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Scaling limit of the staggered six-vertex model with Uq (sl(2)) invariant boundary conditions

Frahm, Holger, Gehrmann, Sascha, Andreevich Kotousov, Gleb

Published Jan 1, 2024• 0 citations
Publisher

Abstract

We study the scaling limit of a statistical system, which is a special case of the integrable inhomogeneous six-vertex model. It possesses Uq(sl(2)) invariance due to the choice of open boundary conditions imposed. An interesting feature of the lattice theory is that the spectrum of scaling dimensions contains a continuous component. By applying the ODE/IQFT correspondence and the method of the Baxter Q operator the corresponding density of states is obtained. In addition, the partition function appearing in the scaling limit of the lattice model is computed, which may be of interest for the study of nonrational CFTs in the presence of boundaries. As a side result of the research, a simple formula for the matrix elements of the Q operator for the general, integrable, inhomogeneous six-vertex model was discovered, that has not yet appeared in the literature. It is valid for a certain one parameter family of diagonal open boundary conditions in the sector with the z-projection of the total spin operator being equal to zero.

DOI
10.15488/19934

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We study the scaling limit of a statistical system, which is a special case of the integrable inhomogeneous six-vertex model. It possesses Uq(sl(2)) invariance due to the choice of open boundary conditions imposed. An interesting feature of the lattice theory is that the spectrum of scaling dimensions contains a continuous component. By applying the ODE/IQFT correspondence and the method of the Baxter Q operator the corresponding density of states is obtained. In addition, the partition function appearing in the scaling limit of the lattice model is computed, which may be of interest for the study of nonrational CFTs in the presence of boundaries. As a side result of the research, a simple formula for the matrix elements of the Q operator for the general, integrable, inhomogeneous six-vertex model was discovered, that has not yet appeared in the literature. It is valid for a certain one parameter family of diagonal open boundary conditions in the sector with the z-projection of the total spin operator being equal to zero.
Scaling limit of the staggered six-vertex model with Uq(sl(2))U_q\big(\mathfrak{sl}(2)\big)Uq​(sl(2)) invariant boundary conditions
We study the scaling limit of a statistical system, which is a special case of the integrable inhomogeneous six-vertex model. It possesses U_q\big(\mathfrak{sl}(2)\big) U q ( 𝔰 𝔩 ( 2 ) ) invariance due to the choice of open boundary conditions imposed. An interesting feature of the lattice theory is that the spectrum of scaling dimensions contains a continuous component. By applying the ODE/IQFT correspondence and the method of the Baxter Q Q operator the corresponding density of states is obtained. In addition, the partition function appearing in the scaling limit of the lattice model is computed, which may be of interest for the study of nonrational CFTs in the presence of boundaries. As a side result of the research, a simple formula for the matrix elements of the Q Q operator for the general, integrable, inhomogeneous six-vertex model was discovered, that has not yet appeared in the literature. It is valid for a certain one parameter family of diagonal open boundary conditions in the sector with the z z -projection of the total spin operator being equal to zero.
The inhomogeneous six-vertex model is a 2D multiparametric integrable statistical system. In the scaling limit it is expected to cover different classes of critical behaviour, which have remained mostly unexplored. For general values of the parameters and twisted boundary conditions, the model possesses U(1) invariance. We discuss the restrictions imposed on the parameters for which additional global symmetries that are consistent with the integrable structure arise. We also describe the Hermitian structures, which are consistent with the integrable one. The aforementioned general study laid the foundation for performing a detailed study of the scaling limit of a certain critical, integrable inhomogeneous six-vertex model subject to twisted boundary conditions. It is mainly based on a numerical analysis of the Bethe ansatz equations accompanied by the powerful analytic technique of the ODE/IQFT correspondence. The outcome of our study indicates that the critical behaviour of the lattice system is described by the gauged SL(2) WZW model with certain boundary and reality conditions imposed on the fields.
Finite size spectrum of the staggered six-vertex model with Uq(sl \mathfrak{sl} sl(2))-invariant boundary conditions
Abstract The finite size spectrum of the critical ℤ2-staggered spin-1/2 XXZ model with quantum group invariant boundary conditions is studied. For a particular (self-dual) choice of the staggering the spectrum of conformal weights of this model has been recently been shown to have a continuous component, similar as in the model with periodic boundary conditions whose continuum limit has been found to be described in terms of the non-compact SU(2, ℝ)/U(1) Euclidean black hole conformal field theory (CFT). Here we show that the same is true for a range of the staggering parameter. In addition we find that levels from the discrete part of the spectrum of this CFT emerge as the anisotropy is varied. The finite size amplitudes of both the continuous and the discrete levels are related to the corresponding eigenvalues of a quasi-momentum operator which commutes with the Hamiltonian and the transfer matrix of the model.
The work contains a detailed study of the scaling limit of a certain critical, integrable inhomogeneous six-vertex model subject to twisted boundary conditions. It is based on a numerical analysis of the Bethe ansatz equations as well as the powerful analytic technique of the ODE/IQFT correspondence. The results indicate that the critical behaviour of the lattice system is described by the gauged SL(2) WZW model with certain boundary and reality conditions imposed on the fields. Our proposal revises and extends the conjectured relation between the lattice system and the Euclidean black hole non-linear sigma model that was made in the 2011 paper of Ikhlef, Jacobsen and Saleur.
The inhomogeneous six-vertex model is a 2D multiparametric integrable statistical system.In the scaling limit it is expected to cover different classes of critical behaviour which, for the most part, have remained unexplored.For general values of the parameters and twisted boundary conditions the model possesses U(1) invariance.In this paper we discuss the restrictions imposed on the parameters for which additional global symmetries arise that are consistent with the integrable structure.These include the lattice counterparts of C, P and T as well as translational invariance.The special properties of the lattice system that possesses an additional Z r invariance are considered.We also describe the Hermitian structures, which are consistent with the integrable one.The analysis lays the groundwork for studying the scaling limit of the inhomogeneous six-vertex model.
The subject matter of this work is a 1D quantum spin - 12 chain associated with the inhomogeneous six-vertex model possessing an additional Zr symmetry. The model is studied in a certain parametric domain, where it is critical. Within the ODE/IQFT approach, a class of ordinary differential equations and a quantization condition are proposed which describe the scaling limit of the system. Some remarkable features of the CFT underlying the critical behaviour are observed. Among them is an infinite degeneracy of the conformal primary states and the presence of a continuous component in the spectrum in the case of even r.