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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 III von Neumann algebra (with the type III0 component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type IIIλ 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.

Barreto, S., Fidaleo, F. (2011). Disordered fermions on lattices and their spectral properties. JOURNAL OF STATISTICAL PHYSICS, 143(4), 657-684 [10.1007/s10955-011-0197-8].

Disordered fermions on lattices and their spectral properties

FIDALEO, FRANCESCO
2011-01-01

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 III von Neumann algebra (with the type III0 component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type IIIλ 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.
2011
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Noncommutative dynamical systems; Disordered systems; Classification of C∗-algebras; Factors
Mathscinet Review: This paper gives a detailed study of fermionic systems on a lattice with random (even) interactions through their dynamics and KMS states. The starting point is a Z2-graded C∗-algebra A which includes the physical observables. The simplest interesting example is given by A=CAR(Zd) where CAR(Zd) is the algebra of the canonical anticommutation rules on the lattice Zd generated by the family {a†j| j∈Zd} of the single site fermionic creators. This case, which covers among others the Hubbard model, is discussed in detail in Section 7. The disorder is encoded in the model in a standard way considering the product algebra A⊗L2(Ω,μ). Here, the probability space (Ω,μ) is the sample space for the random coupling constants. Sections 2, 3, and 4 contain a description of general properties of such systems and the corresponding states. Because of the anticommutation properties of local algebras, some of the known results valid for disordered spin lattice systems do not automatically go over to the disordered fermion lattice systems. This is mainly due to the loss of asymptotic Abelianness caused by the presence of fermions. However, for random fermion systems described by A⊗L2(Ω,μ) the lattice translations and the time evolution act in a natural way as mutually commuting group actions. The resulting systems fall into the category of so-called Z2-graded asymptotically Abelian systems, namely they are asymptotically Abelian with respect to the the graded commutator. The principal merit of this work consists in the generalization of some standard properties, known for asymptotically Abelian systems, to the case of Z2-graded systems. The investigation of ergodic and spectral properties for Z2-graded asymptotically Abelian dynamical systems is done in Section 5 by means of the study of the Arveson spectrum (i.e., the set of the Bohr frequencies). In particular, the authors show that: (i) the Arveson spectrum is independent of the disorder; (ii) it coincides with the spectrum of the modular Hamiltonian and then with the Borchers Γ-spectrum for the related Z2-graded (nonfactor) W∗-dynamical system. In Section 6 such spectral results are applied to the study of the structure of the von Neumann algebras generated by Z2-graded asymptotically Abelian dynamical systems. The authors obtain that: (iii) a von Neumann algebra with a nontrivial infinite semifinite summand cannot carry an action which is Z2-graded asymptotically Abelian; (iv) a temperature state (i.e., translationary invariant) of such disordered fermionic systems can generate only a type III von Neumann algebra, with the type III0 component excluded. Reviewed by Giuseppe De Nittis
http://link.springer.com/article/10.1007%2Fs10955-011-0197-8
Barreto, S., Fidaleo, F. (2011). Disordered fermions on lattices and their spectral properties. JOURNAL OF STATISTICAL PHYSICS, 143(4), 657-684 [10.1007/s10955-011-0197-8].
Barreto, S; Fidaleo, F
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/91767
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