Dyson's 1969 proof of a finite-temperature phase transition in a one-dimensional Ising chain with long-range ferromagnetic interactions left the marginal regime undecided. Later work settled the power-law case: a transition exists exactly for 1<α≤21<\alpha\le2, including the borderline J(r)∝r−2J(r)\propto r^{-2} (Fröhlich–Spencer), while Rogers–Thompson gave no-order criteria in terms of the first moment, refuting the Kac–Thompson conjecture in full generality. The surviving open core is twofold: (i) the classical marginal window — for decreasing interactions J(r)J(r) with lim inf⁡rr2J(r)=0\liminf_r r^2J(r)=0 but partial first moments ∑r≤nrJ(r)\sum_{r\le n} rJ(r) growing faster than (log⁡n)1/2(\log n)^{1/2} (e.g. J(r)=r−2[log⁡(1+log⁡(1+r))]−1J(r)=r^{-2}[\log(1+\log(1+r))]^{-1}), decide whether spontaneous magnetization can occur and find the sharp order/no-order criterion; (ii) the quantum analogue — whether Dyson-type ordering theorems extend to the one-dimensional long-range quantum Heisenberg ferromagnet. The first is explicitly posed as open in the recent specialist literature (Affonso–Bissacot–Corsini–Welsch 2024); for the second, spontaneous magnetization is already excluded for α≥2\alpha\ge2 by a one-dimensional Mermin–Wagner-type theorem (Nenciu 1971), and on 1<α<21<\alpha<2 no rigorous theorem exists in either direction, although renormalization-group and numerical analyses predict ordered phases.

Background

Consider a one-dimensional chain of Ising spins σi∈{±1}\sigma_i\in\{\pm1\} (classical two-state variables) on Z\mathbb{Z}, coupled by ferromagnetic (alignment-favoring) pair interactions J(∣i−j∣)>0J(|i-j|)>0 that decay with distance, at inverse temperature β\beta. A one-dimensional system with finite-range interactions never orders at finite temperature, but the situation changes for long-range interactions. Ruelle (1968) proved that a one-dimensional lattice gas — and in particular such an Ising chain — has no phase transition when the first moment ∑rrJ(r)\sum_r rJ(r) is finite (for power laws J(r)∝r−αJ(r)\propto r^{-\alpha} this means α>2\alpha>2). Dyson (1969) then proved that sufficiently slowly decaying interactions do produce a low-temperature phase transition: spontaneous magnetization (positive magnetization m(β)=⟨σ0⟩β+>0m(\beta)=\langle\sigma_0\rangle_\beta^{+}>0 in the infinite-volume Gibbs state with plus boundary conditions) exists at large β\beta when JJ is positive, decreasing, summable, and satisfies ∑r(log⁡log⁡r) [r3J(r)]−1<∞\sum_r (\log\log r)\,[r^3J(r)]^{-1}<\infty, in particular for power laws with 1<α<21<\alpha<2. Kac and Thompson had conjectured that the first moment alone controls ordering — order exactly when it diverges — and Dyson left the borderline α=2\alpha=2 undecided, noting that C. J. Thompson privately doubted a transition there. Dyson also introduced a hierarchical (block, renormalization-style) comparison model, proved his ordering theorems only for monotone JJ, and discussed — without proof — the extension to the one-dimensional quantum Heisenberg ferromagnet (spins with continuous rotation symmetry and Hamiltonian H=−∑i<jJ(∣i−j∣) Si⋅SjH=-\sum_{i<j}J(|i-j|)\,\mathbf{S}_i\cdot\mathbf{S}_j), for which the required Griffiths-type monotonicity inequalities (correlation inequalities stating that ferromagnetic couplings increase correlations) were not available.

Later work transformed the power-law picture. Fröhlich and Spencer (1982) proved that the transition exists for J(r)=r−2J(r)=r^{-2} — their multiscale contour (Peierls-type geometric) argument covers decreasing JJ with lim inf⁡rr2J(r)>0\liminf_r r^2J(r)>0 — refuting Thompson's doubt. Imbrie (1982) constructed a low-temperature expansion for the r−2r^{-2} model and proved upper and lower bounds of order ∣i−j∣−2|i-j|^{-2} on the two-point function at large β\beta; Imbrie and Newman (1988) proved an intermediate phase with algebraically decaying correlations, as in the Berezinskii–Kosterlitz–Thouless (BKT) transition of the two-dimensional XY model; Aizenman, Chayes, Chayes and Newman (1988) proved that the spontaneous magnetization jumps discontinuously at the critical point (the Thouless effect), using Fortuin–Kasteleyn (FK) random-cluster representations. Duminil-Copin, Garban and Tassion (2024) revisited these percolation and Potts analogues, proving symmetry breaking for the 1/r21/r^2 interaction at large β\beta and that the transition is necessarily discontinuous. On the no-order side, Rogers and Thompson (1981) proved absence of long-range order whenever the partial first moments ∑r≤nrJ(r)\sum_{r\le n} rJ(r) grow as o((log⁡n)1/2)o((\log n)^{1/2}) (the boundedness variant with lim sup⁡n(log⁡n)−1/2∑r≤nrJ(r)<∞\limsup_n(\log n)^{-1/2}\sum_{r\le n}rJ(r)<\infty is quoted as well-known in the recent literature) — so, for example, J(r)=r−2(log⁡r)−1J(r)=r^{-2}(\log r)^{-1} has divergent first moment but no order, refuting the Kac–Thompson conjecture in full generality. For power-law interactions the classification is thus complete: order for 1<α≤21<\alpha\le2, none for α>2\alpha>2. Affonso, Bissacot, Corsini and Welsch (2024) gave a contour proof of the transition for all 1<α≤21<\alpha\le2 without requiring the nearest-neighbor coupling J(1)J(1) to be large, and Bissacot and Corsini (2026) developed a convergent low-temperature cluster expansion for the same decay range, with algebraic decay of correlations at rate exactly α\alpha.

