Validity of the replica-symmetry-breaking interface calculation and the lower critical dimension $D_{lc}=5/2$ of the Edwards–Anderson Ising spin glass

The Franz–Parisi–Virasoro interface computation, generalized by Astuti, Franz and Parisi in 2019, predicts that forcing an overlap difference $\Delta q$ across a distance $L$ in the low-temperature phase of a $D$-dimensional Edwards–Anderson Ising spin glass costs free energy $\Delta F\propto L^{D-5/2}|\Delta q|^{5/2}$, so that replica symmetry breaking is stable only above a lower critical dimension $D_{lc}=5/2$. The derivation is mean-field based and rests on an unproven variational ansatz, first-order expansions near $T_c$, a restricted class of effective interactions, and one-directional interfaces. The open problem is to establish whether the prediction — and $D_{lc}=5/2$ — holds for the finite-dimensional Edwards–Anderson model below the upper critical dimension $D=6$, notably in $D=3$. Existing determinations cluster around $D_{lc}\approx 2.43$–$2.50$ but do not settle the question.

Background

A spin glass is a magnet whose interactions are randomly ferromagnetic or antiferromagnetic, so that the low-temperature phase is dominated by frustration. The Edwards–Anderson (EA) model places Ising spins $s_i=\pm1$ on a $D$-dimensional lattice with nearest-neighbor couplings $J_{ij}$ that are independent mean-zero random variables, with Hamiltonian $H=-\sum_{\langle ij\rangle}J_{ij}s_is_j$. Its fully connected analogue, the Sherrington–Kirkpatrick (SK) model, is exactly solved: below a critical temperature the naive replica-symmetric description is unstable and the exact solution exhibits replica symmetry breaking (RSB), an infinite hierarchy of broken symmetries between replica copies of the system.

The natural order parameter is the overlap $q$ between two independent equilibrium copies ("real replicas") that share the same disorder, $q=N^{-1}\sum_i\langle s_i^{(1)}s_i^{(2)}\rangle$. In RSB theory the order parameter is not a single number but a function $q(u)$ on $0\le u\le1$ (the Parisi order-parameter function, encoding a hierarchical ultrametric organization of pure states), with an associated probability density $P(q)$ of overlap values.

As in any critical phenomenon, two special dimensions exist. The upper critical dimension of the Ising spin glass is $D=6$: mean field is exact for the SK model and for $D\ge6$. The lower critical dimension $D_{lc}$ is the dimension below which the low-temperature spin-glass phase disappears (the critical temperature vanishes as $D\to D_{lc}$). Simulations show $2<D_{lc}<3$ — a transition exists in $D=3$ but not in $D=2$ — and determining $D_{lc}$ is a standard checkpoint of the field: failure to know it signals missing understanding of the mechanism that destroys the ordered phase at low dimension.

The interface construction probes this directly. Take two real replicas in a $D$-dimensional box of side $L$ and impose, through boundary conditions on the two coupled copies, that their overlap takes one value on one face and a different value (differing by $\Delta q$) on the opposite face, over a distance $L$ in one spatial direction, with free boundary conditions in the other directions. The region interpolating between the two imposed overlaps is an interface, and its free-energy cost $\Delta F(L,\Delta q)$ measures the stability of the RSB phase: if the cost of an interface vanishes in the thermodynamic limit, distinct replica-symmetry-broken phases can coexist and replica symmetry is effectively restored. Franz, Parisi and Virasoro (FPV) computed this cost within the Landau expansion of the replicated effective Hamiltonian near the critical temperature (an expansion in the reduced temperature $\tau=T_c-T$, the "truncated model"), keeping the quartic term responsible for RSB (the "reduced model"), and using a variational ansatz in which the mean-field solution $q(u)$ is perturbed by a smooth polynomial interpolation profile. Their result is

$$\Delta F(L,\Delta q)\;\propto\;L^{D-5/2}\,|\Delta q|^{5/2},$$

so the cost of an extensive interface fails to grow with $L$ exactly for $D\le D_{lc}=5/2$, which is the RSB prediction for the lower critical dimension. Astuti, Franz and Parisi (2019) re-examined this calculation: they enlarged the variational space (higher-order polynomial smoothing, higher-order continuity of the profile — the scaling is robust), extended the computation from the reduced model to general classes of effective Hamiltonians — interactions of the form $\sum_{ab}Q_{ab}^{n}$ (polynomial, "local" in replica space) and trace terms $\mathrm{Tr}(Q^{n})$ as in the EA model truncated at fourth order — and found that for the polynomial class the free-energy density cost is proportional to $P(q)^{3/4}$, while for the EA model the dependence on the overlap is complicated and not expressible through physically interpretable functions. In all cases the $L$-scaling and $D_{lc}=5/2$ are unchanged within the ansatz.

