Extraction of Conformal Data in Critical Spin Chains Using the Koo-Saleur Formula and Periodic Uniform Matrix Product States

At a quantum critical point, the universal properties of a quantum spin chain are captured by an emergent conformal field theory (CFT). We propose and demonstrate new, generic techniques for characterizing the emergent CFT, given a local critical spin chain Hamiltonian, using the Koo-Saleur lattice representations of the Virasoro generators of conformal symmetry. In particular, we develop procedures for identifying the energy eigenstates of the spin chain corresponding to primary operators in the CFT, providing an essential part of the conformal data used to characterize the CFT. Furthermore, we show that periodic uniform Matrix Product States (puMPS), together with puMPS Bloch states, are excellent numerical means of extracting conformal data at large system sizes. Perhaps surprisingly, all low-energy excited states of the circular critical spin chain appear to be well captured by the Bloch-state ansatz.