The inhomogeneous six-vertex model

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 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.

arXiv:2010.10615Symmetry Integrability and Geometry Methods and ApplicationsMar 16, 202111 citations

Scaling limit of the Z2 invariant inhomogeneous six-vertex model

Vladimir V. Bazhanov, Gleb A. Kotousov, Sergii M. Koval +1 more

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 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.

arXiv:2010.10613Nuclear Physics BFeb 10, 202129 citations

Scaling limit of Q-functions for the Z r invariant inhomogeneous XXZ spin- 1 2 chain near free fermion point

Gleb A. Kotousov, Sergei L. Lukyanov, Daria A. Shabetnik

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 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).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 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).

Nuclear Physics BJan 20, 20260 citations

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

Holger Frahm, Sascha Gehrmann, Gleb A. Kotousov

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.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.

arXiv:2312.11238SciPost PhysicsJun 5, 20243 citations

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

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

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.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.

Jan 1, 20240 citations

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

Gleb A. Kotousov, Sergei L. Lukyanov, Daria A. Shabetnik

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).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).

arXiv:2511.20530Nov 25, 20250 citations

Scaling limit of the staggered six-vertex model with

U_q\big(\mathfrak{sl}(2)\big)

invariant boundary conditions

Scaling limit of

Q

-functions for the

{\cal Z}_r

invariant inhomogeneous XXZ spin-

\frac{1}{2}

chain near free fermion point

{\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).