Sharp decay criterion for finite-temperature spontaneous magnetization in the one-dimensional long-range Ising ferromagnet (Dyson's borderline problem)

The ferromagnetic Ising chain on $\mathbb{Z}$ with translation-invariant coupling $J(n)\ge 0$ can break its spin-flip symmetry at finite temperature only because of long-range interactions, and whether it does so depends delicately on the decay of $J$. Dyson (1969) proved that the spontaneous magnetization vanishes at all finite temperatures whenever $\frac{1}{\log\log N}\sum_{n\le N}nJ(n)\to 0$, improving Ruelle's condition $\sum_n nJ(n)<\infty$, and left open the regime between this criterion and his own existence result for $J(n)\sim n^{-\alpha}$ with $1<\alpha<2$. The literature has since moved both boundaries: Rogers and Thompson (1981) strengthened nonexistence to $\sum_{n\le N}nJ(n)=o((\log N)^{1/2})$, and Fröhlich and Spencer (1982) proved that the borderline interaction $J(n)=1/n^{2}$ does order at low temperature, so that for pure power laws the phase diagram in the decay exponent is complete within the summable regime $\alpha>1$ (transition iff $\alpha\le 2$). What remains open is the sharp criterion for general couplings, and specifically the window in between: interactions decaying as $n^{-2}$ times a slowly varying factor tending to zero, such as $J(n)=n^{-2}[\log(1+\log(1+n))]^{-1}$, for which neither existence nor nonexistence of spontaneous magnetization at any finite temperature is known.

