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arXiv:2511.20530

Scaling 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

Published Nov 25, 2025• 0 citations
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Abstract

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).

DOI
10.48550/arxiv.2511.20530

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Scaling limit of QQQ-functions for the Zr{\cal Z}_rZr​ invariant inhomogeneous XXZ spin-12\frac{1}{2}21​ 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 Zr{\cal Z}_rZr​ 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).
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.
It is known that for the Heisenberg XXZ spin - [Formula presented] 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.
It is known that for the Heisenberg XXZ spin - 12 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.
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.
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.