Abstract
We present a collection of simulations of the Edwards–Anderson lattice spin glass at to elucidate the nature of low-energy excitations over a range of dimensions that reach from physically realizable systems to the mean-field limit. Using heuristic methods, we sample the ground states of instances to determine their energies while eliciting excitations through manipulating boundary conditions. We exploit the universality of the phase diagram of bond-diluted lattices to make such a study in higher dimensions computationally feasible. As a result, we obtain a variety of accurate exponents for domain wall stiffness and finite-size corrections, which allow us to examine their dimensional behavior and their connection with predictions from mean-field theory. We also provide an experimentally testable prediction for the thermal-to-percolative crossover exponent in dilute lattice Ising spin glasses.
1 Introduction
Imagining physical systems in non-integer dimensions, such as through -expansion [1] or dimensional regulation [2], to name but two, has provided many important results for the understanding of physics in realistic dimensions. For example, the goal of the -expansion is to establish a connection between the (technically, infinite-dimensional) mean-field solution of a field theory and its real-space behavior. For a disordered system such as a spin glass [3–6], this playbook has proved rather difficult to follow theoretically [7–9]. In contrast, we endeavor to explore the transition between the often well-known mean-field properties and their modifications in real space using numerical means, free of any theoretical preconceptions. In this task, on top of the computationally extensive disorder averages, the complexity of spin glasses reveals itself through increasingly slower convergence in thermal simulations, while the deeper one pushes into the glassy regime. Going all the way to then makes thermal explorations impossible and renders the problem of finding the ground-state NP-hard in general [10]. However, simulations at also avail us considerable conceptual clarity and an entirely new suit of techniques, albeit for just a few, yet important, observables. Some equilibrium properties of spin glasses below can be obtained from merely determining ground-state energies, such as domain wall stiffness, finite-size corrections, and thermal–percolative crossover exponents. To keep systematic errors low while also creating enough statistics for the disorder average, we need to use fast but ultimately inexact heuristic methods to overcome NP-hardness. To reach a sensible scaling regime in system sizes, , especially in higher dimensions, requires clever exploitation of the phase diagram of a spin glass. In this study, we discuss the results obtained from large-scale simulations conducted over several years and spread over a number of papers [11–15].1
To be specific, we simulate the Ising spin glass model due to Edwards and Anderson (EA) with the Hamiltonian [16].The dynamic variables are binary (Ising) spins placed on a hypercubic lattice in the integer dimension with couplings between nearest neighbors via random bonds drawn from some distribution of zero mean and unit variance. The lattices are periodic with base length in all directions, i.e., each such instance has spins. To relate real-world behavior in (which is explored experimentally and theoretically in other articles in this collection) with mean-field behavior, which manifests itself above the upper critical dimension [3], we found ground states of EA on lattices in . In each , we need to simulate instances over a wide range of to extrapolate our results to the thermodynamic limit . At each size , we further need to measure a large number of instances with independently drawn random bonds for the disorder average inherent to obtain observables in spin glasses. Each instance entails approximating its ground state, which is an NP-hard combinatorial problem.
To sample ground-state of the Hamiltonian in Equation 1 at high throughput and with minimal systematic errors, heuristics can only be relied on for systems with not more than spins coupled together. This would appear to limit the “dynamic range” in size up to approximately in but limited to in , and even to in , definitely insufficient to extract any limit. However, the phase diagram for a bond-diluted EA system (with such that ) shown in Figure 1 suggests that universal scaling behavior extends across the entire spin-glass (SG) phase down to the scaling window near the bond-percolation threshold for low enough , i.e., most definitely for . Thus, our strategy is to find ground states for EA instances at bond density with sufficient dynamic ranges in for just above that scaling window to be within the SG phase, using exact reduction methods [12, 17] (see Supplementary Appendix SA) to remove a large number of spins, followed by heuristic optimization of remainder systems with [18, 19] (see Supplementary Appendix SB). These reduction methods recursively trace out all spins with fewer than four connected neighbors, at least, and are particularly effective near since each spin in the EA system has at most potential neighbors while in large such that for just above , lattices remain sparse, each spin being connected to barely more than one other spin, on average, albeit with large variations. For example, in for and , an EA system with spins typically reduces to a remainder graph with spins, each connected to 5.3 neighbors, on average, to be optimized heuristically.
