Finite Volume Spectrum of 2D Field Theories from Hirota Dynamics

TL;DR

This work develops a universal framework for solving integrable two-dimensional quantum field theories in finite volume by combining Y-system/TBA with Hirota dynamics to yield a single nonlinear integral equation for a gauge function g(x) that encodes all states. The authors apply the method to the O(4) sigma model / SU(2) principal chiral field, derive exact finite-volume spectrum equations for arbitrary states, and reproduce Luscher-type corrections as well as the Destri–de Vega equation in related models. They provide detailed numerical implementations and demonstrate agreement with known results across volumes, validating the approach and its potential applicability to broader classes of integrable theories. The framework promises broad utility for computing finite-volume spectra in models such as SU(N) PCF, O(n) sigma models, and even AdS/CFT contexts, by unifying ABA, Luscher corrections, and DdV-like equations under the Hirota/Y-system formalism.

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.

Figures (8)

• Figure 1: Physical channel, cross-channel and finite volume vs finite temperature.
• Figure 2: Plots of energies $E$ of a few excited states of $O(4)$ model on a circle of a circumference $L$. The vertical axis corresponds to the values of $\frac{L}{2\pi}E$, the horizontal axis - to the values of $L$ in the logarithmic scale. The lowest curve depicts the vacuum energy. The next one, labeled as $\theta_{0}$, shows the mass gap energy. The corresponding state is in the $U(1)$ sector, with a single particle at rest, hence with the mode number $=0$. The next states in the $U(1)$ sector are denoted by $\theta_{n_{1}n_{2}n_{3},\cdots }$, according to the mode numbers $n_{1},n_{2},n_{3},\dots$ excited for the 1-st, 2-nd, 3-rd, etc., particles. For all these states the $SU(2)_L$ and $SU(2)_R$ spins of the several particles are pointing in the same direction, say they are spin "up". The dashed line represents a state having a polarization out of the $U(1)$ sector, with left and right "magnons" excited - it corresponds to the quantum state of two particles where both $SU(2)_L$ and $SU(2)_R$ spins are in the singlet $s=0$ state. The qualitative explanation of these graphs will be given in subsection \ref{['NUMexpl']}.
• Figure 3: Domains of applicability of different descriptions of an integrable field theory at a finite volume $L$. In the ultra-violet regime, for small volume measured in units of a dynamically generated mass, the theory could be described by a conformal theory. In the infrared, at large volume, one can use the asymptotic Bethe equations. The leading order finite size corrections are governed by the (generalized) Lüscher corrections. At any volume but for the ground state energy only one can use Thermodynamical Bethe ansatz. Hirota equation, equivalent to Y-system but more efficient when it comes to imposing appropriate analyticity properties, is a universal tool covering the whole diagram.
• Figure 4: Dynkin diagram (three central nodes) and its extension for the magnon bound states (grey nodes) reflecting the structure of the Y-system. The central, black node corresponds to the U(1) sector excitations of the model ($\theta$-roots), the upper and lower nodes correspond to the more general states for magnon excitations for the $SU(2)_L$ wing ($u$-roots) and the $SU(2)_R$ wing ($v$-roots).
• Figure 5: The function $F(x)$ in (\ref{['ansatz']}) can be recast as a contour integral as in (\ref{['ansatz2']}) with the contours as represented in this figure.