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Journal of Physics A Mathematical and General

Non-Abelian gauge theory for quantum Heisenberg antiferromagnetic chain

Chigak Itoi, Hisamitsu Mukaida

Published Jul 7, 1994• 14 citations
Publisher

Abstract

A continuum SU(2S)*U(1) gauge theory for a Heisenberg antiferromagnet with arbitrary spin S is obtained. We show that the level-1 SU(2) WZW model with a certain marginally irrelevant perturbation describes the low-energy phenomena of the Heisenberg model with spin-1/2. The model with spins other than 1/2 can be described in the level-2S SU(2) WZW model with a certain relevant perturbation. We give a qualitative explanation of the Haldane conjecture for Heisenberg antiferromagnets.

DOI
10.1088/0305-4470/27/13/042

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A continuum SU(2S)*U(1) gauge theory for a Heisenberg antiferromagnet with arbitrary spin S is obtained. We show that the level-1 SU(2) WZW model with a certain marginally irrelevant perturbation describes the low-energy phenomena of the Heisenberg model with spin-1/2. The model with spins other than 1/2 can be described in the level-2S SU(2) WZW model with a certain relevant perturbation. We give a qualitative explanation of the Haldane conjecture for Heisenberg antiferromagnets.
We study the long-wavelength limit of a spin SSS Heisenberg antiferromagnetic chain. The fermionic Lagrangian obtained corresponds to a perturbed level 2S2S2S SU(2)\mathrm{SU}(2)SU(2) Wess-Zumino-Witten (WZW) model. This effective theory is then mapped into a compact U(1)U(1)U(1) boson interacting with Z2S{Z}_{2S}Z2S​ parafermions. The analysis of this effective theory allows us to show that when SSS is an integer there is a mass gap to all excitations, whereas this gap vanishes in the half-odd-integer spin case and the SU(2)2S{\mathrm{SU}(2)}_{2S}SU(2)2S​ WZW model flows towards the SU(2)1{\mathrm{SU}(2)}_{1}SU(2)1​ stable fixed point. This gives a field theory treatment of the so-called Haldane's conjecture for arbitrary values of the spin SSS.
The low-energy spin excitations in the flux phase of the (2+1)-dimensional large-nnn SU(n)\mathrm{SU}(n)SU(n)Heisenberg antiferromagnet are relativistic fermions coupled to a U(1) gauge field. These excitations are studied in the physical n=2n=2n=2 case with the use of a 1n\frac{1}{n}n1​ expansion. A Chern-Simons term is included to remove gauge configurations forbidden in the n=2n=2n=2 theory.
We find that the nature of the gapless low-energy excitations of the spin 1/2 isotropic Heisenberg chain with J < 0 (antiferromagnetic chain at magnetic field H = 0) is different in the usual SU(2) "phase" and in the finite-magnetic field "phase" of symmetry U(1). This is shown to have relevant consequences for the microscopic description of the low-energy physics, which is different from the usual "spinon" two-kink picture. The use of the until now hidden "Landau-liquid" character of the model reveals a number of unexpected aspects of its physics and deepens the microscopic understanding of this fundamental quantum system.
In this thesis we study various one-dimensional quantum spin systems with SU(2) and SU(N) symmetry. We investigate the short-distance behavior of the SU(2) Heisenberg model in the limit of large spin and show that there exists an extended regime where perturbation theory, in the form of spin-wave theory, can be successfully applied. The reason for this perturbative regime stems from the asymptotic freedom of the nonlinear sigma model onto which the Heisenberg model can be mapped. When considering observables which respect the rotational invariance of the model, we observe a cancellation of infrared divergences in the perturbative expansions, leading to a meaningful description of correlation functions. We then turn to the study of SU(N) models. Building on the representation theory of the SU(N) group and on the matrix product state (MPS) formalism, we introduce a generic method to construct Affleck-Kennedy-Lieb-Tasaki (AKLT) models having edge states described by any self-conjugate irreducible representation (irrep) of SU(N). A simple example is given by a spin-1 AKLT model having spin-1 edge states. The phase transition between this model and the original AKLT model is shown to be continuous and to correspond to a topological phase transition described by the SU(2) level-1 Wess-Zumino-Witten conformal field theory universality class. In addition we study an SU(3) AKLT model for the 3-box symmetric irrep at each site. We demonstrate that the edge states are adjoint edge irreps, we extract its correlation length and provide a useful construction as an optimal MPS. Finally, we develop a density matrix renormalization group algorithm based on standard Young tableaus and subduction coefficients to make full use of the non-abelian symmetry and to investigate the SU(3) Heisenberg model with 3-box symmetric irrep at each site. We show that the model has a finite gap above the singlet ground state, in agreement with an extension of the Haldane conjecture to SU(3) chains in the fully symmetric irreps. We also argue that there are five branches of elementary excitations living in four different irreps, each of which is gapped.
According to Haldane’s conjecture, there is a spin gap of the excitation spectrum in integer Heisenberg spin chains whereas half-integer spin chains have a gapless excitation spectrum. In an earlier publication (Hung and Gong 2005 Phys. Rev. B 71 054413), the Haldane gap and other properties of the spin-1 system are explained using a spin-1/2 alternating bond chain model. The ferromagnetic–ferromagnetic–antiferromagnetic Heisenberg spin-1/2 chain is expected to behave similarly to the uniform spin-3/2 Heisenberg chain for strong ferromagnetic coupling. And the general nonuniform spin-1/2 chain may be used to explain the properties of the integer and half-integer high spin- S chains, consistently. In this paper, we provide a detailed numerical verification for the above problems.
The double chain Heisenberg antiferromagnet with anisotropy is studied using the mapping onto the coupled anisotropic nonlinear σ model. This model consists of two anisotropic antiferromagnetic quantum Heisenberg chains with half-integer spin coupled with each other via the ferromagnetic interchain coupling J '. For small J ', our analysis suggests that J ' is an irrelevant parameter and consequently the two chains behave independently and no gap is generated. On the other hand, for large enough J ', the two chains behave as a single Heisenberg chain with integer spin which has a gap in the excitation spectrum. The expected behavior in the isotropic limit is also discussed.