FIGURE 1
2 Domain wall stiffness exponents
A quantity of fundamental importance for the modeling of amorphous magnetic materials through spin glasses [3, 20–23] is the domain wall or “stiffness” exponent , often also labeled . As Hook’s law describes the response to increasing elastic energy imparted to a system for increasing displacement from its equilibrium position, the stiffness of a spin configuration describes the typical increase in magnetic energy due to an induced defect interface of a domain of size . However, unlike uniform systems with a convex potential energy function over its configuration space (say, a parabola for the single degree of freedom in Hook’s law, or a high-dimensional funnel for an Ising ferromagnet), an amorphous many-body system exhibits a function more reminiscent of a high-dimensional mountain landscape. Any defect-induced displacement of size in such a complicated energy landscape may move a system through numerous undulations in energy . Averaging over many incarnations of such a system results in a typical energy scalefor the standard deviations of the domain wall energy .
The importance of this exponent for small excitations in disordered spin systems has been discussed in many contexts [22, 24–28]. Spin systems with provide resistance (“stiffness”) against the spontaneous formation of defects at sufficiently low temperatures , an indication that a phase transition to an ordered state exists. For instance, in an Ising ferromagnet, the energy is always proportional to the size of the interface, i.e., , which is consistent with the fact that only when . For , the state of a system is unstable with respect to defects, and spontaneous fluctuations may proliferate, preventing any ordered state. Thus, determining the exact “lower critical dimension” , where , is of singular importance, and understanding the mechanism leading to , however unnatural its value, provides clues to the origin of order [13, 29–33].
Instead of waiting for a thermal fluctuation to spontaneously induce a domain wall, it is expedient to directly impose domains of size through reversed boundary conditions on the system and measure the energy needed to determine . To wit, in a system with periodic boundary conditions, we first obtain its ground state unaltered and obtain it again as after reversing the signs on all bonds within a -dimensional hyperplane, resulting in a complex domain of spins of size that are reversed between both ground states such that is the energy due to the interface of that domain. Since is equally likely to be positive or negative, it is its deviation, , which sets the energy scale in Equation 2. Notably, this problem places an even higher demand on the ground-state heuristic than described in the introduction. Here, the domain wall energy has a minute, sub-extensive difference between two almost identical, extensive energies, and , each of which is NP-hard to find. Thus, any systematic error would escalate rapidly with , the size of the remainder graph.
As shown in Figure 2, using bond-diluted lattices for the EA system, in contrast, not only affords us a larger dynamic range in but also allows for an extended scaling regime due to the additional parameter of ranging over an entire interval. Instead of one set of data for increasing at a fixed (typically, [34]) leading to the scaling in Equation 2, we can scale multiple independent sets for such a range of into a collective scaling variable, , which collapses the data according to . Although the extension to an interval in makes simulations more laborious, it typically yields an extra order of magnitude in scaling compared to the prohibitive effort of confronting the NP-hard problem of reaching large at fixed alone. For instance, in at , attainable sizes span , at best, while we obtain a perfect data collapse for about for (note that while and has some relation to the correlation-length exponent in percolation [see below], it is necessary to allow these to be a free parameter for the bimodal bonds used in these simulations, as was argued by [12]). The fitted values for for each , as obtained from Figure 2, are listed in Table 1.
