Journal of Statistical Physics, Vol. 21, No. 3, 1979 Linear Response and Relaxation in Quantum Lattice Systems J. Naudts, 1 J. V. Pul6, 2 and A. Verbeure 2 Received October 9, 1978; revised February 6, 1979 For spin-lattice systems, the Kubo formula, expressing the relaxation function in terms of the linear response function, is found to be exact in the thermodynamic limit. In addition, analyticity properties are obtained. KEY WORDS: Quantum lattice; local algebra of observables; linear response function ; relaxation function ; thermodynamic limit. 1. INTRODUCTION Thermodynamics when derived from statistical mechanics is concerned with time-dependent correlation functions. The calculation of these functions is in general far too complex. However, one is usually interested in the long-time behavior of correlations between extensive variables, and in this regime the simplifications are considerable. In the study of time-dependent correlations, the relaxation function plays a very important role. This function describes the manner in which the fluctuations of a system in statistical equilibrium relax. It has been proposed by Kubo (1>that the relaxation can be expressed as a time integral of another function which is much more readily calculated--the linear response function. This gives the response of the system to small perturbations. Several objections have been put forward regarding the validity of Kubo's relation connecting relaxation and linear response (see, e.g., Ref. 2). One of these is the validity of the linear approximation in the perturbation expansion, and another, more serious, perhaps, is the conflict arising from the fact that the perturbation expansion is only valid for short times, while one is interested in the long-time behavior of the relaxation function (see, e.g., Ref. 3). 1 Universiteit Antwerpen, Antwerp, Belgium. 2 Instituut voor Theoretische Fysica, Universiteit Leuven, Leuven, Belgium. 279 0022-4715/79[0900.0279503.00/0 9 1979 Plenum Publishing Corporation