Benford analysis of quantum critical phenomena: First digit provides high finite-size scaling exponent while first two and further are not much better

TL;DR

This study uses Benford's law violation analysis on quantum many-body systems to identify quantum phase transitions, finding that the first digit alone provides a high finite-size scaling exponent, simplifying experimental detection.

Contribution

It demonstrates that analyzing only the first significant digit of physical observables effectively detects quantum phase transitions with high scaling exponents, reducing experimental complexity.

Findings

First digit analysis captures quantum phase transitions effectively.

Higher significant digits do not significantly improve detection.

First digit analysis is practical for noisy experimental environments.

Abstract

Benford's law is an empirical edict stating that the lower digits appear more often than higher ones as the first few significant digits in statistics of natural phenomena and mathematical tables. A marked proportion of such analyses is restricted to the first significant digit. We employ violation of Benford's law, up to the first four significant digits, for investigating magnetization and correlation data of paradigmatic quantum many-body systems to detect cooperative phenomena, focusing on the finite-size scaling exponents thereof. We find that for the transverse field quantum XY model, behavior of the very first significant digit of an observable, at an arbitrary point of the parameter space, is enough to capture the quantum phase transition in the model with a relatively high scaling exponent. A higher number of significant digits do not provide an appreciable further advantage, in particular, in terms of an increase in scaling exponents. Since the first significant digit of a physical quantity is relatively simple to obtain in experiments, the results have potential implications for laboratory observations in noisy environments.