Background

One-dimensional spin systems with short-range interactions cannot break a discrete symmetry at finite temperature: for the nearest-neighbour Ising chain the Gibbs state is unique at every positive temperature, and the same absence of order holds for all finite-range one-dimensional systems. The picture changes when the pair coupling between spins at distance $n$ decays slowly, because a flipped interval of spins (a domain misaligned with its surroundings) then has an interface energy growing with the interval's length rather than staying constant. The model of interest is the ferromagnetic Ising chain on the integer lattice $\mathbb{Z}$ with formal Hamiltonian $H=-\sum_{i<j}J(j-i)\,\sigma_i\sigma_j,\qquad \sigma_i\in\{-1,+1\},\quad J(n)\ge 0,$ where the coupling sequence $J$ is translation-invariant and summable and $\beta=1/(k_BT)$ is the inverse temperature. Its equilibrium states are the Gibbs measures, i.e. probability measures on $\{-1,1\}^{\mathbb{Z}}$ whose finite-volume conditional probabilities follow the Boltzmann weights of $H$; the chain has spontaneous magnetization at $\beta$ if the plus and minus states (infinite-volume limits taken with all-plus and all-minus boundary conditions) differ, equivalently if $m^{+}(\beta)=\lim_{h\downarrow 0}\langle\sigma_0\rangle_{\beta,h}>0$, where $h$ is a uniform external field. The basic mechanism is an energy-entropy competition: a flipped interval of length $\ell$ costs interface energy of order $\sum_{n\le\ell} nJ(n)$ (a spin at depth $t$ from the boundary interacts with the opposite phase with weight about $\sum_{m\ge t}J(m)$), while its positional entropy is of order $\log\ell$. Ruelle proved in 1968 that there is no phase transition when the first moment converges, $\sum_n nJ(n)<\infty$, and it was conjectured (in correspondence summarized as the Kac-Thompson conjecture) that this condition was also necessary for the absence of order. In two companion 1969 papers Dyson settled the rough outlines of the problem. In the existence paper he proved, by comparing the chain with the Dyson hierarchical model (an exactly solvable, tree-structured approximation in which blocks of spins interact self-similarly) and using the Griffiths correlation inequalities (standard monotonicity inequalities for ferromagnets), that a phase transition does exist at low temperature when $J(n)\sim n^{-\alpha}$ with $1<\alpha<2$. In Dyson's nonexistence paper he proved the complementary theorem: the spontaneous magnetization is zero at every finite temperature whenever $\frac{1}{\log\log N}\sum_{n\le N}nJ(n)\longrightarrow 0 \quad (N\to\infty),$ a condition strictly weaker than Ruelle's, thereby refuting the necessity direction of the conjecture; the intermediate regimes, notably the borderline power law $J(n)=n^{-2}$ that Thouless had predicted to be the marginal decay of a chain that still orders, were left unresolved. The borderline has since been transformed by three developments. First, Rogers and Thompson strengthened the nonexistence criterion to $\sum_{n\le N}nJ(n)=o((\log N)^{1/2})$, and Simon and Sokal recast Dyson-type energy-entropy arguments into a rigorous general framework; no sharper nonexistence condition for the Ising chain is known as of 2026. Second, Fröhlich and Spencer proved in 1982, by a multiscale contour argument (organizing spin-flip clusters at dyadically nested scales subject to a separation condition), that the borderline interaction $J(n)=1/n^{2}$ does have spontaneous magnetization at low temperature, confirming Thouless' prediction; Imbrie then proved sharp decay of correlations for that model, Aizenman, Chayes, Chayes and Newman proved that its magnetization is discontinuous at the transition (the Thouless effect), Imbrie and Newman exhibited an intermediate phase with slow (power-law) decay of correlations, and Duminil-Copin, Garban and Tassion gave in 2024 a probabilistic proof via long-range Bernoulli and random-cluster percolation (models of random edges between sites $i,j$ present independently with probability about $\beta J(|i-j|)$), showing that the transition is necessarily discontinuous. Third, a contour-based program (Cassandro, Ferrari, Merola and Presutti; Cassandro, Merola, Picco and Rozikov; Littin and Picco; Bissacot, Endo, van Enter, Kimura and Ruszel; Affonso, Bissacot, Endo and Handa; Affonso, Bissacot, Corsini and Welsch) re-proved and extended the existence side to all polynomial decays $J(n)\asymp n^{-\alpha}$ with $1<\alpha\le 2$, removing the earlier requirement of a large nearest-neighbour coupling $J(1)$. Consequently, for pure power laws the phase diagram in the decay exponent is complete: within the summable regime a transition exists if and only if $\alpha\le 2$. What is not known is the sharp criterion for general coupling sequences, and in particular the window between the two boundaries. The existence techniques require essentially $J(r)\ge c/r^{2}$: Fröhlich and Spencer's energy-entropy argument needs a decreasing $J$ with $\liminf_{r\to\infty}r^{2}J(r)>0$, and Affonso, Bissacot, Corsini and Welsch (2024) stress that the multiscale-contour estimates break down precisely at $J(r)=r^{-2}$. Nonexistence is proved only when $\sum_{n\le N}nJ(n)=o((\log N)^{1/2})$ (i.e. grows slower than order $(\log N)^{1/2}$). For interactions decaying as $n^{-2}$ multiplied by a slowly varying factor tending to zero (a function $L$ with $L(\lambda n)/L(n)\to 1$ for every fixed $\lambda>0$, such as $1/\log\log n$) whose weighted harmonic partial sums $\sum_{n\le N}L(n)/n$ are not $o((\log N)^{1/2})$ (in particular still divergent), for example $J(n)=n^{-2}[\log(1+\log(1+n))]^{-1}$, for which $\sum_{n\le N}nJ(n)$ grows like $\log N/\log\log N$, neither side applies, and it is undecided whether the chain orders at some finite temperature.