FIGURE 2
TABLE 1
| 2 | −0.282(2) | 1.141(1) | −0.993(3) | 1.497(2) | 1.323(4) | |||
| 3 | 0.24(1) | 0.920(4) | 0.915(4) | −1.289(6) | 1.429(3) | 0.2488126 | 0.87436(46) | 1.127(5) |
| 4 | 0.61(1) | 0.847(3) | 0.82(1) | −1.574(6) | 1.393(2) | 0.1601314 | 0.70(3) | 1.1(1) |
| 5 | 0.88(5) | 0.824(10) | 0.81(1) | −1.84(2) | 1.37(1) | 0.118172 | 0.571(3) | 1.05(2) |
| 6 | 1.1(1) | 0.82(2) | 0.82(2) | −2.01(4) | 1.34(1) | 0.0942019 | 1.00(2) | |
| 7 | 1.24(5) | 0.823(7) | 0.91(5) | −2.28(6) | 1.33(1) | 0.0786752 | 1.14(3) | |
| 8 | 1.2(1) | 0.85(2) | ||||||
Stiffness exponents for Edwards–Anderson spin glasses [11, 12] for dimensions obtained numerically from domain wall excitations of ground states, as shown in Figure 2. The next column contains the measured values for finite-size corrections, denoted as , from the fit of the data shown in Figure 3. The stiffness exponents obtained by [14] refer to EA at the bond–percolation threshold , with values of obtained from [36] for and [37] for . The correlation–length exponents for percolation are from [38] in and from [39] for , where is exactly above the upper critical dimension, .
The values for are listed in Table 1 and plotted in Figure 4 as . That quantity has been obtained in the mean-field case by [35], yielding above the upper critical dimension, . That value is clearly consistent with our high-dimensional data, providing a rare direct comparison between the mean field theory (RSB) and real-world spin glasses. As shown in Figure 4, the exponent varies continuously with dimension and allows for a simple cubic fit of the numerical data between , weighted by the statistical errors [13]. The fit independently reproduces the exact known result outside the fitted domain at , , to less than (not shown here). The fit has a zero at and yields at ; there is strong evidence that , which has also been suggested by theory [30, 33] and is consistent with the experiment [32].
In the following section, we consider some other uses of the domain wall excitations.
3 Ground-state finite-size correction exponents
Since simulations of statistical systems are bound to be conducted at system sizes typically quite far from the thermodynamic limit , it becomes essential to understand the corrections entailed by such limitations. This is especially pertinent for spin glasses beset with extra complexities such as NP-hardness at (or, similarly, the lack of equilibration at low but finite ) or the additional burden of disorder averaging over many random samples severely limiting . Only rarely do such corrections decay fast enough to reveal the thermodynamic behavior of an observable in a simulation at a single, “large enough” . Instead, as already observed for the stiffness in Section 2, typically, sets of data need to be generated to glean the asymptotic behavior for large sizes. To extrapolate the value of an intensive observable (like the ground-state energy density), it is then necessary to have a handle on the nature of the finite-size corrections (FSCs) that have to be expected for the generated data [25, 40, 41]. However, FSCs are not only a technical necessity. Their behavior is often closely related to other physical properties in the thermodynamic limit via scaling relations [27]. They can also be instrumentalized, for instance, to assess the scalability of optimization heuristics [42, 43].
For the ground-state energy densities in the EA system, [27] argued that such FSCs should be due to locked-in domain walls of energy , which would lead to the scaling correction for the extensive energies of for large , defining as the limit of the average ground-state energy density . This is consistent with Equation 2, where we purposefully created such a domain wall because the same system freed from that domain wall (or locked into another one) would have and, thus, . Dividing by system size, we obtainwhere the FSC exponent is conjectured to beIndeed, our direct evaluation of ground-state energy densities at some fixed bond density in dimensions , as shown in Figure 3, is convincingly in agreement with this picture for the dominant contributions to FSCs. However, that leaves us with somewhat of a conundrum when compared with mean-field simulations, where FSCs for the Sherrington–Kirkpatrick (SK) spin glass model [44–46] appear to yield for , which is not close to from RSB theory [35].
