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The Lee-Yang property of isotropic vector ferromagnets and lattice fields

Yuri Kozitsky

Abstract

The Lee-Yang property of a given spin model means that its partition function has purely imaginary zeros as a function of an external magnetic field. A similar property is also used in the theory of quantum anharmonic crystals and quantum lattice fields. A number of powerful analytic methods of the mathematical theory of such models employ this property. Its suitable generalization is used in the theory of models of isotropic $D$-dimensional spins (rotors) or $D$-component quantum lattice fields. So far, the (generalized) Lee-Yang property has been established only for two-dimensional isotropic models. In this work, we prove that isotropic spin and field models living on $\mathds{Z}$ have this property for all even $D$.

The Lee-Yang property of isotropic vector ferromagnets and lattice fields

Abstract

The Lee-Yang property of a given spin model means that its partition function has purely imaginary zeros as a function of an external magnetic field. A similar property is also used in the theory of quantum anharmonic crystals and quantum lattice fields. A number of powerful analytic methods of the mathematical theory of such models employ this property. Its suitable generalization is used in the theory of models of isotropic -dimensional spins (rotors) or -component quantum lattice fields. So far, the (generalized) Lee-Yang property has been established only for two-dimensional isotropic models. In this work, we prove that isotropic spin and field models living on have this property for all even .
Paper Structure (5 sections, 5 theorems, 60 equations)

This paper contains 5 sections, 5 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The Result
  3. Strongly isotropic measures
  4. The statement
  5. The Proof

Key Result

Lemma 2.2

For some $D\in \mathds{N}$, let $v_\tau(\cdot, D)\in \mathcal{L}$. Then $v_\tau(\cdot, D+2m)\in \mathcal{L}^{(m)}$ for all $m\in \mathds{N}_0$.

Theorems & Definitions (7)

  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Lemma 3.1
  • proof