Structure of the low-temperature phase of the hierarchical Edwards-Anderson spin glass in a magnetic field and its relation to non-perturbative effects

Monte Carlo simulations of the hierarchical Edwards-Anderson Ising spin glass in a magnetic field (HEA) by Castellana and Parisi (Scientific Reports 5, 8697, 2015) found evidence for a finite-temperature spin-glass transition in a field in the non-mean-field region $2/3<\sigma<1$ of the coupling-decay parameter $\sigma$, precisely where the companion renormalization-group analysis of Castellana and Barbieri (Physical Review B 91, 024202, 2015) shows that no perturbative fixed point exists - implying that the transition, if genuine, is governed by strong non-perturbative effects. The source paper leaves open the structure of the resulting low-temperature phase: does the transition signal the replica-symmetry-breaking (RSB) scenario, with an exponential multiplicity of low-lying energy states, or are the physical features of the non-mean-field phase markedly different from mean-field RSB, aligning with the droplet picture or with intermediate scenarios? How the low-temperature phase relates to the non-perturbative effects remains to be clarified. The question remains untouched: no work since 2015 has studied the HEA in a field, while the broader question of whether spin-glass order survives a field below the upper critical dimension remains actively contested, with 2024-2026 numerical evidence on both sides.

Background

Spin glasses and the fate of order in a field. A spin glass is a magnet whose exchange couplings are randomly ferromagnetic and antiferromagnetic; the canonical lattice model is the Edwards-Anderson Ising spin glass, with Hamiltonian $H=-\sum_{ij}J_{ij}S_iS_j$, Ising spins $S_i=\pm1$, and independent random couplings $J_{ij}=\pm1$. Its fully connected mean-field version, the Sherrington-Kirkpatrick model, is solved by Parisi's replica-symmetry-breaking (RSB) theory: below the critical temperature the system freezes into a hierarchy (technically, an ultrametric organization) of infinitely many pure thermodynamic states, with an exponential-in-system-size multiplicity of low-lying energy states. Within mean-field theory an applied magnetic field does not destroy the transition: below the de Almeida-Thouless (AT) line in the field-temperature plane the system still enters an RSB phase. Whether the AT line survives at finite dimension - where fluctuations renormalize the theory - is one of the longest-standing open problems of disordered-systems physics, because it decides whether real spin glasses are described by RSB or by the droplet theory.

Competing pictures of the ordered phase. In the droplet theory of Fisher and Huse there are only two pure states, related by a global spin flip. The low-energy landscape consists of droplet excitations - connected clusters of spins flipped relative to the ground state - whose excitation energy grows with linear size $L$ as $L^\theta$ (the stiffness exponent $\theta$) and whose boundary is a fractal of surface dimension $d_s$. In a magnetic field the droplet picture predicts no transition: the energy gained by aligning a large droplet with the random background grows faster with $L$ than the stiffness cost (the Imry-Ma argument, which introduces a crossover length that diverges as the field vanishes), so arbitrarily large droplets cost vanishing energy and no field-induced spin-glass order can form. Intermediate 'trivial-nontrivial' (TNT) scenarios mix the two pictures: as in the droplet theory there are only two pure states, so the order parameter is trivial (Palassini & Young, 1999), but as in mean-field RSB the spectrum of large-scale excitations is nontrivial, with large droplets costing only O(1) energy rather than growing as $L^\theta$ (Houdayer, Krzakala & Martin, 2000). Measuring excitation stiffness, surface dimension of flipped clusters, and the multiplicity of low-lying states in a field is therefore precisely what discriminates the scenarios.

Why the problem is hard for ordinary lattices. For short-range spin glasses in a field the renormalization-group (RG) fixed-point structure - the set of scale-invariant theories reached by iterating the coarse-graining transformation that integrates out short-distance fluctuations - is out of reach: the perturbative treatment is an epsilon-expansion (an expansion in $\varepsilon=6-d$, the deviation from the upper critical dimension $d_u=6$, the dimension above which mean-field theory is exact), whose series diverges and yields no predictions below $d_u$. Even the situation at and just below six dimensions is disputed: Moore and Bray argued that the AT line disappears already at $d=6$, while Parisi and Temesvari found the expansion around six dimensions consistent with RSB surviving in and around six dimensions.

The hierarchical laboratory. Hierarchical models make the RG tractable while retaining non-mean-field fluctuations. The hierarchical Edwards-Anderson model (HEA) places $2^k$ Ising spins on the leaves of a binary hierarchical tree: two spins at hierarchical distance $d_{ij}$ (the number of levels one must ascend the tree to reach a common ancestor of the two spins) interact with a probability that decays with $d_{ij}$ at a rate controlled by a parameter $\sigma\in(1/2,1)$, the nonzero couplings being $\pm1$; an independent diluted binary random field is added, with $h_i$ nonzero with probability $p_h$ (the field concentration) and nonzero $h_i$ equal to $\pm1$ with equal probability. The parameter $\sigma$ plays the role of an inverse dimension: for $1/2<\sigma\le 2/3$ order-parameter fluctuations vanish and the critical behavior is mean-field-like, the boundary $\sigma=2/3$ playing the role of the upper critical dimension, while for $2/3<\sigma<1$ fluctuations change the critical behavior (the non-mean-field, NMF, region). The HEA in a field is the only spin-glass-in-a-field setting where the complete set of perturbative RG fixed points has been determined (Castellana and Barbieri): a stable critical fixed point exists in the mean-field region, but in the NMF region every potentially stable fixed point has a nonzero imaginary part, so no perturbative fixed point - and hence no perturbatively accessible transition - exists there.

