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Spin chains and vertex models based on superalgebras

Konstantin Hobuß

Published Jan 1, 2019• 0 citations
Publisher

Abstract

The thermodynamic limit of superspin chains can show several intriguing properties, including the emergence of continua of scaling dimensions and the appearance of discrete states when imposing toroidal boundary conditions. Nevertheless, the former’s exhaustive characterization in terms of Conformal Field Theories is still lacking. In order to study the thermodynamic limit of the U_q[sl(2|1)] 3 ⊗ 3̄ superspin chain in the antiferromagnetic regime, we analyze the low lying excitations by means of the model’s exact solution using the Algebraic Bethe Ansatz. In the isotropic limit, this model may be used as a toy model for the description of plateau transitions in Quantum Hall systems. The definition of a quasimomentum operator allows for a characterization of the continua of scaling dimensions, thereby giving rise to a quantum number for the corresponding non-compact component of the spectrum in the thermodynamic limit. The associated degeneracies are lifted on the lattice by logarithmic fine structures. Based on the extrapolation of our finite size data we find that under a variation of the boundary conditions from periodic to antiperiodic for the fermionic degrees of freedom, levels from the continuous part of the spectrum flow into discrete levels and vice versa. Investigating the thermodynamic limit of the q-deformed osp(3|2) superspin chain, corresponding to an intersecting loop model in the rational limit, we seek to uncover its low-lying critical exponents. We present evidences that the latter are built in terms of composites of anomalous dimensions of two Coulomb gases with distinct radii and exponents associated to Z(2) degrees of freedom. This view is supported by the fact that the S = 1 XXZ integrable chain spectrum is present in some of the sectors of the superspin chain at a particular value of the deformation parameter. We find that the fine structure of finite-size effects is very rich for a typical anisotropic spin chain. In fact, we argue on the existence of a family of states with the same conformal dimension whose lattice degeneracies are lifted by logarithmic corrections. On the other hand, we also report on states of the spectrum whose finite-size corrections seem to be governed by a power law behaviour. We finally observe that under toroidal boundary conditions the ground state dependence on the twist angle has two distinct analytical structures.

