  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.

Statistical Mechanics;Strongly Correlated Electrons;High Energy Physics - Theory;Quantum Physics