Finite Volume Spectrum of 2D Field Theories from Hirota Dynamics

TL;DR

This paper introduces a novel method based on Hirota dynamics to compute the finite volume spectrum of 2D integrable quantum field theories, exemplified by the O(4) sigma model, providing a unified approach for arbitrary states.

Contribution

It develops a general framework using the Y-system and Hirota dynamics to solve finite size effects in 2D integrable models, including deriving a nonlinear integral equation and finite size Bethe equations.

Findings

Numerical data for energy levels as a function of system size.

Derived Luscher-type formulas for finite size corrections.

Re-derived the Destri-deVega equation for the SU(2) chiral Gross-Neveu model.

Abstract

We propose, using the example of the O(4) sigma model, a general method for solving integrable two dimensional relativistic sigma models in a finite size periodic box. Our starting point is the so-called Y-system, which is equivalent to the thermodynamic Bethe ansatz equations of Yang and Yang. It is derived from the Zamolodchikov scattering theory in the cross channel, for virtual particles along the non-compact direction of the space-time cylinder. The method is based on the integrable Hirota dynamics that follows from the Y-system. The outcome is a nonlinear integral equation for a single complex function, valid for an arbitrary quantum state and accompanied by the finite size analogue of Bethe equations. It is close in spirit to the Destri-deVega (DdV) equation. We present the numerical data for the energy of various states as a function of the size, and derive the general Luscher-type formulas for the finite size corrections. We also re-derive by our method the DdV equation for the SU(2) chiral Gross-Neveu model.