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Journal of Physics A Mathematical and General

The modified XXZ Heisenberg chain, conformal invariance and the surface exponents of c<1 systems

Francisco Castilho Alcaraz, Michael Baake, Uwe Grimm, V. Rittenberg

Published Jan 7, 1989• 33 citations
Publisher

Abstract

The authors present the operator content of the XXZ chain in the presence of external fields. The finite-size scaling spectra can be expressed in terms of irreps of shifted U(1) Kac-Moody algebras, the shift depending on the external fields. Using an explicit Feigin-Fuchs construction (1982), they project out from these spectra the operator content of c<1 systems with free, fixed and mixed boundary conditions. They also show how this projection mechanism works for a finite number of sites.

DOI
10.1088/0305-4470/22/1/002

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The authors present the operator content of the XXZ chain in the presence of external fields. The finite-size scaling spectra can be expressed in terms of irreps of shifted U(1) Kac-Moody algebras, the shift depending on the external fields. Using an explicit Feigin-Fuchs construction (1982), they project out from these spectra the operator content of c<1 systems with free, fixed and mixed boundary conditions. They also show how this projection mechanism works for a finite number of sites.
The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.
Based on numerical analysis, the authors conjecture the operator content of the finite-size limit of the spectra of the Ashkin-Teller model with free boundary conditions. The same operator content is obtained from a Hamiltonian with a four-fermion interaction and a U(1) Kac-Moody Sugawara structure. For some special values of the coupling constant the model exhibits N=2 superconformal and Zamolodchikov-Fateev invariance. The operator content in these cases is expressed in terms of irreducible representations of the corresponding algebras.
The finite-size energy spectrum of the anisotropic Heisenberg chain in an external magnetic field is calculated for free and twisted boundary conditions. As with periodic boundary conditions, it is found that the spectra exhibit non-analytical terms which do not fit into the form predicted on the basis of conformal invariance unless extra commensurability conditions between the size of the system and the external field are introduced. Taking these conditions into account some scaling dimensions for associated models are derived.
Eigenspectra of the critical quantum Ashkin-Teller and Potts chains with free boundaries can be obtained from that of the XXZ chain with free boundaries and a complex surface field. By deriving and solving numerically the Bethe ansatz equations for such boundaries the authors obtain eigenenergies of XXZ chains of up to 512 sites. The conformal anomaly and surface exponents of the quantum XXZ, Ashkin-Teller, and Potts chains are calculated by exploiting their relations with the mass gap amplitudes as predicted by conformal invariance.