Quantum vorticity at positive temperature for spins systems with continuous symmetry

D Minenkov,M Rouleux

doi:10.1088/1742-6596/804/1/012031

Abstract

We propose a definition of vorticity at inverse temperature β for Gibbs states in quantum XY or Heisenberg spin systems on the lattice by testing exp[−βH] on a complete set of observables (“one-point functions”). Imposing a compression of Pauli matrices at the boundary, which stands for the classical environment, we perform some numerical simulations on finite lattices in case of XY model, which exhibit usual vortex patterns.

Highlights

Consider the quantum XY or Heisenberg spin model for S = 1/2 on the 2-D lattice Z2, with nearest neighbor interactions
Marmin-Wagner, and Hohenberg theorems tell us that Gibbs states, for all inverse temperature β, are invariant under simultaneous rotation of spins
We know a bit more : there is a unique Gibbs state, with rotational symmetry, which rules out the existence of first order transitions, a particular form for phase transition exists, characterized by a change of behavior in the correlation functions

Summary

Introduction

We shall define vorticity at inverse temperature β by decomposing the linear form ωβ on a canonical (orthonormal) basis of observables. 1. Vorticity matrices Gibbs state (1) for spin S = 1/2 systems, as a linear form on the C∗-algebra of observables. The simplest way is to restrict to “one-point functions”, i.e. the set O ⊂ O of 2N × 2N , block-diagonal 2 × 2 matrices (Pauli matrices are spin representations of SU(2) of dimension 2S + 1 = 2), supported on individual sites of Λ ∪ ∂Λ, N = |Λ ∪ ∂Λ|. Example 2: OR is the (real) algebra generated by Pauli matrices (Di)i∈Λ∪∂Λ with diagonal block supported on site i that takes values in {Id, iσx, iσy, iσz}. Definition 1.2: We call vorticity matrix at site i, relative to the basis b, at inverse temperature β, the matrix : Ωiβ (b) tr1(e−βH Bi) tr(e−βH )

The traceless matrix

We have

Given degree