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arXiv:2610.24943

Scaling Dimensions and Two-Point Amplitudes from Operator Covariance Matrices

You-Chiuan Chen, Ribhu K. Kaul

Published Oct 18, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $N_\ell$ 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 $N_\ell$ 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.

DOI
10.48550/arxiv.2610.24943

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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.
We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ with mass parameter ${\ensuremath{\mu}}^{2}$ and lattice spacing aaa, we demonstrate a double data collapse for the correlation length $\ensuremath{\delta}\ensuremath{\xi}(\ensuremath{\mu},\ensuremath{\lambda},D)=\stackrel{\texttildelow{}}{\ensuremath{\xi}}((\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c})(\ensuremath{\delta}/a{)}^{\ensuremath{-}1/\ensuremath{\nu}})$ with DDD the bond dimension, $\ensuremath{\delta}$ the gap between eigenvalues of the transfer matrix, and ${\ensuremath{\alpha}}_{c}={\ensuremath{\mu}}_{R}^{2}/\ensuremath{\lambda}$ the parameter which fixes the critical quantum field theory.
The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.
We propose a new class of universal amplitude ratios which involve the first terms of the short distance expansion of the correlators of a statistical model in the vicinity of a critical point. We will describe the critical system with a conformal field theory (UV fixed point) perturbed by an appropriate relevant operator. In two dimensions the exact knowledge of the UV fixed point allows for accurate predictions of the ratios and in many nontrivial integrable perturbations they can even be evaluated exactly. In three-dimensional O ( N ) scalar systems, feasible extensions of some existing results should allow perturbative expansions for the ratios to be obtained. By construction these universal ratios are a perfect tool to explore the short distance properties of the underlying quantum field theory even in regimes where the correlation length and one-point functions are not accessible in experiments or simulations.
We compute the spectrum and several critical amplitudes of the two dimensional Ising model in a magnetic field with the transfer matrix method. The three lightest masses and their overlaps with the spin and the energy operators are computed on lattices of a width up to L=21. In extracting the continuum results we also take into account the corrections to scaling due to irrelevant operators. In contrast with previous Monte Carlo simulations our final results are in perfect agreement with the predictions of S-matrix and conformal field theory. We also obtain the amplitudes of some of the subleading corrections, for which no S-matrix prediction has yet been obtained.