Equilibrium states for mean field models

Fannes, Mark; Spohn, Herbert; Verbeure, André

doi:10.1063/1.524422

Cited by 97 publications

(92 citation statements)

References 9 publications

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“…There have been many applications of the de Finetti theorem to topics including foundational issues [7,26], mathematical physics [17,27] and quantum information theory [10,18,28,29,30,31]; there have also been various generalisations [3,4,5,6,7,9,10,15,16,17]. We…”

Section: Discussionmentioning

confidence: 99%

One-and-a-Half Quantum de Finetti Theorems

Christandl

König

Mitchison

et al. 2007

Commun. Math. Phys.

174

294

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When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by 2 kd 2 n , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation Uµ+ν ⊂ Uµ ⊗ Uν of the unitary group U(d), for highest weights µ, ν and µ + ν. Let ξµ be the state obtained by tracing out Uν. Then ξµ is close to a convex combination of the coherent states Uµ(g)|vµ , where g ∈ U(d) and |vµ is the highest weight vector in Uµ.For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our "half" theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.

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Section: Discussionmentioning

confidence: 99%

One-and-a-Half Quantum de Finetti Theorems

Christandl

König

Mitchison

et al. 2007

Commun. Math. Phys.

174

294

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“…The paper [10] addresses general mean field quantum spin systems in their C * algebraic representation. Relying on an operator theoretic version of deFinetti's Theorem (i.e.…”

Section: Introductionmentioning

confidence: 99%

The Phase Diagram of the Quantum Curie-Weiss Model

et al. 2008

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This paper studies a generalization of the Curie-Weiss model (the Ising model on a complete graph) to quantum mechanics. Using a natural probabilistic representation of this model, we give a complete picture of the phase diagram of the model in the parameters of inverse temperature and transverse field strength. Further analysis computes the critical exponent for the vanishing of the order parameter in the approach to the critical curve and gives useful stability properties for a variational problem associated with the representation.

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“…All results in this paper concern the BCS-model, and rely very much on the meanfield character [14] of this model: all the states under consideration are chosen within the set of symmetric product states, the extension of these results to other mean-field models [12] is straight-forward. However, this work should be considered as a prototype model of a scheme which can be generalised to bona fide interacting systems.…”

Section: Stability-instability Properties Of the Intermediate Statesmentioning

confidence: 99%

“…It is immediately checked that L φ is dissipative (14) and satisfies the detailed balance conditions in the state ω φ with fixed phase φ, i.e.…”

Section: The Perturbed Statesmentioning

confidence: 99%

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Metastability in the BCS model

Lauwers¹,

Verbeure²

2001

J. Phys. A: Math. Gen.

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Abstract. We discuss metastable states in the mean-field version of the strong coupling BCS-model and study the evolution of a superconducting equilibrium state subjected to a dynamical semi-group with Lindblad generator in detailed balance w.r.t. another equilibrium state. The intermediate states are explicitly constructed and their stability properties are derived. The notion of metastability in this genuine quantum system, is expressed by means of energy-entropy balance inequalities and canonical coordinates of observables.

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Equilibrium states for mean field models

Abstract: We rigorously characterize the KMS and the limiting Gibbs states for mean field models. As an application we prove the convergence of the Gibbs states for the Dicke Maser model in the infinite volume limit.

Cited by 97 publications

References 9 publications

One-and-a-Half Quantum de Finetti Theorems

One-and-a-Half Quantum de Finetti Theorems

The Phase Diagram of the Quantum Curie-Weiss Model

Metastability in the BCS model

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