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arXiv:1701.03111 · Physical review. B./Physical review. B

Spectrum of the Wilson-Fisher conformal field theory on the torus

Seth Whitsitt, Michael Schuler, Louis-Paul Henry, Andreas Martin Läuchli, Subir Sachdev A

Published Jul 24, 2017• 24 citations
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Abstract

The authors study the finite-size energy spectrum of the O($N$) symmetric Wilson-Fisher conformal field theory (CFT) on the spatial torus. They first perform an analytic calculation of the spectrum using the $ϵ$ expansion with a careful treatment of the zero-momentum modes. The authors also perform exact diagonalization on several lattice Hamiltonians whose critical points are known to be described by the Wilson-Fisher CFT. Comparison of the analytic and numerical data demonstrates that the energy spectrum is a useful diagnostic for determining the universality class of a critical point. These results have applications in determining the critical behavior of more complicated many-body systems.

DOI
10.1103/physrevb.96.035142

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Conformal data for the O(2)O(2)O(2) Wilson-Fisher CFT in (2+1)(2+1)(2+1)-dimensional spacetime from exact diagonalization and matrix product states on the fuzzy sphere
We study at zero temperature a microscopic quantum spin-1 model on the fuzzy sphere that realizes the O(2) O ( 2 ) Wilson-Fisher conformal field theory (CFT) in (2+1) ( 2 + 1 ) -dimensional spacetime at a quantum critical point. Here, we use the fuzzy-sphere regularization as it preserves the full spatial SO(3) S O ( 3 ) rotational symmetry of the CFT, enabling the state-operator correspondence that maps energy eigenstates directly to CFT operators. Using exact diagonalization (ED) and matrix product state (MPS) techniques combined with conformal perturbation theory (CPT), we extract conformal data including scaling dimensions and operator product expansion (OPE) coefficients. We identify 48 primary operators and their descendants, organized by the conserved O(2) O ( 2 ) charge S^{z} S z and spatial angular momentum L L . Our numerical results for the scaling dimensions of the lowest primary operators show good agreement with conformal bootstrap predictions. We verify predictions from the large charge expansion, which provides systematic predictions for operators carrying large U(1) U ( 1 ) charge, connecting the Goldstone mode physics in the ordered phase to phonon primaries at the critical point.