### Abstract

We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the non-commutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. Its proof applies to spin lattice systems as well, yielding a vast improvement over known results. All formulations are in terms of a C*-dynamical systems for the Fermion (CAR) algebra A with all or a part of the following assumptions: (I) The interaction is even, namely, the dynamics α_{t} commutes with the even-oddness automorphism Θ. (Automatically satisfied when (IV) is assumed.) (II) The domain of the generator δ_{α} of α_{t} contains the set A_{o} of all strictly local elements of A. (III) The set A_{o} is the core of δ_{α}. (IV) The dynamics at commutes with lattice translation automorphism group τ of A. A major technical tool is the conditional expectation from A onto its C*-subalgebras A(I) for any subset I of the lattice, which induces a system of commuting squares. This technique overcomes the lack of tensor product structures for Fermion systems and even simplifies many known arguments for spin lattice systems. In particular, this tool is used for obtaining the isomorphism between the real vector space of all *-derivations with their domain A_{o}, commuting with Θ, and that of all Θ-even standard potentials which satisfy a specific norm convergence condition for the one point interaction energy. This makes it possible to associate a unique standard potential to every dynamics satisfying (I) and (II). The convergence condition for the potential is a consequence of its definition in terms of the *-derivation and not an additional assumption. If translation invariance is imposed on *-derivations and potentials, then the isomorphism is kept and the space of translation co-variant standard potentials becomes a separable Banach space with respect to the norm of the one point interaction energy. This is a crucial basis for an application of convex analysis to the equivalence proof in the major result. Everything goes in parallel for spin lattice systems without the evenness assumption (I).

Original language | English |
---|---|

Pages (from-to) | 93-198 |

Number of pages | 106 |

Journal | Reviews in Mathematical Physics |

Volume | 15 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2003 Apr |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Reviews in Mathematical Physics*,

*15*(2), 93-198. https://doi.org/10.1142/S0129055X03001606

**Equilibrium statistical mechanics of Fermion lattice systems.** / Araki, Huzihiro; Moriya, Hajime.

Research output: Contribution to journal › Article

*Reviews in Mathematical Physics*, vol. 15, no. 2, pp. 93-198. https://doi.org/10.1142/S0129055X03001606

}

TY - JOUR

T1 - Equilibrium statistical mechanics of Fermion lattice systems

AU - Araki, Huzihiro

AU - Moriya, Hajime

PY - 2003/4

Y1 - 2003/4

N2 - We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the non-commutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. Its proof applies to spin lattice systems as well, yielding a vast improvement over known results. All formulations are in terms of a C*-dynamical systems for the Fermion (CAR) algebra A with all or a part of the following assumptions: (I) The interaction is even, namely, the dynamics αt commutes with the even-oddness automorphism Θ. (Automatically satisfied when (IV) is assumed.) (II) The domain of the generator δα of αt contains the set Ao of all strictly local elements of A. (III) The set Ao is the core of δα. (IV) The dynamics at commutes with lattice translation automorphism group τ of A. A major technical tool is the conditional expectation from A onto its C*-subalgebras A(I) for any subset I of the lattice, which induces a system of commuting squares. This technique overcomes the lack of tensor product structures for Fermion systems and even simplifies many known arguments for spin lattice systems. In particular, this tool is used for obtaining the isomorphism between the real vector space of all *-derivations with their domain Ao, commuting with Θ, and that of all Θ-even standard potentials which satisfy a specific norm convergence condition for the one point interaction energy. This makes it possible to associate a unique standard potential to every dynamics satisfying (I) and (II). The convergence condition for the potential is a consequence of its definition in terms of the *-derivation and not an additional assumption. If translation invariance is imposed on *-derivations and potentials, then the isomorphism is kept and the space of translation co-variant standard potentials becomes a separable Banach space with respect to the norm of the one point interaction energy. This is a crucial basis for an application of convex analysis to the equivalence proof in the major result. Everything goes in parallel for spin lattice systems without the evenness assumption (I).

AB - We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the non-commutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. Its proof applies to spin lattice systems as well, yielding a vast improvement over known results. All formulations are in terms of a C*-dynamical systems for the Fermion (CAR) algebra A with all or a part of the following assumptions: (I) The interaction is even, namely, the dynamics αt commutes with the even-oddness automorphism Θ. (Automatically satisfied when (IV) is assumed.) (II) The domain of the generator δα of αt contains the set Ao of all strictly local elements of A. (III) The set Ao is the core of δα. (IV) The dynamics at commutes with lattice translation automorphism group τ of A. A major technical tool is the conditional expectation from A onto its C*-subalgebras A(I) for any subset I of the lattice, which induces a system of commuting squares. This technique overcomes the lack of tensor product structures for Fermion systems and even simplifies many known arguments for spin lattice systems. In particular, this tool is used for obtaining the isomorphism between the real vector space of all *-derivations with their domain Ao, commuting with Θ, and that of all Θ-even standard potentials which satisfy a specific norm convergence condition for the one point interaction energy. This makes it possible to associate a unique standard potential to every dynamics satisfying (I) and (II). The convergence condition for the potential is a consequence of its definition in terms of the *-derivation and not an additional assumption. If translation invariance is imposed on *-derivations and potentials, then the isomorphism is kept and the space of translation co-variant standard potentials becomes a separable Banach space with respect to the norm of the one point interaction energy. This is a crucial basis for an application of convex analysis to the equivalence proof in the major result. Everything goes in parallel for spin lattice systems without the evenness assumption (I).

UR - http://www.scopus.com/inward/record.url?scp=0037737896&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037737896&partnerID=8YFLogxK

U2 - 10.1142/S0129055X03001606

DO - 10.1142/S0129055X03001606

M3 - Article

AN - SCOPUS:0037737896

VL - 15

SP - 93

EP - 198

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 2

ER -