Problem Statement

Consider the ferromagnetic Ising chain on $\mathbb{Z}$ with formal Hamiltonian $H=-\sum_{i<j}J(j-i)\sigma_i\sigma_j$, where $J(n)\ge 0$ is summable and translation-invariant, $\sigma_i\in\{-1,+1\}$, and $\beta$ is the inverse temperature; the chain has spontaneous magnetization at $\beta$ if $m^{+}(\beta)=\lim_{h\downarrow 0}\lim_{\Lambda\uparrow\mathbb{Z}}\langle\sigma_0\rangle_{\Lambda,\beta,h}>0$, where $\langle\cdot\rangle_{\Lambda,\beta,h}$ is the finite-volume Gibbs expectation at inverse temperature $\beta$ with a uniform external field $h\ge 0$ (positive $h$ selects the plus phase, and $\Lambda$ runs over finite intervals growing to $\mathbb{Z}$). (1) Borderline window. For couplings with $\liminf_{n\to\infty}n^{2}J(n)=0$ (not bounded below by any positive multiple of $n^{-2}$, so that the Fröhlich-Spencer existence argument does not apply) while the first-moment partial sums escape the known nonexistence criteria ($\sum_{n\le N}nJ(n)$ is not $o((\log N)^{1/2})$, hence also not $o(\log\log N)$ and not convergent, so the Dyson and Ruelle criteria do not apply either), decide whether $m^{+}(\beta)>0$ at some finite $\beta$, or whether $m^{+}(\beta)=0$ at all finite $\beta$. Canonical open cases are $J(n)=n^{-2}L(n)$ with $L$ a slowly varying function (i.e. $L(\lambda n)/L(n)\to 1$ for every fixed $\lambda>0$) tending to zero and $\sum_{n\le N}L(n)/n$ not $o((\log N)^{1/2})$ (which implies $\sum_{n}L(n)/n=\infty$), for instance $J(n)=n^{-2}[\log(1+\log(1+n))]^{-1}$ and $J(n)=n^{-2}/\log\log n$ for $n\ge 3$ (with $J(1),J(2)$ arbitrary positive values). (2) Sharp criterion. Determine the necessary and sufficient asymptotic condition $C(J)$ on non-negative, summable, translation-invariant couplings such that the chain has spontaneous magnetization at some finite temperature exactly when $C(J)$ holds; equivalently, complete the phase diagram of the one-dimensional Ising ferromagnet as a function of the interaction decay, on the stretch between the Rogers-Thompson nonexistence regime and the Fröhlich-Spencer existence regime. The criterion must be tight against explicit coupling sequences on both sides, and the answer should make explicit which auxiliary hypotheses (such as monotonicity of $J$) genuinely matter. The boundaries to be improved upon: nonexistence is known when $\sum_n nJ(n)<\infty$ (Ruelle), when $\sum_{n\le N}nJ(n)=o(\log\log N)$ (Dyson), and when $\sum_{n\le N}nJ(n)=o((\log N)^{1/2})$ (Rogers-Thompson); existence is known for $J(n)\asymp n^{-\alpha}$ with $1<\alpha\le 2$ (Dyson; Fröhlich-Spencer for $\alpha=2$; modern extensions without a large nearest-neighbour coupling) and, more generally, for decreasing $J$ with $\liminf_{n}n^{2}J(n)>0$ (Fröhlich-Spencer).

The two demands are not independent: establishing the sharp criterion (2) is strictly stronger than deciding the borderline window (1) and implies a decision of (1), whereas a decision of (1) does not imply (2). Within (1), the stated class is the target and the couplings listed as canonical are illustrative examples only.

