The one-dimensional Ising chain in the fixed-magnetization (canonical) ensemble was solved exactly by Dantchev and Rudnick in 2022, who derived closed-form hypergeometric partition functions for periodic, antiperiodic, and Dirichlet boundary conditions, used them to define and compute the Helmholtz force — the canonical-ensemble counterpart of the critical Casimir force — and explicitly noted that the Ising model with fixed magnetization is unsolved in every dimension higher than one. Later work by the same group rigorously completed the one-dimensional case (all boundary conditions, Legendre relation between the ensembles, exact ln⁡N/N\ln N/N finite-size corrections), solved the Nagle–Kardar chain and, exactly, the three-dimensional Gaussian model in the canonical ensemble, while mean-field, ϵ\epsilon-expansion, functional-renormalization-group, and Monte Carlo studies had already established that canonical and grand-canonical critical forces genuinely differ. What remains open is the exact canonical treatment of genuinely interacting and higher-dimensional models: most prominently the two-dimensional Ising model with fixed magnetization — whose partition function is a magnetization sector of the unsolved field-dependent model — together with the interacting (non-Gaussian) fixed points and higher-dimensional solvable models such as the spherical model in d≥2d\ge 2; the concluding remark of the 2026 Gaussian-model study that exact canonical results seem feasible for further models of interest (including the one-dimensional Potts and spin-1 Ising chains) is a general feasibility observation and does not bring those one-dimensional chains within the present beyond-one-dimension scope. This record states that surviving open core and the structural obstruction behind it.

Background

The Ising model — spins si=±1s_i=\pm 1 on a lattice coupled through a nearest-neighbor interaction JJ, with dimensionless coupling K=βJK=\beta J (β=1/kBT\beta=1/k_BT) — is the reference model of statistical mechanics. It is exactly solved in one dimension at arbitrary temperature and magnetic field, and in two dimensions only at zero field. In the grand-canonical ensemble (GCE) one fixes the external field hh; in the canonical ensemble (CE), also called the fixed-MM ensemble, one instead fixes the total magnetization M=∑isiM=\sum_i s_i, i.e. the difference between the numbers of up and down spins. The CE is the rigorous description of any closed system with a conserved scalar order parameter — a binary alloy or lattice gas of fixed composition, a fluid film of fixed total mass — and the two ensembles are inequivalent for finite systems.

Near a critical point, confined order-parameter fluctuations generate the critical Casimir force between the confining boundaries: minus the derivative, with respect to the film thickness LL, of the excess free energy (the free energy minus the bulk contribution) of a film of geometry ∞d−1×L\infty^{d-1}\times L. Dantchev and Rudnick (2022) introduced the canonical counterpart, which they named the Helmholtz force: minus the LL-derivative of the excess Helmholtz free energy at fixed magnetization, with the total magnetization MM or the magnetization density m=M/Nm=M/N held constant during the differentiation. Their starting observation was that the Ising chain in the fixed-MM ensemble — unlike the field ensemble — had never been solved: they noted that the solution of the Ising model with fixed magnetization 'is not available' in one or any higher dimension, and they filled that gap for the finite chain under periodic, antiperiodic, and Dirichlet (free-end) boundary conditions.

The one-dimensional solution rests on domain-wall combinatorics. A domain wall is a bond between two adjacent unlike spins; with periodic boundary conditions domain walls occur in pairs, so a configuration of 2k2k alternating domains has energy cost 4kK4kK relative to the aligned state. Counting the arrangements of N+=(N+M)/2N_+=(N+M)/2 up spins and N−=(N−M)/2N_-=(N-M)/2 down spins into kk domains of each sign produces products ∏p=1k−1[(N−2p)2−M2]\prod_{p=1}^{k-1}[(N-2p)^2-M^2], which sum to a Gauss hypergeometric function 2F1_2F_1. The resulting closed form is exact when NN and MM have the same parity and ∣M∣<N|M|<N (the formula smoothly interpolates otherwise, and M=±NM=\pm N is trivial). From it one finds that the Helmholtz force of the chain can be attractive or repulsive depending on temperature and mm — whereas the one-dimensional Casimir force is only attractive — and that the fixed-MM free energy contains a logarithmic contribution from the translational degeneracy of the domain-wall positions, a non-scaling term that violates naive finite-size scaling (the statement that the excess free energy of a critical system is a universal function of L/ξL/\xi, with ξ\xi the correlation length).

