arXiv subjects

Gleb A. Kotousov

Publications and source records attributed to Gleb A. Kotousov.

15 recordsLinked to original sources

arXiv2025

Scaling limit of $Q$-functions for the ${\cal Z}_r$ invariant inhomogeneous XXZ spin-$\frac{1}{2}$ chain near free fermion point

At the beginning of the 70's, Baxter introduced a multiparametric generalization of the six-vertex model. This integrable system has been found to exhibit a remarkable variety of critical behaviors. The work is part of a series of papers devoted to their systematic study. We focus on the case when the lattice model possesses an additional ${\cal Z}_r$ symmetry and consider the critical behavior near the so-called free fermion point. Among other things, discussed is the algebra of extended conformal symmetry underlying the universal behavior. The main result of the paper is the class of differential equations that describe the scaling limit of the solutions to the Bethe Ansatz equations. This is an instance of the correspondence between Ordinary Differential Equations and Integrable Quantum Field Theory (ODE/IQFT correspondence).

hep-th↗

arXiv2024

Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin-$\frac{1}{2}$ chain

It is known that for the Heisenberg XXZ spin-$\frac{1}{2}$ chain in the critical regime, the scaling limit of the vacuum Bethe roots yields an infinite set of numbers that coincide with the energy spectrum of the quantum mechanical 3D anharmonic oscillator. The discovery of this curious relation, among others, gave rise to the approach referred to as the ODE/IQFT (or ODE/IM) correspondence. Here we consider a multiparametric generalization of the Heisenberg spin chain, which is associated with the inhomogeneous six-vertex model. When quasi-periodic boundary conditions are imposed the lattice system may be explored within the Bethe Ansatz technique. We argue that for the critical spin chain, with a properly formulated scaling limit, the scaled Bethe roots for the ground state are described by second order differential equations, which are multi-parametric generalizations of the Schrödinger equation for the anharmonic oscillator.

math-ph↗

arXiv2024

Scaling limit of the staggered six-vertex model with $U_q\big(\mathfrak{sl}(2)\big)$ 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 $U_q\big(\mathfrak{sl}(2)\big)$ 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.

hep-th↗

arXiv2023

On the scaling behaviour of an integrable spin chain with ${\cal Z}_r$ symmetry

The subject matter of this work is a 1D quantum spin - $\frac{1}{2}$ chain associated with the inhomogeneous six-vertex model possessing an additional ${\cal Z}_r$ 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 $r$.

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arXiv2022

Integrable sigma models at RG fixed points: quantisation as affine Gaudin models

The goal of this paper is to make first steps towards the quantisation of integrable non-linear sigma models using the formalism of affine Gaudin models, by approaching these theories through their conformal limits. We focus mostly on the example of the Klimčík model, which is a two-parameter deformation of the Principal Chiral Model on a Lie group $G$. We show that the UV fixed point of this theory is described classically by two decoupled chiral affine Gaudin models, encoding its left- and right-moving degrees of freedom, and give a detailed analysis of the chiral and integrable structures of these models. Their quantisation is then explored within the framework of Feigin and Frenkel. We study the quantum local integrals of motion using the formalism of quantised affine Gaudin models and show agreement of the first two integrals with known results in the literature for $G={\rm SU}(2)$. Evidence is given for the existence of a monodromy matrix satisfying the Yang-Baxter algebra for this model, thus paving the way for the quantisation of the non-local integrals of motion. We conclude with various perspectives, including on generalisations of this program to a larger class of integrable sigma models and applications of the ODE/IQFT correspondence to the description of their quantum spectrum.

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arXiv2022

On the Yang-Baxter Poisson algebra in non-ultralocal integrable systems

A common approach to the quantization of integrable models starts with the formal substitution of the Yang-Baxter Poisson algebra with its quantum version. However it is difficult to discern the presence of such an algebra for the so-called non-ultralocal models. The latter includes the class of non-linear sigma models which are most interesting from the point of view of applications. In this work, we investigate the emergence of the Yang-Baxter Poisson algebra in a non-ultralocal system which is related to integrable deformations of the Principal Chiral Field.

