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Explicit conserved operators for a class of integrable bosonic networks from the classical Yang-Baxter equation

Phillip S. Isaac, Jon Links, Inna Lukyanenko, Jason L. Werry

Abstract

Let $B$ denote the weighted adjacency matrix of a balanced, symmetric, bipartite graph. We define a class of bosonic networks given by Hamiltonians whose hopping terms are determined by $B$. We show that each quantum Hamiltonian is Yang-Baxter integrable, admitting a set of mutually commuting operators derived through a solution of the classical Yang-Baxter equation. We discuss some applications and consequences of this result.

Explicit conserved operators for a class of integrable bosonic networks from the classical Yang-Baxter equation

Abstract

Let denote the weighted adjacency matrix of a balanced, symmetric, bipartite graph. We define a class of bosonic networks given by Hamiltonians whose hopping terms are determined by . We show that each quantum Hamiltonian is Yang-Baxter integrable, admitting a set of mutually commuting operators derived through a solution of the classical Yang-Baxter equation. We discuss some applications and consequences of this result.

Paper Structure

This paper contains 13 sections, 106 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie algebras and related Poisson algebras
  4. Classical Yang-Baxter equation for generating commutative Poisson subalgebras
  5. Shift elements and realisation of the commutative Poisson subalgebra
  6. Quantisation
  7. Results
  8. Solution of the classical Yang-Baxter equation, shift element, and the Hamiltonian
  9. Commutative Poisson subalgebra
  10. Conserved operators from the commutative Poisson subalgebra
  11. An example
  12. Discussion
  13. Appendix - Commutativity of the transfer matrices

Figures (1)

  • Figure 1: Graphical presentation of the complete bipartitie graph $K_{2,2}$. The vertex sets coloured blue and teal are assigned labels 1 and 2. Weights are assigned to the edges such that the associated matrix $\mathcal{B}$ given by (\ref{['m2b']}) is symmetric, leading to the Hamiltonian (\ref{['4ham']}).

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Definition 3