The source result. Castellana and Parisi simulated the HEA with binary couplings and diluted binary fields ($2^k$ spins, $k=6,\dots,13$; field concentration $p_h=0.36$ at $\sigma=0.6$ and $p_h=0.16$ at $\sigma=0.68$), extracting the spin-glass susceptibility (the spatial sum of squared correlations between two independent copies, or replicas, of the system) and the associated correlation length from its long-wavelength Fourier modes. In the mean-field region ($\sigma=0.6$) Monte Carlo and RG agree, both predicting a transition with the classical exponent $\nu=1/(2\sigma-1)=5$ (Monte Carlo: $5.0\pm0.4$). In the NMF region at $\sigma=0.68$ - only about 2% above the upper-critical value $\sigma=2/3$ - the two approaches conflict: the Monte Carlo data indicate a transition (diverging correlation length and susceptibility) while perturbation theory forbids one, and the fitted correlation-length exponent $\nu\approx5.0\pm1.0$ is nearly twice the classical value $2.78$. The conclusion is that the NMF transition, if it survives the thermodynamic limit, is governed by strong non-perturbative effects - physics that cannot be reached by any expansion around the mean-field fixed point. What kind of low-temperature phase such non-perturbative physics builds is exactly the question the source paper explicitly leaves open.

Problem Statement

For the hierarchical Edwards-Anderson spin glass in a magnetic field (binary couplings, diluted binary random fields of concentration $p_h$, hierarchical interaction decay at rate $\sigma$), in the non-mean-field region $2/3<\sigma<1$ - where Monte Carlo evidence at $\sigma=0.68$ indicates a spin-glass transition in a field that the complete perturbative renormalization-group analysis cannot capture - determine the structure of the low-temperature phase and its relation to the non-perturbative effects. A resolution requires all three of the following, which may be addressed by any sound method:

1. Characterize the low-energy excitations above the ground state: the scaling of excitation energy with excitation size (the stiffness exponent $\theta$), the surface dimension $d_s$ of the flipped clusters, and the multiplicity of low-lying energy states - exponentially many, as in the mean-field replica-symmetry-breaking (RSB) scenario, or few, as in the droplet scenario.

2. Establish whether the observed transition in a field survives the thermodynamic limit at non-mean-field $\sigma$ (rather than being a finite-size or crossover artifact of the near-critical regime $\sigma\approx2/3$), and hence whether it is governed by a genuinely non-perturbative fixed point.

3. Identify which scenario the results realize - RSB-like, droplet-like, or intermediate - and explain how the measured excitation properties account for, or rule out, the non-perturbative effects revealed by the disagreement between Monte Carlo numerics and perturbative renormalization group.

