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Physical review. E

Using operator covariance to disentangle scaling dimensions in lattice models

Anders W. Sandvik

Published Jul 22, 2026• 0 citations
Publisher

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.

DOI
10.1103/jq2v-wjkf

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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.
In critical lattice models, distance (rrr) dependent correlation functions contain power laws r−2Δr^{-2Δ}r−2Δ governed by scaling dimensions ΔΔΔ of an underlying continuum field theory. In Monte Carlo simulations, the leading dimensions can be extracted by data fitting, which is 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 is demonstrated explicitly for conformal field theories. The scheme is first tested on the critical point of the 2D Ising model, where the two primary scaling dimensions and their respective two lowest descendant dimensions are extracted. The 3D Ising model is studied next, revealing the two relevant primaries and their lowest descendants to high precision. The 2D tricritical Ising point is studied with the Blume-Capel model. Here the scaling dimensions of all three symmetric primary operators are successfully isolated along with the leading descendants. The eigenvectors 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 criticality, in particular conformal critical points.
The long distance behavior of two-point functions of critical lattice statistical mechanics models depend on detailed information of the underlying quantum field theory. Yet these universal properties can be difficult to extract because generic lattice operators couple to the infinite hierarchy of continuum scaling operators with unknown coefficients. Recently in Physical Review E 114, 024146 (2026), one approach to disentangle some of this information was proposed using the eigen-decomposition of the operator covariance matrix constructed from cross correlations of a finite set of NℓN_\ellNℓ​ lattice operators. In this work we analyze this method in detail. We provide an explicit proof that under the assumption of scale invariance the leading decay of the eigenvalues are controlled by the smallest NℓN_\ellNℓ​ scaling dimensions, even after taking irrelevant perturbations into account. Based on this understanding, we propose a new method by which the eigenvectors of the covariance matrix can be used to construct the amplitudes of the continuum two-point correlation functions. This allows for a complete determination of the structure of the continuum two-point functions from the lattice covariance matrix. We demonstrate our ideas both on synthetic data sets (drawn from the scaling hypothesis) and numerically exact data for the two dimensional Ising model.
Finite-size effects limit the accuracy with which conformal data can be extracted from lattice simulations of critical systems. While action improvement suppresses some corrections to scaling, it does not address operator-dependent effects arising from imperfect lattice representations of continuum conformal fields. In this work, we propose a data-driven method for improving lattice operators themselves, constructing estimators with enhanced overlap with the corresponding primary operators of the continuum conformal field theory. We identify improved lattice representations of leading spin and energy operators in three two-dimensional critical systems: the Ising model, the q = 3 Potts model, and the dilute q = 3 Potts model. In all cases, the resulting operators exhibit reduced corrections to scaling and yield more accurate estimates of scaling dimensions compared to conventional lattice choices. The code and analysis workflows used to produce these results are made available in an accompanying GitHub repository.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
Using a Monte Carlo renormalization-group (MCRG) method we study tricritical behavior in two very different two-dimensional spin models: the Blume-Capel ferromagnet and the next-nearest-neighbor Ising antiferromagnet. The MCRG method is used to locate accurately tricritical points independent of the convergence of eigenvalue estimates. Despite the different symmetries and the different renormalization transformations used for the two models, in both cases we find four relevant tricritical eigenvalues which are essentially identical for both models. For the antiferromagnet, there is no indication of any decomposition of the tricritical point as predicted by mean-field theory, for ratios of intrasublattice to intersublattice coupling as small as (1/8).
We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to L=1600L=1600L=1600. As a check, also simulations of the spin-12\frac{1}{2}21​ Ising model are performed. We find ${f}_{\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\epsilon}}=1.051(1)$ and ${f}_{\ensuremath{\epsilon}\ensuremath{\epsilon}\ensuremath{\epsilon}}=1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of NNN-point functions. Finite size corrections vanish with ${L}^{\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}}$, where LLL is the linear size of the lattice and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}$ is the scaling dimension of the leading Z2{Z}_{2}Z2​-even scalar $\ensuremath{\epsilon}$.