Exactly solved two-dimensional frustrated Ising models — the centered square, centered honeycomb, and generalized kagomé lattices, mapped onto free-fermion vertex models — exhibit reentrant paramagnetic phases: disordered phases interposed between incompatible ordered phases along the temperature axis, occurring in finite regions of parameter space. A long-standing conjecture holds that ground-state partial disorder (the coexistence of ordered and free spins at zero temperature) is necessary for reentrance, but the centered honeycomb lattice supplies three cases with identical residual entropy of which only one is reentrant, so partial disorder is not sufficient. No theoretical criterion, necessary and sufficient and computable from microscopic or ground-state data, exists for when a frustrated spin system develops a reentrant paramagnetic phase rather than a direct first-order transition between the two ordered phases. The problem is to establish such a criterion for periodically frustrated spin systems, with the exactly solved phase diagrams of these lattices as the testbed.

Background

A spin system is said to be frustrated when competing interactions cannot be simultaneously minimized: on a triangle of antiferromagnetic Ising bonds, or on a lattice mixing ferromagnetic and antiferromagnetic couplings, no configuration satisfies all bonds (a concept introduced by Toulouse in 1977). Frustrated Ising systems can display a counterintuitive phase-diagram topology: on heating, the system may leave an ordered phase, pass through a paramagnetic phase, enter a different ordered phase, and only become paramagnetic again at still higher temperature. The interposed disordered phase is called a reentrant paramagnetic phase. The centered square lattice — a square lattice with an additional spin at the center of each square, coupled to its four nearest neighbors — was introduced and solved exactly by Vaks, Larkin and Ovchinnikov in 1966 and was the first exactly soluble model found to exhibit a reentrant paramagnetic phase. Between 1987 and 1992, Azaria, Diep, Giacomini and co-workers exactly solved a family of further such models by mapping them onto vertex models: the kagomé lattice (a lattice of corner-sharing triangles) with nearest- and next-nearest-neighbor interactions, the anisotropic centered honeycomb lattice (a honeycomb lattice with an additional spin at the center of each hexagon, carrying three competing couplings), the generalized kagomé lattice, and periodically dilute centered square lattices. A vertex model is a lattice statistical model whose Boltzmann weights are attached to local configurations of arrows or bonds around each site; when the weights satisfy certain algebraic 'free-fermion' conditions, the partition function reduces to a fermionic (determinantal) form and the free energy follows in closed form. The relevant solvable classes here are the free-fermion 8-vertex model, the free-fermion 16-vertex model, and the 32-vertex model on the triangular lattice solved by Sacco and Wu. These exact solutions established three linked phenomena: (i) ground states with partial disorder, in which some sublattices order while other spins remain free down to zero temperature, producing a residual entropy at zero temperature (for example S0=ln⁡2/3S_{0}=\ln 2/3 per spin in three anisotropic centered-honeycomb cases); (ii) reentrant paramagnetic phases occupying finite regions of the coupling-temperature phase diagram, including cases with two successive reentrant phases; and (iii) disorder lines, first identified by Stephenson in 1970 — loci where a nearest-neighbor correlation function vanishes and changes sign inside the paramagnetic phase, separating regimes of different fluctuation character. The central empirical finding motivating the open problem is that partial disorder accompanies reentrance in most known cases, yet the centered honeycomb lattice provides three ground-state cases with identical residual entropy in which only one exhibits reentrance: partial disorder is not sufficient, and its necessity remains a conjecture. Additional qualitative ingredients were identified — the coordination number of the disordered sites (four in the known reentrant cases; two appears too small to induce reentrance, six appears to contaminate the whole system with disorder), the anisotropy of the interactions (which enlarges reentrant regions, making them infinite in a generalized kagomé model), and sufficient-but-not-excessive frustration — but none was shown to be decisive. In three dimensions, a body-centered-cubic generalization of the centered square lattice shows the reentrant region shrinking into a first-order transition line terminated by a multicritical point from which two second-order lines emerge, illustrating that reentrance competes directly with first-order transitions between incompatible orders. Numerical work on the two-dimensional random-bond Ising model by Thomas and Katzgraber further found reentrant behavior to be generic whenever frustration is present, even without periodic ground-state partial disorder, complicating the scope of any criterion. Despite more than three decades of exactly solved examples, numerical studies, and recent exact decorated-lattice solutions, the mechanism has remained without a theoretical explanation: there is no known criterion that predicts, from the interaction pattern or the ground state, whether a given frustrated system will display a reentrant paramagnetic phase.

