Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories

TL;DR

This paper demonstrates that the finite-size torus spectra at quantum critical points reveal universal signatures of the underlying conformal field theories in 2+1D systems, linking spectral data to operator content.

Contribution

It introduces a method to identify universal spectral signatures at quantum critical points using analytical and numerical techniques, connecting spectra to conformal field theory operator content.

Findings

Spectra at criticality are universal and scale as 1/L.

The spectrum encodes the operator content of the critical theory.

Implications of neighboring topological phases on the spectrum are elucidated.

Abstract

The low-energy spectra of many body systems on a torus, of finite size $L$, are well understood in magnetically ordered and gapped topological phases. However, the spectra at quantum critical points separating such phases are largely unexplored for 2+1D systems. Using a combination of analytical and numerical techniques, we show that the low-energy torus spectrum at criticality provides a universal fingerprint of the underlying quantum field theory, with the energy levels given by universal numbers times $1/L$. We highlight the implications of a neighboring topological phase on the spectrum by studying the Ising* transition (i.e. the transition between a $Z_2$ topological phase and a trivial paramagnet), in the example of the toric code in a longitudinal field, and advocate a phenomenological picture that provides insight into the operator content of the critical field theory.