Minimal Models of Entropic Order

TL;DR

This work demonstrates that entropic order—spontaneous translation-symmetry breaking at arbitrarily high temperatures—can arise from minimal, physically plausible models. By analyzing the Arithmetic Ising Model (AIM) and its quantum counterpart (qAIM), along with classical gas analogs, the authors show that repulsive quadratic interactions induce a high-T checkerboard solid phase, with a 2D Ising universality class transition and robustness under large-k generalizations. They extend the idea to continuous-space gases (polymers) where entropy suppresses overlaps at high T, leading to solid ordering, and provide a grand-canonical field-theoretic perspective that predicts clustering instabilities for suitable short-range repulsions. Collectively, the results highlight entropic order as a robust mechanism potentially realizable in experiments (e.g., Rydberg atom arrays) and offer pathways to high-temperature ordered phases in both lattice and continuum settings.

Abstract

Due to entropic effects, it is possible that generic high-energy states of a quantum or classical system are ordered. This leads to spontaneous symmetry breaking at arbitrarily high temperatures. We present minimal models of entropic order that arise from very simple interactions. Our main examples are the Arithmetic Ising Model (AIM) and its quantum analogue, where usual Ising spins are replaced by non-negative integers. Using a large-flavor expansion together with numerical simulations, we find that the high-temperature phase is ordered in the classical and quantum models. We also introduce classical gas models whose interactions drive the system to a crystal at high temperatures.

Figures (5)

• Figure 1: A schematic phase diagram of the model \ref{['eq:Hn+nn']}. The insets show typical Monte Carlo configurations of the two phases.
• Figure 2: Typical Monte Carlo configurations of the AIM at various values of $U$ (with $\mu=1$).
• Figure 3: Top: The MC measurements of the order parameter as function of $U$ at $T=1000$ (given in units of $\mu$), for lattice sizes $L=20,40,60$ and $80$. The results are compared to a MFT prediction (see SM for details). Bottom: Monte Carlo comparison of transitions at different temperatures, showing a slow drift towards $U/\mu=1/2$.
• Figure 4: Order parameter susceptibility scaling for the 2D AIM with 2D Ising critical exponens $\nu=1,\gamma=7/4$, and $U_c=0.63$.
• Figure 5: Monte Carlo simulation at system size $L=80$ results for the order parameter, for multiple values of $k$. All dimensionful quantities are in units of $\mu$.