Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula

Abstract

We study the emergence of two-dimensional conformal symmetry in critical quantum spin chains on the finite circle. Our goal is to characterize the conformal field theory (CFT) describing the universality class of the corresponding quantum phase transition. As a means to this end, we propose and demonstrate automated procedures which, using only the lattice Hamiltonian $H = \sum_j h_j$ as an input, systematically identify the low-energy eigenstates corresponding to Virasoro primary and quasiprimary operators, and assign the remaining low-energy eigenstates to conformal towers. The energies and momenta of the primary operator states are needed to determine the primary operator scaling dimensions and conformal spins -- an essential part of the conformal data that specifies the CFT. Our techniques use the action, on the low-energy eigenstates of $H$, of the Fourier modes $H_n$ of the Hamiltonian density $h_j$. The $H_n$ were introduced as lattice representations of the Virasoro generators by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper we demonstrate that these operators can be used to extract conformal data in a nonintegrable quantum spin chain.