FIGURE 3
We conducted a corresponding ground-state study at the edge of the SG regime (see Figure 1) by choosing the percolation point exactly. Since the fractal percolation cluster cannot sustain an ordered state, we found that the stiffness exponent defined in Equation 2 is negative there, . Numerical studies of ground states at (using Gaussian bonds in this case) are computationally quite efficient since the fractals embedded in the lattice often reduce completely or so substantially that heuristics produce little systemic error. Large lattice sizes can be achieved, limited only by rare large remainder graphs or the lack of memory needed to build the original, unreduced EA lattice. The values for thus obtained [14] are also listed in Table 1. Although the hypothesis for FSCs from Equation 4, , leads to large values for when and it becomes hard to test numerically, the corrections found are well consistent with the hypothesis [15]. In particular, it appears that for , which would be consistent with FSCs in percolating random graphs [47]. Although this provides an argument that Equation 4 should also hold in the mean-field limit for the EA system in the spin glass phase, the SK model might be a poor representation of that limit for the EA system. In the EA system, we first let for a fixed number of neighbors before , while in the SK model, both system size and neighborhood diverge simultaneously. Unfortunately, sparse mean-field spin glasses on regular graphs (“Bethe lattices”) appear to have FSCs with [48], but those results might depend, to some extent, on the structural details of the mean-field networks [45, 49, 50], and which structure most closely resembles a mean-field version of EA at remains unclear.
4 Thermal–percolative crossover exponents
Having already determined the percolative stiffness exponents in the previous section, we can utilize it to make an interesting—and potentially experimentally testable—prediction about the behavior of the phase transition line, as shown in Figure 1. For diluted lattices at variable bond density , Equation 2 generalizes to [51, 52]Here, we assume that for the surface tension and is the conventional correlation length for percolation. The scaling function is defined to be constant for , where percolation (and hence, ) plays no role, and we regain Equation 2 for . For , Equation 5 requires for to satisfy with some power of , needed to cancel the dependence at . Thus, , and we obtain to mark the dependence of at , as mentioned before, which yields . Finally, at the crossover , where the range of the excitations reaches the percolation length beyond which spin glass order ensues, Equation 5 providesAssociating a temperature with the energy scale of the crossover in Equation 6 by (since, for , thermal fluctuations destroy order at a length-scale ) leads todefining [51] the “thermal–percolative crossover exponent” . All data for are listed in Table 1, and the results for are also shown in Figure 4. It appears that decreases with increasing for , has a minimum of at , and increases as above .
FIGURE 4
Of particular experimental interest is the result for , , predicting with [38]. This exponent provides a non-trivial, experimentally testable prediction derived from scaling arguments of equilibrium theory at low temperatures (since bond and site percolation are typically in the same universality class, it should make little difference whether an experiment varies the site concentration of atoms with dipolar spin or the bonds between them). Such tests are few as disordered materials by their very nature fall out of equilibrium when entering the glassy state. The phase boundary itself provides the perfect object for such a study. It can be approached by theory from below and by experiments from above where equilibration is possible. [53] already provided highly accurate results for the freezing temperature as a function of dilution for a doped, crystalline glass, (Gdx)80, proposing a linear dependence, . The tabulated data are equally well fitted using Equation 7 in that regime. [54] determined a phase diagram for , an amorphous alloy, for a wide range of temperatures and site concentrations but did not discuss its near-linear behavior at low . A similar phase diagram for the insulator In2(1-x)S4 is shown in Figure 1A of [55]. New experiments dedicated to the limit should provide the results of sufficient accuracy to test our prediction for .
5 Conclusions
We summarized a collection of simulation data pertaining to the lattice spin glass EA over a range of dimensions, providing a comprehensive description of low-energy excitations from experimentally accessible systems to the mean-field level, where exact results can be compared with. Putting all those results side-by-side paints a self-consistent picture of domain wall excitations, their role in the stability of the ordered glass state, and their role for finite-size corrections. Extending to the very physical concept of bond density made simulations in high dimensions feasible, added accuracy, and opened up the spin-glass phase diagram, which makes new observables experimentally accessible, such as the thermal–percolative crossover exponent.
