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arXiv:1709.06602 · Journal of Physics A Mathematical and Theoretical

The bilinear–biquadratic model on the complete graph

Dávid Jakab, Gergely Szirmai, Zoltán Zimborás

Published Jan 19, 2018• 9 citations
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Abstract

Abstract We study the spin-1 bilinear–biquadratic model on the complete graph of N sites, i.e. when each spin is interacting with every other spin with the same strength. Because of its complete permutation invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir operators of s u (3) and s u (2). Using group representation theory, we explicitly diagonalize the Hamiltonian and map out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, is obtained analytically for any number of sites.

DOI
10.1088/1751-8121/aaa92b

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Abstract We study the spin-1 bilinear–biquadratic model on the complete graph of N sites, i.e. when each spin is interacting with every other spin with the same strength. Because of its complete permutation invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir operators of s u (3) and s u (2). Using group representation theory, we explicitly diagonalize the Hamiltonian and map out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, is obtained analytically for any number of sites.
We study the spin-1 bilinear–biquadratic model on the complete graph of N sites, i.e. when each spin is interacting with every other spin with the same strength. Because of its complete permutation invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir operators of su(3) and su(2). Using group representation theory, we explicitly diagonalize the Hamiltonian and map out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, is obtained analytically for any number of sites.
Abstract We study a two-parameter family of quantum spin systems on the complete graph, which is the most general model invariant under the complex orthogonal group. In spin S=12S=\frac {1}{2}S=21​ it is equivalent to the XXZ model, and in spin S=1S=1S=1 to the bilinear-biquadratic Heisenberg model. The paper is motivated by the work of Björnberg, whose model is invariant under the (larger) complex general linear group. In spin S=12S=\frac {1}{2}S=21​ and S=1S=1S=1 we give an explicit formula for the free energy for all values of the two parameters, and for spin S>1S>1S>1 for when one of the parameters is non-negative. This allows us to draw phase diagrams and determine critical temperatures. For spin S=12S=\frac {1}{2}S=21​ and S=1S=1S=1, we give the left and right derivatives as the strength parameter of a certain magnetisation term tends to zero, and we give a formula for a certain total spin observable, and heuristics for the set of extremal Gibbs states in several regions of the phase diagrams, in the style of a recent paper of Björnberg, Fröhlich, and Ueltschi. The key technical tool is expressing the partition function in terms of the irreducible characters of the symmetric group and the Brauer algebra. The parameters considered include, and go beyond, those for which the systems have probabilistic representations as interchange processes.
We show that the ground state of the bilinear-biquadratic exchange Hamiltonian is always singlet on a connected finite lattice with an even number of sites, while it is always triplet on one with an odd number of sites, if coefficients of the bilinear term $\ensuremath{-}J$ and that of the biquadratic term $\ensuremath{-}{J}^{\ensuremath{'}}$ satisfy the condition ${J}^{\ensuremath{'}}>J>0.$ We also find that the total-spin eigenvalue Stot{S}_{\mathrm{tot}}Stot​ of the lowest-energy state in each subspace classified by M,M,M, an eigenvalue of zzz component of the total spin, depends on the parity of M,M,M, i.e., Stot=∣M∣{S}_{\mathrm{tot}}=|M|Stot​=∣M∣ for even MMM and Stot=∣M∣+1{S}_{\mathrm{tot}}=|M|+1Stot​=∣M∣+1 for odd MMM on an even number site lattice and vice versa on an odd number site lattice.
The Heisenberg model for spin-1 bosons in one dimension presents many different quantum phases, including the famous topological Haldane phase. Here we study the robustness of such phases in front of a SU(2) symmetry-breaking field as well as the emergence of unique phases. Previous studies have analyzed the effect of such uniaxial anisotropy in some restricted relevant points of the phase diagram. Here we extend those studies and present the complete phase diagram of the spin-1 chain with uniaxial anisotropy. To this aim, we employ the density-matrix renormalization group together with analytical approaches. The complete phase diagram can be realized using ultracold spinor gases in the Mott insulator regime under a quadratic Zeeman effect.