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Journal of Physics A Mathematical and General

Test of modular invariance for finite XXZ chains

B -D Dorfel, Burkhard Militzer

Published Oct 7, 1993• 1 citation
Publisher

Abstract

The central charge c of the Virasoro algebra is determined using its relation to the set of anomalous dimensions of the scaling fields for the critical anisotropic Heisenberg chain. The results obtained by exactly solving the Bethe ansatz for N<or=4 are compared with the standard procedure for calculating c in the whole region of anisotropy. For odd N the system is no longer modular invariant and the analogous averaging leads to an anisotropy-dependent function which has been calculated.

DOI
10.1088/0305-4470/26/19/020

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The central charge c of the Virasoro algebra is determined using its relation to the set of anomalous dimensions of the scaling fields for the critical anisotropic Heisenberg chain. The results obtained by exactly solving the Bethe ansatz for N<or=4 are compared with the standard procedure for calculating c in the whole region of anisotropy. For odd N the system is no longer modular invariant and the analogous averaging leads to an anisotropy-dependent function which has been calculated.
The eigenspectrum of the critical anisotropic Heisenberg model (or XXZ model), with arbitrary spin S in its disordered-ferroelectric regime (0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}\ensuremath{\pi}/2S) is solved by the Bethe-Ansatz method. The amplitudes of the leading finite-size corrections are calculated analytically and numerically. Using conformal invariance we give exact results for the conformal anomaly and scaling dimensions. Our results indicate that, for all spin S, the critical behavior is governed by a conformal field theory with central charge c=1.
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.
In two dimensions conformal invariance has important implications for the finite-size scaling properties of the spectra of transfer matrices and quantum chains at the critical point. Some relations between the finite-size scaling amplitudes are obtained which can be used as a test of conformal invariance and implicitly to distinguished between second-order and first-order phase transitions. The numerical values of the scaling amplitudes give the scale dimensions of various operators. Six- and eight-state self-dual quantum chains with cubic symmetries are considered at the critical point for three values of the coupling constant. The systems are found to be conformal invariant and estimates for several critical exponents are obtained. Based on the approximate values of the critical exponent one tries to find the values of the corresponding central charges of the Virasoro algebras.
The author shows that the central charge c of the Virasoro algebra is determined by the spectrum of the Hamiltonian L 0 +NL 0 corresponding to a partition function which is invariant under the subgroup of modular transformations generated by S. Using this result he discusses in detail a new possibility of determining c for a given system at criticality which turns out to give excellent estimates even if the lattices accessible to numerical calculation are very small. This enables him to predict the central charge of some spin systems. Furthermore his approach to the determination of c leads to new universal functions interpolating between criticality and off-criticality.
The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically exchange-coupled centres are studied numerically by using finite-size calculations and conformal invariance. These models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism. Most of our results are centred in the mixed Heisenberg chain where we have at even (odd) sites a spin- SU(2) operator interacting with a XXZ-like interaction (anisotropy ). Our results indicate universal properties for all these chains. The whole phase, , where the models change from ferromagnetic to ferrimagnetic behaviour is critical. Along this phase the critical fluctuations are ruled by a c = 1 conformal field theory of Gaussian type. The conformal dimensions and critical exponents, along this phase, are calculated by studying these models with several boundary conditions.
The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically exchange-coupled centers are studied numerically by using finite-size calculations and conformal invariance. These models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism. Most of our results are centered in the mixed Heisenberg chain where we have at even (odd) sites a spin-S (S ′) SU(2) operator interacting with a XXZ like interaction (anisotropy ∆). Our results indicate universal properties for all these chains. The whole phase, 1> ∆> −1, where the models change from ferromagnetic ( ∆ = 1) to ferrimagnetic ( ∆ = −1) behaviour is critical. Along this phase the critical fluctuations are ruled by a c = 1 conformal field theory of Gaussian type. The conformal dimensions and critical exponents, along this phase, are calculated by studying these models with several boundary conditions. 1