What remains open is twofold. First, the classical marginal window: interactions decaying just faster than r−2r^{-2}. Affonso et al. (2024) state explicitly that the Fröhlich–Spencer contour method "breaks down precisely at J(r)=r−2J(r)=r^{-2}" — its energy–entropy balance requires decreasing JJ with lim inf⁡rr2J(r)>0\liminf_r r^2J(r)>0 — while the Rogers–Thompson criterion stops at the (log⁡n)1/2(\log n)^{1/2} first-moment scale; the canonical test interaction J(r)=r−2[log⁡(1+log⁡(1+r))]−1J(r)=r^{-2}[\log(1+\log(1+r))]^{-1} defeats both criteria, and the authors "wonder" whether the magnetization is always trivial for decreasing JJ with lim inf⁡rr2J(r)=0\liminf_r r^2J(r)=0. Second, the quantum analogue: whether Dyson-type ordering theorems extend to the one-dimensional quantum Heisenberg ferromagnet with 1/rα1/r^\alpha interactions, 1<α<21<\alpha<2 (for α≥2\alpha\ge2 a one-dimensional Mermin–Wagner-type theorem of Nenciu (1971) already rules out spontaneous magnetization). No rigorous finite-temperature ordering theorem for the latter is known in either direction; renormalization-group and numerical analyses of long-range quantum spin chains predict ordered phases in this regime (the interactions are beyond the reach of the Mermin–Wagner–Hohenberg theorem, which forbids finite-temperature breaking of a continuous symmetry for sufficiently short-range systems in one and two dimensions), and this is the regime probed by quantum-simulation platforms with tunable long-range couplings.

Problem Statement

For the one-dimensional ferromagnetic Ising chain on Z\mathbb{Z} (spins σi∈{±1}\sigma_i\in\{\pm1\}, pair interactions J(∣i−j∣)>0J(|i-j|)>0, inverse temperature β\beta), the power-law classification of finite-temperature spontaneous magnetization — positive magnetization m(β)=⟨σ0⟩β+>0m(\beta)=\langle\sigma_0\rangle_\beta^{+}>0 in the infinite-volume Gibbs state with plus boundary conditions — is complete: order holds for decreasing JJ with lim inf⁡rr2J(r)>0\liminf_r r^2J(r)>0 (Fröhlich–Spencer, proved for J(r)=r−2J(r)=r^{-2}), and absence of order when the partial first moments are o((log⁡n)1/2)o((\log n)^{1/2}) (Rogers–Thompson) or finite (Ruelle). Two parts of Dyson's original problem survive. (A) The classical marginal window: for decreasing interactions JJ with lim inf⁡rr2J(r)=0\liminf_r r^2J(r)=0 whose partial first moments M(n)=∑r≤nrJ(r)M(n)=\sum_{r\le n} rJ(r) grow faster than (log⁡n)1/2(\log n)^{1/2} (the canonical interaction J(r)=r−2[log⁡(1+log⁡(1+r))]−1J(r)=r^{-2}[\log(1+\log(1+r))]^{-1} has M(n)M(n) growing on the scale log⁡n/log⁡log⁡n\log n/\log\log n), decide whether spontaneous magnetization at sufficiently low temperature can occur at all, and determine the sharp criterion — the necessary and sufficient condition on JJ separating order from absence of order in this window — without assuming more monotonicity or regularity than the argument genuinely requires; the canonical interaction is a mandatory case that the criterion must settle. (B) The quantum analogue: prove or disprove the existence of a finite-temperature ordering phase transition for the one-dimensional quantum Heisenberg ferromagnet with Hamiltonian H=−∑i<j∣i−j∣−α Si⋅SjH=-\sum_{i<j}|i-j|^{-\alpha}\,\mathbf{S}_i\cdot\mathbf{S}_j (where Si\mathbf{S}_i are quantum spin operators of fixed spin length, with continuous rotation symmetry) for 1<α<21<\alpha<2 (for α≥2\alpha\ge2 spontaneous magnetization is already excluded by the one-dimensional Mermin–Wagner-type theorem of Nenciu (1971), reference 12): either establish spontaneous magnetization (continuous-symmetry breaking) at low temperature, or a rigorous obstruction to it.

References

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