Two obstructions block the extension of this exact program to interacting and higher-dimensional systems. First, in the two-dimensional Ising model the exact solution (Onsager, Kaufman) exists only at h=0h=0: the fixed-MM partition function Z(Nx,Ny,M,K)Z(N_x,N_y,M,K) is precisely the magnetization-sector (Fourier) component, in hh, of the field-dependent partition function, and extracting it requires either solving the model in a nonzero field — a famous open problem — or summing interacting Peierls contours (closed domain-boundary loops on the dual lattice) with fixed enclosed area and fixed energy, for which no technique is known. Second, in the continuum Ginzburg–Landau–Wilson (GLW) field theory of criticality, the Gaussian (free-field) approximation remains exactly solvable under the fixed-order-parameter constraint, because the constraint merely fixes the zero-momentum mode; but at the interacting Wilson–Fisher fixed point that governs real critical behavior, only mean-field, ϵ=4−d\epsilon=4-d expansion, functional-renormalization-group, and Monte Carlo results for canonical critical forces exist. Prior to the 2022 Letter, the Dietrich group had already established, at mean-field, ϵ\epsilon-expansion, and three-dimensional-Ising-Monte-Carlo level, that canonical and grand-canonical critical Casimir forces in films genuinely differ (for example, a sign reversal under strong adsorption); the recent review by Dantchev and Dietrich (Physics Reports 1005 (2023)) surveys exact critical Casimir results. What remained, and remains, open is exact treatment.

Problem Statement

Extend the exact solution of the fixed-magnetization (canonical) ensemble — and with it the exact theory of the Helmholtz fluctuation-induced force — beyond the one-dimensional Ising chain and beyond the free Gaussian fixed point, to at least one admissible model that is genuinely interacting or higher-dimensional (either one of these two criteria suffices on its own; the model need not satisfy both), doing for that model what the 2022 solution did for the chain. Admissible target models are short-range statistical-mechanical models with a finite-temperature critical point in spatial dimension d≥2d\ge 2 or with genuinely interacting (non-Gaussian) critical behavior, whose canonical ensemble is not already exactly solved: the flagship instance is the two-dimensional nearest-neighbor Ising model on a finite Nx×NyN_x\times N_y lattice with periodic boundary conditions and fixed total magnetization MM (other standard boundary conditions admissible), and the spherical model in d≥2d\ge 2 — flagged as a feasible exact target in the follow-up literature — is likewise admissible. Models whose canonical ensemble is already exactly solved (the one-dimensional Ising and Nagle–Kardar chains, and the Gaussian model) do not qualify, and neither do one-dimensional short-range chains without a finite-temperature critical point, such as the qq-state Potts chain and the spin-1 Ising chain: the 2026 Gaussian-model study mentions them only in a general feasibility remark about extending ensemble-dependence calculations to other models, and solving them would not go beyond one dimension. For the chosen model the answer must deliver, in exact form:

  1. The fixed-MM partition function Z(N,M,K)Z(N,M,K), with NN the total number of spins of the chosen finite system (for the two-dimensional flagship, Z(Nx,Ny,M,K)Z(N_x,N_y,M,K) on the Nx×NyN_x\times N_y lattice) — a closed form in standard special functions, an exact finite sum, or an equivalently explicit exact representation — covering the full range of the coupling KK (not only the critical point) and the parity and ∣M∣=N|M|=N limit cases.

  2. The Helmholtz force derived from it: minus the derivative with respect to system size of the excess Helmholtz free energy, taken in both conventions — at fixed total magnetization MM and at fixed magnetization density m=M/Nm=M/N (for a finite system these give different forces, so both are required) — including the finite-size scaling function and the leading non-scaling finite-size corrections of each (the analog of the one-dimensional ln⁡N/N\ln N/N terms).

  3. The precise comparison with the grand-canonical Casimir force of the same model and boundary conditions: where the two forces coincide, where and why they differ, and how the canonical and grand-canonical free energies are related in both the thermodynamic limit and the finite-size regime.

These three deliverables form one chain — partition function, free energy and force, ensemble comparison — and a solution is complete only when the whole chain is exact for the chosen model.

