Spectral signatures of nonstabilizerness and criticality in infinite matrix product states

TL;DR

This paper introduces a spectral transfer-matrix framework to analyze nonstabilizerness in infinite matrix product states, revealing universal features and a new correlation length that diverges at critical points, linking quantum resources to phase transitions.

Contribution

It develops a novel spectral transfer-matrix approach for stabilizer Rènyi entropy in infinite MPS, uncovering universal subleading information and a new correlation length related to nonstabilizerness near criticality.

Findings

Identifies an SRE correlation length that diverges at phase transitions.

Derives exact SRE expressions for the cluster-Ising model at bond dimension 2.

Numerically demonstrates universal scaling of SRE along critical lines.

Abstract

While nonstabilizerness (''magic'') is a key resource for universal quantum computation, its behavior in many-body quantum systems, especially near criticality, remains poorly understood. We develop a spectral transfer-matrix framework for the stabilizer R\'enyi entropy (SRE) in infinite matrix product states, showing that its spectrum contains universal subleading information. In particular, we identify an SRE correlation length -- distinct from the standard correlation length -- which diverges at continuous phase transitions and governs the spatial response of the SRE to local perturbations. We derive exact SRE expressions for the bond dimension $\chi=2$ MPS ''skeleton'' of the cluster-Ising model, and we numerically probe its universal scaling along the $\mathbb{Z}_2$ critical lines in the phase diagram. These results demonstrate that nonstabilizerness captures signatures of criticality and local perturbations, providing a new lens on the interplay between computational resources and emergent phenomena in quantum many-body systems.