Going forward, the methods developed here could be extended to study, say, ground-state entropy and their overlaps [56] or the fractal nature of domain walls [57, 58]. Our method might also inspire new ways of using dilution as a gadget to make simulations more efficient [59].
Statements
Author contributions
SB: conceptualization, data curation, formal analysis, investigation, methodology, writing–original draft, and writing–review and editing.
Funding
The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.
Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2024.1466987/full#supplementary-material
References
1.
WilsonKGFisherME. Critical exponents in 3.99 dimensions. Phys Rev Lett (1972) 28:240–3. 10.1103/physrevlett.28.240
2.
t’HooftGVeltmanMJG. Regularization and renormalization of gauge fields. Nucl Phys B (1972) 44:189–213. 10.1016/0550-3213(72)90279-9
3.
FischerKHHertzJA. Spin glasses, Cambridge studies in magnetism. Cambridge: Cambridge University Press (1991).
4.
MézardMParisiGVirasoroMA. Spin glass theory and beyond. Singapore: World Scientific (1987).
5.
SteinDLNewmanCM. Spin glasses and complexity. Princeton: Princeton University Press (2013).
6.
CharbonneauPMarinariEMezardM. Editors. Spin glass theory and far beyond. Singapore: World Scientific (2023).
7.
de DominicisCKondorITemesáriT. Spin glasses and random fields. In: YoungA, editor. Series on directions in condensed matter physics: volume 12. Singapore: World Scientific (1998).
8.
MooreMAReadN. Multicritical Point on the de Almeida–Thouless Line in Spin Glasses in d > 6 Dimensions. Phys Rev Lett (2018) 120:130602. 10.1103/physrevlett.120.130602
9.
MooreMA. Droplet-scaling versus replica symmetry breaking debate in spin glasses revisited. Phys Rev E (2021) 103:062111. 10.1103/physreve.103.062111
10.
BarahonaF. On the computational complexity of Ising spin glass models. J Phys A: Math Gen (1982) 15:3241–53. 10.1088/0305-4470/15/10/028
11.
BoettcherS. Stiffness exponents for lattice spin glasses in dimensions d = 3, . . . , 6. The Eur Phys J B - Condensed Matter (2004) 38:83–91. 10.1140/epjb/e2004-00102-5
12.
BoettcherS. Low-temperature excitations of dilute lattice spin glasses. Europhys Lett (2004) 67:453–9. 10.1209/epl/i2004-10082-0
13.
BoettcherS. Stiffness of the Edwards-Anderson model in all dimensions. Phys Rev Lett (2005) 95:197205. 10.1103/physrevlett.95.197205
14.
BoettcherSMarchettiE. Low-temperature phase boundary of dilute-lattice spin glasses. Phys Rev B (2008) 77:100405(R). 10.1103/physrevb.77.100405
15.
BoettcherSFalknerS. Finite-size corrections for ground states of Edwards-Anderson spin glasses. EPL (Europhysics Letters) (2012) 98:47005. 10.1209/0295-5075/98/47005
16.
EdwardsSFAndersonPW. Theory of spin glasses. J Phys F (1975) 5:965–74. 10.1088/0305-4608/5/5/017
17.
BoettcherSDavidheiserJ. Reduction of dilute Ising spin glasses. Phys Rev B (2008) 77:214432. 10.1103/physrevb.77.214432
18.
BoettcherSPercusAG. Optimization with extremal dynamics. Phys Rev Lett (2001) 86:5211–4. 10.1103/physrevlett.86.5211
19.In:
HartmannARiegerH, editors. New optimization algorithms in physics. Berlin: Wiley VCH (2004).
20.
SouthernBWYoungAP. Real space rescaling study of spin glass behaviour in three dimensions. J Phys C: Solid State Phys (1977) 10:2179–95. 10.1088/0022-3719/10/12/023
21.
McMillanWL. Scaling theory of Ising spin glasses. J Phys C: Solid State Phys (1984) 17:3179–87. 10.1088/0022-3719/17/18/010
22.