The authors are explicit about the limits of the derivation, which define the open problem. Mean field is exact only for the SK model; its validity below the upper critical dimension $D=6$ is debated. An epsilon-expansion treatment of the RSB phase is blocked by strongly divergent series, unreliable even at $D=5$, so no controlled renormalization-group check exists. The variational maximization is performed over a restricted finite-dimensional space of polynomial profiles: stability within this family was checked, but optimality within the full solution space was not proven. The extensions beyond the reduced model are first order in the replica-symmetry-breaking coupling (with a temperature-correction term) and rely on the near-$T_c$ expansion. Completeness of the interaction class $\sum_{ab}Q_{ab}^{n}\cup\{\mathrm{Tr}(Q^{n})\}$ for all RSB interactions is not shown. Only interfaces forced in a single spatial direction are considered; other excitation geometries could have different costs and could shift $D_{lc}$. The paper concludes that there is no strong evidence that mean-field results hold down to the lower critical dimension and that one must rely heavily on simulation and experiment.

A competing description, the droplet theory of Fisher and Huse, posits only two pure states with domain-wall free energies growing as $L^{y_D}$, where $y_D$ is the stiffness exponent ($y_3\approx 0.24$ in $D=3$); its lower critical dimension is where $y(D)=0$, which zero-temperature stiffness numerics also place near $2.5$. Discriminating RSB from the droplet picture in $D=3$ is the central open question of finite-dimensional spin-glass physics, and the interface cost is one of its most direct probes. The interface scaling is also accessible numerically: constraining the overlap at the two ends of a long prism whose longitudinal dimension greatly exceeds its transverse size $L$, with free boundary conditions in the transverse directions, makes the longitudinal correlation length grow with transverse size as $\xi(L)\propto L^{a_D}$, and the lower critical dimension then follows from $D_{lc}=1+(D-1)/a_D$ (Bernaschi et al., 2026).

Problem Statement

Central question. For the $D$-dimensional Edwards–Anderson Ising spin glass at zero magnetic field below its critical temperature, consider an interface forced by constraining the overlap between two real replicas to differ by $\Delta q$ over a distance $L$ in one spatial direction (free boundary conditions in the others). The Franz–Parisi–Virasoro interface computation, as generalized by Astuti, Franz and Parisi (2019), predicts

$$\Delta F(L,\Delta q)\propto L^{D-5/2}\,|\Delta q|^{5/2},$$

so that the cost of an extensive interface fails to grow with $L$ for $D\le D_{lc}=5/2$ — replica symmetry breaking is stable only above the lower critical dimension $D_{lc}=5/2$. Does this prediction hold for the finite-dimensional Edwards–Anderson model below the upper critical dimension $D=6$, in particular in $D=3$?

Open points. Below the upper critical dimension $D=6$ — in the range $5/2<D<6$ and at $D=3$ — the status of the interface scaling and of $D_{lc}$ turns on several open points of the FPV/AFP derivation: optimality of the variational solution within the full solution space (not merely stability within polynomial families); the effect of higher orders in the replica-symmetry-breaking coupling and of the expansion in reduced temperature near $T_c$; completeness of the interaction class $\sum_{ab}Q_{ab}^{n}$ and $\mathrm{Tr}(Q^{n})$ for all replica-symmetry-breaking interactions; and interface/excitation geometries beyond the single-direction setting. A different admissible variational solution that changed the scaling exponent would change $D_{lc}$, so a refutation of the variational solution would also bear directly on the validity question.

Discriminating the replica-symmetry-breaking prediction ($a_3=4/3$, $D_{lc}=5/2$) from the competing droplet-model prediction ($a_3=1+y_3\approx 1.24$, with $y_3$ the stiffness exponent) in $D=3$ (and ideally in $D=4$) requires the finite-dimensional interface/correlation-length scaling to be determined with statistical and systematic errors small enough to distinguish the two, and would decide whether the mean-field interface prediction is quantitatively valid below $D=6$. The question is thereby what the interface scaling implies for $D_{lc}$ and for the stability of replica symmetry breaking in $D=3$.

