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Annales De L Institut Henri Poincare-physique Theorique

Translation invariant quasi-free states and Bogoliubov transformations

F. Rocca, M. Sirugue, D. Testard

Published Jan 1, 1969• 2 citations
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Related papers

Quasi-free states of the C.C.R.—Algebra and Bogoliubov transformations

J. Manuceau, André F. Verbeure

Communications in Mathematical Physics•Dec 1, 1968•145 citations

The invariant quasi-free states

Sergio A Albeverio, Raphael J. Høegh-Krohn

Lecture notes in mathematics•Jan 1, 1976•0 citations

Among Quadratic Hamiltonians, Bogoliubov Transformations and Non-Regular States on CCRs *-Algebra. I. Pure and Invariant States. (In the Mood for the Manuceau Verbeure Theorems about Quasi-free States and Automorphisma of the CCR Algebra)

Sergej A. Choroszavin

The features of the paper are these: 1) we discuss {\bf especially} quadratic (alias bilinear) Bose-Hamiltonians, the related Bogoliubov transformations and {\bf especially} quasi-free-like (alias coherent or Fock-like) states 2) we discuss {\bf any} quadratic Bose-Hamiltonians and Bogoliubov transformations, whether diagonalizable or not, whether proper or improper, and {\bf arbitrary} quasi-free-like states, whether regular or non-regular they are 3) we associate notions and terms of the CCRs theory {CCRs = Canonical Commutation Relations} with notions and terms of the indefinite inner product spaces theory. Then, we apply the corresponding `bilingual dictionary' so as to construct invariant states of some of the quadratic Hamiltonians.

arXiv:math-ph/0301027•Jan 20, 2003•0 citations

TRANSLATION-INVARIANT QUASI-FREE STATES AND BOGOLYUBOV TRANSFORMATIONS.

F. Rocca, M. Sirugue, D. Testard

•Jan 1, 1968•0 citations

Entropy Maximization and Weak Gibbsianity of Quasi-Free Fermionic States

Vojkan Jakšić, Claude-Alain Pillet, Anna Szczepanek

In their 1972 study of approach to equilibrium, Lanford and Robinson showed that gauge-invariant quasi-free states of lattice fermions maximize entropy among all translation-invariant states with a fixed two-point function, and suggested that the maximizer is unique. In subsequent work on this topic, the uniqueness question re-emerged, together with the problem of whether such quasi-free states are weak Gibbs states. We provide a positive answer to both questions within a class of states whose momentum-space two-point function $\widehat C$ satisfies $0<\widehat C(k)<1$ and belongs to the Wiener algebra of the Brillouin zone. The proof reveals that both the entropy maximization principle and weak Gibbsianity follow directly from the thermodynamic formalism for lattice fermions.

math-phCross-listedarXiv:2603.14020•Mar 17, 2026

Translation-invariant quasi-free states for fermionic systems and the BCS approximation

Gerhard Bräunlich, Christian Hainzl, Robert Seiringer

We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but also includes direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity.

arXiv:1305.5135•Reviews in Mathematical Physics•Jul 6, 2014•17 citations
The features of the paper are these: 1) we discuss {\bf especially} quadratic (alias bilinear) Bose-Hamiltonians, the related Bogoliubov transformations and {\bf especially} quasi-free-like (alias coherent or Fock-like) states 2) we discuss {\bf any} quadratic Bose-Hamiltonians and Bogoliubov transformations, whether diagonalizable or not, whether proper or improper, and {\bf arbitrary} quasi-free-like states, whether regular or non-regular they are 3) we associate notions and terms of the CCRs theory {CCRs = Canonical Commutation Relations} with notions and terms of the indefinite inner product spaces theory. Then, we apply the corresponding `bilingual dictionary' so as to construct invariant states of some of the quadratic Hamiltonians.
In their 1972 study of approach to equilibrium, Lanford and Robinson showed that gauge-invariant quasi-free states of lattice fermions maximize entropy among all translation-invariant states with a fixed two-point function, and suggested that the maximizer is unique. In subsequent work on this topic, the uniqueness question re-emerged, together with the problem of whether such quasi-free states are weak Gibbs states. We provide a positive answer to both questions within a class of states whose momentum-space two-point function C^\widehat CC satisfies 0<C^(k)<10<\widehat C(k)<10<C(k)<1 and belongs to the Wiener algebra of the Brillouin zone. The proof reveals that both the entropy maximization principle and weak Gibbsianity follow directly from the thermodynamic formalism for lattice fermions.
We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but also includes direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity.