Closed product representation of the residual determinant of the square-lattice Ising model on the open L×M rectangle

The exact solution of the anisotropic square-lattice Ising model on an $L\times M$ rectangle with open boundary conditions in both directions (Hucht, 2017) factorizes the partition function into an $M$-fold product over transfer-matrix eigenvalues times a residual determinant $Z^{res}_{strip}=\det(1+Y)$ of size $M/2\times M/2$, which carries all nontrivial finite-aspect-ratio contributions including the critical Casimir potential. Hucht could not find a closed product representation of this determinant for arbitrary finite $L$ and $M$, and showed that it equals the partition function of an effective one-dimensional spin model with $M$ spins, long-range frustrated pair couplings, and an external field term whose strength is the rectangle length $L$ — a model solved in closed form only at $L=0$. No published work provides such a factorization: paper III reformulated the full partition function as a Hankel/Toeplitz determinant and described a closed product representation as the 'ultimate goal'. The scaling-limit questions deferred in the source were resolved by the companion paper II, and the below-$T_c$ product formulas for the singular bulk, surface and corner free energies were derived rigorously by Baxter, but the paramagnetic-phase ($T>T_c$) product formulas of the source's Appendix A also remain without derivation. The open problem is the closed-product completion of the open-rectangle Ising solution: factorize $\det(1+Y)$ for arbitrary finite $L$ and $M$, equivalently solve the mapped long-range frustrated spin model at nonzero field. A separate open item, stated below as the secondary objective, is to place the conjectured paramagnetic-phase $q$-products of the source's Appendix A on rigorous footing.

Background

The Ising model places spins $\sigma_{\ell,m}=\pm 1$ on the sites of a lattice, with Boltzmann weight $\exp(K\sigma\sigma')$ for each nearest-neighbor pair, where the reduced coupling is $K=J/(k_B T)$. Its two-dimensional version is the fundamental exactly solved model of statistical mechanics: Onsager (1944) computed the free energy of the infinite square lattice with periodic boundary conditions, and a long subsequent program has extended exact results to finite geometries, because the non-extensive remainder of the free energy of a finite near-critical system controls finite-size scaling and fluctuation-induced boundary forces. The latter are called critical Casimir forces, in analogy with the electromagnetic Casimir effect: near a critical point, order-parameter fluctuations mediate a universal force between the confining boundaries whose strength and sign depend on the geometry and on the surface universality classes (the universal boundary-condition classes of surface critical behavior).

For the square-lattice Ising model on an $L\times M$ rectangle with open (free) boundary conditions in both directions, an exact solution for arbitrary size and temperature was obtained only in 2016–2017. Hucht starts from the Pfaffian (dimer) representation of the partition function introduced by Kasteleyn and developed by McCoy and Wu — the partition function written as a Pfaffian, i.e. a signed square root of a determinant, of a dimer (perfect-matching) problem — organizes it as a product of $2\times 2$ block transfer matrices with $M\times M$ blocks (the transfer matrix being the operator propagating one column of spins to the next), and factorizes the resulting determinant. The final partition function has the form

$Z = C\,d_{o,e}^{2}\prod_{\mu=1}^{M}\Phi(\lambda_\mu)\;Z^{res}_{strip},$

where the factors $\Phi(\lambda_\mu)$ are explicit elementary functions of the dominant eigenvalues $\lambda_\mu$ of an $M\times M$ transfer matrix (the eigenvalues are the zeros of an explicit characteristic polynomial), $o$ and $e$ denote the sets of odd and even eigenvalue indices, $\sigma_\mu=(-1)^{\mu-1}$ is the eigenvalue parity, and the residual determinant

$Z^{res}_{strip}=\det(1+Y),\qquad Y=-\Lambda_e^L V_e T_{e,o}\Lambda_o^{-L}V_o T_{o,e}$

is the determinant of an $M/2\times M/2$ matrix built from diagonal matrices $\Lambda^L$ and $V$ of eigenvalue factors (all dependence on the length $L$ sits in $\Lambda^L$) and blocks $T_{e,o}$ of the eigenvector matrix. The free energy splits as $F=F_{strip}+F^{res}_{strip}$, and the residual part contains all nontrivial finite-aspect-ratio contributions, including the critical Casimir potential. Because the matrix $Y$ involves the full zero set of the characteristic polynomial simultaneously, no entry-wise simplification is available.

Closed product representations are known only in limiting cases. For $L\to\infty$ at fixed $M$ (infinite strip) the residual determinant tends to 1, and the strip contribution was written as a product by Baxter in the ordered phase $T<T_c$. For $L=0$ the residual determinant reduces to a Cauchy-type matrix — a matrix with entries of the form $1/(c_\mu-c_\nu)$, whose determinant has a classical product formula — and factorizes into eigenvalue products (eq. (85) of the source, arXiv v5). In the thermodynamic limit $L,M\to\infty$ the free energy decomposes as $LM f_b+L f_s^{\leftrightarrow}+M f_s^{l}+f_c$ into bulk, surface, and corner free energies — the corner free energy being the $O(1)$ contribution induced by the sharp $90^\circ$ corners — for which periodic infinite products in a natural variable $q$, built from $q$-Pochhammer symbols $(a;q)_\infty=\prod_{k\geq 0}(1-aq^k)$, were conjectured by Vernier and Jacobsen below $T_c$ and derived there by Baxter's calculation with Kaufman's spinor method, in which the transfer matrix is represented through free-fermion operators. In the finite-size scaling limit $L,M\to\infty$, $T\to T_c$ at fixed scaling variable and aspect ratio, the residual determinant at criticality reduces to an expression in the Dedekind eta function (the modular form related to $(q;q)_\infty$), as established in the companion paper II, which also computed the full universal Casimir potential and force scaling functions and thereby closed the scaling-limit questions that paper I had deferred.

Two further structures demarcate what remains open. First, the source exhibits an exact mapping of the residual determinant onto an effective one-dimensional spin model: $Z^{res}_{strip}=\mathrm{Tr}\,e^{-H_{eff}}$ with

$H_{eff}=-\sum_{\mu<\nu}K_{\mu\nu}s_\mu s_\nu+L\sum_{\mu}\hat\gamma_\mu s_\mu+b\Big[\sum_{\mu}\sigma_\mu s_\mu\Big]^2,\qquad b\to\infty,$

with $M$ occupation variables $s_\mu\in\{0,1\}$ (a $C(M,M/2)$-dimensional trace) constrained to equally many occupied sites in the odd and even sublattices; the pair couplings $K_{\mu\nu}=-\sigma_\mu\sigma_\nu\log(v_\mu v_\nu/(c_\mu-c_\nu)^2)$ are ferromagnetic within each sublattice and antiferromagnetic between them, so the model is frustrated, the positive $\hat\gamma_\mu$ act as magnetic moments in a homogeneous field of strength equal to the rectangle length $L$, and the $c_\mu$ are eigenvalue-related quantities. A factorization of $Z^{res}_{strip}$ for arbitrary $L$ would amount to an exact solution of this long-range frustrated model in a field, of which only the zero-field case $L=0$ is solved. Second, the product formulas of Appendix A of the source for the singular parts of $f_b$, $f_s$, $f_c$ were given there without derivation: they were obtained by generating long high- and low-temperature series with the finite-lattice method (exact enumeration of finite sublattices) and re-summing them into periodic $q$-products whose periodic coefficient pattern is conjectured to persist to all orders, with the high-temperature variable $z=\tanh K$ identified through the self-duality of the model. Below $T_c$ these coincide with the Vernier–Jacobsen conjectures that Baxter derived; for the paramagnetic phase $T>T_c$ the products have remained conjectural. In its conclusions the author states that for the residual determinant 'we could not find a closed product representation', pointing also to a MathOverflow question on the underlying block Vandermonde determinant, and notes that the effective couplings are products of symmetric functions of the $c_\mu$, which 'might be utilized to find a factorization'.

Problem Statement

The problem is the closed-product completion of the exact solution of the square-lattice Ising model on the open $L\times M$ rectangle, as left open by Hucht's paper I (2017).

Primary objective. Find a closed product representation — a complete factorization into products of explicitly given elementary factors (eigenvalue products, theta or elliptic functions, or convergent infinite products of the $q$-Pochhammer type) — of the residual determinant

$Z^{res}_{strip}(L,M,T)=\det(1+Y)=\mathrm{Tr}\,e^{-H_{eff}}$

for arbitrary finite $L$ and (even) $M$ and arbitrary temperature $T$. Equivalently: give an exact closed-form evaluation of the partition function of the effective spin model $H_{eff}$ ($M$ spins, long-range frustrated pair couplings $K_{\mu\nu}$, sublattice-occupation constraint, field term of strength $L$) at nonzero field, whose $L=0$ case is already solved by the Cauchy-type factorization of paper I, eq. (85). Any answer must reproduce the known limiting product forms: $Z^{res}_{strip}\to 1$ for $L/M\to\infty$, the $L=0$ factorization (which holds at arbitrary temperature), Baxter's infinite-strip and thermodynamic-limit products (both in the ordered phase $T<T_c$), and the Dedekind-eta form at criticality in the scaling limit. The route suggested in the source — exploiting that the effective couplings are products of symmetric functions of the quantities $c_\mu$ — is admissible but not required.

Secondary objective. Provide rigorous derivations (not series-based conjectures) of the periodic $q$-product formulas of Appendix A of paper I for the singular parts of the bulk, surface and corner free energies in the paramagnetic phase $T>T_c$; below $T_c$ these product formulas were derived rigorously by Baxter's spinor-method calculation, confirming the Vernier–Jacobsen conjectures. A closed product representation of $Z^{res}_{strip}$ for arbitrary finite $L$ and $M$ resolves the problem on its own; the secondary objective is an independent open item, not a precondition for that resolution.

Explicitly outside this problem: the finite-size scaling limit and the universal Casimir potential/force scaling functions, which were deferred in paper I and resolved in the companion paper II (2017).

References

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