References

  1. D. Dantchev and J. Rudnick, Exact expressions for the partition function of the one-dimensional Ising model in the fixed-MM ensemble, Physical Review E 106, L042103 (2022). DOI: 10.1103/PhysRevE.106.L042103; arXiv:2207.01134; https://arxiv.org/abs/2207.01134
  2. D. M. Dantchev, N. S. Tonchev, and J. Rudnick, Casimir versus Helmholtz forces: Exact results, Annals of Physics 459, 169533 (2023). DOI: 10.1016/j.aop.2023.169533; arXiv:2308.09796; https://arxiv.org/abs/2308.09796
  3. D. M. Dantchev, N. S. Tonchev, and J. Rudnick, Casimir and Helmholtz forces in one-dimensional Ising model with Dirichlet (free) boundary conditions, Annals of Physics 464, 169647 (2024). DOI: 10.1016/j.aop.2024.169647; arXiv:2402.04459; https://arxiv.org/abs/2402.04459
  4. D. Dantchev, N. S. Tonchev, and J. Rudnick, Fluctuation induced force in the Nagle–Kardar model in ensemble with conserved magnetization, Symmetry 17(1), 22 (2024). DOI: 10.3390/sym17010022; https://doi.org/10.3390/sym17010022
  5. D. Dantchev, N. S. Tonchev, and J. Rudnick, Casimir versus Helmholtz fluctuation-induced force in the Nagle–Kardar model: exact results, European Physical Journal Special Topics 235, 3111–3118 (2025). DOI: 10.1140/epjs/s11734-025-02017-9; arXiv:2508.15430; https://arxiv.org/abs/2508.15430
  6. D. M. Dantchev and J. Rudnick, Helmholtz versus Casimir forces in the Gaussian model: Exact results, Annals of Physics 494, 170623 (2026). DOI: 10.1016/j.aop.2026.170623; arXiv:2603.16030; https://arxiv.org/abs/2603.16030
  7. M. Gross, O. Vasilyev, A. Gambassi, and S. Dietrich, Critical adsorption and critical Casimir forces in the canonical ensemble, Physical Review E 94, 022103 (2016). DOI: 10.1103/PhysRevE.94.022103; https://doi.org/10.1103/PhysRevE.94.022103
  8. M. Gross, A. Gambassi, and S. Dietrich, Statistical field theory with constraints: Application to critical Casimir forces in the canonical ensemble, Physical Review E 96, 022135 (2017). DOI: 10.1103/PhysRevE.96.022135; https://doi.org/10.1103/PhysRevE.96.022135
  9. C. M. Rohwer, A. Squarcini, O. Vasilyev, S. Dietrich, and M. Gross, Ensemble dependence of critical Casimir forces in films with Dirichlet boundary conditions, Physical Review E 99, 062103 (2019). DOI: 10.1103/PhysRevE.99.062103; https://doi.org/10.1103/PhysRevE.99.062103
  10. F. Rose, A. Rançon, and I. Balog, Constraint effective action and critical correlation functions at fixed magnetization, Physical Review E 113, 054106 (2026). DOI: 10.1103/yqjz-kdlj; arXiv:2509.05195; https://arxiv.org/abs/2509.05195
  11. I. Balog and A. Rançon, Constraint correlation functions of the one-dimensional Ising model in the scaling limit, Physical Review E 112, 054127 (2025). DOI: 10.1103/x84b-6yhv; arXiv:2509.08557; https://arxiv.org/abs/2509.08557
  12. C. Carlson, E. Davies, A. Kolla, and W. Perkins, Computational thresholds for the fixed-magnetization Ising model (2021). arXiv:2111.03033; https://arxiv.org/abs/2111.03033
  13. Z. Wang, F. F. Assaad, and F. Parisen Toldin, Finite-size effects in canonical and grand-canonical quantum Monte Carlo simulations for fermions, Physical Review E 96, 042131 (2017). DOI: 10.1103/PhysRevE.96.042131; arXiv:1706.01874; https://arxiv.org/abs/1706.01874
  14. D. Dantchev, On Casimir and Helmholtz Fluctuation-Induced Forces in Micro- and Nano-Systems: Survey of Some Basic Results, Entropy 26(6), 499 (2024). DOI: 10.3390/e26060499; https://doi.org/10.3390/e26060499
  15. D. Dantchev and S. Dietrich, Critical Casimir effect: Exact results, Physics Reports 1005, 1–130 (2023). DOI: 10.1016/j.physrep.2022.12.004; https://doi.org/10.1016/j.physrep.2022.12.004