Non - Ergodicity of the 1 D Heisenberg Model

The relevance of zero-energy functions, coming from zero-energy modes and present in the structure of bosonic Green's functions, is often underestimated. Usually, their values are fixed by assuming the ergodicity of the dynamics, but it can be shown that this is not always correct. As the zero-energy functions are connected to fundamental response properties of the system under analysis (specific heat, compressibility, susceptibility , etc.), their correct determination is not an irrelevant issue. In this paper we present some results regarding the zero-energy functions for the Heisenberg chain of spin-1/2 with periodic boundary conditions as functions of the number of sites, temperature and magnetic field. Calculations are pursued for finite chains, using equations of motion, exact diagonalization and Lanczos technique, and the extrapolation to thermodynamic limit is studied. Introduction The issue of ergodicity of physical systems is not new. A system is ergodic if it goes through every point of phase space during its time evolution. In such systems, the equilibrium averages (i.e., the time averages) are equal to the ensemble averages, which are much easier to compute. Anyway, we have to face the problem of non-ergodicity of many physical systems (e.g., the existence of even one integral of motion divides the phase space into separate subspaces not connected by the dynamics). The lack of ergodicity has measurable effects as the well-known difference between the static isolated (or Kubo [1]) susceptibility and the isothermal one [2, 3]. In the widely used formalism of Green's functions the issue of ergodicity appears as a difficulty in the determination of the zero-frequency functions present in the bosonic propagators [4]-[11]. The problem reappears also in e.g., composite operator method [12]. The equations of motion do not uniquely determine causal Green's functions and correlation functions but only up to some momentum function which severely affects the self-consistent calculation of the retarded Green's functions too. Usually, these zero-frequency functions are fixed by assigning them their ergodic values, but this can not be justified a priori. Wrong determination of them dramatically affects the values of directly measurable quantities like compressibility, specific heat, magnetic susceptibility. According to this, they should be calculated case by case. In this paper, we calculate the zero-frequency functions for the Heisenberg chain of spin-1/2 and show that they take non-ergodic values for finite lengths and in the bulk limit too.