The Ising ferromagnet on the Bethe lattice (the loop-free infinite regular tree) is exactly solvable when every bond is present. When bonds are independently removed with probability 1−p1-p (quenched bond dilution), exact results are known only in special limits: the pure model at p=1p=1, the one-dimensional chain (q=2q=2), the critical line and percolation threshold, the T=0T=0 magnetization (the percolation probability), and the infinite-coupling limit. Semkin and Smagin (2014), computing correlation functions of pure and bond-diluted Ising magnets in an effective-field approximation, noted that their self-consistent equations, while exact at p=1p=1 and reproducing the exact percolation threshold, cannot be regarded as the exact solution of the diluted Bethe-lattice Ising problem; their 2020 follow-up states that an exact solution cannot be constructed for the Bethe lattice with q>2q>2. The cavity method characterizes the solution exactly but only implicitly, as the fixed point of a functional recursion for the cavity-field distribution, which at finite coupling is evaluated numerically by population dynamics. The problem is to construct the exact solution at finite coupling — explicit analytic expressions for the quenched free energy, magnetization, and spin correlation functions for general p<1p<1 and q>2q>2 — going beyond all known special limits and beyond the implicit cavity characterization. No exact solution of this problem is known in the literature to date.

Background

The Ising model assigns to each site of a lattice a spin σi=±1\sigma_i=\pm 1 and to each nearest-neighbor pair an energy −Jσiσj-J\sigma_i\sigma_j; its thermodynamics is the prototype of phase-transition theory. The Bethe lattice is the infinite regular tree on which every site has the same coordination number qq and there are no closed loops; operationally it is the bulk (deep-interior) limit of a finite Cayley tree, whose large boundary is sent away before bulk quantities are extracted. Because loops are absent, the pure ferromagnet (J>0J>0 on every bond) is exactly solvable: magnetization, free energy per site, and correlation functions all follow from a one-dimensional self-consistency recursion for the ratio of partial partition functions of a rooted subtree — the Bethe–Peierls solution, presented in Baxter's book as one of the exactly solved models. In particular, the spin–spin correlation function decays exponentially with lattice distance, with an amplitude fixed by the nearest-neighbor correlation.

Bond dilution introduces quenched disorder: each bond is present independently with probability pp and absent with probability 1−p1-p, the dilution pattern is frozen while thermal averages are taken, and thermodynamic quantities are then averaged over patterns (a quenched average). For p<1p<1 the model is no longer exactly solvable in the same sense. Some pieces are exact: the bond-percolation threshold of the tree (the bond concentration above which an infinite connected cluster of present bonds first appears) is pc=1/(q−1)p_c=1/(q-1); linearizing the exact recursion gives the exact critical line (q−1) p tanh⁡Kc=1(q-1)\,p\,\tanh K_c=1 with K=J/kTK=J/kT; and at T=0T=0 the spontaneous magnetization equals the exact probability that a site belongs to the infinite cluster. But at finite coupling an exact solution requires the full distribution Q(h)Q(h) of the effective field acting on a site, produced by the random configuration of its subtrees. In the cavity method (a message-passing technique that is exact on trees precisely because there are no loops), Q(h)Q(h) is characterized as the fixed point of a distributional recursion Q=H[Q]Q=\mathcal{H}[Q]: the cavity field at a site is the sum over its q−1q-1 forward neighbors of contributions that are zero for an absent bond and tanh⁡−1[tanh⁡K tanh⁡h′]\tanh^{-1}[\tanh K\,\tanh h'] for a present bond with subfield h′h'. This characterization is exact but implicit: at finite KK the fixed point lives on a complicated set of field values and is computed numerically by population dynamics (iterating a large population of sampled fields). Explicit exact results exist only in special limits: p=1p=1; the chain q=2q=2, solvable by the transfer matrix (the 2×22\times 2 matrix of Boltzmann weights of a single bond); T=0T=0; and the infinite-coupling limit K→∞K\to\infty, in which each percolation cluster behaves as a rigid ferromagnet, the magnetization series over clusters is exactly summable, and the exact density of Lee–Yang zeros (zeros of the partition function in the complex magnetic-field plane, which encode the Griffiths–McCoy singularities — the anomalously slow, non-perturbative response of the paramagnet caused by rare ordered clusters) can be extracted, as shown by Laumann, Scardicchio and Sondhi.

The effective-field school approaches the same model with cluster approximations: a small cluster of spins is treated exactly and its surroundings are replaced by self-consistent average fields (a fixed-scale renormalization-group transformation, i.e. one that embeds the cluster in a self-consistently determined field rather than rescaling the couplings). Semkin and Smagin (2014) used this method to compute the magnetization and correlation functions of pure and bond-diluted Ising magnets. For the bond-diluted model their self-consistency equation reduces to the exact Bethe result at p=1p=1 and yields the exact percolation threshold, yet, as they emphasize, it cannot be regarded as the exact solution of the Ising problem for a diluted magnet on the Bethe lattice; their 2020 follow-up states that an exact solution cannot be constructed for the diluted model even on the Bethe lattice with q>2q>2, with only the q=2q=2 chain exactly solvable as a benchmark.

