An explicit injective mapping between magnetization sectors at fixed energy in the one-dimensional Ising chain

Zheng, Lin and Zhang (Eur. J. Phys. 47, 035103 (2026)) proved that the density of states of the open one-dimensional Ising chain satisfies $\mathcal{D}(E,M)\geq\mathcal{D}(E,M+2)$ for $0\leq M\leq L-4$ at every fixed interaction energy $E$, by a formal manipulation of binomial coefficients, and posed as an open challenge the construction of an explicit injective mapping from the configurations with magnetization $M+2$ to those with magnetization $M$ that would make the inequality manifest as a direct combinatorial correspondence. Such a mapping is required to preserve the interaction energy $E$, i.e. the number of domain walls. The construction of an explicit, structural, energy-preserving injection between magnetization sectors remains open.

Background

The one-dimensional Ising model is a chain of $L$ spins $s_i\in\{+1,-1\}$; with open boundary conditions its interaction energy and magnetization are $E=-\sum_{i=1}^{L-1}s_i s_{i+1}$ and $M=\sum_{i=1}^{L}s_i$. A bond between two antiparallel neighbouring spins is called a domain wall and costs an energy penalty of 2, so $E=-(L-1)+2N_{dw}$ where $N_{dw}$ is the number of domain walls; a configuration is thus an alternating sequence of maximal blocks of parallel spins (domains), and the magnetization fixes the number of up spins $N_{\uparrow}=(L+M)/2$ and down spins $N_{\downarrow}=(L-M)/2$. The density of states $\mathcal{D}(E,M)$ is the number of configurations with interaction energy $E$ and magnetization $M$. Following Ising's original combinatorial solution of his model, $\mathcal{D}(E,M)$ is obtained by counting the ways to split $N_{\uparrow}$ and $N_{\downarrow}$ into a prescribed number of positive integer parts (an ordered sum of $n$ into $k$ positive integers is counted by the stars-and-bars identity as $\binom{n-1}{k-1}$): for $N_{dw}=2K-1$ there are $K$ up-domains and $K$ down-domains and $\mathcal{D}(E,M)=2\binom{N_{\uparrow}-1}{K-1}\binom{N_{\downarrow}-1}{K-1}$, while for $N_{dw}=2K$ the chain begins and ends with domains of the same type and $\mathcal{D}(E,M)=\binom{N_{\uparrow}-1}{K}\binom{N_{\downarrow}-1}{K-1}+\binom{N_{\uparrow}-1}{K-1}\binom{N_{\downarrow}-1}{K}$.

Zheng, Lin and Zhang (2026) revisited the classic fact that the one-dimensional chain shows no spontaneous magnetization at any finite temperature from the Landau free energy perspective. There the Landau free energy $F(T,M)=-T\ln Z(T,M)$ is defined through the partial partition function $Z(T,M)$, which sums the Boltzmann weights only over configurations with magnetization $M$; spontaneous magnetization would require the free-energy curve to develop the double-well shape of a Landau second-order transition, with $M=0$ turning into a local maximum of $F(T,M)$ — equivalently, a local minimum of $Z(T,M)$ — flanked by two degenerate minima at $\pm M_0\neq 0$. The linchpin of their argument is the monotonicity inequality

$\mathcal{D}(E,M)\ \geq\ \mathcal{D}(E,M+2),\qquad 0\leq M\leq L-4,$

valid at every fixed interaction energy $E$ (the upper restriction on $M$ is needed because the fully polarized sector $M=L$ contains a single configuration at the ground-state energy, while the sector $M=L-2$ has none there). Summing over $E$ with Boltzmann weights then shows that $Z(T,M)$ decreases with $M\geq 0$, so the Landau free energy is increasing in $|M|$ and its U-shaped curve can never develop the double-well shape of a second-order phase transition. The authors prove the inequality formally: as $M\to M+2$ one spin moves from the down population to the up population ($N_{\uparrow}\to N_{\uparrow}+1$, $N_{\downarrow}\to N_{\downarrow}-1$), and the ratio $\mathcal{D}(E,M+2)/\mathcal{D}(E,M)$ reduces to a product of binomial ratios whose numerator-minus-denominator difference is $-(K-1)(N_{\uparrow}-N_{\downarrow}+1)\leq 0$. They contrast this with the trivial global bound $\sum_E\mathcal{D}(E,M)=\binom{L}{N_{\uparrow}}\geq\binom{L}{N_{\uparrow}+1}=\sum_E\mathcal{D}(E,M+2)$, emphasizing that the monotonicity holds term by term at every specific energy level, not just after summing over energies. Classical combinatorics does provide explicit injections between adjacent binomial-rank levels of the set of all $2^L$ spin configurations — for instance through symmetric chain decompositions of the Boolean lattice (the partition of all subsets of an $L$-element set into inclusion chains symmetric about the middle rank, a standard tool of Sperner theory) — but such bit-flipping maps do not preserve the number of domain walls and therefore do not respect the fixed-energy constraint. What is missing, and what the source poses as an open challenge, is an explicit correspondence that preserves the energy.

