For the Ising model on a Bethe lattice (the infinite loop-free regular graph of coordination qq), the Bethe approximation is exact for the pure model in the deep interior of the lattice. Semkin and Smagin (2014) extended a cluster-mean-field (fixed-scale renormalization-group) scheme to bond-diluted magnets, deriving closed-form self-consistent equations for the magnetization and correlation functions that reduce to the Bethe result at full bond occupation p=1p=1 and reproduce the Bethe-lattice bond-percolation threshold pc=1/(q−1)p_c=1/(q-1); the authors presented these equations as approximate, not as the exact solution of the Ising problem for a diluted magnet on the Bethe lattice. The resulting open problem is to determine whether such single-effective-field closures are exact for the quenched bond-diluted Ising model on the Bethe lattice and, where they are not, to identify the correlations and distributional information they miss (the exact solution being the fixed point of the cavity-field distribution). The exact cavity-method critical line coincides with the source equations' critical coupling, and the exact solution exists in distributional form.

Background

Consider the Ising model — spins σi=±1\sigma_i=\pm 1 with exchange energy −J∑⟨ij⟩σiσj-J\sum_{\langle ij\rangle}\sigma_i\sigma_j in an external field HH — on a Bethe lattice, the infinite loop-free regular graph in which every site has qq neighbors (distinct from a finite Cayley tree, which carries boundary sites). A classical result is that the Bethe (Bethe–Peierls) approximation, a self-consistent one-spin-cluster equation, becomes exact for the deep interior of the Bethe lattice; Katsura and Takizawa [4] gave the classical analysis of the correspondence between the Bethe lattice and the Bethe approximation. Semkin and Smagin [1,2,3] developed a cluster-mean-field scheme — a fixed-scale renormalization-group transformation built by averaging over local exchange fields in clusters of one and two spins — that reproduces the Bethe approximation for pure magnets and extends it to bond dilution, where each exchange bond is present independently with probability pp (quenched disorder); for pp below the Bethe-lattice bond-percolation threshold pc=1/(q−1)p_c=1/(q-1) no infinite cluster exists and no ferromagnetic order is possible at finite coupling. Their diluted self-consistent equations (their Eq. (18)) read tanh⁡(Kqpμ+h)=(1−p)tanh⁡(K(q−1)pμ+h)+p sinh⁡(2K(q−1)pμ+2h)cosh⁡(2K(q−1)pμ+2h)+e−2K,\tanh(Kqp\mu + h) = (1-p)\tanh(K(q-1)p\mu + h) + p\,\frac{\sinh(2K(q-1)p\mu + 2h)}{\cosh(2K(q-1)p\mu + 2h) + e^{-2K}}, M=tanh⁡(Kqpμ+h),M = \tanh(Kqp\mu + h), with K=J/kBTK=J/k_BT, h=H/kBTh=H/k_BT, an effective cluster field μ\mu, and per-site magnetization MM; at zero field a nonzero solution appears only above the critical coupling Kc(p)=12ln⁡p+pcp−pc,pc=1q−1.K_c(p) = \tfrac{1}{2}\ln\frac{p+p_c}{p-p_c},\qquad p_c=\frac{1}{q-1}. The same paper derives closed-form nearest-neighbor and longer-range connected correlation functions gij=⟨σiσj⟩−M2g_{ij}=\langle\sigma_i\sigma_j\rangle - M^2 and the correlation length within the same closure. The authors emphasize that although the equations reduce to the Bethe result at p=1p=1 and reproduce the percolation threshold, the equations are approximate and cannot be regarded as the exact solution of the Ising problem for a diluted magnet on the Bethe lattice.

The exact solution of this diluted problem is known in distributional form through the cavity method (belief propagation), which is exact on trees because the Gibbs measure factorizes over branches: the exact solution is the fixed point of the cavity-field distribution Q(h)Q(h) under a recursive distributional equation combining the q−1q-1 incoming messages, each forwarded through a bond that is present with probability pp, with u(J,h)=tanh⁡−1(tanh⁡J tanh⁡h)u(J,h)=\tanh^{-1}(\tanh J\,\tanh h); it is evaluated numerically by population dynamics (Laumann, Scardicchio and Sondhi [7]; Ohta [8]). At p=1p=1 the fixed-point distribution collapses to a delta function and the single Bethe self-consistency equation is recovered exactly; at p<1p<1 the distribution is non-degenerate and also encodes the Griffiths-type rare-region singularities of the diluted ferromagnet. Linearizing the distributional recursion about the paramagnetic fixed point yields the exact critical line (q−1)ptanh⁡Kc=1(q-1)p\tanh K_c=1, i.e. Kc(p)=tanh⁡−1(pc/p)K_c(p)=\tanh^{-1}(p_c/p), which is arithmetically identical to the cluster-mean-field value above. Rigorously, Lyons [5] computed the exact phase-transition temperature of the Ising model on arbitrary infinite trees, including diluted and random-interaction cases. For the pure model, systematically improvable closure hierarchies that are exact in equilibrium are known (Semerjian and Weigt [6]). What is not known is whether the cluster-mean-field equations of [1] coincide with the exact quenched solution away from their exactly reproduced limits, and which correlations or distributional information they miss where they do not.

