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