Problem Statement

For the open one-dimensional Ising chain of $L$ spins, let $\mathcal{C}_L(E,M)$ denote the set of spin configurations with interaction energy $E=-\sum_{i=1}^{L-1}s_i s_{i+1}$ and magnetization $M=\sum_{i=1}^{L}s_i$. Construct, for every chain length $L$, every attainable energy $E\in\{-L+1,-L+3,\ldots,L-1\}$ and every $M$ with $0\leq M\leq L-4$, an explicitly specified injective mapping

$\varphi_{L,E,M}:\ \mathcal{C}_L(E,M+2)\ \longrightarrow\ \mathcal{C}_L(E,M),$

together with a proof that (i) $\varphi$ is well defined on every input configuration, (ii) $\varphi(s)$ has the same interaction energy $E$ and magnetization exactly $M$ whenever $s$ has magnetization $M+2$, and (iii) $\varphi$ is injective. The construction thereby yields a direct combinatorial proof of the term-by-term monotonicity $\mathcal{D}(E,M)\geq\mathcal{D}(E,M+2)$ of the density of states, which is currently established only by a formal binomial-coefficient computation.

Here 'explicitly specified' is meant in the standard sense of bijective-proof combinatorics: the rule must be given as a concrete structural operation on configurations — on their domain decomposition, wall positions, or comparable intrinsic structure — uniform in $L$, $E$, $M$ (or covering the sectors by explicit case rules), and specified without invoking the closed-form binomial counts $\mathcal{D}(E,M)$, without enumerating or ranking the configurations within the two sectors, and without otherwise re-encoding the counting argument. This requirement is the source's own objective: the existing proof is rigorous but 'formal and unilluminating', and the desired construction should make the monotonicity manifest through the correspondence itself. An injection that merely matches configurations by lexicographic or rank order within the sectors satisfies (i)–(iii) trivially but does not answer the question.

A construction for the open chain is the required core. The source derives the analogous monotonicity for the periodic chain from the open-chain case (its Eq. (16) relates the two densities of states), so the periodic boundary condition is a natural secondary extension of the same construction rather than a separate requirement.

References

1. Zheng Z F, Lin R K, Zhang J M (2026). Landau free energy and the absence of spontaneous magnetization of the one-dimensional Ising model. European Journal of Physics 47(3), 035103. DOI: 10.1088/1361-6404/ae6b1c. https://doi.org/10.1088/1361-6404/ae6b1c (preprint arXiv:2605.06691, https://arxiv.org/abs/2605.06691, v1 of 2 May 2026)
2. Ising E (1925). Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik 31, 253–258. DOI: 10.1007/BF02980577. https://doi.org/10.1007/BF02980577
3. Seth S (2017). Combinatorial approach to exactly solve the 1D Ising model. European Journal of Physics 38, 015104. DOI: 10.1088/1361-6404/38/1/015104. https://doi.org/10.1088/1361-6404/38/1/015104
4. Kryzhanovsky B V, Egorov V I (2025). The generalized density of states in a one-dimensional Ising model with ferrromagnetic and antiferromagnetic interactions. arXiv:2512.12640. https://arxiv.org/abs/2512.12640
5. Sitarachu K, Zia R K P, Bachmann M (2023). Exact microcanonical statistical analysis of transition behavior in Ising chains and strips. arXiv:2306.16994. https://arxiv.org/abs/2306.16994
6. Greene C, Kleitman D J (1976). Strong versions of Sperner's theorem. Journal of Combinatorial Theory, Series A 20, 80–88. DOI: 10.1016/0097-3165(76)90079-0. https://doi.org/10.1016/0097-3165(76)90079-0
7. Zhong Y (2021). A symmetric chain decomposition of N(m,n) of composition. arXiv:2107.11715. https://arxiv.org/abs/2107.11715