Problem Statement

For periodically frustrated spin systems with competing interactions — translationally periodic lattices with competing ferromagnetic and antiferromagnetic couplings whose ground state is frustrated, while non-periodic and randomly disordered models lie outside the class — with the exactly solved periodically frustrated Ising lattices (centered square, centered honeycomb, generalized kagomé, and their dilute and three-dimensional generalizations) as the paradigm class and testbed — establish a theoretical criterion, both necessary and sufficient, for the occurrence of a reentrant paramagnetic phase, that is, an intermediate disordered phase between two incompatible ordered phases on the temperature axis. The criterion should be formulated in terms of microscopic or ground-state data — such as the presence and structure of partial disorder, the coordination number of the disordered sites, the anisotropy and relative strengths of the competing couplings, and the entropies of the competing phases — and must jointly answer three linked questions that together define the criterion: (1) prove, or refute by an explicit counterexample, the conjecture that ground-state partial disorder is necessary for reentrance; (2) identify the additional ingredients that, combined with partial disorder, are sufficient — explaining in particular why the three centered-honeycomb cases with identical residual entropy S0=ln⁡2/3S_{0}=\ln 2/3 per spin do not all reenter; and (3) determine under which microscopic circumstances a system traverses an intermediate reentrant paramagnetic phase rather than undergoing a direct first-order transition between the two incompatible ordered phases. Any proposed criterion must be consistent with the exactly solved phase diagrams of this class, correctly predicting where reentrant phases, where successive reentrances, and where direct first-order transitions occur.