References

1. V. Astuti, S. Franz, G. Parisi, 'New analysis of the free energy cost of interfaces in spin glasses', J. Phys. A: Math. Theor. 52, 294001 (2019). DOI: 10.1088/1751-8121/ab2744; arXiv:1810.13013. https://doi.org/10.1088/1751-8121/ab2744
2. S. Franz, G. Parisi, M. A. Virasoro, 'The replica method on and off equilibrium', J. Physique I 2, 1869 (1992). DOI: 10.1051/jp1:1992115. https://doi.org/10.1051/jp1:1992115
3. S. Franz, G. Parisi, M. A. Virasoro, 'Interfaces and lower critical dimension in a spin glass model', J. Physique I 4, 1657 (1994). DOI: 10.1051/jp1:1994213; arXiv:cond-mat/9405007. https://doi.org/10.1051/jp1:1994213
4. S. Franz, T. Jörg, G. Parisi, 'Overlap interfaces in hierarchical spin-glass models', J. Stat. Mech. (2009) P02002. DOI: 10.1088/1742-5468/2009/02/P02002; arXiv:0809.0106. https://doi.org/10.1088/1742-5468/2009/02/P02002
5. A. Maiorano, G. Parisi, 'Support for the value 5/2 for the spin glass lower critical dimension at zero magnetic field', Proc. Natl. Acad. Sci. USA 115, 5129 (2018). DOI: 10.1073/pnas.1720832115; arXiv:1711.05590. https://doi.org/10.1073/pnas.1720832115
6. M. Bernaschi, L. A. Fernández, I. González-Adalid Pemartín, V. Martín-Mayor, G. Parisi, F. Ricci-Tersenghi, 'Low energy excitations in a long prism geometry: testing the lower critical dimension of the Ising spin glass' (2026, preprint). arXiv:2601.07926. https://arxiv.org/abs/2601.07926
7. M. Aguilar-Janita, V. Martín-Mayor, J. Moreno-Gordo, J. J. Ruiz-Lorenzo, 'Evidence of de Almeida–Thouless line below six dimensions', J. Stat. Mech. (2025) 113301. DOI: 10.1088/1742-5468/ae1a4a; arXiv:2510.14446. https://doi.org/10.1088/1742-5468/ae1a4a
8. S. Boettcher, 'Stiffness of the Edwards-Anderson Model in all Dimensions', Phys. Rev. Lett. 95, 197205 (2005). DOI: 10.1103/PhysRevLett.95.197205. https://doi.org/10.1103/PhysRevLett.95.197205
9. S. Boettcher, 'Physics of the Edwards–Anderson spin glass in dimensions d = 3, …, 8 from heuristic ground state optimization', Front. Phys. 12, 1466987 (2024). DOI: 10.3389/fphy.2024.1466987; arXiv:2407.14646. https://doi.org/10.3389/fphy.2024.1466987
10. P. Contucci, C. Giardinà, C. Giberti, G. Parisi, C. Vernia, 'Interface Energy in the Edwards-Anderson Model', J. Stat. Phys. 142, 1 (2011). DOI: 10.1007/s10955-010-0100-z; arXiv:1007.3679. https://doi.org/10.1007/s10955-010-0100-z
11. S. Franz, G. Parisi, F. Ricci-Tersenghi, 'Overlap locking and nonperturbative effects in spin glasses', Proc. Natl. Acad. Sci. USA 123, e2606171123 (2026). DOI: 10.1073/pnas.2606171123; arXiv:2602.18399. https://doi.org/10.1073/pnas.2606171123
12. C. M. Newman, D. L. Stein, 'Free Energy Difference Fluctuations in Short-Range Spin Glasses', J. Stat. Phys. 191, 128 (2024). DOI: 10.1007/s10955-024-03334-4; arXiv:2407.17506. https://doi.org/10.1007/s10955-024-03334-4
13. G. Parisi, T. Temesvári, 'Replica symmetry breaking in and around six dimensions', Nucl. Phys. B 858, 293 (2012). DOI: 10.1016/j.nuclphysb.2012.01.014; arXiv:1111.3313. https://doi.org/10.1016/j.nuclphysb.2012.01.014
14. M. C. Angelini, 'Real-Space Renormalization group for spin glasses', contribution to 'Spin Glass Theory and Far Beyond: Replica Symmetry Breaking After 40 Years' (World Scientific, 2023). arXiv:2302.05292. https://arxiv.org/abs/2302.05292
15. V. Martín-Mayor, J. J. Ruiz-Lorenzo, B. Seoane, A. P. Young, 'Numerical Simulations and Replica Symmetry Breaking', contribution to 'Spin Glass Theory and Far Beyond: Replica Symmetry Breaking After 40 Years' (World Scientific, 2023); preprint arXiv:2205.14089 (2022). https://arxiv.org/abs/2205.14089
16. B. Atalay, A. N. Berker, 'Lower lower-critical spin-glass dimension from quenched mixed-spatial-dimensional spin glasses', Phys. Rev. E 98, 042125 (2018). DOI: 10.1103/PhysRevE.98.042125; arXiv:1809.06029. https://doi.org/10.1103/PhysRevE.98.042125
17. M. Okuyama, M. Ohzeki, 'Absence of Spin-Glass Order on Migdal–Kadanoff Hierarchical Lattices near Three Dimensions' (2026, preprint). arXiv:2607.15676. https://arxiv.org/abs/2607.15676
18. D. S. Fisher, D. A. Huse, 'Ordered Phase of Short-Range Ising Spin-Glasses', Phys. Rev. Lett. 56, 1601 (1986). DOI: 10.1103/PhysRevLett.56.1601. https://doi.org/10.1103/PhysRevLett.56.1601