Problem Statement

Construct the exact solution of the quenched bond-diluted Ising ferromagnet on the Bethe lattice — the infinite regular tree of coordination number q>2q>2 in which each bond is present independently with probability p<1p<1 and absent with probability 1−p1-p, the dilution pattern being quenched — at finite coupling K=J/kTK=J/kT and arbitrary external field hh. Here an exact solution has the standard content of an exact lattice-model solution: explicit analytic expressions (closed forms, exactly summable series, or equivalent representations that do not rest on the implicit fixed point of the functional recursion Q=H[Q]Q=\mathcal{H}[Q]), together with a derivation establishing exactness, for the joint set of quantities that constitutes the Bethe–Peierls solution at p=1p=1: (i) the quenched free energy per site, (ii) the magnetization, and (iii) the spin–spin correlation functions, at least the nearest-neighbor correlation and the correlation as a function of lattice distance. The solution must reduce to the known exact special cases — the pure-model results at p=1p=1, the transfer-matrix solution of the diluted chain at q=2q=2, the critical line (q−1)ptanh⁡Kc=1(q-1)p\tanh K_c=1 with percolation threshold pc=1/(q−1)p_c=1/(q-1), the T=0T=0 magnetization equal to the percolation probability, and the K→∞K\to\infty cluster-expansion results — and it must go beyond the present status quo, in which the solution is only characterized implicitly as the fixed point of the cavity-field distributional recursion Q=H[Q]Q=\mathcal{H}[Q] and its thermodynamic consequences are evaluated numerically by population dynamics.

References

  1. S. V. Semkin, V. P. Smagin, Correlation functions of pure and diluted Ising magnets in the mean-field approximation (English translation of the Russian original «Корреляционные функции чистого и разбавленного изинговского магнетика в приближении эффективного поля», Физика твёрдого тела 56(7), 1288–1291 (2014)), Physics of the Solid State 56(7), 1338–1341 (2014), DOI: 10.1134/S1063783414070300, https://doi.org/10.1134/S1063783414070300 (Russian original full text: https://journals.ioffe.ru/articles/26933)
  2. S. V. Semkin, V. P. Smagin, V. I. Lyul'ko, Construction of Approximate Methods within the Ising Model of a Diluted Magnet Using Averaging over Interaction Fields (English translation of the Russian original «Использование усреднения по полям взаимодействия для построения приближенных методов в модели Изинга разбавленного магнетика», Физика твёрдого тела 62(8), 1209–1214 (2020), DOI: 10.21883/FTT.2020.08.49603.021), Physics of the Solid State 62(8), 1355–1360 (2020), DOI: 10.1134/S1063783420080296, https://doi.org/10.1134/S1063783420080296
  3. C. Laumann, A. Scardicchio, S. L. Sondhi, Griffiths-McCoy singularities, Lee-Yang zeros, and the cavity method in a solvable diluted ferromagnet, Physical Review E 77, 061139 (2008), DOI: 10.1103/PhysRevE.77.061139, arXiv:0712.3540, https://arxiv.org/abs/0712.3540
  4. H. Ohta, Multi-step relaxations in the Glauber dynamics of a bond-diluted Ising model on a Bethe lattice, Journal of Physics A: Mathematical and Theoretical 43, 395003 (2010), DOI: 10.1088/1751-8113/43/39/395003, arXiv:1002.4363, https://arxiv.org/abs/1002.4363
  5. R. J. Baxter, Exactly Solved Models in Statistical Mechanics, World Scientific, Singapore (1985; first published by Academic Press, London, 1982), ISBN 978-9971-97-811-2, DOI: 10.1142/9789814415255_0002, https://doi.org/10.1142/9789814415255_0002
  6. M. Seino, The free energy of the random Ising model on the Bethe lattice, Physica A: Statistical Mechanics and its Applications 181, 233–242 (1992), DOI: 10.1016/0378-4371(92)90087-7, https://doi.org/10.1016/0378-4371(92)90087-7
  7. S. V. Semkin, V. P. Smagin, Asymptotic behavior of the susceptibility and magnetization of a diluted Ising magnet (Russian original «Асимптотическое поведение восприимчивости и намагниченности разбавленного изинговского магнетика»), The Territory of New Opportunities. The Herald of Vladivostok State University of Economics and Service (Территория новых возможностей. Вестник Владивостокского государственного университета экономики и сервиса) 12(1), 105–114 (2020), DOI: 10.24866/VVSU/2073-3984/2020-1/105-114, https://doi.org/10.24866/vvsu/2073-3984/2020-1/105-114 (open-access full text: https://science.vvsu.ru/files/1A10D078-C930-44BF-8010-94594E9A9026.pdf)