Equilibrium States of Long Range Interacting Quantum Lattice Systems

Abstract

At this place some thermodynamic aspects of a quantum lattice in thermic equilibrium will be handled. At the center of the discussion stand different variational principles and their connection by generalized Legendre transformation. To convey this program, one restricts to a completely homogeneous lattice that will be realized by choosing the set of permutation invariant states S P (A) as an adequate thermodynamic state space [1]. The generalization of polynomial interactions [2, 3, 1] to equivalence classes of approximately symmetric sequences [4] becomes apparent as a fruitful concept. This large class of possible thermodynamic systems is to be seen as a natural class of mean field interactions. To clarify the behavior of the thermodynamic functional internal energy, entropy and free energy in the thermodynamic limit, the full generality of [4] is not exhausted, especially only finite-dimensional algebras on a lattice point are treated and the absolute entropy is used in contrast to the relative one. The absolute properties allow a direct relation to the measurable expectation values of a macroscopic thermodynamic system. Starting with the principle of minimal free energy density for equilibrium states, the convex duality between free energy and entropy density is proved (Theorem 3.1). With a suited restriction of the variation to extremely permutation invariant states, the duality properties are modified in a way which distinguishes between an interaction in the lattice and an external field (Theorem 3.3). The rigorous treatment uses mainly results of convex analysis [5, 6]. Some ideas can be found in the investigations, presented in [7, 8] for short range interactions on translation invariant systems and the case of quadratic mean field interactions on a permutation invariant lattice in [2]. In contrast to [7], the homogeneity simplifies the calculations. The introduction of the so-called sub gradient allows a geometrical interpretation of states, minimizing the free energy functional. The formalism is suited to treat non-trivial phase transitions of first and second order, [2, 3]. A discussion of models presented, e.g. in [9] seems to be possible and will be worked out in future.