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

TL;DR

The paper develops and validates automated procedures to extract conformal data from finite-size critical quantum spin chains using the Koo-Saleur lattice modes $H_n$ of the Hamiltonian density. By relating lattice spectra to CFT data through $H_n$, the authors identify Virasoro primaries, quasiprimary states, and their towers, and provide robust methods to estimate the central charge $c$ and scaling dimensions $ riangle_ ext{α}$ and spins $S_ ext{α}$. They demonstrate the approach on the Ising and three-state Potts models and extend it to a nonintegrable ANNNI model, confirming the method's applicability away from integrability. The work also outlines paths to determine OPE coefficients from lattice data and discusses finite-size effects and normalization considerations essential for accurate conformal data extraction. This framework offers a scalable route to characterize emergent CFTs in generic critical spin chains using only the lattice Hamiltonian, with potential implications for RG flows and tensor-network discretizations of path integrals.

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.

Figures (13)

• Figure 1: Exact spectrum of the Ising CFT Hamiltonian in terms of $\Delta$ and $s$, color-coded by conformal tower, showing the location of the primary states $|I\rangle$, $|\sigma\rangle$ and $|\varepsilon\rangle$, and the energy-momentum states $|T\rangle$ and $|\overline{T}\rangle$. Note: We shift points horizontally from their allowed values ($S$ is quantized) to avoid overlaps and better show degeneracies in this and subsequent figures.
• Figure 2: Illustration of the action of the ladder operators (Virasoro generators) on the energy eigenstates of the Ising CFT Hamiltonian belonging to the $I$ conformal tower. Two possible paths from $(\Delta\!=\!4,S\!=\!0)$ to $(\Delta\!=\!4,S\!=\!-4)$ are shown, as is the annihilation of the quasiprimary state $|\Delta\!=\!4,S\!=\!0\rangle$ by $\overline{L}_{+1}$ and $L_{+1}$.
• Figure 3: Spectrum of the Ising model at system size $N=14$ with energies and momenta in terms of $\Delta$ and $S$, showing the action of $H^\textsl{\tiny Ising}_{+1}$ and $H^\textsl{\tiny Ising}_{-2}$ on selected energy eigenstates. The empty circles identify the states $|\varphi_\alpha\rangle$ to which the operator is applied and the filled circles indicate the sizes of the matrix elements $\langle \varphi_\beta | H^\textsl{\tiny Ising}_n | \varphi_\alpha \rangle$ with the remaining eigenstates $|\varphi_\beta\rangle$, on a logarithmic scale. Very small matrix elements $< 10^{-12}$ are not plotted.
• Figure 4: Central charge from \ref{['eq:c_lat']}, with linear extrapolation to large $N$ using all visible data. System sizes shown are $N=8\dots 18$ for the Ising model and $N = 8 \dots 14$ for the three-state Potts model. We do not provide an error for the extrapolated $c$ since there are systematic finite-size corrections on each point. The scaling exponent $2$ is consistent with known finite-size corrections present in both models cardy_operator_1986henkel_finite-size_1987reinicke_analytical_1987.
• Figure 5: Ising model spectrum at system size $N=14$, with energies and momenta in terms of $\Delta$ and $S$. States are colored according to their numerically identified conformal towers. Primary candidate states, identified using \ref{['eq:Pstates_Hn']} with $\epsilon_{\max} = 10^{-14}$, are marked with diamonds.