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arXiv2021

ODE/IQFT correspondence for the generalized affine $\mathfrak{ sl}(2)$ Gaudin model

An integrable system is introduced, which is a generalization of the $\mathfrak{sl}(2)$ quantum affine Gaudin model. Among other things, the Hamiltonians are constructed and their spectrum is calculated within the ODE/IQFT approach. The model fits within the framework of Yang-Baxter integrability. This opens a way for the systematic quantization of a large class of integrable non-linear sigma models. There may also be some interest in terms of Condensed Matter applications, as the theory can be thought of as a multiparametric generalization of the Kondo model.

hep-th↗

arXiv2021

Some Algebraic Aspects of the Inhomogeneous Six-Vertex Model

The inhomogeneous six-vertex model is a 2$D$ 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 ${\rm U}(1)$ 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 ${\cal C}$, ${\cal P}$ and ${\cal T}$ as well as translational invariance. The special properties of the lattice system that possesses an additional ${\cal Z}_r$ 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.

math-ph↗

arXiv2021

Equilibrium density matrices for the 2D black hole sigma models from an integrable spin chain

This work concerns the quantum Lorentzian and Euclidean black hole non-linear sigma models. For the Euclidean black hole sigma model an equilibrium density matrix is proposed, which reproduces the modular invariant partition function from the 2001 paper of Maldacena, Ooguri and Son. For the Lorentzian black hole sigma model, using its formulation as a gauged ${\rm SL}(2,\mathbb{R})$ WZW model, we describe the linear and Hermitian structure of its space of states and also propose an expression for the equilibrium density matrix. Our analysis is guided by the results of the study of a certain critical, integrable spin chain. In the scaling limit, the latter exhibits the key features of the Lorentzian black hole sigma model including the same global symmetries, the same algebra of extended conformal symmetry and a continuous spectrum of conformal dimensions.

hep-th↗

arXiv2021

Scaling limit of the ${\cal Z}_2$ invariant inhomogeneous six-vertex 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 ${\rm SL}(2)$ 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.

math-ph↗

arXiv2020

On the scaling behaviour of the alternating spin chain

In this note we report the results of our study of a 1D integrable spin chain whose critical behaviour is governed by a CFT possessing a continuous spectrum of scaling dimensions. It is argued that the computation of the density of Bethe states of the continuous theory can be reduced to the calculation of the connection coefficients for a certain class of differential equations whose monodromy properties are similar to those of the conventional confluent hypergeometric equation. The finite size corrections to the scaling are also discussed.

hep-th↗

arXiv2020

Bethe state norms for the Heisenberg spin chain in the scaling limit

In this paper we discuss the norms of the Bethe states for the spin one-half Heisenberg chain in the critical regime. Our analysis is based on the ODE/IQFT correspondence. Together with numerical work, this has lead us to formulate a set of conjectures concerning the scaling behavior of the norms. Also, we clarify the role of the different Hermitian structures associated with the integrable structure studied in the series of works of Bazhanov, Lukyanov and Zamolodchikov in the mid nineties.

hep-th↗

arXiv2020

Spectrum of the reflection operators in different integrable structures

The reflection operators are the simplest examples of the non-local integrals of motion, which appear in many interesting problems in integrable CFT. For the so-called Fateev, quantum AKNS, paperclip and KdV integrable structures, they are built from the (chiral) reflection S-matrices for the Liouville and cigar CFTs. Here we give the full spectrum of the reflection operators associated with these integrable structures. We also obtained a relation between the reflection S-matrices of the cigar and Liouville CFTs. The results of this work are applicable for the description of the scaling behaviour of the Bethe states in exactly solvable lattice systems and may be of interest to the study of the Generalized Gibbs Ensemble associated with the above mentioned integrable structures.

hep-th↗

arXiv2017

Quantum transfer-matrices for the sausage model

In this work we revisit the problem of the quantization of the two-dimensional O(3) non-linear sigma model and its one-parameter integrable deformation -- the sausage model. Our consideration is based on the so-called ODE/IQFT correspondence, a variant of the Quantum Inverse Scattering Method.The approach allowed us to explore the integrable structures underlying the quantum O(3)/sausage model. Among the obtained results is a system of non-linear integral equations for the computation of the vacuum eigenvalues of the quantum transfer-matrices.

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arXiv2014

Winding vacuum energies in a deformed O(4) sigma model

We consider the problem of calculating the Casimir energies in the winding sectors of Fateev's SS-model, which is an integrable two-parameter deformation of the O(4) non-linear sigma model in two dimensions. This problem lies beyond the scope of all traditional methods of integrable quantum field theory including the thermodynamic Bethe ansatz and non-linear integral equations. Here we propose a solution based on a remarkable correspondence between classical and quantum integrable systems and express the winding energies in terms of certain solutions of the classical sinh-Gordon equation.

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