Exact partition function of arithmetic Ising model

Abstract: We present a compact formula for the exact partition function of the $d$-dimensional arithmetic Ising model (AIM). For a $2\times2$ system, we express it analytically using the $q$-Hurwitz-Lerch zeta function and derive explicit forms for the free energy and entropy. Additionally, we find that the entropy increases at high temperatures, supporting the presence of entropic order.

• The paper computes the exact partition function of the arithmetic Ising model (AIM) using a rigorous upper bound and a novel methdology using an exponential differential operator.
• The authors derive a parametrization free of unnecessary coupling to facilitate calculations and find that partitioning by energy and summation of mixtures allows for simplification.
• The exact formula reveals a deeper mathematical structure of the ARIM, specifically in smaller systems, and computationally implies growth in entropy at high temperatures.

Overview and motivation

The paper by Dhochak, Kikuchi, and Singh computes the exact partition function of the arithmetic Ising model (AIM) on a $d$-dimensional periodic hypercubic lattice of linear size $L$. The AIM replaces Ising spins $n_x=\pm1$ with unbounded natural numbers $n_x\in\mathbb N$, with Hamiltonian

$$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$

where $U,\mu\ge0$ are non-negative couplings. This model is one of the minimal models constructed in prior work on entropic order (Huang et al., 8 Dec 2025), a regime in which ordered phases appear at high temperature, as realized physically in Rochelle salt and the Pomeranchuk effect in $^3$He, and proven for generalized Ising models (Andriolo et al., 10 Apr 2026). What was missing is quantitative control: how the free energy and entropy actually behave as functions of temperature in such models. The present work supplies this by evaluating $Z(\beta,\mu)$ exactly, via a method distinct from Onsager's transfer-matrix approach.

A useful parametrization removes one coupling: since $\beta$ enters only through $\beta U$ and $L$0, one may set $L$1 without loss of generality. The partition function admits the rigorous upper bound

$L$2

which also guarantees absolute convergence of all sums involved.

Exact formula for the partition function

The main result expresses $L$3 as an exponential differential operator acting on a product of geometric series:

$L$4

with $L$5, and analogously in arbitrary dimension $L$6. Three observations combine to yield this formula. First, for irrational chemical potential $L$7, equality of energies $L$8 forces equality of $L$9, so configurations can be grouped by total occupation number $n_x=\pm1$0, of which there are $n_x=\pm1$1; this collapses the original $n_x=\pm1$2-fold sum over natural numbers to a single sum over $n_x=\pm1$3. Second, a monomial correspondence identifies each configuration $n_x=\pm1$4 with $n_x=\pm1$5, and the exponential derivative operator reproduces exactly the factor $n_x=\pm1$6 on each monomial, where $n_x=\pm1$7 is the neighbor-interaction sum. Third, the degree-$n_x=\pm1$8 homogeneous part of $n_x=\pm1$9 collects precisely those monomials, so the remaining $n_x\in\mathbb N$0-sum is just a Taylor expansion resumming to the closed-form product above.

The irrationality assumption on $n_x\in\mathbb N$1 is then removed by a continuity argument: both $n_x\in\mathbb N$2 and the analytic expression are continuous in $n_x\in\mathbb N$3, they agree on the dense set of positive irrationals, hence agree for all $n_x\in\mathbb N$4. Notably, none of these steps depends on the lattice dimension or size; the generalization to $n_x\in\mathbb N$5 dimensions is immediate. In the decoupled limit $n_x\in\mathbb N$6, the operator becomes trivial and the upper bound is recovered, which gives the formula a transparent physical interpretation: the interaction term generates the nontrivial differential structure.

Evaluation for the $n_x\in\mathbb N$7 system

The exponentiated operator is evaluated by diagonalizing the adjacency matrix $n_x\in\mathbb N$8 of the lattice graph, writing the exponent as $n_x\in\mathbb N$9 with $$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$0. For $$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$1, the eigenvalues are $$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$2, and geometric-series expansions apply near the zero mode because $$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$3. The special case $$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$4 is elementary, while $$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$5 requires separate treatment due to doubled edges.

For $$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$6, the adjacency matrix has spectrum $$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$7, and the partition function reduces to the convergent double sum

$$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$8

This admits a compact closed form in terms of the $$H(\mathbf n,U,\mu)=U\sum_{x\sim y}n_xn_y+\mu\sum_x n_x,$$9-Hurwitz-Lerch zeta function,

$U,\mu\ge0$0

where $U,\mu\ge0$1 and $U,\mu\ge0$2. The authors note that the appearance of a $U,\mu\ge0$3-deformed function raises the question whether hidden symmetries or integrable structure underlie the model—an open question the paper does not resolve.

Thermodynamics and entropic order

From the exact expression, free energy $U,\mu\ge0$4 and entropy $U,\mu\ge0$5 follow directly. Since no standard software provides a built-in $U,\mu\ge0$6-Hurwitz-Lerch zeta function, the authors evaluate these numerically (via Mathematica's NSum) for the $U,\mu\ge0$7 system and use the plots only qualitatively, given machine-precision errors. The results show that at high temperature (small $U,\mu\ge0$8), the free energy decreases and the entropy increases with temperature. This enhanced entropy in the high-temperature regime is consistent with the mechanism proposed for entropic order—the ordered phase minimizing free energy through maximized entropy—though the authors are careful to state that no phase transition occurs at finite $U,\mu\ge0$9, so this constitutes qualitative support rather than a demonstration of a thermodynamic transition.

Limitations and open questions

Several caveats bound the scope of the results. The analytic closed form in terms of $^3$0 is established only for the $^3$1 system; for larger lattices the exact formula remains an unevaluated differential operator whose evaluation grows combinatorially with $^3$2. The thermodynamic analysis relies on numerical differentiation subject to floating-point error, so claims about the entropy's behavior are qualitative rather than controlled asymptotics. Most importantly, finite-volume systems cannot exhibit genuine phase transitions, so the connection between the observed high-temperature entropy growth and entropic order in the thermodynamic limit remains indirect. Whether the $^3$3-deformed structure signals integrability or a hidden symmetry is left open, as is the extension of these exact techniques to extract order parameters or correlation functions.

Conclusion

The paper provides an exact, closed-form expression for the partition function of the arithmetic Ising model in arbitrary dimension and volume, derived through a grouping-by-energy argument combined with a monomial/differential-operator representation of the interaction term. For the smallest two-dimensional system, the result reduces analytically to a $^3$4-Hurwitz-Lerch zeta function, from which free energy and entropy follow explicitly. The computed entropy increases at high temperatures, providing concrete quantitative support for the entropic-order picture in a minimal model where such behavior was previously only argued indirectly.