FisherDSHuseDA. Ordered phase of short-range Ising spin-glasses. Phys Rev Lett (1986) 56:1601–4. 10.1103/physrevlett.56.1601
23.
BrayAJMooreMA. Heidelberg colloquium on glassy dynamics and optimization In: Van HemmenLMorgensternI, editors. Proceedings of a colloquium on spin glasses, optimization and neural networks held at the University of Heidelberg. New York: Springer (1986). p. 121.
24.
KrzakalaFMartinO. Spin and link overlaps in three-dimensional spin glasses. Phys Rev Lett (2000) 85:3013–6. 10.1103/physrevlett.85.3013
25.
PalassiniMYoungAP. Nature of the spin glass state. Phys Rev Lett (2000) 85:3017–20. 10.1103/physrevlett.85.3017
26.
PalassiniMLiersFJüngerMYoungAP. Interface energies in Ising spin glasses. Phys Rev B (2003) 68:064413. 10.1103/physrevb.68.064413
27.
BouchaudJ-PKrzakalaFMartinOC. Energy exponents and corrections to scaling in Ising spin glasses. Phys Rev B (2003) 68:224404. 10.1103/physrevb.68.224404
28.
AspelmeierTMooreMAYoungAP. Interface energies in ising spin glasses. Phys Rev Lett (2003) 90:127202. 10.1103/physrevlett.90.127202
29.
BrayAJMooreMA. Lower critical dimension of Ising spin glasses: a numerical study. J Phys C: Solid State Phys (1984) 17:L463–8. 10.1088/0022-3719/17/18/004
30.
FranzSParisiGVirasoroMA. Interfaces and lower critical dimension in a spin glass model. J Phys (France) (1994) 4:1657–67. 10.1051/jp1:1994213
31.
HartmannAKYoungAP. Lower critical dimension of Ising spin glasses. Phys Rev B (2001) 64:180404(R). 10.1103/physrevb.64.180404
32.
GuchhaitSOrbachR. Direct dynamical evidence for the spin glass lower critical dimension 2 < dℓ < 3. Phys Rev Lett (2014) 112:126401. 10.1103/physrevlett.112.126401
33.
MaioranoAParisiG. Support for the value 5/2 for the spin glass lower critical dimension at zero magnetic field. Proc Natl Acad Sci U S A (2018) 115:5129–34. 10.1073/pnas.1720832115
34.
HartmannAK. Ground-state clusters of two-three-and four-dimensional ± JIsing spin glasses. Phys Rev E (2000) 63:016106. 10.1103/physreve.63.016106
35.
ParisiGRizzoT. Large deviations in the free energy of mean-field spin glasses. Phys Rev Lett (2008) 101:117205. 10.1103/physrevlett.101.117205
36.
LorenzCDZiffRM. Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices. Phys Rev E (1998) 57:230–6. 10.1103/physreve.57.230
37.
GrassbergerP. Critical percolation in high dimensions. Phys Rev E (2003) 67:036101. 10.1103/physreve.67.036101
38.
DengYBlöteHWJ. Monte Carlo study of the site-percolation model in two and three dimensions. Phys Rev E (2005) 72:016126. 10.1103/PhysRevE.72.016126
39.
HughesBD. Random walks and random environments. Oxford: Oxford University Press (1996).
40.
PalKF. The ground state of the cubic spin glass with short-range interactions of Gaussian distribution. Physica A (1996) 233:60–6. 10.1016/s0378-4371(96)00241-5
41.
YoungAP. Finite-size scaling. World Scientific (2024). p. 599–615.
42.
BoettcherS. Analysis of the relation between quadratic unconstrained binary optimization and the spin-glass ground-state problem. Phys Rev Res (2019) 1:033142. 10.1103/physrevresearch.1.033142
43.
BoettcherS. Deep reinforced learning heuristic tested on spin-glass ground states: the larger picture. Nat Commun (2023) 14:5658. 10.1038/s41467-023-41106-y
44.
BoettcherS. Extremal optimization for Sherrington-Kirkpatrick spin glasses. The Eur Phys J B (2005) 46:501–5. 10.1140/epjb/e2005-00280-6
45.