Problem Statement

For the Ising model on the Bethe lattice of coordination qq with quenched bond dilution (each exchange bond occupied independently with probability pp, ferromagnetic coupling J>0J>0, arbitrary temperature K=J/kBTK=J/k_BT and field h=H/kBTh=H/k_BT), determine the exactness of closed-form, single-effective-field self-consistent equations of the cluster-mean-field type — the diluted equations of Semkin and Smagin reproduced in the Background being the concrete instance under test — for the magnetization and spatial correlation functions of the bond-diluted Ising model on the Bethe lattice. Concretely: (1) establish whether these closures coincide with the exact quenched solution — the fixed point of the cavity-field distribution of the diluted model — away from the critical line and at finite field, given that they already reproduce the pure Bethe result at p=1p=1 and the exact critical coupling Kc(p)=12ln⁡p+pcp−pcK_c(p)=\tfrac{1}{2}\ln\frac{p+p_c}{p-p_c}, pc=1/(q−1)p_c=1/(q-1); and (2) where they do not coincide, identify which correlations or distributional information — beyond the single effective field μ\mu — is responsible for the failure of exactness, i.e. characterize the discrepancy between these closures and the exact fixed-point cavity-field distribution of the diluted model, at least qualitatively and where possible quantitatively.

References

  1. S. V. Semkin and V. P. Smagin, "Correlation functions of pure and diluted Ising magnets in the mean-field approximation", Physics of the Solid State 56(7), 1338-1341 (2014). DOI 10.1134/S1063783414070300, https://doi.org/10.1134/S1063783414070300
  2. S. V. Semkin and V. P. Smagin, "Methods of derivation of self-consistent equations for an Ising magnet", Russian Physics Journal 56(2), 118-124 (2013). DOI 10.1007/s11182-013-0008-6, https://doi.org/10.1007/s11182-013-0008-6
  3. S. V. Semkin and V. P. Smagin, "Method of averaging over interaction fields for constructing the fixed-scale renormalization group transformation", Physics of the Solid State 55(5), 970-973 (2013). DOI 10.1134/S1063783413050302, https://doi.org/10.1134/S1063783413050302
  4. S. Katsura and M. Takizawa, "Bethe lattice and the Bethe approximation", Progress of Theoretical Physics 51(1), 82-98 (1974). DOI 10.1143/PTP.51.82, https://doi.org/10.1143/PTP.51.82
  5. R. Lyons, "The Ising model and percolation on trees and tree-like graphs", Communications in Mathematical Physics 125(2), 337-353 (1989). DOI 10.1007/BF01217911, https://doi.org/10.1007/BF01217911
  6. G. Semerjian and M. Weigt, "Approximation schemes for the dynamics of diluted spin models: the Ising ferromagnet on a Bethe lattice", Journal of Physics A: Mathematical and General 37(21), 5525-5546 (2004). DOI 10.1088/0305-4470/37/21/003, arXiv:cond-mat/0402451, https://arxiv.org/abs/cond-mat/0402451
  7. C. Laumann, A. Scardicchio and 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
  8. H. Ohta, "Multi-step relaxations in Glauber dynamics of a bond-diluted Ising model on a Bethe lattice", Journal of Physics A: Mathematical and Theoretical 43(39), 395003 (2010). DOI 10.1088/1751-8113/43/39/395003, arXiv:1002.4363, https://arxiv.org/abs/1002.4363
  9. S. V. Semkin and V. P. Smagin, "Diluted Ising Magnet on the Bethe Lattice", Russian Physics Journal 58(12), 1848-1858 (2016). DOI 10.1007/s11182-016-0726-7, https://doi.org/10.1007/s11182-016-0726-7
  10. S. V. Semkin and V. P. Smagin, "Bethe Approximation for Pure and Diluted Magnets as Averaging over Local Exchange Fields", Russian Physics Journal 62(1), 172-178 (2019). DOI 10.1007/s11182-019-01698-9, https://doi.org/10.1007/s11182-019-01698-9
  11. S. V. Semkin, V. P. Smagin and E. G. Gusev, "Ising Model with Nonmagnetic Dilution on Recursive Lattices", Theoretical and Mathematical Physics 202(2), 265-271 (2020). DOI 10.1134/S0040577920020099, https://doi.org/10.1134/S0040577920020099
  12. S. V. Semkin, V. P. Smagin and V. S. Tarasov, "Frustrations in a Diluted Ising Magnet on the Bethe Lattice", Journal of Experimental and Theoretical Physics 134(6), 713-719 (2022). DOI 10.1134/S1063776122060048, https://doi.org/10.1134/S1063776122060048
  13. S. V. Semkin and V. P. Smagin, "Construction of an approximate solution for a dilute magnet based on the solution for a pure magnet on the same lattice", Physics of the Solid State 64(12), 1919 (2022). DOI 10.21883/pss.2022.12.54386.456, https://journals.ioffe.ru/articles/54386