Conformal invariance, the XXZ chain and the operator content of two-dimensional critical systems

Abstract

The massless regime of the quantum XXZ chain is an example of a conformally invariant (1 + 1)-dimensional Hamiltonian with conformal anomaly c = 1. In this paper, Bethe ansatz equations are formulated and solved numerically for eigenstates of the XXZ Hamiltonian on a finite chain with periodic boundary conditions and with a generalized class of “twisted” boundary conditions. The resulting spectra are found to be in accord with predictions of conformal invariance and the corresponding operator content is identified. With periodic boundary conditions, eight-vertex and Gaussian model operators are found. With the twisted boundary conditions, operators from the operator algebras of the Ashkin-Teller and q-state Potts models are identified. This identification is achieved by constructing exact equivalences between eigenergies of the quantum Ashkin-Teller and Potts Hamiltonians with periodic boundary conditions and levels of the XXZ Hamiltonian with modified boundary conditions. In the Potts case, states in the ground-state sector correspond exactly to states of the XXZ chain with a “defect seam.” The effect of this seam on the ground-state energy is shown to generate the conformal anomaly of the Potts model. For the 4-state model, the XXZ equivalence is used to perform very large lattice calculations and, thereby, to obtain direct confirmation of the expected values for its critical exponents. Finally, the leading finite-size corrections to the predictions of conformal invariance are analyzed and the dominant irrelevant operators governing these corrections identified.