Finite-Size Scaling in the Energy-Entropy Plane for the 2D +- J Ising Spin Glass

TL;DR

This study investigates the relationship between energy and entropy in 2D ±J Ising spin glasses on small lattices, revealing scaling behaviors and potential implications for low-temperature specific heat anomalies.

Contribution

It provides the first detailed analysis of finite-size scaling in the energy-entropy plane for 2D ±J Ising spin glasses, highlighting anomalous entropy scaling and its implications.

Findings

Entropy scales as L^{1.78} at x=0.5

Energy scales as L^2, showing no anomalous behavior

Crossover to L^2 entropy scaling at L=12 for x=0.25

Abstract

For $L \times L$ square lattices with $L \le 20$ the 2D Ising spin glass with +1 and -1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For $x = 0.5$ (where $x$ is the fraction of negative bonds), over this range of $L$, the characteristic entropy defined by the energy-entropy correlation scales with size as $L^{1.78(2)}$. Anomalous scaling is not found for the characteristic energy, which essentially scales as $L^2$. When $x= 0.25$, a crossover to $L^2$ scaling of the entropy is seen near $L = 12$. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small $L$.