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arXiv:2512.07980

Minimal Models of Entropic Order

Huang, Xiaoyang, Komargodski, Zohar, Andrew Lucas, Popov, Fedor K., Tin Sulejmanpašić

Published Dec 8, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2512.07980

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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.
High temperature is usually expected to destroy order: as the Gibbs state approaches the infinite-temperature limit, it becomes an equal-weight ensemble over all states and the system is generically disordered. Recent works showed that entropic order can violate this expectation through coupling to bosons in classical lattice models and quantum field theories, where the ordered states have higher entropy. Here we present new analytic methods for constructing quantum lattice models that exhibit entropic orders. In particular, we construct quantum lattice models with continuous symmetry breaking at high temperature in 1+1 dimensions and clarify how entropic order can evade the Hohenberg-Mermin-Wagner theorems. We also construct high-temperature entropic p+ipp+ipp+ip chiral topological superconducting states in 2+1 dimensions with temperature-independent anyon correlation functions. In addition, we obtain a broad family of high-temperature entropic non-chiral topological orders. We show that the entropic topological orders have strong higher form symmetries at high temperature unlike the conventional topological orders, and the symmetry is spontaneously broken. These results follow from two general constructions that couple a given lattice model with a low-temperature ordered phase either to ordered bosons or, for local commuting-projector Hamiltonians, to more general bosonic degrees of freedom.
Ordered phases of matter, such as solids, ferromagnets, superfluids, or quantum topological order, typically only exist at low temperatures. Despite this conventional wisdom, we present explicit local models in which all such phases persist to arbitrarily high temperature. This is possible since order in one degree of freedom can enable other degrees of freedom to strongly fluctuate, leading to "entropic order", whereby typical high energy states are ordered. Our construction, which utilizes interacting bosons, avoids existing no-go theorems on long-range order or entanglement at high temperature. We propose a simple model for high-temperature superconductivity using these general principles.
Ordered phases of matter, such as solids, ferromagnets, superfluids, or quantum topological order, typically only exist at low temperatures. Despite this conventional wisdom, we present explicit local models in which all such phases persist to arbitrarily high temperature. This is possible since order in one degree of freedom can enable other degrees of freedom to strongly fluctuate, leading to entropic order, whereby typical high energy states are ordered. Our construction, which utilizes interacting bosons, avoids existing no-go theorems on long-range order or entanglement at high temperature. We propose a simple model for high-temperature superconductivity using these general principles.
Slow mixing and emergent one-form symmetries in three-dimensional Z2\mathbb{Z}_2Z2​ gauge theory
Symmetry-breaking order at low temperatures is often accompanied by slow relaxation dynamics, due to diverging free-energy barriers arising from interfaces between different ordered states. Here, we extend this correspondence to classical topological order, where the ordered states are locally indistinguishable, so there is no notion of interfaces between them. We study the relaxation dynamics of the three-dimensional (3D) classical Z2\mathbb{Z}_2Z2​ lattice gauge theory (LGT) as a canonical example. We prove a lower bound on the mixing time in the deconfined phase, tmix=exp⁡[Ω(L)]t_{\text{mix}} = \exp [Ω(L)]tmix​=exp[Ω(L)], where L is the linear system size. This bound applies even in the presence of perturbations that explicitly break the one-form symmetry between different long-lived states. This perturbation destroys the energy barriers between ordered states, but we show that entropic effects nevertheless lead to diverging free-energy barriers at nonzero temperature. Our proof establishes the LGT as a robust finite-temperature classical memory. We further prove that entropic effects lead to an emergent one-form symmetry, via a notion that we make precise. We argue that the exponential mixing time follows from universal properties of the deconfined phase, and numerically corroborate this expectation by exploring mixing time scales at the Higgs and confinement transitions out of the deconfined phase. These transitions are found to exhibit markedly different dynamic scaling, even though both have the static critical exponents of the 3D Ising model. We expect this novel entropic mechanism for memory and emergent symmetry to also bring insight into self-correcting quantum memories.
The concept of symmetry breaking and the emergence of corresponding local order parameters constitute the pillars of modern day many body physics. We demonstrate that the existence of symmetry breaking is a consequence of the geometric structure of the convex set of reduced density matrices of all possible many body wavefunctions. The surfaces of these convex bodies exhibit non-analyticities, which signal the emergence of symmetry breaking and of an associated order parameter and also show different characteristics for different types of phase transitions. We illustrate this with three paradigmatic examples of many body systems exhibiting symmetry breaking: the quantum Ising model, the classical q -state Potts model in two-dimensions at finite temperature and the ideal Bose gas in three-dimensions at finite temperature. This state based viewpoint on phase transitions provides a unique novel tool for studying exotic many body phenomena in quantum and classical systems.
Spontaneous symmetry breaking generally circumvents one-dimensional systems with local interactions in thermal equilibrium. Here, we analyze a category of one-dimensional Hermitian models via local non-Hermitian constructions. Notably, spontaneous symmetry breaking and long-range order may emerge at finite temperatures in such systems under periodic boundary conditions, in sharp contrast to Hermitian constructions. We demonstrate clear numerical evidence, such as order parameters and specific heat, supporting phase diagrams with robust ordered phases. Non-Hermitian physics plays a vital role in prohibiting domain-wall proliferation and promoting spontaneous symmetry breaking. The fermions exhibit an exotic topological nature in their path-integral windings, which uphold nonzero integers-commonly a non-Hermitian signature-in the ordered phases, thus offering a novel and spontaneous origin for both symmetry breaking and non-Hermiticity.