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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

TL;DR

The paper builds a detailed ODE/IQFT bridge for the $\mathcal{Z}_r$-invariant inhomogeneous XXZ chain near the free-fermion point, proposing a new class of ODEs whose spectral determinants $D_\pm(\mu)$ capture the scaling limit of lattice $Q$-functions. By identifying ODE parameters with lattice quantum numbers and mapping Bethe states to a CFT basis built from $r$-component free fermions, the authors connect the scaling data to an extended conformal symmetry algebra $\widehat{\mathfrak{su}}_1(r)\oplus\widehat{\mathfrak{u}}(1)$, yielding explicit expressions for vacuum and low-lying states and validating the construction through extensive numerical checks (sum rules, quasi-shifts, and product relations). The work also clarifies the r=1 case, linking the proposed ODEs to the Schrödinger equation with monster potentials, and discusses how the framework adapts to softly broken $\mathcal{Z}_r$ symmetry. Overall, the results advance the understanding of how lattice integrable structures scale to conformal field theories and highlight concrete procedures to identify the corresponding ODEs from finite-size data.

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).

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

TL;DR

The paper builds a detailed ODE/IQFT bridge for the -invariant inhomogeneous XXZ chain near the free-fermion point, proposing a new class of ODEs whose spectral determinants capture the scaling limit of lattice -functions. By identifying ODE parameters with lattice quantum numbers and mapping Bethe states to a CFT basis built from -component free fermions, the authors connect the scaling data to an extended conformal symmetry algebra , yielding explicit expressions for vacuum and low-lying states and validating the construction through extensive numerical checks (sum rules, quasi-shifts, and product relations). The work also clarifies the r=1 case, linking the proposed ODEs to the Schrödinger equation with monster potentials, and discusses how the framework adapts to softly broken symmetry. Overall, the results advance the understanding of how lattice integrable structures scale to conformal field theories and highlight concrete procedures to identify the corresponding ODEs from finite-size data.

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 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).

Paper Structure

This paper contains 22 sections, 278 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Class of ODEs describing the scaling
  3. The free fermion point $(\delta=0)$
  4. Scaling limit of $A_+(\zeta)$
  5. Scaling limit of the Bethe states
  6. CFT space of states for $|\delta|<\frac{1}{r}$
  7. Identification of parameters
  8. Numerical checks
  9. Sum rules
  10. Quasi-shift operators
  11. Product rules
  12. Enumerating solutions of the algebraic system
  13. Solutions of the algebraic system for ${\tt J}=1,2$
  14. Solutions of the algebraic system in the vicinity of $\delta=0$
  15. The case $r=1$
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Presented is numerical data for a certain RG trajectory of the ${\cal Z}_r$ invariant inhomogeneous XXZ spin-$\frac{1}{2}$ chain with $r=5$, $\delta=\frac{1}{15}$ and ${\tt k}=\frac{1}{50}$. The corresponding quantum numbers are $S^z=\frac{3}{2}$$({\tt s}=1)$, ${\tt w}=0$, ${\tt m}=1$, ${\tt L}=2$, $\bar{\tt L}=1$. It follows from formula \ref{['ajks8723hj']} that $\mathfrak{j}_{{\tt m}}=1$ and $\bar{\mathfrak{j}}_{{\tt m}}=2$. The sets of Bethe roots corresponding to the RG trajectory for different $N$ can be found in the online repository rep. From this data, it was computed $H_1^{(N)}$ and $\bar{H}_1^{(N)}$, which are defined as the l.h.s. of \ref{['ksai23jnnbjsas']} and \ref{['ksai23jnnbjsasA']}, respectively, considered at finite fixed $N$ without taking the limit $N\to\infty$. The obtained values are depicted by the black crosses in the plots. The black solid lines represent the predicted limiting values $H_1=2.41386-0.48188\,{\rm i}$ and $\bar{H}_1=0.35948-1.41215\,{\rm i}$. This was computed by means of eq. \ref{['kjsa89jhjh12']}, where ${\tt K}=-\frac{7}{25}$, ${\tt J}=5$ and using a certain solution set of the algebraic system \ref{['kasj892jhsa']} satisfying \ref{['9023jdjnkjsd']} with ${\tt M}=1$, while for the barred counterpart $\bar{\tt K}=-\frac{8}{25}$, $\bar{\tt J}=5$ and $\bar{\tt M}=3$. The solution sets $\{(c_i,\varpi_i)\}_{i=1}^{5}$ and $\{(\bar{c}_i,\bar{\varpi}_i)\}_{i=1}^{5}$ are also stored in the repository rep. The dashed blue lines are plots of the real and imaginary parts of $2.41257-0.48021\,{\rm i}+(3.19121-2.33556\,{\rm i})\,N^{-0.4}$ and $0.35959-1.41237\,{\rm i}-(4.05034+1.63060\,{\rm i})\,N^{-0.4}$, which were obtained via a fit of the last three data points for $H_1^{(N)}$ and $\bar{H}_1^{(N)}$, respectively.
  • Figure 2: Presented is numerical data for $b_{\pm1}^{(N)}$, defined as the r.h.s. of \ref{['kjsa23j12io']} with $m=1$, where the number of lattice sites $N$ is kept fixed. The data corresponds to the same RG trajectory as in fig.\ref{['fig1']}, i.e., $r=5$, $\delta=\frac{1}{15}$, ${\tt k}=\frac{1}{50}$, $S^z=\frac{3}{2}$$({\tt s}=1)$, ${\tt w}=0$, ${\tt m}=1$, ${\tt L}=2$ and $\bar{\tt L}=1$. Additional details can be found in the caption to that figure. The crosses come from the solutions to the Bethe Ansatz equations. The black solid lines represent the limiting values $b_{+1}=-1.42756-0.13571\,{\rm i}$ and $b_{-1}=-0.65829+0.92530\,{\rm i}$ calculated by means of eq. \ref{['aksjhdsfi234']} with ${\tt K}=-\frac{7}{25}$, $\bar{\tt K}=-\frac{8}{25}$ as well as the same sets $\{(c_i,\varpi_i)\}_{i=1}^5$ and $\{(\bar{c}_i,\bar{\varpi}_i)\}_{i=1}^5$ that were used to compute $H_1$ and $\bar{H}_1$ for fig. \ref{['fig1']}. The dashed blue lines are plots of the real/imaginary parts of $-1.42733-0.13562\,{\rm i}-(0.01428+0.05120\,{\rm i})\,N^{-0.4}$ for $b_{+1}^{(N)}$ and $-0.65874+0.92504\,{\rm i}-(0.08395-0.23694\,{\rm i})\,N^{-0.4}$ for $b_{-1}^{(N)}$, which were obtained via a fit of the numerical data. The fit was performed in a rather rough way, using only the last three points with the largest values of $N$.