References

1. Castellana, M. & Parisi, G. (2015). Non-perturbative effects in spin glasses. Scientific Reports 5, 8697. DOI 10.1038/srep08697. https://doi.org/10.1038/srep08697 (preprint arXiv:1503.02103)
2. Castellana, M. & Barbieri, C. (2015). Hierarchical spin glasses in a magnetic field: A renormalization-group study. Physical Review B 91, 024202. DOI 10.1103/PhysRevB.91.024202. https://doi.org/10.1103/PhysRevB.91.024202 (preprint arXiv:1503.00066)
3. de Almeida, J. R. L. & Thouless, D. J. (1978). Stability of the Sherrington-Kirkpatrick solution of a spin glass model. Journal of Physics A: Mathematical and General 11, 983-990. DOI 10.1088/0305-4470/11/5/028. https://doi.org/10.1088/0305-4470/11/5/028
4. Fisher, D. S. & Huse, D. A. (1986). Ordered phase of short-range Ising spin-glasses. Physical Review Letters 56, 1601. DOI 10.1103/PhysRevLett.56.1601. https://doi.org/10.1103/PhysRevLett.56.1601
5. Palassini, M. & Young, A. P. (1999). Triviality of the ground state structure in Ising spin glasses. Physical Review Letters 83, 5126-5129. DOI 10.1103/PhysRevLett.83.5126. https://doi.org/10.1103/PhysRevLett.83.5126
6. Houdayer, J., Krzakala, F. & Martin, O. C. (2000). Large-scale low-energy excitations in 3-d spin glasses. European Physical Journal B 18, 467-477. DOI 10.1007/s100510070035. https://doi.org/10.1007/s100510070035 (preprint arXiv:cond-mat/0009382)
7. Katzgraber, H. G., Larson, D. & Young, A. P. (2009). Study of the de Almeida-Thouless line using power-law diluted one-dimensional Ising spin glasses. Physical Review Letters 102, 177205. DOI 10.1103/PhysRevLett.102.177205. https://doi.org/10.1103/PhysRevLett.102.177205
8. Leuzzi, L., Parisi, G., Ricci-Tersenghi, F. & Ruiz-Lorenzo, J. J. (2009). Ising spin-glass transition in a magnetic field outside the limit of validity of mean-field theory. Physical Review Letters 103, 267201. DOI 10.1103/PhysRevLett.103.267201. https://doi.org/10.1103/PhysRevLett.103.267201
9. Alvarez Baños, R., Cruz, A., Fernandez, L. A., Gil-Narvion, J. M., Gordillo-Guerrero, A., Guidetti, M., Iñiguez, D., Maiorano, A., Marinari, E., Martin-Mayor, V., Monforte-Garcia, J., Muñoz Sudupe, A., Navarro, D., Parisi, G., Perez-Gaviro, S., Ruiz-Lorenzo, J. J., Schifano, S. F., Seoane, B., Tarancon, A., Tellez, P., Tripiccione, R. & Yllanes, D. (2012). Thermodynamic glass transition in a spin glass without time-reversal symmetry. Proceedings of the National Academy of Sciences USA 109, 6452-6456. DOI 10.1073/pnas.1203295109. https://doi.org/10.1073/pnas.1203295109
10. Moore, M. A. & Bray, A. J. (2011). Disappearance of the de Almeida-Thouless line in six dimensions. Physical Review B 83, 224408. DOI 10.1103/PhysRevB.83.224408. https://doi.org/10.1103/PhysRevB.83.224408
11. Parisi, G. & Temesvári, T. (2012). Replica symmetry breaking in and around six dimensions. Nuclear Physics B 858, 293-316. DOI 10.1016/j.nuclphysb.2012.01.014. https://doi.org/10.1016/j.nuclphysb.2012.01.014
12. Aspelmeier, T., Katzgraber, H. G., Larson, D., Moore, M. A., Wittmann, M. & Yeo, J. (2016). Finite-size critical scaling in Ising spin glasses in the mean-field regime. Physical Review E 93, 032123. DOI 10.1103/PhysRevE.93.032123. https://doi.org/10.1103/PhysRevE.93.032123 (preprint arXiv:1509.05372)
13. Castellana, M. (2019). Real-space renormalization-group methods for hierarchical spin glasses. Journal of Physics A: Mathematical and Theoretical 52, 445002. DOI 10.1088/1751-8121/ab2f46. https://doi.org/10.1088/1751-8121/ab2f46 (preprint arXiv:1910.04521)
14. Angelini, M. C., Parisi, G. & Ricci-Tersenghi, F. (2020). Comment on 'Real-space renormalization-group methods for hierarchical spin glasses'. Journal of Physics A: Mathematical and Theoretical 53, 418001. DOI 10.1088/1751-8121/ab8414. https://doi.org/10.1088/1751-8121/ab8414 (preprint arXiv:1911.02323)
15. Castellana, M. (2020). Reply to comment on 'Real-space renormalization-group methods for hierarchical spin glasses'. Journal of Physics A: Mathematical and Theoretical 53, 418002. DOI 10.1088/1751-8121/ab8671. https://doi.org/10.1088/1751-8121/ab8671 (preprint arXiv:2102.06263)
16. Vedula, B., Moore, M. A. & Sharma, A. (2024). Evidence that the de Almeida-Thouless transition disappears below six dimensions. Physical Review E 110, 054131. DOI 10.1103/PhysRevE.110.054131. https://doi.org/10.1103/PhysRevE.110.054131 (preprint arXiv:2402.03711, titled 'Evidence that the AT transition disappears below six dimensions')
17. Vedula, B., Moore, M. A. & Sharma, A. (2025). Nature of spin glass order in physical dimensions. Physical Review E 111, 034102. DOI 10.1103/PhysRevE.111.034102. https://doi.org/10.1103/PhysRevE.111.034102 (preprint arXiv:2410.19069)
18. Aguilar-Janita, M., Martin-Mayor, V., Moreno-Gordo, J. & Ruiz-Lorenzo, J. J. (2025). Evidence of de Almeida-Thouless line below six dimensions. Journal of Statistical Mechanics: Theory and Experiment 2025, 113301. DOI 10.1088/1742-5468/ae1a4a. https://doi.org/10.1088/1742-5468/ae1a4a (preprint arXiv:2510.14446)
19. Miranda-Filho, L. H. & Jin, Y. (2026). Evidence of the de Almeida-Thouless transition in three-dimensional spin glasses. arXiv:2601.19192. https://arxiv.org/abs/2601.19192
20. Franz, S., Parisi, G. & Ricci-Tersenghi, F. (2026). Overlap locking and non-perturbative effects in spin glasses. arXiv:2602.18399. https://arxiv.org/abs/2602.18399
21. Castellana, M. (2025). Order parameter for non-mean-field spin glasses. Physical Review B 112, 224206. DOI 10.1103/7lm2-ljrd. https://doi.org/10.1103/7lm2-ljrd (preprint arXiv:2512.08691)