Murray T. Batchelor

Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models

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.

q-Deformations of the O(3) symmetric spin-1 Heisenberg chain

The authors present the general expression for the spin-1 Heisenberg chain invariant under the Uq(SO(3)) quantum algebra. Several physical and mathematical implications are discussed.

Central charges of the 6- and 19-vertex models with twisted boundary conditions

A new and general analytic method for calculating finite-size corrections and central charges is applied to the 6- and 19-vertex models and their related spin-1/2 and spin-1 XXZ chains with twisted boundary conditions. Nonlinear integral equations are derived from which the central charge c can be extracted in terms of Rogers dilogarithms. For twist angle phi , the central charge is c=3S/S+1 (1-4(S-1) phi 2/ pi ( pi -2S gamma )) where gamma is the crossing parameter or chain anisotropy and spin S=1/2 or 1. For periodic boundary conditions ( phi =0) this reduces to the known results c=1 and c=3/2, respectively.

Fermi gases in one dimension: From Bethe ansatz to experiments

This article reviews theoretical and experimental developments for one-dimensional Fermi gases. Specifically, the experimentally realized two-component delta-function interacting Fermi gas-the Gaudin-Yang model-and its generalizations to multicomponent Fermi systems with larger spin symmetries is discussed. The exact results obtained for Bethe ansatz integrable models of this kind enable the study of the nature and microscopic origin of a wide range of quantum many-body phenomena driven by spin population imbalance, dynamical interactions, and magnetic fields. This physics includes Bardeen-Cooper-Schrieffer-like pairing, Tomonaga-Luttinger liquids, spin-charge separation, Fulde-Ferrel-Larkin-Ovchinnikov-like pair correlations, quantum criticality and scaling, polarons, and the few-body physics of the trimer state (trions). The fascinating interplay between exactly solved models and experimental developments in one dimension promises to yield further insight into the exciting and fundamental physics of interacting Fermi systems.

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.

Conformal Anomaly and Surface Energy for Potts and Ashkin-teller Quantum Chains

Exact equivalences between the critical quantum Potts and Ashkin-Teller chains and a modified XXZ Heisenberg chain have recently been derived by Alcaraz et al (1987). The leading finite-size corrections to the ground-state energies of these chains are derived using the methods of de Vega and Woynarovich (1985) and Eckle. Exact results are then obtained for the conformal anomaly of each model, and for the surface energy in the case of free boundaries.

Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices

Abstract The procedure for obtaining integrable vertex models via reflection matrices on the square lattice with open boundaries is reviewed and explicitly carried out for a number of two- and three-state vertex models. These models include the six-vertex model, the 15-vertex A 2 (1) model and the 19-vertex models of Izergin-Korepin and Zamolodchikov-Fateev. In each case the eigenspectra is determined by application of either the algebraic or the analytic Bethe ansatz with inhomogeneities. With suitable choices of reflection matrices, these vertex models can be associated with integrable loop models on the same lattice. In general, the required choices do not coincide with those which lead to quantum-group-invariant spin chains. The exact solution of the integrable loop models — including an O( n ) model on the square lattice with open boundaries — is of relevance to the surface critical behaviour of two-dimensional polymers.

Analytical eigenstates for the quantum Rabi model

We develop a method to find analytical solutions for the eigenstates of the quantum Rabi model. These include symmetric, anti-symmetric and asymmetric analytic solutions given in terms of the confluent Heun functions. Both regular and exceptional solutions are given in a unified form. In addition, the analytic conditions for determining the energy spectrum are obtained. Our results show that conditions proposed by Braak [Phys. Rev. Lett. \textbf{107}, 100401 (2011)] are a type of sufficiency condition for determining the regular solutions. The well-known Judd isolated exact solutions appear naturally as truncations of the confluent Heun functions.

Solutions of the reflection equation for face and vertex models associated with An(1), Bn(1), Cn(1), Dn(1) and An(2)

Abstract We present new diagonal solutions of the reflection equation for elliptic solutions of the star-triangle relation. The models considered are related to the affine Lie algebras A n (1) , B n (1) , C n (1) , D n (1) and A n (2) . We recover all known diagonal solutions associated with these algebras and find how these solutions are related in the elliptic regime. Furthermore, new solutions of the reflection equation follow for the associated vertex models in the trigonometric limit.

Exact solution for the spin-s XXZ quantum chain with non-diagonal twists

Abstract We study integrable vertex models and quantum spin chains with toroidal boundary conditions. An interesting class of such boundaries is associated with non-diagonal twist matrices. For such models there are no trivial reference states upon which a Bethe ansatz calculation can be constructed, in contrast to the well-known case of periodic boundary conditions. In this paper we show how the transfer matrix eigenvalue expression for the spin- s XXZ chain twisted by the charge-conjugation matrix can in fact be obtained. The technique used is the generalization to spin- s of the functional relation method based on “pair propagation through a vertex”. The Bethe ansatz-type equations obtained reduce, in the case of lattice size N = 1, to those recently found for the Hofstadter problem of Bloch electrons on a square lattice in a magnetic field.