An eigenvalue problem related to the non-linear sigma-model: analytical and numerical results

TL;DR

This paper investigates an eigenvalue problem linked to the non-linear sigma model with singular metrics, providing analytical and numerical insights into its spectrum across different regimes, and confirming results with the Thermodynamic Bethe Ansatz.

Contribution

It establishes the existence of a non-degenerate spectrum for all system sizes and derives convergent power series expansions in the IR regime, connecting spectral properties with the central charge.

Findings

Spectrum is non-degenerate and pure point for all finite R

Power series expansion of eigenvalues converges for all R

Spectrum transitions to a continuum with residual bound states in the UV

Abstract

An eigenvalue problem relevant for non-linear sigma model with singular metric is considered. We prove the existence of a non-degenerate pure point spectrum for all finite values of the size R of the system. In the infrared (IR) regime (large R) the eigenvalues admit a power series expansion around IR critical point R\to\infty. We compute high order coefficients and prove that the series converges for all finite values of R. In the ultraviolet (UV) limit the spectrum condenses into a continuum spectrum with a set of residual bound states. The spectrum agrees nicely with the central charge computed by the Thermodynamic Bethe Ansatz method