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 = j hj 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 Hn of the Hamiltonian density hj. The Hn 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.

Conformal Fields and Operator Product Expansion in Critical Quantum Spin Chains

“…The specific assignment x j = j + 1/2 for O 1 (j) is important when combining O 1 (j) = −X(j)X(j + 1) with O 2 (j) = −Z(j) to form the Hamiltonian density h(j) = O 1 (j) + O 2 (j). In order for the Fourier mode h s to correspond to a linear combination of Virasoro generators L CFT −s +L CFT s , it has been shown numerically [25] that the correct choice is x j = j + 1/2 for O 1 (j) and x j = j for O 2 (j). If we have chosen a different x j for O 1 (j), the Fourier mode h s (s = 0) would connect states in identity tower and tower.…”

“…For B we choose all states |ψ β with scaling dimension ∆ α ≤ 3 + 1/8, namely the 23 lowest energy states. After normalizing H such that ∆ 1 = 0 and ∆ T = ∆T = 2 [25], our estimates for the scaling dimensions of primaries are ∆ σ = 0.1249995 and ∆ = 0.9999994, together with exact values s 1 = s σ = s = 0, s T = −sT = 2 for the conformal spins. Scaling dimensions and conformal spins of derivative descendants are obtained by adding integers to these values.…”

“…We can then exploit two classic results: (i) As first pointed out by Cardy [13][14][15][16][17], the low energy states |ψ α of a critical quantum spin chain on the circle are in one-to-one correspondence with CFT states, |ψ α ∼ |ψ CFT α , and approximately reproduce the spectrum of energies and momenta (1)- (2), from which we can estimate c CFT and ∆ CFT α and s CFT α (or h CFT α andh CFT α ); (ii) Following Koo and Saleur [18][19][20][21][22][23][24], a Fourier expansion of the lattice Hamiltonian term h(j) results in an approximate lattice realization of the CFT Virasoro generators. These can then be used [25] to identify the specific low energy states of the spin chain, denoted |φ α , that correspond to CFT primary states |φ CFT α , which are the ones contributing to the conformal data. In addition, in order to significantly reduce finite size errors in the resulting numerical estimates of c CFT and h CFT α ,h CFT α , in Ref.…”

Emergence of conformal symmetry in quantum spin chains: Antiperiodic boundary conditions and supersymmetry

“…where |0 CFT is the ground state of the CFT and the state |T CFT corresponds to the stress tensor of the CFT. Primary fields φ CFT α correspond to primary states |φ CFT α , which are characterized by [12,14] L CFT n |φ CFT α = 0,L CFT n |φ CFT α = 0 (n > 0).…”

Critical Scaling Behaviors of Entanglement Spectra*

“…which gives the spectrum of the primary operators and corresponding descendants, each expansion term mq n represents m-fold degenerate scaling operators with conformal dimension ∆ i = n. The operator-state correspondence suggests that the eigenspectrum of (entanglement) Hamiltonian shares the structure of scaling operators, and this relation has been well studied in the energy spectrum of spin chains. [59][60][61][62].…”