Quantum Jacobi-Trudi Formula and $E_8$ Structure in the Ising Model in a Field

TL;DR

This paper develops a quantum transfer matrix approach to analyze the Ising model in a field, revealing an $E_8$ structure through functional relations and a quantum Jacobi-Trudi formula, aligning with known $E_8$ exponents.

Contribution

It introduces a quantum Jacobi-Trudi formula and functional relations among fusion QTMs, uncovering the $E_8$ structure in the Ising model in a field without relying on string solutions.

Findings

Functional relations among fusion QTMs characterized by skew Young tableaux.

Numerical eigenvalue analysis matches $E_8$ exponents.

Recovered $E_8$ Thermodynamic Bethe ansatz equations without specific string solutions.

Abstract

We investigate a 1D quantum system associated with the Ising model in a field(the dilute $A_3$ model) by the recently developed quantum transfer matrix (QTM) approach. A closed set of functional relations is found among variants of fusion QTMs which are characterized by skew Young tableaux. These relations are proved by using a quantum analogue of Jacobi-Trudi formula, together with special features at "root of unity" . The numerical analysis on their eigenvalues shows a remarkable coincidence with exponents characteristic to $E_8$. From these findings, we have successfully recovered the $E_8$ Thermodynamic Bethe ansatz equation by Bazhanov et al, however, without specific choice of strings solutions.