Ground-state manifold and residual entropy of the checkerboard-lattice Ising model with crossing and four-spin interactions beyond its three integrable points

The Ising model on a checkerboard lattice whose elementary crossed squares carry all six two-spin pairings of coupling $J$ plus a four-spin coupling $\Delta$ maps onto square ice, and its ground-state degeneracy reproduces Lieb's exact residual entropy $(3/2)\ln(4/3)$ per molecule whenever the ice rules select the ground states. Li, Zhao, Yao, and Yang evaluated the residual entropy at three integrable parameter points: the free-fermion condition in zero field (which forces a temperature-dependent coupling $\Delta(\beta)$ through $\exp(4\beta\Delta)=\cosh(4\beta J)$ and gives $S/k_B\approx 1.07052$ , larger than Lieb's value because the ground states then include configurations violating the ice rules), the $\Delta\to-\infty$ limit in zero field, and $\Delta=0$ in an imaginary field $i(\pi/2)k_BT$ ; the latter two reproduce Lieb's result exactly. This problem asks for the completion of that picture over the remaining parameter space: the exact ground-state manifold and residual entropy in real external magnetic fields and on the zero-field degeneracy boundary $\Delta=J$ where ice-rule and non-ice-rule configurations compete, the classification of which couplings select purely ice-rule ground states, and the relation of the temperature-dependent free-fermion trajectory value $\approx 1.07052$ to the $\Delta=J$ ground-state entropy. These cases remain unsolved in the literature.

Background

Residual entropy is the entropy that a frustrated many-body system retains at zero temperature, fixed by the degeneracy of its ground states. The paradigm is water ice: the ice rules — exactly one hydrogen on every bond between two neighbouring oxygens and exactly two hydrogens adjacent to each oxygen — are satisfied by a macroscopic number of hydrogen arrangements, so ice keeps a non-zero entropy as $T\to 0$ . In the simplest idealization, square ice, the oxygens sit on a square lattice and each hydrogen bond carries one of two orientations; counting the allowed orientations is equivalent to counting the vertex configurations of the six-vertex model, a lattice model in which each vertex carries one of the arrow patterns with two arrows in and two arrows out. Lieb solved this counting problem exactly in 1967, obtaining the residual entropy $S/k_B=(3/2)\ln(4/3)\approx 0.4315$ per molecule.

The model examined here is a classical Ising model of $N$ spins $s_i=\pm 1$ on a checkerboard lattice — a square lattice in which only every second square, called a crossed square, carries interactions; each spin belongs to exactly two crossed squares, and there are $N/2$ crossed squares in total. Within one crossed square the interaction energy is

E(s_1,s_2,s_3,s_4)=J(s_1s_2+s_2s_3+s_3s_4+s_4s_1+s_1s_3+s_2s_4)+\Delta\,s_1s_2s_3s_4,

where $J>0$ couples all six spin pairs of the square (four edges and two crossing diagonals — hence 'crossing interactions') and $\Delta$ is a four-spin coupling; an external field $H_{ex}$ enters through $-H_{ex}\sum_i s_i$ . Each crossed square is then of one of three types: type A with two spins up and two down ( $E_A=-2J+\Delta$ , six arrangements), type B with a three-to-one imbalance ( $E_B=-\Delta$ , eight arrangements), and type C with all four spins equal ( $E_C=6J+\Delta$ , two arrangements).

Identifying each crossed square with an oxygen and its four corner spins with the orientations of the surrounding hydrogen bonds, the spin configurations map onto the hydrogen configurations of square ice, and the configurations obeying the ice rules are exactly those in which every crossed square is of type A. The residual entropy of the Ising model is $S/k_B=\lim_{N\to\infty}(1/(N/2))\ln g(E_0)$ with $g(E_0)$ the ground-state degeneracy.

Exact evaluation proceeds through vertex models. In zero field, Giacomini (1985) mapped this Ising model onto an eight-vertex model — whose vertices carry the eight arrow patterns with an even number of arrows in — with weights depending on $\beta J$ and $\beta\Delta$ , where $\beta=1/k_BT$ . The general eight-vertex model is not solvable, but two submanifolds are: the free-fermion condition $\omega_1\omega_2+\omega_3\omega_4=\omega_5\omega_6+\omega_7\omega_8$ of Fan and Wu, under which the partition function reduces to a two-dimensional integral of free-fermion (dimer) form, and the symmetric zero-field weights $\omega_1=\omega_2$ , $\omega_3=\omega_4$ , $\omega_5=\omega_6$ , $\omega_7=\omega_8$ (a symmetry of the vertex weights under global reversal of all arrows, unrelated to the Ising field $H_{ex}$ ) solved by Baxter. For the present model the free-fermion condition amounts to $\exp(4\beta\Delta)=\cosh(4\beta J)$ , which forces the four-spin coupling to depend on temperature, with $\Delta\to J$ as $T\to 0$ ; the Baxter case requires $\Delta=-\infty$ . Wu (1986) additionally solved the model in the pure-imaginary field $H_{ex}=i(\pi/2)k_BT$ , a value at which two-dimensional Ising partition functions remain exactly computable in the spirit of the Lee–Yang solution, for $\Delta=0$ .

