Disordered Fermions on Lattices and Their Spectral Properties

Abstract

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 for disordered quantum spin systems. We discuss the Arveson spectrum and its connection with the Connes and Borchers Γ-invariants for such W * -dynamical systems. In the case of KMS states exhibiting a natural property of invariance with respect to the spatial translations, some interesting properties, associated with standard spinglass-like behaviour, emerge naturally. It covers infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice Z d . In particular, we show that a temperature state of the systems under consideration can generate only a type III von Neumann algebra (with the type III 0 component excluded). Moreover, in the case of the pure thermodynamic phase, the associated von Neumann is of type III λ for some λ ∈ (0, 1], independent of the disorder. Such a result is in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The present approach 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.Mathematics Subject Classification: 46L55, 82B44, 46L35.

Symmetric states for $C^*$-Fermi systems I: De Finetti theorem

“…where M is any mean obtained by using a Følner sequence if G is amenable. 4 It is possible to see that (iii) above is equivalent to ϕ being weakly clustering, see e.g. [13].…”

“…The early investigation of such discrete model was systematically started in [2], and continued in a long series of papers. We also mention the recent investigation in [4], of disordered models (i.e. including the Fermi counterpart of a spin-glass) based on the Z 2 -graded C * -algebras.…”

De Finetti Theorem on the CAR Algebra

“…The connection with the Markov Property, Quantum Statistical Systems, Quantum Information Theory and Entanglement, of chains of Fermi systems [2,16]. Disordered systems based on Fermions [6].…”

On the Thermodynamics of the q-Particles

“…In view of theoretical and concrete application of this interesting fact, it might be of interest to extend the investigation of quons living in inhomogeneous networks following the lines of [ 31 , 34 , 35 ]. It is also of interest to investigate the magnetic properties of systems of quons on lattices analogously to the celebrated Ising and similar models, including the disordered ones, see e.g., [ 36 , 37 , 38 ] and the literature cited therein. We leave out these interesting arguments, postponing these for future investigation.…”