References

1. F. J. Dyson, Non-existence of spontaneous magnetization in a one-dimensional Ising ferromagnet, Communications in Mathematical Physics 12, 212-215 (1969). DOI: 10.1007/BF01661575, https://doi.org/10.1007/BF01661575
2. F. J. Dyson, Existence of a phase-transition in a one-dimensional Ising ferromagnet, Communications in Mathematical Physics 12, 91-107 (1969). DOI: 10.1007/BF01645907, https://doi.org/10.1007/BF01645907
3. D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Communications in Mathematical Physics 9, 267-278 (1968). DOI: 10.1007/BF01654281, https://doi.org/10.1007/BF01654281
4. D. J. Thouless, Long-Range Order in One-Dimensional Ising Systems, Physical Review 187, 732-733 (1969). DOI: 10.1103/PhysRev.187.732, https://doi.org/10.1103/PhysRev.187.732
5. J. B. Rogers and C. J. Thompson, Absence of long-range order in one-dimensional spin systems, Journal of Statistical Physics 25, 669-678 (1981). DOI: 10.1007/BF01022361, https://doi.org/10.1007/BF01022361
6. B. Simon and A. D. Sokal, Rigorous entropy-energy arguments, Journal of Statistical Physics 25, 679-694 (1981). DOI: 10.1007/BF01022362, https://doi.org/10.1007/BF01022362
7. J. Fröhlich and T. Spencer, The phase transition in the one-dimensional Ising Model with $1/r^{2}$ interaction energy, Communications in Mathematical Physics 84, 87-101 (1982). DOI: 10.1007/BF01208373, https://doi.org/10.1007/BF01208373
8. J. Z. Imbrie, Decay of correlations in the one dimensional Ising model with $J_{ij}=|i-j|^{-2}$, Communications in Mathematical Physics 85, 491-515 (1982). DOI: 10.1007/BF01403501, https://doi.org/10.1007/BF01403501
9. M. Aizenman, J. T. Chayes, L. Chayes and C. M. Newman, Discontinuity of the magnetization in one-dimensional $1/|x-y|^{2}$ Ising and Potts models, Journal of Statistical Physics 50, 1-40 (1988). DOI: 10.1007/BF01022985, https://doi.org/10.1007/BF01022985
10. J. Z. Imbrie and C. M. Newman, An intermediate phase with slow decay of correlations in one dimensional $1/|x-y|^{2}$ percolation, Ising and Potts models, Communications in Mathematical Physics 118, 303-336 (1988). DOI: 10.1007/BF01218582, https://doi.org/10.1007/BF01218582
11. M. Cassandro, P. A. Ferrari, I. Merola and E. Presutti, Geometry of contours and Peierls estimates in d=1 Ising models with long range interactions, Journal of Mathematical Physics 46, 053305 (2005). DOI: 10.1063/1.1897644, https://doi.org/10.1063/1.1897644
12. M. Cassandro, I. Merola, P. Picco and U. Rozikov, One-dimensional Ising models with long range interactions: cluster expansion, phase-separating point, Communications in Mathematical Physics 327, 951-991 (2014). DOI: 10.1007/s00220-014-1957-5, https://doi.org/10.1007/s00220-014-1957-5
13. J. Littin and P. Picco, Quasi-additive estimates on the Hamiltonian for the one-dimensional long range Ising model, Journal of Mathematical Physics 58, 073301 (2017). DOI: 10.1063/1.4994034, https://doi.org/10.1063/1.4994034
14. M. Cassandro, E. Orlandi and P. Picco, Phase Transition in the 1d Random Field Ising Model with Long Range Interaction, Communications in Mathematical Physics 288, 731-744 (2009). DOI: 10.1007/s00220-009-0778-4, https://doi.org/10.1007/s00220-009-0778-4
15. R. Bissacot, E. O. Endo, A. C. D. van Enter, B. Kimura and W. M. Ruszel, Contour methods for long-range Ising models: weakening nearest-neighbor interactions and adding decaying fields, Ann. Henri Poincaré 19, 2557-2574 (2018). DOI: 10.1007/s00023-018-0693-3, arXiv:1710.02986, https://arxiv.org/abs/1710.02986
16. A. C. D. van Enter and A. Le Ny, Decimation of the Dyson-Ising Ferromagnet, Stochastic Processes and their Applications 127, 3776-3791 (2017). DOI: 10.1016/j.spa.2017.03.007, arXiv:1603.05409, https://arxiv.org/abs/1603.05409
17. L. Affonso, R. Bissacot, E. O. Endo and S. Handa, Long-range Ising models: Contours, phase transitions and decaying fields, Journal of the European Mathematical Society 27 (4), 1679-1714 (2024). DOI: 10.4171/jems/1529, arXiv:2105.06103, https://arxiv.org/abs/2105.06103
18. L. Affonso, R. Bissacot, H. Corsini and K. Welsch, Phase Transitions on 1d Long-Range Ising Models with Decaying Fields: A Direct Proof via Contours (2024). arXiv:2412.07098, https://arxiv.org/abs/2412.07098
19. H. Duminil-Copin, C. Garban and V. Tassion, Long-range models in 1D revisited, Ann. Inst. Henri Poincaré Probab. Stat. 60 (1) (2024). DOI: 10.1214/22-aihp1355, arXiv:2011.04642, https://arxiv.org/abs/2011.04642
20. M. Okuyama and M. Ohzeki, Existence of a phase transition in the one-dimensional Ising spin glass model with long-range interactions on the Nishimori line, Journal of Physics A: Mathematical and Theoretical 59, 305204 (2026). DOI: 10.1088/1751-8121/ae8c95, arXiv:2604.07130, https://arxiv.org/abs/2604.07130
21. P. Hulse, Unique Bernoulli Gibbs states and g-measures (2025). arXiv:2504.15750, https://arxiv.org/abs/2504.15750