Physical Combinatorics and Quasiparticles

TL;DR

This paper explores the combinatorial structure of critical lattice models and their conformal field theories, introducing quasiparticles as features of RSOS paths and linking them to transfer matrix eigenvalues for energy calculations.

Contribution

It presents a novel combinatorial framework for analyzing quasiparticles in critical models, connecting patterns of zeros to transfer matrix eigenvalues and deriving excitation energies via TBA.

Findings

Identification of fermionic quasiparticles as features of RSOS paths

Establishment of a bijection between particles and zeros of transfer matrix eigenvalues

Derivation of excitation energies using the Thermodynamic Bethe Ansatz

Abstract

We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 sl(2) Wess-Zumino-Witten (WZW) model. The Hamiltonian of the WZW model is the $U_q(sl(2))$ invariant XXX spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through an energy preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated (m,n) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the Thermodynamic Bethe Ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings.