Exact finite-temperature characterization of the disordered regions of the generalized kagomé-lattice Ising model with pair and triplet interactions

The Ising model on the kagomé lattice (a lattice of corner-sharing triangles) with a nearest-neighbor pair coupling $J_2$ , a three-spin (triplet) coupling $J_3$ on every elementary triangle, and magnetic field $h$ is one of the few two-dimensional spin systems with multi-spin interactions that admits exact treatment — but only on special loci: it maps exactly onto the symmetric eight-vertex model on the honeycomb lattice, which is equivalent to an Ising model in a nonzero magnetic field, a famously unsolved problem away from those loci. Barry and Muttalib (2019) obtained exact finite-field phase boundaries, coexistence curves, and critical constants only in the ferromagnetic ordered region ( $J_2>0$ , $|J_3/J_2|<3$ ), leaving the disordered regions — antiferromagnetic pair coupling ( $J_2<0$ ) and the triplet-dominated regime ( $|J_3/J_2|\geq 3$ , where no long-range order develops at the coexistence field) — uncharacterized; these regions may host distinct physics including geometrical frustration. The same authors (2022) then determined the exact ground-state phase diagram of these regions: geometrical frustration when the antiferromagnetic pair coupling dominates, disorder without frustration when the triplet coupling dominates, with exact boundaries at $J_3=\pm J_2$ ; a companion study (2024) determined the correlation functions of the pure pair-coupling frustrated axis at all temperatures. The finite-temperature thermodynamics of the full three-parameter model, however, is known only through a closed-form approximation that is exact at zero temperature and restricted to a two-parameter subspace. The remaining problem is to characterize the disordered regions exactly at finite temperature for the full three-parameter model, at the field values where the disordered ground states are realized. The ground-state part of the question is thus resolved; the exact finite-temperature characterization of the disordered regions remains open and is the subject of this problem.

Background

The kagomé lattice is a two-dimensional net of corner-sharing triangles, each site having four nearest neighbors and belonging to exactly two elementary triangles. When Ising spins ( $\sigma_i=\pm 1$ ) on this lattice interact antiferromagnetically in pairs, each triangle cannot satisfy all three of its bonds simultaneously — the defining situation of geometrical frustration, which produces massive ground-state degeneracy and the absence of long-range order at any temperature. Kano and Naya (1953) solved this prototype kagomé antiferromagnet exactly: it remains disordered at all temperatures, with a residual entropy (the zero-temperature entropy per spin left by the degenerate ground states) of approximately $0.5018\,k_B$ . Its ferromagnetic counterpart was solved by Naya (1954), who obtained the exact spontaneous magnetization of the honeycomb and kagomé Ising ferromagnets.

Beyond pair interactions, Ising models with three-spin (triplet) couplings — where the energy of each elementary triangle depends on the product $\sigma_1\sigma_2\sigma_3$ of its three spins — have an exact-solution lineage of their own: Hintermann and Merlini (1972) and Baxter and Wu (1973) solved the pure-triplet model on the triangular lattice. On the kagomé lattice the pure-triplet model is disordered at all temperatures but unfrustrated, since the triplet coupling alone is satisfied by an even number of down-spins in every triangle (Muttalib, Khatun, and Barry 2017 computed its exact thermodynamics, along with the exact perpendicular susceptibility — the linear response of an Ising system to a weak field applied transverse to its quantization axis — of the frustrated pure-pair antiferromagnet, and proposed a temperature-scaled susceptibility as a quantitative measure of frustration).

The generalized model studied in this problem combines both ingredients, $H = -J_2\sum_{\langle ij\rangle}\sigma_i\sigma_j \; - \; J_3\sum_{\Delta}\sigma_1\sigma_2\sigma_3 \; - \; h\sum_i\sigma_i,$ with the pair sum over nearest-neighbor kagomé bonds, the triplet sum over all elementary triangles, and $h$ a uniform magnetic field. Its solvability rests on a remarkable structure: by a decoration transformation — a mapping that sums out auxiliary decorating spins into effective interaction weights — the generalized kagomé Ising (or lattice-gas) model maps exactly onto the symmetric eight-vertex model on the honeycomb lattice (a vertex model whose local Boltzmann weights take four values $a,b,c,d$ related by symmetry). Wu (1974) showed that this symmetric eight-vertex model is in turn equivalent to an Ising model in a nonzero (real or pure-imaginary) magnetic field, and re-derived the equivalence directly in 1990. Exploiting the same equivalence, Barry and Sullivan (1993) obtained the exact liquid-vapor phase diagrams of the pure-triplet kagomé lattice gas; and certain special kagomé Ising Hamiltonians are exactly soluble in a field — super-exchange models, in which interactions between lattice spins are mediated by decorating spins on the bonds (Lu and Wu 2005), and lattices with competing pair interactions (Debauche, Diep, Azaria, and Giacomini 1991) — because their vertex-model images satisfy the free-fermion condition, an algebraic condition on the vertex weights under which the partition function reduces to that of free fermions. None of these solvable cases determines the finite-temperature thermodynamics of the pair-plus-triplet Hamiltonian studied here at generic parameter values. The exact transition temperature is known only in the ferromagnetic case; the general free energy of the Ising model in a field — and hence of the symmetric honeycomb eight-vertex model — is a long-standing unsolved problem, since the commuting-transfer-matrix integrability — the exact-solution technique in which transfer matrices taken along different lattice directions commute and share eigenstates — that solves Baxter's square-lattice eight-vertex model has no known analogue on the honeycomb lattice. Exact results therefore exist only on measure-zero loci: zero-field-type conditions, the coexistence line, and critical loci.

Working on these loci, Barry and Muttalib (2019) obtained exact finite-field phase boundaries, coexistence curves, and critical constants of the generalized kagomé Ising ferromagnet — but only in the ferromagnetic ordered region $J_2>0$ with $\alpha'=-J_3/J_2$ in the finite interval $(-3,3)$ : at the ground-state field $h=-(2/3)J_3$ the ground state is long-range ordered precisely for $|\alpha'|<3$ , and the exact finite-field phase boundary terminates, as $\alpha'\to 3$ , at a critical point collapsing to $(h/J_2=2,\,T=0)$ with $m_c=-1$ . Outside this region the model is disordered at the coexistence field, and its characterization — including where frustration appears — was left open. Their 2022 follow-up answered the ground-state part of that question exactly (frustrated versus unfrustrated disorder, boundaries $J_3=\pm J_2$ , crossover curves persisting to finite temperature), which sharpens, but does not close, the finite-temperature question that remains the subject of this problem record. A 2024 companion work by the same authors determined the correlation functions of the pure pair-coupling frustrated kagomé antiferromagnet at all temperatures, yielding exact solutions of its inelastic neutron scattering, but likewise left the finite-temperature interior of the disordered regions, where both couplings are nonzero, untouched.

Problem Statement

Consider the Ising model on the kagomé lattice with Hamiltonian $H=-J_2\sum_{\langle ij\rangle}\sigma_i\sigma_j - J_3\sum_{\Delta}\sigma_1\sigma_2\sigma_3 - h\sum_i\sigma_i$ ( $\sigma_i=\pm 1$ ; the pair sum runs over nearest-neighbor kagomé bonds, the triplet sum over all elementary triangles, and $h$ is a uniform magnetic field). Barry and Muttalib's exact solutions (2019) cover only the ferromagnetic ordered region $J_2>0$ , $|J_3/J_2|<3$ ; the disordered regions — antiferromagnetic pair interactions ( $J_2<0$ , any $J_3$ ) and the triplet-dominated regime ( $|J_3/J_2|\ge 3$ , where the ground state at the coexistence field $h=-(2/3)J_3$ develops no long-range order) — were left uncharacterized at finite temperature. The ground-state phase diagram of these regions is now exactly known (Muttalib and Barry 2022: frustrated disorder when the antiferromagnetic pair coupling dominates, non-frustrated disorder when the triplet coupling dominates, boundary $J_3=\pm J_2$ at the appropriately tuned field, with crossover curves persisting to finite temperature, and a $T=0$ threshold field argued to indicate a gap). The remaining open problem is the exact finite-temperature characterization of these disordered regions for the three-parameter model, at the field values where the disordered ground states are realized: the appropriately tuned field of Muttalib and Barry (2022) — the coexistence field $h=-(2/3)J_3$ in the triplet-dominated regime ( $|J_3/J_2|\ge 3$ ), and $h=0$ in the $J_2<0$ regime with $|J_3/J_2|<3$ , split into frustrated disorder for $|J_3/J_2|<1$ and non-frustrated disorder for $1\le|J_3/J_2|<3$ (the pure $J_3=0$ and $J_2=0$ axes are prototype limits with exactly known zero-field thermodynamics). Specifically, the goal is to determine the exact finite-temperature thermodynamics of the disordered regions — the free energy and, from it, the induced magnetization and parallel susceptibility at the coexistence field $h=-(2/3)J_3$ (triplet-dominated regime) and at $h=0$ (the $J_2<0$ regime), as functions of temperature — going beyond the currently available closed-form approximation, which is exact only at $T=0$ and restricted to a two-parameter subspace; equivalently, to obtain exact finite-temperature results for the corresponding parameter regimes of the symmetric honeycomb-lattice eight-vertex model. Whether any true finite-temperature phase transitions occur inside the disordered regions, beyond the crossover behavior identified in the 2022 analysis, forms part of this characterization.

References

J. H. Barry and K. A. Muttalib, 'Finite-field phase transitions and criticality in a generalized kagomé Ising ferromagnet: Exact solutions', Physica A: Statistical Mechanics and its Applications 527, 121326 (2019). DOI: 10.1016/j.physa.2019.121326, https://doi.org/10.1016/j.physa.2019.121326 (exact finite-field phase boundaries, coexistence curves, and critical constants of the generalized kagomé Ising ferromagnet in the ferromagnetic ordered region $J_2>0$ , $|J_3/J_2|<3$ ).
K. A. Muttalib and J. H. Barry, 'Frustration in a generalized kagomé Ising antiferromagnet: Exact results', Physical Review E 106, 014149 (2022). DOI: 10.1103/PhysRevE.106.014149, https://doi.org/10.1103/PhysRevE.106.014149 (exact ground-state phase diagram of the generalized kagomé antiferromagnet, together with a closed-form approximation for the finite-field induced magnetization and parallel susceptibility on a two-parameter subspace).
K. A. Muttalib and J. H. Barry, 'Inelastic neutron scattering from a geometrically frustrated Ising antiferromagnet: Exact solutions', Physical Review B 110, 024414 (2024). DOI: 10.1103/PhysRevB.110.024414, https://doi.org/10.1103/PhysRevB.110.024414 (same authors' exact determination of all-temperature correlations and neutron scattering on the pure pair-coupling frustrated axis).
K. A. Muttalib, M. Khatun, and J. H. Barry, 'Perpendicular susceptibility and geometrical frustration in two-dimensional Ising antiferromagnets: Exact solutions', Physical Review B 96, 184411 (2017). DOI: 10.1103/PhysRevB.96.184411, https://doi.org/10.1103/PhysRevB.96.184411 (exact results for the pure pair-antiferromagnet and pure-triplet kagomé limits).
J. H. Barry, T. Tanaka, M. Khatun, and C. H. Múnera, 'Exact solutions for Ising-model odd-number correlations on planar lattices', Physical Review B 44, 2595 (1991). DOI: 10.1103/PhysRevB.44.2595, https://doi.org/10.1103/PhysRevB.44.2595 (exact solutions for odd-number and multi-spin correlations on planar lattices).
J. H. Barry and N. S. Sullivan, 'Exact phase diagrams for the condensation of a kagomé lattice gas with three-particle interactions', International Journal of Modern Physics B 7, 2831 (1993). DOI: 10.1142/S0217979293003061, https://doi.org/10.1142/S0217979293003061 (exact liquid-vapor phase-diagram loci of the pure-triplet kagomé lattice gas, via the honeycomb Ising-in-a-field equivalence).
K. Kano and S. Naya, 'Antiferromagnetism. The Kagome Ising Net', Progress of Theoretical Physics 10, 158 (1953). DOI: 10.1143/ptp/10.2.158, https://doi.org/10.1143/ptp/10.2.158 (prototype frustrated kagomé antiferromagnet).
S. Naya, 'On the Spontaneous Magnetizations of Honeycomb and Kagomé Ising Lattices', Progress of Theoretical Physics 11, 53 (1954). DOI: 10.1143/PTP.11.53, https://doi.org/10.1143/PTP.11.53 (exact spontaneous magnetization of the honeycomb and kagomé Ising ferromagnets).
F. Y. Wu, 'Eight-Vertex Model on the Honeycomb Lattice', Journal of Mathematical Physics 15, 687 (1974). DOI: 10.1063/1.1666712, https://doi.org/10.1063/1.1666712 (symmetric honeycomb eight-vertex model as an Ising model in a nonzero field).
F. Y. Wu, 'Eight-vertex model and Ising model in a non-zero magnetic field: honeycomb lattice', Journal of Physics A: Mathematical and General 23, 375 (1990). DOI: 10.1088/0305-4470/23/3/021, https://doi.org/10.1088/0305-4470/23/3/021 (direct rederivation of the equivalence underpinning the model's solvability).
R. J. Baxter and F. Y. Wu, 'Exact Solution of an Ising Model with Three-Spin Interactions on a Triangular Lattice', Physical Review Letters 31, 1294 (1973). DOI: 10.1103/PhysRevLett.31.1294, https://doi.org/10.1103/PhysRevLett.31.1294 (three-spin-interaction exact-solution lineage).
A. Hintermann and D. Merlini, 'Exact solution of a two dimensional Ising model with pure 3 spin interactions', Physics Letters A 41, 208 (1972). DOI: 10.1016/0375-9601(72)90261-7, https://doi.org/10.1016/0375-9601(72)90261-7 (pure three-spin Ising model).
D. Sivý and J. Strečka, 'Continuous and discontinuous transitions in the Ising-Heisenberg model on the extended Lieb lattice in a magnetic field', arXiv:2606.10821 (2026). https://arxiv.org/abs/2606.10821 (an Ising-Heisenberg model on the extended Lieb lattice in a magnetic field).
E. Jurčišinová and M. Jurčišin, 'Geometric frustration effects in the spin-1 antiferromagnetic Ising model on the kagome-like recursive lattice: exact results', Journal of Statistical Mechanics: Theory and Experiment (2016) 093207. DOI: 10.1088/1742-5468/2016/09/093207, https://doi.org/10.1088/1742-5468/2016/09/093207 (geometric-frustration effects in the spin-1 antiferromagnetic Ising model on a kagomé-like recursive lattice).
W. T. Lu and F. Y. Wu, 'Soluble kagome Ising model in a magnetic field', Physical Review E 71, 046120 (2005). DOI: 10.1103/PhysRevE.71.046120, https://doi.org/10.1103/PhysRevE.71.046120 (an exactly soluble kagomé Ising model in a field: a super-exchange Hamiltonian whose vertex-model image satisfies the free-fermion condition).
M. Debauche, H. T. Diep, P. Azaria, and H. Giacomini, 'Exact phase diagram of a generalized Kagomé Ising lattice: Reentrance and disorder lines', Physical Review B 44, 2369 (1991). DOI: 10.1103/PhysRevB.44.2369, https://doi.org/10.1103/PhysRevB.44.2369 (a generalized kagomé Ising lattice with competing pair interactions, solved via a free-fermion 16-vertex mapping).