustrative examples, we then provide in Section 4 explicit results, both numerical and analytical, on the application of the Mazur-Suzuki results in different physical systems. These include few body systems such as an oscillator and a two coupled spin system, as well as many body systems such as systems described by quadratic Hamiltonians, and finally the Toda chain. We conclude with a discussion in Section 5.