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Universal properties of a critical quantum spin chain are encoded in theunderlying conformal field theory (CFT). This underlying CFT is fullycharacterized by its conformal data. We propose a method to extract theconformal data from a critical quantum spin chain with both periodic andanti-periodic boundary conditions (PBC and APBC) based on low-energyeigenstates, generalizing previous work on spin chains with only PBC. First,scaling dimensions and conformal spins are extracted from the energies andmomenta of the eigenstates. Second, the Koo-Saleur formula of lattice Virasorogenerators is generalized to APBC and used to identify conformal towers. Third,local operators and string operators on the lattice are identified with CFToperators with PBC and APBC, respectively. Finally, operator product expansioncoefficients are extracted by computing matrix elements of lattice primaryoperators in the low-energy subspaces with PBC and APBC. To go beyond exactdiagonalization, tensor network methods based on periodic uniform matrixproduct states are used. We illustrate our approach with critical andtricritical Ising quantum spin chains. In the latter case, we propose latticeoperators that correspond to supervirasoro generators and verify their actionon low-energy eigenstates. In this way we explore the emergence ofsuperconformal symmetry in the quantum spin chain.

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