Equilibrium Statistical Mechanics of Fermion Lattice Systems

Araki, Huzihiro; Moriya, Hajime

doi:10.1142/s0129055x03001606

Cited by 83 publications

(212 citation statements)

References 45 publications

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“…Since the reference [4] does not seem to be widely available, we present a complete proof in § 3. The if part of the first part of Theorem 1 (1) is also given in Theorem 11.2 of [1] which plays a crucial role in that paper.…”

Section: Introduction and Resultsmentioning

confidence: 93%

Joint Extension of States of Subsystems for a CAR System

Araki

Moriya²

2003

Commun. Math. Phys.

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The problem of existence and uniqueness of a state of a joint system with given restrictions to subsystems is studied for a Fermion system, where a novel feature is non-commutativity between algebras of subsystems.For an arbitrary (finite or infinite) number of given subsystems, a product state extension is shown to exist if and only if all states of subsystems except at most one are even (with respect to the Fermion number). If the states of all subsystems are pure, then the same condition is shown to be necessary and sufficient for the existence of any joint extension. If the condition holds, the unique product state extension is the only joint extension.For a pair of subsystems, with one of the given subsystem states pure, a necessary and sufficient condition for the existence of a joint extension and the form of all joint extensions (unique for almost all cases) are given. For a pair of subsystems with non-pure subsystem states, some classes of examples of joint extensions are given where non-uniqueness of joint extensions prevails.

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Section: Introduction and Resultsmentioning

confidence: 93%

Joint Extension of States of Subsystems for a CAR System

Araki

Moriya²

2003

Commun. Math. Phys.

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“…If A, B ∈ A with B(ω) = B a constant field, we see that {a x (A), B} ǫ → 0 when |x| → +∞. By reasoning as in Lemma 8.2 of [6], we see that…”

Section: The Disordered Algebra Of the Observablesmentioning

confidence: 92%

“…the spatial translations. The proposition now follows by applying the reasoning in the proof of proposition 8.1 of [6] to the time translations and spatial translations on the disorder algebra A.…”

Section: The Disordered Algebra Of the Observablesmentioning

confidence: 99%

“…the spatial translations if it is not automatically specified. 6 For continuous dynamical systems, one uses in (4.2) the natural modification M of the Cesaro mean given on bounded measurable functions, given by…”

Section: Statesmentioning

confidence: 99%

“…Investigations concerning the existence of dynamics have been made in the past and, more recently, the equilibrium statistical mechanics of such systems including the thermodynamic limits have been studied. We refer the reader to [6,7,37,38] and the literature cited therein, for a systematic treatment of the topic.…”

Section: Introductionmentioning

confidence: 99%

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Disordered Fermions on Lattices and Their Spectral Properties

Barreto¹,

Fidaleo

2011

J Stat Phys

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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.

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Heat Production of Noninteracting Fermions Subjected to Electric Fields

Bru

Pedra

Hertling

2014

Comm Pure Appl Math

122

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Electric resistance in conducting media is related to heat (or entropy) production in the presence of electric fields. In this paper, by using Araki's relative entropy for states, we mathematically define and analyze the heat production of free fermions within external potentials. More precisely, we investigate the heat production of the nonautonomous C -dynamical system obtained from the fermionic second quantization of a discrete Schrödinger operator with bounded static potential in the presence of an electric field that is time-and spacedependent. It is a first preliminary step towards a mathematical description of transport properties of fermions from thermal considerations. This program will be carried out in several papers. The regime of small and slowly varying in space electric fields is important in this context and is studied the present paper. We use tree-decay bounds of the n-point, n 2 2N, correlations of the many-fermion system to analyze this regime. We verify below the first law of thermodynamics for the system under consideration. The latter implies, for systems doing no work, that the heat produced by the electromagnetic field is exactly the increase of the internal energy resulting from the modification of the (infinite volume) state of the fermion system. The identification of heat production with an energy increment is, among otheg things, technically convenient. We initially focus our study on noninteracting (or free) fermions, but our approach will be later applied to weakly interacting fermions.

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Equilibrium Statistical Mechanics of Fermion Lattice Systems

Cited by 83 publications

References 45 publications

Joint Extension of States of Subsystems for a CAR System

Joint Extension of States of Subsystems for a CAR System

Disordered Fermions on Lattices and Their Spectral Properties

Heat Production of Noninteracting Fermions Subjected to Electric Fields

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