Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finite Size Spectrum of SU(N) Principal Chiral Field from Discrete Hirota Dynamics

Vladimir Kazakov, Sebastien Leurent

TL;DR

The paper develops a finite-volume framework for the SU(N)×SU(N) principal chiral field by translating the Y-system/Hirota dynamics into a finite set of nonlinear integral equations (NLIEs) via Wronskian determinant representations. It provides explicit construction for the U(1) sector, reveals how central node equations and large-volume ABA data feed into the NLIEs, and demonstrates through N=3 numerics that the method captures the full spectrum from infrared to ultraviolet, including Lüscher corrections and conformal limits. The work clarifies subtle bound-state effects for N>2, derives finite-size Bethe equations, and shows how the energy of excited states can be computed with carefully chosen contours; it also discusses the generalization to more complex states and potential applications to AdS/CFT Y-systems. Overall, the approach offers a powerful, broadly applicable route to exact finite-volume spectra in integrable sigma-models and paves the way for extensions to higher N and holographic duals. $

Abstract

Using recently proposed method of discrete Hirota dynamics for integrable (1+1)D quantum field theories on a finite space circle of length L, we derive and test numerically a finite system of nonlinear integral equations for the exact spectrum of energies of SU(N)xSU(N) principal chiral field model as functions of m L, where m is the mass scale. We propose a determinant solution of the underlying Y-system, or Hirota equation, in terms of determinants (Wronskians) of NxN matrices parameterized by N-1 functions of the spectral parameter, with the known analytical properties at finite L. Although the method works in principle for any state, the explicit equations are written for states in the U(1) sector only. For N>2, we encounter and clarify a few subtleties in these equations related to the presence of bound states, absent in the previously considered N=2 case. In particular, we solve these equations numerically for a few low-lying states for N=3 in a wide range of m L.

Finite Size Spectrum of SU(N) Principal Chiral Field from Discrete Hirota Dynamics

TL;DR

The paper develops a finite-volume framework for the SU(N)×SU(N) principal chiral field by translating the Y-system/Hirota dynamics into a finite set of nonlinear integral equations (NLIEs) via Wronskian determinant representations. It provides explicit construction for the U(1) sector, reveals how central node equations and large-volume ABA data feed into the NLIEs, and demonstrates through N=3 numerics that the method captures the full spectrum from infrared to ultraviolet, including Lüscher corrections and conformal limits. The work clarifies subtle bound-state effects for N>2, derives finite-size Bethe equations, and shows how the energy of excited states can be computed with carefully chosen contours; it also discusses the generalization to more complex states and potential applications to AdS/CFT Y-systems. Overall, the approach offers a powerful, broadly applicable route to exact finite-volume spectra in integrable sigma-models and paves the way for extensions to higher N and holographic duals. $

Abstract

Using recently proposed method of discrete Hirota dynamics for integrable (1+1)D quantum field theories on a finite space circle of length L, we derive and test numerically a finite system of nonlinear integral equations for the exact spectrum of energies of SU(N)xSU(N) principal chiral field model as functions of m L, where m is the mass scale. We propose a determinant solution of the underlying Y-system, or Hirota equation, in terms of determinants (Wronskians) of NxN matrices parameterized by N-1 functions of the spectral parameter, with the known analytical properties at finite L. Although the method works in principle for any state, the explicit equations are written for states in the U(1) sector only. For N>2, we encounter and clarify a few subtleties in these equations related to the presence of bound states, absent in the previously considered N=2 case. In particular, we solve these equations numerically for a few low-lying states for N=3 in a wide range of m L.

Paper Structure

This paper contains 29 sections, 79 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The principal chiral field model in the large volume
  3. The PCF model, its S-matrix and the large L ABA
  4. TBA, Y-system and Hirota equation
  5. Central node equations
  6. The large L, "spin chain" limit of Y-system and its relation to ABA
  7. Expressions for T-functions in the large volume, spin chain limit
  8. Asymptotic Bethe ansatz (ABA)
  9. Expressions for the energy of excited states
  10. Energy of state in the U(1) sector at odd N's
  11. Energy of state in the U(1) sector at even N's
  12. Wronskian solution for Hirota equation equivalent to Y-system
  13. Solution of the Y-system for PCF at a finite volume L
  14. Definition of the jump densities
  15. Relation to the analyticity of T functions
  16. ...and 14 more sections

Figures (4)

  • Figure 1: The $(a,s)$-strip for Y-system and T-system
  • Figure 2: Analyticity of the integrand $\cosh(\frac{2\pi}{3} \theta)\log\left( (1+Y_{1,0}) (1+Y_{2,0}) \right)$ and manipulations with the contours when $N=3$
  • Figure 3: Analyticity of the integrand for the Energy and choice of the contours for $N=4$
  • Figure 4: Mass gap $\Delta E=E_{\theta_0}-E_{vacuum}$. The numeric results (red crosses) are compared to the analytic Lüscher correction \ref{['eq:MGLuscher']} for $E^{mass gap}_{L\to\infty}$ [blue curve], to the 1-loop expression $\frac{L}{2\pi}[E_{\theta_{0}}(L)-E_{0}(L)]\approx\frac{8 }{9}\frac{1}{\log \frac{c}{L}}$ [orange curve], and to the 2-loop expression $\frac{L}{2\pi}[E_{\theta_{0}}(L)-E_{0}(L)]\approx\frac{8 }{9}\frac{1}{\log \frac{c}{L}+\frac{1}{2}\log\log \frac{c}{L}}$ [green curve] \ref{['smallLmassgap']}, where $c$ is chosen as the best fit for the $L<10^{-1}$ data$^{\ref{['ft:c']}}$.