Finding critical points using improved scaling Ansaetze

TL;DR

This paper introduces accelerated methods for accurately determining critical points from finite size data, outperforming existing techniques and applicable to various systems, including challenging quantum models.

Contribution

The authors develop new scaling ansaetze and a Homogeneity Condition Method that rapidly converge to critical points using ground-state data, even in complex quantum systems.

Findings

Methods outperform Phenomenological Renormalization Group.

Effective in quantum and classical systems across dimensions.

Able to locate BKT transition with small system sizes.

Abstract

Analyzing in detail the first corrections to the scaling hypothesis, we develop accelerated methods for the determination of critical points from finite size data. The output of these procedures are sequences of pseudo-critical points which rapidly converge towards the true critical points. In fact more rapidly than previously existing methods like the Phenomenological Renormalization Group approach. Our methods are valid in any spatial dimensionality and both for quantum or classical statistical systems. Having at disposal fast converging sequences, allows to draw conclusions on the basis of shorter system sizes, and can be extremely important in particularly hard cases like two-dimensional quantum systems with frustrations or when the sign problem occurs. We test the effectiveness of our methods both analytically on the basis of the one-dimensional XY model, and numerically at phase transitions occurring in non integrable spin models. In particular, we show how a new Homogeneity Condition Method is able to locate the onset of the Berezinskii-Kosterlitz-Thouless transition making only use of ground-state quantities on relatively small systems.