References

  1. H. T. Diep and H. Giacomini, 'Frustration — Exactly Solved Frustrated Models', in Frustrated Spin Systems, 3rd ed. (World Scientific, 2020), pp. 1–62; DOI 10.1142/9789811214141_0001; arXiv:1912.13042; https://arxiv.org/abs/1912.13042
  2. V. G. Vaks, A. I. Larkin and Yu. N. Ovchinnikov, 'Ising Model with Interaction Between Nonnearest Neighbors', Sov. Phys. JETP 22, 820 (1966); https://www.semanticscholar.org/paper/5fe8fae95c9adc67d352ea1733bf796061a05e89
  3. P. Azaria, H. T. Diep and H. Giacomini, 'Coexistence of order and disorder and reentrance in an exactly solvable model', Phys. Rev. Lett. 59, 1629 (1987); DOI 10.1103/PhysRevLett.59.1629; https://doi.org/10.1103/PhysRevLett.59.1629
  4. P. Azaria, H. T. Diep and H. Giacomini, 'Ordering in the centered-square-lattice Ising model', Phys. Rev. B 39, 740 (1989); DOI 10.1103/PhysRevB.39.740; https://doi.org/10.1103/PhysRevB.39.740
  5. P. Azaria, H. T. Diep and H. Giacomini, 'First-Order Transition, Multicriticality and Re-entrance in a b.c.c. Lattice with Ising Spins', Europhys. Lett. 9, 755 (1989); DOI 10.1209/0295-5075/9/8/003; https://doi.org/10.1209/0295-5075/9/8/003
  6. H. T. Diep, M. Debauche and H. Giacomini, 'Exact solution of an anisotropic centered honeycomb Ising lattice: Reentrance and partial disorder', Phys. Rev. B 43, 8759 (1991); DOI 10.1103/PhysRevB.43.8759; https://doi.org/10.1103/PhysRevB.43.8759
  7. M. Debauche, H. T. Diep, P. Azaria and H. Giacomini, 'Exact phase diagram of a generalized Kagomé Ising lattice: Reentrance and disorder lines', Phys. Rev. B 44, 2369 (1991); DOI 10.1103/PhysRevB.44.2369; https://doi.org/10.1103/PhysRevB.44.2369
  8. M. Debauche and H. T. Diep, 'Successive reentrances and phase transitions in exactly solved dilute centered square Ising lattices', Phys. Rev. B 46, 8214 (1992); DOI 10.1103/PhysRevB.46.8214; https://doi.org/10.1103/PhysRevB.46.8214
  9. H. T. Diep, 'Frustrated spin systems: history of the emergence of a modern physics', Comptes Rendus. Physique 26, 225–251 (2025); DOI 10.5802/crphys.235; https://doi.org/10.5802/crphys.235
  10. C. K. Thomas and H. G. Katzgraber, 'Simplest model to study reentrance in physical systems', Phys. Rev. E 84, 040101(R) (2011); DOI 10.1103/PhysRevE.84.040101; arXiv:1104.2582; https://arxiv.org/abs/1104.2582
  11. R. Pohle, O. Benton and L. D. C. Jaubert, 'Reentrance of disorder in the anisotropic shuriken Ising model', Phys. Rev. B 94, 014429 (2016); DOI 10.1103/PhysRevB.94.014429; arXiv:1602.06115; https://arxiv.org/abs/1602.06115
  12. J. Strečka, K. Karl'ová, T. Verkholyak, N. Caci, S. Wessel and A. Honecker, 'Thermal first-order phase transitions, Ising critical points, and reentrance in the Ising-Heisenberg model on the diamond-decorated square lattice in a magnetic field', Phys. Rev. B 107, 134402 (2023); DOI 10.1103/PhysRevB.107.134402; arXiv:2212.01278; https://arxiv.org/abs/2212.01278
  13. F. A. Kassan-Ogly and A. V. Zarubin, 'Frustration in the Ising model on a decorated square lattice', Phys. Rev. E 111, 044133 (2025); DOI 10.1103/PhysRevE.111.044133; https://doi.org/10.1103/PhysRevE.111.044133
  14. A. V. Zarubin and F. A. Kassan-Ogly, 'The formation of magnetic reentrancy in the Ising model on a decorated square lattice' (2026); arXiv:2606.12572; https://arxiv.org/abs/2606.12572
  15. R. Kaneko, Y. Douda, S. Goto and I. Danshita, 'Reentrance of the Disordered Phase in the Antiferromagnetic Ising Model on a Square Lattice with Longitudinal and Transverse Magnetic Fields', J. Phys. Soc. Jpn. 90, 073001 (2021); DOI 10.7566/jpsj.90.073001; arXiv:2103.12364; https://arxiv.org/abs/2103.12364
  16. D. R. Yahne, D. Pereira, L. D. C. Jaubert, L. D. Sanjeewa, M. Powell, J. W. Kolis, G. Xu, M. Enjalran, M. J. P. Gingras and K. A. Ross, 'Understanding Reentrance in Frustrated Magnets: The Case of the Er2Sn2O7 Pyrochlore', Phys. Rev. Lett. 127, 277206 (2021); DOI 10.1103/PhysRevLett.127.277206; arXiv:2101.08361; https://arxiv.org/abs/2101.08361
  17. F.-F. Song and G.-M. Zhang, 'Tensor network approach to the two-dimensional fully frustrated XY model and a chiral ordered phase', Phys. Rev. B 105, 134516 (2022); DOI 10.1103/PhysRevB.105.134516; arXiv:2112.03550; https://arxiv.org/abs/2112.03550
  18. J. E. Sacco and F. Y. Wu, '32-vertex model on the triangular lattice', J. Phys. A: Math. Gen. 8, 1780 (1975); DOI 10.1088/0305-4470/8/11/013; https://doi.org/10.1088/0305-4470/8/11/013
  19. J. Stephenson, 'Ising-Model Spin Correlations on the Triangular Lattice. IV. Anisotropic Ferromagnetic and Antiferromagnetic Lattices', J. Math. Phys. 11, 420–431 (1970); DOI 10.1063/1.1665155; https://doi.org/10.1063/1.1665155
  20. D. Loison, 'Phase Transitions in Frustrated Vector Spin Systems: Numerical Studies', in Frustrated Spin Systems, 3rd ed. (World Scientific, 2020), pp. 191–244; DOI 10.1142/9789811214141_0004; https://doi.org/10.1142/9789811214141_0004