Extrapolation-CAM Theory for Critical Exponents

TL;DR

This paper introduces a novel extrapolation method based on intentionally underestimating convergence rates to accurately determine critical exponents in finite systems, demonstrated on Ising models.

Contribution

It presents a new approach using the Coherent Anomaly Method with controlled underestimation to improve critical exponent estimation accuracy.

Findings

Accurately estimated $ u=1$ for 2D Ising model.

Estimated $ u \\approx 0.629$ for 3D Ising model.

Method enhances clarity of coherent anomalies in finite-size data.

Abstract

By intentionally underestimating the rate of convergence of exact-diagonalization values for the mass or energy gaps of finite systems, we form families of sequences of gap estimates. The gap estimates cross zero with generically nonzero linear terms in their Taylor expansions, so that $\nu = 1$ for each member of these sequences of estimates. Thus, the Coherent Anomaly Method can be used to determine $\nu$. Our freedom in deciding exactly how to underestimate the convergence allows us to choose the sequence that displays the clearest coherent anomaly. We demonstrate this approach on the two-dimensional ferromagnetic Ising model, for which $\nu = 1$. We also use it on the three-dimensional ferromagnetic Ising model, finding $\nu \approx 0.629$, in good agreement with other estimates.