Disordered Fermions on Lattices and Their Spectral Properties

Abstract

We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra some standard results concerning the spectral properties exhibited by temperature states of disordered quantum spin systems. We investigate the Arveson spectrum, known to physicists as the set of the Bohr frequencies. We also establish its connection with the Connes and Borchers spectra, and with the associated invariants for such W ∗-dynamical systems which determine the type of von Neumann algebras generated by a temperature state. We prove that all such spectra are independent of the disorder. Such results cover infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice ℤ d , including models exhibiting some standard spin-glass-like behavior. As a natural application, we show that a temperature state can generate only a type $\mathop {\rm {III}}$ von Neumann algebra (with the type $\mathop {\rm {III_{0}}}$ component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type $\mathop {\rm {III_{\lambda }}}$ for some λ∈(0,1], independent of the disorder. All such results are in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The approach pursued in the present paper can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.