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Using operator covariance to disentangle scaling dimensions in lattice models

Anders W. Sandvik*

  • *Contact author: sandvik@bu.edu
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Phys. Rev. E 114, 024146 – Published 24 August, 2026

DOI: https://doi.org/10.1103/jq2v-wjkf

Abstract

In critical lattice models, distance (r)-dependent correlation functions contain power laws r−2Δ governed by scaling dimensions Δ of an underlying continuum field theory. In Monte Carlo simulations and other numerical approaches, the leading dimensions can be extracted by data fitting, which can be difficult when two or more powers contribute significantly. Here a method utilizing covariance between multiple lattice operators is developed where the r-dependent eigenvalues of the covariance matrix reflect scaling dimensions of individual field operators. This disentangling of scaling dimensions is demonstrated explicitly for conformal field theories. The computational scheme is first tested on the critical point of the two-dimensional Ising model, where the two primary scaling dimensions and their respective two lowest descendant dimensions are extracted. The three-dimensional Ising model is studied next, revealing the two relevant primaries and their lowest descendants to high precision. For a more challenging case, the tricritical Ising point in two dimensions is studied with the Blume-Capel (diluted Ising) model. Here the scaling dimensions of all three fully symmetric (under lattice point group and spin-inversion transformations) primary operators are successfully isolated along with the leading descendants. The eigenvectors in the space of the two relevant primary operators are also studied and give useful information on the boundary between the ordered and disordered phases in the neighborhood of the tricritical point. Finally, the crossover from regular to tricritical Ising scaling is investigated on several points on the phase boundary of the Blume-Capel model away from its tricritical point. The scaling of the eigenvalues corresponding to tricritical descendant operators are found to be remarkably stable even far from the tricritical point. The covariance method represents a simple extension of standard analysis of correlation functions and can significantly enhance the utility of Monte Carlo simulations and other computational methods in studies of classical and quantum criticality.

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Phys. Rev. E 114, 024146– Published 24 August, 2026
  • Received 14 April 2025
  • Revised 11 July 2026
  • Accepted 22 July 2026
Article part of CHORUS
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