Uniqueness versus multiplicity of low-temperature Gibbs states for the one-dimensional long-range Ising ferromagnet under structured external fields

The one-dimensional Ising ferromagnet with algebraically decaying interactions (the Dyson model, decay exponent $1<\gamma\le 2$) has a low-temperature phase transition in zero field, while a nonzero uniform field restores uniqueness of the Gibbs state. A. Kerimov (2009) proved uniqueness at low temperature for weak, sparse external fields by a contour method that requires a unique, Peierls-stable ground state, and posed the general question of which deterministic structured fields permit multiple low-temperature Gibbs states and which force uniqueness. The open issues comprise three linked problems: the conjecture, raised in 2007 and restated in 2009, that the two critical block sizes $R_1$ and $R_2$ of the alternating-block field coincide; extension of the uniqueness criterion to all periodic fields with nonzero period sum; and adaptation of the contour/percolation technique to fields whose ground-state problem has multiple solutions or fails Peierls stability. Later work by the same author resolved the periodic-field criterion in the weak-amplitude regime and produced a disagreement-percolation uniqueness condition that does not need ground-state uniqueness, but the sharp block-size threshold, periodic fields beyond the weak-amplitude regime (including an explicit two-ground-state periodic construction), and the method extension to extreme field realizations remain open.

Background

The Dyson model is the one-dimensional Ising ferromagnet with algebraically decaying interactions: spins $\sigma_x\in\{-1,+1\}$ sit on the integer lattice $\mathbb{Z}$ and interact through the formal Hamiltonian

$H(\sigma)=-\sum_{\{x,y\}}\frac{J}{|x-y|^{\gamma}}\sigma_x\sigma_y-\sum_{x}h_x\sigma_x,\qquad J>0,\;1<\gamma\le 2,$

where $(h_x)$ is an external magnetic field, taken here to be deterministic. A Gibbs state is any infinite-volume limit of the finite-volume probability measures defined by this Hamiltonian together with a boundary condition; it satisfies the DLR (Dobrushin–Lanford–Rivas) consistency equations, and a low-temperature phase transition means that more than one such state exists. Dyson (1969) proved that in zero field the model has a phase transition for $1<\gamma<2$ and no spontaneous magnetization for $\gamma>2$; the borderline decay $\gamma=2$ was settled by Fröhlich and Spencer (1982), so $1<\gamma\le 2$ is exactly the interesting range. At the opposite pole, a classical result of Lebowitz and Martin-Löf (1972) shows that a nonzero uniform field forces uniqueness of the equilibrium state for Ising systems, and counterpart uniqueness criteria for long-range interactions are now known. Between these poles lies the question of structured fields: deterministic fields that are neither uniform nor zero.

Two structured families drive the question. Kerimov (2007) studied the alternating block field, which takes the value $+\epsilon$ on blocks of $R$ consecutive lattice sites and $-\epsilon$ on the following $R$ sites, so the field has mean zero and period $2R$. He proved a block effect: when $R$ is small the field oscillates too fast to be seen at large scales and at least two Gibbs states coexist at low temperature, while for $R$ large the field acts locally like a uniform field and the Gibbs state is unique. His argument produced two critical block sizes, $R_1$ and $R_2$, with multiplicity for $R<R_1$ and uniqueness for $R>R_2$ and $R_1\le R_2$, and he conjectured that in fact $R_1=R_2$, that is, that there is a single sharp threshold.

Kerimov (2009), the source of this problem, treated weak and sparse fields — fields of small amplitude supported on a sparse set of sites. The proof analyzes the ground state (the configuration minimizing the energy, here uniquely the all-plus or all-minus phase according to the sign of the net field bias) and verifies a Peierls stability condition: a constant $\tau>0$ such that any finite droplet of the opposite phase costs at least $\tau$ per unit of boundary length. With a unique, Peierls-stable ground state, a contour expansion — contours being the connected regions occupied by the minority phase, organized and weighted in the manner of Pirogov–Sinai theory, the general low-temperature scheme that classifies phases by ground states and treats excitations as dilute contours — combined with the percolation-based uniqueness criterion the same author developed in 2002, controls all large deviations and yields a unique Gibbs state at low temperature. The paper closes by identifying what this method cannot yet do, and those gaps are the problem posed here.

Subsequent work sharpened the picture. Kerimov (2012, Physica A) proved that every periodic field of period $L$ with nonzero period sum $\sum_{x=0}^{L-1}h_x\neq 0$ and sufficiently small amplitude admits a unique Gibbs state at low temperature; the same paper constructs an explicit $8L$-periodic field — a pattern of segments of differing signs with one single-site sign flip per period — which has two distinct ground states when $L$ is large and a nonzero period sum; whether it supports two Gibbs states could not be decided. Kerimov (2014) gave a disagreement-percolation uniqueness condition — a criterion comparing the interaction couplings with an absorbing percolation process on sites where two coupled copies of the system disagree — which, unlike the contour method, does not require the ground state to be unique. A parallel line concerns fields of other kinds: fields decaying at infinity, where the phase transition provably survives (Bissacot et al. 2018 for summable fields; Affonso et al. 2024, whose one-dimensional contour proof covers all decays $1<\alpha\le 2$ and extends to sufficiently fast-decaying nonsummable fields), and random fields, where it also survives (Cassandro, Orlandi and Picco 2009). None of these addresses deterministic periodic or block fields at intermediate or large amplitudes. The 2019 survey of Le Ny records that a general study of inhomogeneous or alternating external fields is still to be done.

