On Quasifree States of CAR and Bogoliubov Automorphisms

Araki, Huzihiro

doi:10.2977/prims/1195193913

Cited by 221 publications

(411 citation statements)

References 13 publications

Supporting

Mentioning

409

Contrasting

Unclassified

Order By: Relevance

“…Quasi-free states on self-dual CAR algebras have been introduced in [2]. We briefly review these notions.…”

Section: A Quasi-free States On Self-dual Car Algebrasmentioning

confidence: 99%

“…introduced in [2] in the framework of self-dual CAR algebras (see Appendix A). It has the properties…”

Section: Remarkmentioning

confidence: 99%

“…is built on the parts Z L , Z ✷ , and Z R according to (2). The infinite pieces Z L and Z R will play the role of thermal reservoirs to which the finite system Z ✷ is coupled by means of the perturbation H − H 0 .…”

Section: The Non-equilibrium Setting For the Xy Chainmentioning

confidence: 99%

See 2 more Smart Citations

Non-zero Entropy Density in the XY Chain Out of Equilibrium

Aschbacher

2006

Lett Math Phys

View full text Add to dashboard Cite

The von Neumann entropy density of a block of n spins is proved to be non-zero for large n in the non-equilibrium steady state of the XY chain constructed by coupling a finite cutout of the chain to the two infinite parts to its left and right which act as thermal reservoirs at different temperatures. Moreover, the non-equilibrium density is shown to be strictly greater than the density in thermal equilibrium. Mathematics Subject Classifications (2000). 46L60, 47B35, 82C10, 82C23.

show abstract

“…Quasi-free states on self-dual CAR algebras have been introduced in [2]. We briefly review these notions.…”

Section: A Quasi-free States On Self-dual Car Algebrasmentioning

confidence: 99%

“…introduced in [2] in the framework of self-dual CAR algebras (see Appendix A). It has the properties…”

Section: Remarkmentioning

confidence: 99%

See 1 more Smart Citation

Non-zero Entropy Density in the XY Chain Out of Equilibrium

Aschbacher

2006

Lett Math Phys

View full text Add to dashboard Cite

show abstract

“…The necessary and sufficient condition for 0_©+6L to be inner is that G>+-1 is in the trace class and det(W+ = l or (o 2 + + l is in the trace class and det ( -af+} = -1 by Theorem 5 of [10]. We shall exclude the first case by showing that (a?…”

Section: (6 16) W-lim Fifl(v T a T (A)) =0mentioning

confidence: 99%

“…Since 0_ is a Bogolubov transformation given by 6-, and since 1±0_ is not in the trace class (they are twice infinite projections), ©_ is not inner (Theorem 5 and Definition 8. 1 of [10]). Thus the alternative t = Q is impossible.…”

Section: (6 16) W-lim Fifl(v T a T (A)) =0mentioning

confidence: 99%

On the $XY$-Model on Two-Sided Infinite Chain

Araki¹

1984

Publ. Res. Inst. Math. Sci.

Self Cite

View full text Add to dashboard Cite

The ^Y-model on the one-dimensional lattice, infinitely extended to both directions, is studied by a method of C*-algebras. Return to equilibrium is found for any vector state in the cyclic representation of the equilibrium state.A known relation between the algebras of Pauli spins and the algebra of canonical anticommutation relations (CARs) is used to obtain an explicit solution. However the C*-algebras generated by the two sets of operators become dissociated in the thermodynamic limit of an infinite one-dimensional lattice extending in both directions (in contrast to onesided chain) and this causes a mathematical complication.In particular, we find three features different from the case of one-sided infinite chain: (1) There are no non-trivial constant observables.

show abstract

Linear Fermion Systems, Molecular Field Models, and the KMS Condition

Hemmen

1978

Fortschr. Phys.

View full text Add to dashboard Cite

We take advantage of the simplifications made possible by assuming the system to be infinite from the beginning, to study some exactly solvable models in statistical mechanics. (1) For fermi lattice systems with quadratic hamiltonians the unifying concept of our treatment is a precise formulation of the idea of a linear system. This concept makes it possible to define a dynamical matrix which incorporates all the information one needs for equilibrium and non‐equilibrium problems. As an example, we consider the alternating X Y chain. Furthermore we pay some attention to ergodicity, pointing out the similarity to the infinite harmonic crystal. (2) For molecular field models, including the BCS model, the method is to consider an arbitrary extremal homogeneous state, to construct the dynamics in the representation determined by that state, and then to use the KMS condition to single out the equilibrium state of the infinite system. A synthesis of these methods – (1) and (2) – enables us to give a new simple treatment of the BCS model and a direct analysis of its mathematical structure. In addition we show the equivalence of Kac and van der Waals (spin) systems in the so‐called van der Waals limit. Finally we indicate briefly how to deal with molecular field models with polynomial and more general interactions.

show abstract

On Quasifree States of CAR and Bogoliubov Automorphisms

Abstract: A necessary and sufficient condition for two quasifree states of CAR to be quasiequivalent is obtained. Quasif ree states is characterized as the unique KMS state of a Bogoliubov automorphism of CAR. The structure of the group of all inner Bogoliubov automorphisms of CAR is clarified.

Cited by 221 publications

References 13 publications

Non-zero Entropy Density in the XY Chain Out of Equilibrium

Non-zero Entropy Density in the XY Chain Out of Equilibrium

On the $XY$-Model on Two-Sided Infinite Chain

Linear Fermion Systems, Molecular Field Models, and the KMS Condition

Contact Info

Product

Resources

About