BoettcherS. Simulations of ground state fluctuations in mean-field Ising spin glasses. J Stat Mech Theor Exp (2010) 2010:P07002. 10.1088/1742-5468/2010/07/p07002
46.
AspelmeierTBilloireAMarinariEMooreMA. Finite-size corrections in the Sherrington–Kirkpatrick model. J Phys A: Math Theor (2008) 41:324008. 10.1088/1751-8113/41/32/324008
47.
BollobasB. Random graphs. London: Academic Press (1985).
48.
BoettcherS. Numerical results for ground states of spin glasses on Bethe lattices. Eur Phys J B - Condensed Matter (2003) 31:29–39. 10.1140/epjb/e2003-00005-y
49.
ZdeborováLBoettcherS. A conjecture on the maximum cut and bisection width in random regular graphs. J Stat Mech Theor Exp (2010) 2010:P02020. 10.1088/1742-5468/2010/02/p02020
50.
BoettcherS. Ground state properties of the diluted Sherrington-Kirkpatrick spin glass. Phys Rev Lett (2020) 124:177202. 10.1103/physrevlett.124.177202
51.
BanavarJRBrayAJFengS. Critical behavior of random spin systems at the percolation threshold. Phys Rev Lett (1987) 58:1463–6. 10.1103/physrevlett.58.1463
52.
BrayAJFengS. Percolation of order in frustrated systems: the dilute J spin glass. Phys Rev B (1987) 36:8456–60. 10.1103/physrevb.36.8456
53.
PoonSJDurandJ. Magnetic-cluster description of spin glasses in amorphous La-Gd-Au alloys. Phys Rev B (1978) 18:6253–64. 10.1103/physrevb.18.6253
54.
BeckmanOFigueroaEGrammKLundgrenLRaoKVChenHS. Spin wave and scaling law Analysis of amorphous (FexNix)75P16B6Al3by magnetization measurements. Phys Scr (1982) 25:726–30. 10.1088/0031-8949/25/6a/017
55.
VincentE. Ageing and the glass transition. In: HenkelMPleimlingMSanctuaryR, editors. Ageing and the glass transition. Heidelberg: Springer (2007). 716 of Springer Lecture Notes in Physics, condmat/063583.
56.
BoettcherS. Reduction of spin glasses applied to the Migdal-Kadanoff hierarchical lattice. Eur Phys J B - Condensed Matter (2003) 33:439–45. 10.1140/epjb/e2003-00184-5
57.
WangWMooreMAKatzgraberHG. Fractal dimension of interfaces in Edwards-Anderson and long-range Ising spin glasses: determining the applicability of different theoretical descriptions. Phys Rev Lett (2017) 119:100602. 10.1103/physrevlett.119.100602
58.
VedulaBMooreMASharmaA. Evidence that the AT transition disappears below six dimensions (2024). 10.48550/arXiv.2402.03711
59.
JörgTRicci-TersenghiF. Entropic effects in the very low temperature regime of diluted Ising spin glasses with discrete couplings. Phys Rev Lett (2008) 100:177203. 10.1103/physrevlett.100.177203
Summary
Keywords
Edwards–Anderson spin glass, critical dimension, domain wall excitations, ground-state energies, percolation, heuristic algorithms
Citation
Boettcher S (2024) Physics of the Edwards–Anderson spin glass in dimensions d = 3, … ,8 from heuristic ground state optimization. Front. Phys. 12:1466987. doi: 10.3389/fphy.2024.1466987
Received
18 July 2024
Accepted
13 August 2024
Published
20 September 2024
Volume
12 - 2024
Edited by
Konrad Jerzy Kapcia, Adam Mickiewicz University, Poland
Reviewed by
Francisco Welington Lima, Federal University of Piauí, Brazil
Ekrem Aydiner, Istanbul University, Türkiye
Updates
Copyright
© 2024 Boettcher.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Stefan Boettcher, sboettc@emory.edu
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.