DOI
10.15488/8827

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The thermodynamic limit of superspin chains can show several intriguing properties, including the emergence of continua of scaling dimensions and the appearance of discrete states when imposing toroidal boundary conditions. Nevertheless, the former’s exhaustive characterization in terms of Conformal Field Theories is still lacking. In order to study the thermodynamic limit of the U_q[sl(2|1)] 3 ⊗ 3̄ superspin chain in the antiferromagnetic regime, we analyze the low lying excitations by means of the model’s exact solution using the Algebraic Bethe Ansatz. In the isotropic limit, this model may be used as a toy model for the description of plateau transitions in Quantum Hall systems. The definition of a quasimomentum operator allows for a characterization of the continua of scaling dimensions, thereby giving rise to a quantum number for the corresponding non-compact component of the spectrum in the thermodynamic limit. The associated degeneracies are lifted on the lattice by logarithmic fine structures. Based on the extrapolation of our finite size data we find that under a variation of the boundary conditions from periodic to antiperiodic for the fermionic degrees of freedom, levels from the continuous part of the spectrum flow into discrete levels and vice versa. Investigating the thermodynamic limit of the q-deformed osp(3|2) superspin chain, corresponding to an intersecting loop model in the rational limit, we seek to uncover its low-lying critical exponents. We present evidences that the latter are built in terms of composites of anomalous dimensions of two Coulomb gases with distinct radii and exponents associated to Z(2) degrees of freedom. This view is supported by the fact that the S = 1 XXZ integrable chain spectrum is present in some of the sectors of the superspin chain at a particular value of the deformation parameter. We find that the fine structure of finite-size effects is very rich for a typical anisotropic spin chain. In fact, we argue on the existence of a family of states with the same conformal dimension whose lattice degeneracies are lifted by logarithmic corrections. On the other hand, we also report on states of the spectrum whose finite-size corrections seem to be governed by a power law behaviour. We finally observe that under toroidal boundary conditions the ground state dependence on the twist angle has two distinct analytical structures.
This paper is concerned with the investigation of the massless regime of an integrable spin chain based on the quantum group deformation of the O S p ( 3 | 2 ) superalgebra. The finite-size properties of the eigenspectra are computed by solving the respective Bethe ansatz equations for large system sizes allowing us to uncover the low-lying critical exponents. We present evidences that critical exponents appear to be built in terms of composites of anomalous dimensions of two Coulomb gases with distinct radii and the exponents associated to Z ( 2 ) degrees of freedom. This view is supported by the fact that the S = 1 XXZ integrable chain spectrum is present in some of the sectors of our superspin chain at a particular value of the deformation parameter. We find that the fine structure of finite-size effects is very rich for a typical anisotropic spin chain. In fact, we argue on the existence of a family of states with the same conformal dimension whose lattice degeneracies are apparently lifted by logarithmic corrections. On the other hand we also report on states of the spectrum whose finite-size corrections seem to be governed by a power law behaviour. We finally observe that under toroidal boundary conditions the ground state dependence on the twist angle has two distinct analytical structures.
The low energy spectrum of a spin chain with OSp(3|2) supergroup symmetry is studied based on the Bethe ansatz solution of the related vertex model. This model is a lattice realization of intersecting loops in two dimensions with loop fugacity z=1 which provides a framework to study the critical properties of the unusual low temperature Goldstone phase of the O(N) sigma model for N=1 in the context of an integrable model. Our finite-size analysis provides strong evidence for the existence of continua of scaling dimensions, the lowest of them starting at the ground state. Based on our data we conjecture that the so-called watermelon correlation functions decay logarithmically with exponents related to the quadratic Casimir operator of OSp(3|2). The presence of a continuous spectrum is not affected by a change to the boundary conditions although the density of states in the continua appears to be modified.
The D3(2) {D}_3^{(2)} D3(2)​ spin chain and its finite-size spectrum
Abstract Using the analytic Bethe ansatz, we initiate a study of the scaling limit of the quasi-periodic D3(2) {D}_3^{(2)} D3(2)​ D 3 2 spin chain. Supported by a detailed symmetry analysis, we determine the effective scaling dimensions of a large class of states in the parameter regime γ ∈ (0, π4 \frac{\pi }{4} 4π​ π 4 ). Besides two compact degrees of freedom, we identify two independent continuous components in the finite-size spectrum. The influence of large twist angles on the latter reveals also the presence of discrete states. This allows for a conjecture on the central charge of the conformal field theory describing the scaling limit of the lattice model.
In this paper we investigate the spectrum of OSp(n|2m) quantum spin chains with free boundary conditions. We compute the surface free energy of these models which, similar to other properties in the thermodynamic limit including the effective central charge of the underlying conformal field theory, depends on n−2m only. For several models in the regime n−2m<2 we have studied the finite-size properties including the subleading logarithmic corrections to scaling. As in the case of periodic boundary conditions we find the existence of a tower of states with the same conformal dimension as the identity operator. As expected the amplitudes of the corresponding logarithmic corrections differ from those found previously for the models with periodic boundary conditions. We point out however the existence of simple relations connecting such amplitudes for free and periodic boundaries. Based on our findings we formulate a conjecture on the long distance behaviour of the bulk and surface watermelon correlators.
Abstract Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They are characterised, in particular, by the appearance of a continuum of conformal dimensions. Surprisingly, such CFTs have been identified as the continuum limits of lattice models with a finite number of degrees of freedom per site. However, results have so far been restricted to the case of periodic boundary conditions, precluding the exploration via lattice models of aspects of non-compact boundary CFTs and the corresponding D-brane constructions. The present paper follows a series of previous works on a ℤ2-staggered XXZ spin chain, whose continuum limit is known to be a non-compact CFT related with the Euclidian black hole sigma model. By using the relationship of this spin chain with an integrable D22 {D}_2^2 D22​ D 2 2 vertex model, we here identify integrable boundary conditions that lead to a continuous spectrum of boundary exponents, and thus correspond to non-compact branes. In the context of the Potts model on a square lattice, they correspond to wired boundary conditions at the physical antiferromagnetic critical point. The relations with the boundary parafermion theories are discussed as well. We are also able to identify a boundary renormalisation group flow from the non-compact boundary conditions to the previously studied compact ones.