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arXiv:2506.15769 · Physical Review X

Breakdown of the Thermodynamic Limit in Quantum Spin and Dimer Models

Jeet Shah, Laura Shou, Jeremy Shuler, Victor Galitski

Published Feb 27, 2026• 3 citations

Abstract

The thermodynamic limit is foundational to statistical mechanics, underlying our understanding of many-body phases. It assumes that, as the system size grows infinitely at fixed density of particles, unambiguous macroscopic phases emerge that are independent of the system’s boundary shape. We present explicit quantum spin and dimer Hamiltonians whose ground states violate this principle. Our construction relies on the previous mathematical work on classical dimers on the Aztec diamond and the square-octagon fortress, where geometry-dependent phase behaviors are observed in the infinite-size limit. We reverse engineer quantum spin Hamiltonians on the square and the square-octagon lattices whose ground states at the Rokhsar-Kivelson points are described by classical dimer coverings. On diamond-shaped domains, we find macroscopic boundary regions exhibiting distinct quantum phases from those on square-shaped domains. We study the nature of these phases by calculating the dimer-dimer and vison correlators and adapt Kasteleyn matrix based analytical and numerical methods for computing the vison correlator, which are significantly more efficient than standard Monte Carlo techniques. Our results show that the square-octagon lattice supports a single gapped short-range entangled phase, with exponentially decaying dimer correlators and a constant vison correlator. When the same model is considered on a diamond-shaped domain, two additional macroscopic regions emerge, with one near the corners and exhibiting staggered dimer order, and another exhibiting critical correlations.

DOI: 10.1103/ckrx-wbct

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The thermodynamic limit is foundational to statistical mechanics, underlying our understanding of many-body phases. It assumes that, as the system size grows infinitely at fixed density of particles, unambiguous macroscopic phases emerge that are independent of the system’s boundary shape. We present explicit quantum spin and dimer Hamiltonians whose ground states violate this principle. Our construction relies on the previous mathematical work on classical dimers on the Aztec diamond and the square-octagon fortress, where geometry-dependent phase behaviors are observed in the infinite-size limit. We reverse engineer quantum spin Hamiltonians on the square and the square-octagon lattices whose ground states at the Rokhsar-Kivelson points are described by classical dimer coverings. On diamond-shaped domains, we find macroscopic boundary regions exhibiting distinct quantum phases from those on square-shaped domains. We study the nature of these phases by calculating the dimer-dimer and vison correlators and adapt Kasteleyn matrix based analytical and numerical methods for computing the vison correlator, which are significantly more efficient than standard Monte Carlo techniques. Our results show that the square-octagon lattice supports a single gapped short-range entangled phase, with exponentially decaying dimer correlators and a constant vison correlator. When the same model is considered on a diamond-shaped domain, two additional macroscopic regions emerge, with one near the corners and exhibiting staggered dimer order, and another exhibiting critical correlations.

We consider a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We focus on a quantum dimer model on the triangular lattice, with doubly periodic edge weights. For simplicity we consider a

2\times1

periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled

α

. We analytically show that the model exhibits a continuous quantum phase transition at

α=3

, changing from a topological

\mathbb{Z}_2

quantum spin liquid (

α<3

) to a columnar ordered state (

α>3

). The dimer-dimer correlator decays exponentially on both sides of the transition with the correlation length

ξ\propto1/|α-3|

and as a power-law at criticality. The vison correlator exhibits an exponential decay in the spin liquid phase, but becomes a constant in the ordered phase, which we explain in terms of loop statistics of the double-dimer model. Using finite-size scaling of the vison correlator, we extract critical exponents consistent with the 2D Ising universality class. Additionally, we analytically show that the topological Rényi entropy of order

\infty

(topological min-entropy) changes from

\log2

for the quantum spin liquid phase

α<3

, to

0

for the ordered phase

α>3

, thereby analytically confirming the topological nature of the phase transition.

