Correlation Inequalities for Quantum Spin Systems with Quenched Centered Disorder

Quantum spin system with quenched randomness are both important and theoretically challenging. They are widely used as models for metallic alloys in condensed matter physics, see [2] for a review. They are also important in combinatorial optimization problems especially in relation to quantum annealing procedures [3] and quantum error correcting codes [4]. One approach to studying such macroscopic systems is via correlation inequalities. These are useful in many areas of statistical mechanics. They have been used to prove that the free energy, correlation functions and surface free energy have well defined thermodynamic limits. They further show that some of these quantities approach their limit monotonically at increasing volumes. They have also proved important for computing bounds on critical temperatures and critical exponents by comparing lattices in different dimensions. In a previous work [1] we have shown that, despite the presence of competing interactions, a general classical spin glass model with quenched centered disorder has a family of positive correlation functions.This positivity implies a monotone behavior of the pressure with respect to the strength of the random interaction. From the monotonicity one can deduce sub-additivity of the free energy (which implies existence of its thermodynamic limit) and also the sign of and bounds on the surface pressure for different boundary conditions. See also [5] for further applications of the inequalities. The extension to the quantum case of classical correlation inequalities may not be possible or require further conditions. Examples were found in [8] where some correlations violate the GKS inequality of type II for the isotropic Heisenberg model with ferromagnetic interactions, and in [10] where the GKS inequality of type I is violated for the anisotropic ferromagnetic Heisenberg model. In order to reestablish the validity of the GKS inequalities in quantum systems it is then necessary to impose further conditions on the interaction coefficients beyond positivity (see for instance [10] for conditions to prove GKS I in the anisoptropic Heisenberg model with zero magnetic field and [11] for the conditions to prove GKS II). Here we show that quantum systems with quenched centered disorder do fulfill the same family of correlation inequalities of the classical case without any further restriction with respect to the classical case. The result is obtained as follows. For each finite set of points Λ let us consider the quantum spin system with Hamiltonian