The Gibbs Variational Principle for General BCS-Type Models

Raggio, G. A.; Werner, R. F.

doi:10.1209/0295-5075/9/7/004

Cited by 14 publications

(32 citation statements)

References 8 publications

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“…It is maybe surprising that, in comparison, very little is known about its quantum counterpart: We are only aware of refs. 4,5,15,21,27,29,30,32,33, and none of these results are very general in nature. The ones that are closest to the present work are refs.…”

Section: Introductionmentioning

confidence: 93%

Large Deviations in Quantum Lattice Systems: One-Phase Region

Lenci

Rey-Bellet

2005

J Stat Phys

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Section: Introductionmentioning

confidence: 93%

Large Deviations in Quantum Lattice Systems: One-Phase Region

Lenci

Rey-Bellet

2005

J Stat Phys

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“…See, e.g., Chapter VII, Section 4 in [8]. Even though the analysis of the thermodynamics of the BCS Hamiltonian was rigorously performed in the eighties [9,10] (see also the innovating work of Bernadskii and Minlos in 1972 [11]), generalizations of the strong coupling approximation of the BCS model are still subject of research. For instance, strong coupling-BCS-type models with superconducting phases at arbitrarily high temperatures are treated in [12].…”

Section: Introductionmentioning

confidence: 99%

“…Note that D ω|U N is the density matrix associated to the state ω restricted on the local CAR C * -algebra U N ≃ B C ΛN ×{↑,↓} (isomorphism). Such a derivation of the pressure as a minimization problem over states on a C * -algebras are also performed for various quantum spin systems, see, e.g., [19,20,21,22,23].…”

Section: Introductionmentioning

confidence: 99%

“…8 A face F of a compact convex set K is subset of K with the property that if ω = Σ m n=1 λnωn ∈ F with Σ m n=1 λn = 1 and {ωn} m n=1 ⊂ K, then {ωn} m n=1 ⊂ F. 9 This means that there is no isomorphism between h j 1 and h j 2 whenever h j 1 and h j 2 are the Hilbert spaces corresponding to two different irreducible representations. 10 This means that the Hamiltonian can be seen as an operator acting on several Hilbert spaces {h j } j∈J with no (non-trivial) invariant subspace.…”

Section: Introductionmentioning

confidence: 99%

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EFFECT OF A LOCALLY REPULSIVE INTERACTION ON s-WAVE SUPERCONDUCTORS

2010

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The thermodynamic impact of the Coulomb repulsion on s-wave superconductors is analyzed via a rigorous study of equilibrium and ground states of the strong coupling BCS-Hubbard Hamiltonian. We show that the one-site electron repulsion can favor superconductivity at fixed chemical potential by increasing the critical temperature and/or the Cooper pair condensate density. If the one-site repulsion is not too large, a first or a second order superconducting phase transition can appear at low temperatures. The Meißner effect is shown to be rather generic but coexistence of superconducting and ferromagnetic phases is also shown to be feasible, for instance near half-filling and at strong repulsion. Our proof of a superconductorMott insulator phase transition implies a rigorous explanation of the necessity of doping insulators to create superconductors. These mathematical results are consequences of "quantum large deviation" arguments combined with an adaptation of the proof of Størmer's theorem [1] to even states on the CAR algebra.

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“…The proofs for the existence of the specific free energy (and the more for the equilibrium states [11][12][13], and dynamics [14]) in the thermodynamic limit [15,16], were originally very involved. A unifying treatment making systematic use of the permutation invariance and the variational principle of the free energy functional seems to have been first rigorously elaborated in the unpublished thesis of Fleig [17], Further developments in [3, 18,19] have enlarged the model class enormously.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamic Formalism and Phase Transitions of Generalized Mean-Field Quantum Lattice Models

Gerisch

Rieckers²,

Volkert³

1998

Zeitschrift Für Naturforschung A

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The general structure of thermodynamic equilibrium states for a class of quantum mechanical (multi-lattice) systems is elaborated, combining quantum statistical and thermodynamical methods. The quantum statistical formulation is performed in terms of recent operator algebraic concepts emphasizing the role of the permutation symmetry due to homogeneous coarse graining and employing the internal symmetries. The variational principle of the free energy functional is derived, which determines together with the symmetries the general form of the limiting Gibbs states in terms of their central decomposition. The limiting minimal free energy density and its possible equilibrium states are analyzed on various levels of the description by means of convex analysis, where the Fenchel transforms of the free energies provide entropy like potentials. On the thermodynamic level a modified entropy surface is obtained, which specifies only in combination with its concave envelope the regions of pure and mixed phase states. The symmetry properties of a certain model allow to specify the (non-) differentiability of the minimal free energy density. A characterization and classification of phase transitions in terms of quantum statistical equilibrium states is proposed, and the connection to the Landau theory is established demonstrating that the latter implies a (continuous) deformation of the sets of equilibrium states along a canonically given curve.

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The Gibbs Variational Principle for General BCS-Type Models

Cited by 14 publications

References 8 publications

Large Deviations in Quantum Lattice Systems: One-Phase Region

Large Deviations in Quantum Lattice Systems: One-Phase Region

EFFECT OF A LOCALLY REPULSIVE INTERACTION ON s-WAVE SUPERCONDUCTORS

Thermodynamic Formalism and Phase Transitions of Generalized Mean-Field Quantum Lattice Models

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