Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice

2000 ◽  
Vol 123 (1) ◽  
pp. 489-493 ◽  
Author(s):  
F. M. Mukhamedov
Keyword(s):  
Ising Model ◽  
Von Neumann Algebras ◽  
Bethe Lattice ◽  
Gibbs States ◽  
Translation Invariant ◽  
Von Neumann

scholarly journals Phase transition and Gibbs measures of Vannimenus model on semi-infinite Cayley tree of order three

2017 ◽  
Vol 31 (13) ◽  
pp. 1750093 ◽  
Author(s):  
Hasan Akın
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Cayley Tree ◽  
Gibbs Measures ◽  
Bethe Lattice ◽  
Nearest Neighbors ◽  
Translation Invariant ◽  
New Approach ◽  
Gibbs Distributions

Ising model with competing nearest–neighbors (NN) and prolonged next–nearest–neighbors (NNN) interactions on a Cayley tree has long been studied, but there are still many problems untouched. This paper tackles new Gibbs measures of Ising–Vannimenus model with competing NN and prolonged NNN interactions on a Cayley tree (or Bethe lattice) of order three. By using a new approach, we describe the translation-invariant Gibbs measures (TIGMs) for the model. We show that some of the measures are extreme Gibbs distributions. In this paper, we try to determine when phase transition does occur.

CONSTRUCTION OF DIRICHLET FORMS ON STANDARD FORMS OF VON NEUMANN ALGEBRAS

2000 ◽  
Vol 03 (01) ◽  
pp. 1-14 ◽  
Author(s):  
YONG MOON PARK
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Admissible Function ◽  
Standard Form ◽  
Dirichlet Forms ◽  
Quantum Spin ◽  
Quantum Spin Systems ◽  
Translation Invariant ◽  
Von Neumann ◽  
Markovian Semigroups

For a von Neumann algebra ℳ acting on a Hilbert space ℋ with a cyclic and separating vector ξ0, we give an explicit expression of Dirichlet forms on the natural standard form [Formula: see text] associated with the pair (ℳ, ξ0). For any self-adjoint analytic element x of ℳ and an admissible function f, we construct a (bounded) Dirichlet form which generates a symmetric Markovian semigroup on ℋ. We then apply our result to construct translation invariant, symmetric, Markovian semigroups for quantum spin systems with finite range interactions.

Lectures on von Neumann Algebras

2019 ◽  
Author(s):  
Serban-Valentin Stratila ◽  
Laszlo Zsido
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann

SPIN ONE ISING MODEL WITH COMPETING INTERACTIONS ON THE BETHE LATTICE

1988 ◽  
Vol 49 (C8) ◽  
pp. C8-1031-C8-1032
Author(s):  
S. Coutinho ◽  
C. R. da Silva
Keyword(s):  
Ising Model ◽  
Bethe Lattice ◽  
Competing Interactions
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