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arXiv:2505.05889 · Physical Review Letters

Topological Devil’s Staircase in a Constrained Kagome Ising Antiferromagnet

Afonso Rufino, Samuel Nyckees, Jeanne Colbois, Frédéric Mila

Published Jan 14, 2026• 1 citation
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Abstract

We show that the constrained Ising model on the kagome lattice with infinite first and third neighbor couplings undergoes an infinite series of thermal first-order transitions at which, as in the Kasteleyn transition, linear defects of infinite length condense. However, their density undergoes abrupt jumps because of the peculiar structure of the low temperature phase, which is only partially ordered and hosts a finite density of zero-energy domain walls. The number of linear defects between consecutive zero-energy domain walls is quantized to integer values, leading to a devil's staircase of topological origin. By contrast to the devil's staircase of the anisotropic next-nearest neighbor Ising and related models, the wave vector is not fixed to commensurate values inside each phase.

DOI
10.1103/dgyr-5y75

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We show that the constrained Ising model on the kagome lattice with infinite first and third neighbor couplings undergoes an infinite series of thermal first-order transitions at which, as in the Kasteleyn transition, linear defects of infinite length condense. However, their density undergoes abrupt jumps because of the peculiar structure of the low temperature phase, which is only partially ordered and hosts a finite density of zero-energy domain walls. The number of linear defects between consecutive zero-energy domain walls is quantized to integer values, leading to a devil's staircase of topological origin. By contrast to the devil's staircase of the anisotropic next-nearest neighbor Ising and related models, the wave vector is not fixed to commensurate values inside each phase.
A Devil's staircase is a sequence of transitions between topological sectors as a control parameter is varied. Such staircases are predicted in frustrated magnets and lattice gauge theories, but observing them is hard: the winding number that labels each sector is topologically protected, so local quantum dynamics alone cannot move the system from one step to the next. Here we propose and analyze an experimentally feasible way to realize a thermal quasi-Devil's staircase in a finite Rydberg-atom array. We study the anisotropic triangular-lattice Ising antiferromagnet using a dimer mapping, a directed-string description, exact thermodynamics, and Monte Carlo simulations. In the defect-free string manifold, the commensurate--incommensurate onset follows from a simple energy--entropy balance, and the exact Ising critical relation reduces to the same condition when triangle-rule defects are rare. On a finite lattice, this onset appears as a quasi-Devil's staircase of winding-sector crossovers, which we resolve through winding distributions, structure factors, and directional correlations. The key point is that thermal fluctuations overcome the topological barriers that block sector changes under purely quantum dynamics, and the staircase exists precisely at the finite sizes and tunable effective temperatures of current Rydberg platforms. This makes the effect directly observable in existing experiments. Our comparison also shows that the finite-size steps are not phase transitions: in the thermodynamic limit only a single phase transition survives, and the system crosses over to a defect-dominated paramagnet.
We study phase transitions and thermodynamic properties in the two-dimensional antiferromagnetic Ising model with next-nearest-neighbor interaction on a Kagomé lattice by Monte Carlo simulations. A histogram data analysis shows that a second-order transition occurs in the model. From the analysis of obtained data, we can assume that next-nearest-neighbor ferromagnetic interactions in two-dimensional antiferromagnetic Ising model on a Kagomé lattice excite the occurrence of a second-order transition and unusual behavior of thermodynamic properties on the temperature dependence.
Abstract We show that a hierarchy of topological phases in one dimension—a topological Devil’s staircase—can emerge at fractional filling fractions in interacting systems, whose single-particle band structure describes a topological or a crystalline topological insulator. Focusing on a specific example in the BDI class, we present a field-theoretical argument based on bosonization that indicates how the system, as a function of the filling fraction, hosts a series of density waves. Subsequently, based on a numerical investigation of the low-lying energy spectrum, Wilczek–Zee phases, and entanglement spectra, we show that they are symmetry protected topological phases. In sharp contrast to the non-interacting limit, these topological density waves do not follow the bulk-edge correspondence, as their edge modes are gapped. We then discuss how these results are immediately applicable to models in the AIII class, and to crystalline topological insulators protected by inversion symmetry. Our findings are immediately relevant to cold atom experiments with alkaline-earth atoms in optical lattices, where the band structure properties we exploit have been recently realized.
We study the quantum phase diagram of a system of hard-core bosons on the kagome lattice with nearest-neighbor repulsive interactions, for arbitrary densities, by means of the hierarchical mean-field theory and exact diagonalization techniques. This system is isomorphic to the spin S=1/2S=1/2S=1/2 XXZ model in presence of an external magnetic field, a paradigmatic example of frustrated quantum magnetism. In the nonfrustrated regime, we find two crystal phases at densities 1/3 and 2/3 that melt into a superfluid phase when increasing the hopping amplitude, in semiquantitative agreement with quantum Monte Carlo computations. In the frustrated regime and away from half-filling, we find a series of plateaux with densities commensurate with powers of 1/3. The broader density plateaux (at densities 1/3 and 2/3) are remnants of the classical degeneracy in the Ising limit. For densities near half-filling, this staircase of crystal phases melts into a superfluid, which displays finite chiral currents when computed with clusters having an odd number of sites. Both the staircase of crystal phases and the superfluid phase prevail in the noninteracting limit, suggesting that the lowest dispersionless single-particle band may be at the root of this phenomenon.
We show that a hierarchy of topological phases in one dimension-a topological Devil's staircase-can emerge at fractional filling fractions in interacting systems, whose single-particle band structure describes a topological or a crystalline topological insulator. Focusing on a specific example in the BDI class, we present a field-theoretical argument based on bosonization that indicates how the system, as a function of the filling fraction, hosts a series of density waves. Subsequently, based on a numerical investigation of the low-lying energy spectrum, Wilczek-Zee phases, and entanglement spectra, we show that they are symmetry protected topological phases. In sharp contrast to the non-interacting limit, these topological density waves do not follow the bulk-edge correspondence, as their edge modes are gapped. We then discuss how these results are immediately applicable to models in the AIII class, and to crystalline topological insulators protected by inversion symmetry. Our findings are immediately relevant to cold atom experiments with alkaline-earth atoms in optical lattices, where the band structure properties we exploit have been recently realized.