Li, Zhao, Yao, and Yang (2023) evaluated the residual entropy at exactly these three points by taking the low-temperature limit of the exact partition functions. At $\Delta=-\infty$ in zero field and at $\Delta=0$ in the imaginary field the ground states are precisely the ice-rule configurations and the residual entropy is Lieb's value. Along the free-fermion trajectory the answer is instead $S/k_B=\frac{1}{8\pi^2}\int_0^{2\pi}\int_0^{2\pi}\ln[9-2\sqrt{2}(\cos\theta+\cos\varphi)]\,d\theta\,d\varphi\approx 1.07052$ , which exceeds Lieb's value because the ground states in the $T\to 0$ limit include type-B (non-ice-rule) crossed squares; in fact the computed limit is a weighted count $g(E_0)+\sum^{*}g(E_j)\,2^{-z_j}$ over configurations without type-C squares, with a factor $2^{-z_j}$ per pair of type-B squares, rather than the plain ground-state degeneracy of a fixed Hamiltonian. Away from these points nothing exact is known: in zero field the eight-vertex weights are generic, and in a real field the mapping produces a sixteen-vertex-type model (all sixteen arrow patterns allowed), for which no general exact solution exists; the only solved field values are imaginary ones. The ground-state and residual-entropy questions at zero temperature do not require the full finite-temperature partition function, but exact degeneracy counts are available in the literature only for the ice manifold itself.

Problem Statement

For the checkerboard-lattice Ising model with crossing and four-spin interactions defined above, with $J>0$ fixed as the energy scale, determine the complete zero-temperature phase structure and residual entropy as a function of the four-spin coupling $\Delta$ and a real external field $H_{ex}$ :

(1) Exact ground-state manifolds and residual entropies. Identify every region and boundary surface of the $(\Delta,H_{ex})$ plane on which the ground-state degeneracy grows extensively with $N$ , and compute the exact residual entropy $S_0(\Delta,H_{ex})$ there. This full-plane classification is required, and must include in particular: (a) real external fields $H_{ex}\neq 0$ , where no exact result of any kind is known; (b) in zero field, the degeneracy boundary $\Delta=J$ , at which type-A and type-B crossed squares cost the same energy and the ground states are all configurations containing no type-C square — the plain, unweighted degeneracy of this manifold has not been computed (the published free-fermion value is a trajectory-weighted count); and (c) in zero field, the region $\Delta>J$ , where the ground states are configurations with a three-to-one imbalance in every crossed square, whose degeneracy is likewise absent from the literature. Known values that must be recovered as limiting cases of $S_0(\Delta,H_{ex})$ are Lieb's $(3/2)\ln(4/3)$ wherever the ground states are exactly the ice-rule configurations. The published free-fermion value $\approx 1.07052$ is not a value of $S_0(\Delta,0)$ : it is the limit along the temperature-dependent trajectory $\exp(4\beta\Delta)=\cosh(4\beta J)$ , which approaches $\Delta=J$ as $T\to 0$ , and its relation to the $\Delta=J$ ground-state entropy is addressed in part (3).

(2) Ice-rule-selection classification. Classify completely which parameter constraints in the $(\Delta,H_{ex})$ plane guarantee that the ground-state manifold consists solely of ice-rule (all type-A) configurations, and which admit ice-rule violations; in zero field, where Giacomini's mapping applies, this classification may equivalently be expressed as vertex-weight relations of the associated eight-vertex model. Locate the zero-temperature boundaries between these regimes in the $(\Delta,H_{ex})$ plane and characterize how the degeneracy behaves on them.

(3) Status of the free-fermion trajectory. Since the free-fermion condition requires a temperature-dependent coupling $\Delta(\beta)$ with $\exp(4\beta\Delta)=\cosh(4\beta J)$ , the published value $\approx 1.07052$ is the low-temperature limit along this trajectory — the weighted count $g(E_0)+\sum^{*}g(E_j)\,2^{-z_j}$ described above, rather than the plain ground-state degeneracy of a fixed Hamiltonian. Determine the precise relation between this trajectory limit and the $\Delta=J$ ground-state entropy requested in part (1b); a complete answer must state the relation explicitly, either deriving it or exhibiting the weighted correction exactly.

These three components, each an independently required part, serve one objective: to complete the exact zero-temperature picture of this model over its full parameter space, of which the three integrable points solved in 2023 are special cases. (The source paper also proposes extending its low-temperature-limit technique to three-dimensional frustrated spin and ice systems; that direction concerns different model families and is outside the scope of this problem.)

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