Problem Statement

For the one-dimensional long-range ferromagnetic Ising model with couplings $J_{xy}=J/|x-y|^{\gamma}$, $J>0$, $1<\gamma\le 2$, at sufficiently low temperature (large inverse temperature $\beta$) under deterministic external fields $(h_x)$, determine the boundary between fields compatible with multiple Gibbs states and fields forcing a unique Gibbs state. The common objective uniting the requested results is the completion of the low-temperature phase diagram of the Dyson ferromagnet under structured external fields, of which the following three source-posed issues are the concrete instances.

(1) Sharp block-size threshold. For the alternating block field taking $+\epsilon$ on $R$ consecutive sites and $-\epsilon$ on the next $R$ sites: prove or refute the conjecture $R_1=R_2$, that is, establish whether the effect of the external field is monotone with respect to the block size, so that the critical block size separating at least two low-temperature Gibbs states from a unique Gibbs state is a single sharp threshold.

(2) Periodic fields without the smallness hypothesis. Decide whether every $L$-periodic field with nonzero period sum $\sum_{x=0}^{L-1}h_x\neq 0$ admits a unique Gibbs state at sufficiently low temperature, with no constraint on the field amplitude; this is the expectation stated in Kerimov (2009).

(3) Method extension to extreme field realizations. Extend the contour/percolation uniqueness method — by the technical modifications of the Kerimov (2002) method anticipated in Kerimov (2009), or by other rigorous means — to field realizations whose induced ground-state problem has multiple ground states or whose ground state fails the Peierls stability condition, so that uniqueness (which the source expects, but does not prove, to hold) or multiplicity can actually be decided for such fields; the 2009 contour method does not apply to them as written.

Parts (1) and (2) are the two concrete field families in which the multiplicity/uniqueness boundary must be located; part (3) is the enabling methodological development needed exactly where the standard tool breaks. Each of the three parts is independently attackable, so settling any single part is a well-defined contribution to the objective; completing the objective, however, requires resolving all three.

References

1. 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
2. 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
3. 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
4. J. L. Lebowitz and A. Martin-Löf, On the uniqueness of the equilibrium state for Ising spin systems, Communications in Mathematical Physics 25, 276-282 (1972). DOI: 10.1007/BF01877686, https://doi.org/10.1007/BF01877686
5. Azer Kerimov, The relationship between phase transitions and percolation in models with ground-state degeneracy, Journal of Physics A: Mathematical and General 35, 5365-5378 (2002). DOI: 10.1088/0305-4470/35/26/301, https://doi.org/10.1088/0305-4470/35/26/301
6. Azer Kerimov, A block effect of external field in the one-dimensional ferromagnetical Ising model with long-range interaction, Journal of Physics A: Mathematical and Theoretical 40, 10407-10414 (2007). DOI: 10.1088/1751-8113/40/34/001, https://doi.org/10.1088/1751-8113/40/34/001
7. Azer Kerimov, One-dimensional long-range ferromagnetic Ising model under weak and sparse external field, International Journal of Modern Physics B 23, 5899-5906 (2009). DOI: 10.1142/S0217979209053291, https://doi.org/10.1142/S0217979209053291
8. Azer Kerimov, The one-dimensional long-range ferromagnetic Ising model with a periodic external field, Physica A: Statistical Mechanics and its Applications 391, 2931-2935 (2012). DOI: 10.1016/j.physa.2012.01.002, https://doi.org/10.1016/j.physa.2012.01.002
9. Azer Kerimov, Ground states of one-dimensional long-range ferromagnetic Ising model with external field, Modern Physics Letters B 26, 1150014 (2012). DOI: 10.1142/S021798491150014X, https://doi.org/10.1142/S021798491150014X
10. Azer Kerimov, A disagreement-percolation type uniqueness condition for Gibbs states in models with long-range interactions, Journal of Statistical Mechanics: Theory and Experiment 2014, P10014 (2014). DOI: 10.1088/1742-5468/2014/10/P10014, https://doi.org/10.1088/1742-5468/2014/10/P10014
11. Ahmet Sensoy, One-dimensional long range Widom-Rowlinson model with periodic particle activities, Modern Physics Letters B 27, 1350218 (2013). DOI: 10.1142/S0217984913502187, https://doi.org/10.1142/S0217984913502187
12. Marzio Cassandro, Enza Orlandi and Pierre 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
13. Rodrigo Bissacot, Eric O. Endo, Aernout C. D. van Enter, Bruno Kimura and Wioletta M. Ruszel, Contour methods for long-range Ising models: weakening nearest-neighbor interactions and adding decaying fields, Annales Henri Poincare 19, 2557-2574 (2018). DOI: 10.1007/s00023-018-0693-3, https://doi.org/10.1007/s00023-018-0693-3
14. Lucas Affonso, Rodrigo Bissacot, Eric O. Endo and Satoshi Handa, Long-range Ising models: contours, phase transitions and decaying fields, Journal of the European Mathematical Society 27, 1679-1714 (2024). DOI: 10.4171/JEMS/1529, https://doi.org/10.4171/JEMS/1529
15. Lucas Affonso, Rodrigo Bissacot, Henrique Corsini and Kelvyn Welsch, Phase transitions on 1d long-range Ising models with decaying fields: a direct proof via contours, arXiv:2412.07098 (2024), https://arxiv.org/abs/2412.07098
16. Arnaud Le Ny, Gibbs measures for long-range Ising models, arXiv:1911.01860 (2019), https://arxiv.org/abs/1911.01860