Operator fusion from wavefunction overlaps: Universal finite-size corrections and application to Haagerup model

TL;DR

This paper demonstrates that wavefunction overlaps in critical quantum spin chains encode all conformal data, including finite-size corrections, enabling numerical extraction of CFT parameters and applying this to the Haagerup model.

Contribution

It establishes a universal method to extract all conformal data from wavefunction overlaps, including finite-size effects, and applies it to the Haagerup model.

Findings

Conformal data can be obtained from wavefunction overlaps with universal finite-size corrections.

Verified the method in Ising and XXZ models.

Estimated the Haagerup model's central charge and operator dimensions.

Abstract

Given a critical quantum spin chain described by a conformal field theory (CFT) at long distances, it is crucial to understand the universal conformal data. One most important ingredient is the operator product expansion (OPE) coefficients, which describe how operators fuse into each other. It has been proposed in [Zou, Vidal, Phys. Rev. B 105, 125125] that the OPE coefficients can be computed from overlaps of low-energy wavefunctions of the spin chain. In this work, we establish that all conformal data including central charge, conformal dimensions, and OPE coefficients are encoded in the wavefunction overlaps, with universal finite-size corrections that depend on the operator content of the cyclic orbifold CFT. Thus this method allows us to numerically compute all the conformal data based solely on the low-energy eigenstates. The predictions are verified in the Ising and XXZ model. As an application, we study the recently proposed Haagerup model built from the Haagerup fusion category. We find that the CFT has central charge $c \approx 2.1$ and the lowest spin-$1$ operator in the twisted sector has scaling dimension $1 < \Delta_J \leq 1.4$.