# Emergence of conformal symmetry in quantum spin chains: anti-periodic boundary conditions and supersymmetry

## TL;DR

This paper introduces a method to extract conformal data from critical quantum spin chains with both periodic and anti-periodic boundary conditions, revealing the emergence of superconformal symmetry.

## Contribution

It generalizes existing techniques to include anti-periodic boundary conditions and identifies lattice operators corresponding to superconformal generators.

## Key findings

- Successfully extracted conformal data for critical and tricritical Ising chains.
- Identified lattice operators corresponding to supervirasoro generators.
- Demonstrated emergence of superconformal symmetry in the quantum spin chain.

## Abstract

Universal properties of a critical quantum spin chain are encoded in the underlying conformal field theory (CFT). This underlying CFT is fully characterized by its conformal data. We propose a method to extract the conformal data from a critical quantum spin chain with both periodic and anti-periodic boundary conditions (PBC and APBC) based on low-energy eigenstates, generalizing previous work on spin chains with only PBC. First, scaling dimensions and conformal spins are extracted from the energies and momenta of the eigenstates. Second, the Koo-Saleur formula of lattice Virasoro generators is generalized to APBC and used to identify conformal towers. Third, local operators and string operators on the lattice are identified with CFT operators with PBC and APBC, respectively. Finally, operator product expansion coefficients are extracted by computing matrix elements of lattice primary operators in the low-energy subspaces with PBC and APBC. To go beyond exact diagonalization, tensor network methods based on periodic uniform matrix product states are used. We illustrate our approach with critical and tricritical Ising quantum spin chains. In the latter case, we propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we explore the emergence of superconformal symmetry in the quantum spin chain.