The spin-1/2 Heisenberg antiferromagnet on the diamond-decorated square lattice in the presence of a magnetic field displays various quantum phases including the Lieb-Mattis ferrimagnetic, dimer-tetramer, monomer-dimer, and spin-canted phases, in addition to the trivial fully saturated state. Thermodynamic properties of this model are investigated using several complementary analytical and numerical methods such as exact diagonalization up to the systems of 40 spins, an effective monomer-dimer description, sign-problem-free quantum Monte Carlo simulations for up to 180 spins, and a decoupling approximation. Our particular attention is focused on the parameter region favoring the dimer-tetramer phase. This ground state can be represented by a classical hard-dimer model on the square lattice and retains a macroscopic degeneracy even under a magnetic field. However, the description of the low-temperature thermodynamics close to the boundary between the macroscopically degenerate dimer-tetramer and the nondegenerate monomer-dimer phases requires an extended classical monomer-dimer lattice-gas model. Anomalous thermodynamic properties emerging in the vicinity of the dimer-tetramer phase are studied in detail. Under the adiabatic demagnetization we detect an enhanced magnetocaloric effect promoting an efficient cooling to absolute zero temperature.

We consider the spin-

1/2

Hamiltonians proposed by Cano and Fendley [Phys. Rev. Lett. 105, 067205 (2010)], which were built to promote the well-known Rokshar-Kivelson (RK) point of quantum dimer models to spin-

1/2

wave functions. We first show that these models, besides the exact degeneracy of RK point, support gapless spinless excitations as well as a spin gap in the thermodynamic limit, signatures of an unusual spin liquid. We then extend the original construction to create a continuous family of SU(2) invariant spin models that reproduces the phase diagram of the quantum dimer model and, in particular, show explicit evidences for existence of columnar and staggered phases. The original models thus appear as multicritical points in an extended phase diagram. Our results are based on the use of a combination of numerical exact simulations and analytical mapping to effective generalized quantum dimer models.

The plaquette phase of the square-lattice quantum dimer model is studied using a continuous-time reptation quantum Monte Carlo method for lattices of sizes up to $48\ifmmode\times\else\texttimes\fi{}48$ sites. We determine the location of the phase transition between the columnar and plaquette phases to occur at ${V}_{c}∕J=0.60\ifmmode\pm\else\textpm\fi{}0.05$ which is significantly larger than inferred from previous exact diagonalization studies on smaller lattices. Off diagonal correlation functions are obtained. They exhibit long-range order in the plaquette phase but not at the Rokhsar-Kivelson point. We also observe significant finite-size corrections to scaling for the transition between the plaquette phase and the critical resonating valence bond liquid. This study demonstrates the importance of understanding finite-size effects when considering critical properties of the square-lattice quantum dimer model.

Dimerization in

O(n)

-invariant quantum spin chains

We establish dimerization in

O(n)

-invariant quantum spin chains with large enough

n

, in a much larger region of the phase diagram than previous results. This entails identifying two distinct infinite volume ground states which are gapped, are translations by one unit of each other, and which both have exponentially decaying correlations. We show these states are the limits of finite-volume, finite temperature states as the graph tends to

\mathbb{Z}

and temperature tends to zero in any order. Our method relies on a probabilistic representation of the quantum system in terms of random loops, and an adaptation of a method developed for loop

O(n)

models on the hexagonal lattice by Duminil-Copin, Peled, Samotij and Spinka.

A family of models is proposed to describe the motion of holes in a fluctuating quantum dimer background on the square lattice. Following Castelnovo et al. [Ann. Phys. (N.Y.) 318, 316 (2005)], a generalized Rokhsar-Kivelson Hamiltonian at finite doping which can be mapped on a doped interacting classical dimer model is constructed. A simple physical extension of this model is also considered. Using numerical computations and simple considerations based on the above exact mapping, we determine the phase diagram of the model showing a number of quantum phases typical of a doped Mott insulator. The two-hole correlation function generically exhibits short-range or long-range algebraic correlations in the solid (columnar) and liquid (critical) phases of the model, respectively. Evidence for an extended region of a doped valence bond solid phase exhibiting holon pairing but no phase separation is given. In contrast, we show that hole deconfinement occurs in the staggered dimer phase.