Non-perturbative effects in spin glasses

Abstract We present a numerical study of an Ising spin glass with hierarchical interactions—the hierarchical Edwards-Anderson model with an external magnetic field (HEA). We study the model with Monte Carlo (MC) simulations in the mean-field (MF) and non-mean-field (NMF) regions corresponding to d ≥ 4 and d < 4 for the d-dimensional ferromagnetic Ising model respectively. We compare the MC results with those of a renormalization-group (RG) study where the critical fixed point is treated as a perturbation of the MF one, along the same lines as in the "Equation missing"-expansion for the Ising model. The MC and the RG method agree in the MF region, predicting the existence of a transition and compatible values of the critical exponents. Conversely, the two approaches markedly disagree in the NMF case, where the MC data indicates a transition, while the RG analysis predicts that no perturbative critical fixed point exists. Also, the MC estimate of the critical exponent ν in the NMF region is about twice as large as its classical value, even if the analog of the system dimension is within only ~2% from its upper-critical-dimension value. Taken together, these results indicate that the transition in the NMF region is governed by strong non-perturbative effects.

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ISING SPIN GLASS: RECENT PROGRESS IN THE FIELD THEORY APPROACH

International Journal of Modern Physics B ◽

10.1142/s0217979293002134 ◽

1993 ◽

Vol 07 (01n03) ◽

pp. 986-992 ◽

Cited By ~ 10

Author(s):

C. DE DOMINICIS ◽

I. KONDOR ◽

T. TEMESVARI

Keyword(s):

Field Theory ◽

Mean Field Theory ◽

Mean Field ◽

Critical Dimension ◽

Zero Field ◽

Theory Approach ◽

Replica Symmetry ◽

Numerical Work ◽

Upper Critical Dimension ◽

Ising Spin

A loop expansion around Parisi's replica symmetry breaking mean field theory is constructed, in zero field. We obtain the equation of state (and associated Parisi's solution) below the upper critical dimension du=6, and, in particular, explicit corrections in εlnt and (εlnt)2 with t=(Tc−T)/Tc and ε=6−d. This allows us to verify that standard scaling is satisfied with β=1+ε/2+0(ε2). We have also investigated the transverse (replicon) correlation function, particularly at zero overlap. Near du, G00 (p) ~ t/P4 (for p~0) is the most obvious obstacle to a meaningful theory in d=3. If we make the twofold assumption that (i) scaling applies, and (ii) the replicon propagator is dominated by the spectrum of the (small) transverse masses, we obtain the softer behavior G00(p) ~ (1/p2–η). (t/p1/v)β and a prediction for a soft replicon spectrum in tγk2γ/β γ/β instead of tk2 at du. We have checked tγ to one loop and work is in progress to check k2γ/β to the same order. Taking the above divergence in p−(d+2−η)/2 as the leading divergence defines the lower critical dimension d ℓ by d ℓ=2−η (d ℓ). Known values of η (at d=6, from the ε expansion at Tc, or from numerical work at d=4, 3) are compatible with d ℓ ~ 2.5±.3.

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Random field Ising model and Parisi-Sourlas supersymmetry. Part II. Renormalization group

Journal of High Energy Physics ◽

10.1007/jhep03(2021)219 ◽

2021 ◽

Vol 2021 (3) ◽

Author(s):

Apratim Kaviraj ◽

Slava Rychkov ◽

Emilio Trevisani

Keyword(s):

Fixed Point ◽

Renormalization Group ◽

Ising Model ◽

Random Field ◽

Dimensional Reduction ◽

Critical Dimension ◽

Random Field Ising Model ◽

Upper Critical Dimension ◽

Anomalous Dimensions ◽

Weakly Coupled

Abstract We revisit perturbative RG analysis in the replicated Landau-Ginzburg description of the Random Field Ising Model near the upper critical dimension 6. Working in a field basis with manifest vicinity to a weakly-coupled Parisi-Sourlas supersymmetric fixed point (Cardy, 1985), we look for interactions which may destabilize the SUSY RG flow and lead to the loss of dimensional reduction. This problem is reduced to studying the anomalous dimensions of “leaders” — lowest dimension parts of Sn-invariant perturbations in the Cardy basis. Leader operators are classified as non-susy-writable, susy-writable or susy-null depending on their symmetry. Susy-writable leaders are additionally classified as belonging to superprimary multiplets transforming in particular OSp(d|2) representations. We enumerate all leaders up to 6d dimension ∆ = 12, and compute their perturbative anomalous dimensions (up to two loops). We thus identify two perturbations (with susy- null and non-susy-writable leaders) becoming relevant below a critical dimension dc ≈ 4.2 - 4.7. This supports the scenario that the SUSY fixed point exists for all 3 < d ⩽ 6, but becomes unstable for d < dc.

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Finite-size critical scaling in Ising spin glasses in the mean-field regime

Physical Review E ◽

10.1103/physreve.93.032123 ◽

2016 ◽

Vol 93 (3) ◽

Cited By ~ 7

Author(s):

T. Aspelmeier ◽

Helmut G. Katzgraber ◽

Derek Larson ◽

M. A. Moore ◽

Matthew Wittmann ◽

...

Keyword(s):

Spin Glasses ◽

Mean Field ◽

Finite Size ◽

Critical Scaling ◽

Ising Spin ◽

The Mean ◽

Mean Field Regime

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Lower critical dimension of Ising spin glasses: a numerical study

Journal of Physics C Solid State Physics ◽

10.1088/0022-3719/17/18/004 ◽

1984 ◽

Vol 17 (18) ◽

pp. L463-L468 ◽

Cited By ~ 315

Author(s):

A J Bray ◽

M A Moore

Keyword(s):

Spin Glasses ◽

Numerical Study ◽

Critical Dimension ◽

Ising Spin

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Hyperscaling Violation in Ising Spin Glasses

Entropy ◽

10.3390/e21100978 ◽

2019 ◽

Vol 21 (10) ◽

pp. 978

Author(s):

Ian A. Campbell ◽

Per H. Lundow

Keyword(s):

Critical Exponents ◽

Spin Glasses ◽

Critical Dimension ◽

Ising Models ◽

Upper Critical Dimension ◽

Random Interactions ◽

Ising Spin ◽

Ising Systems ◽

Binder Cumulant ◽

Hyperscaling Violation

In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models, which are in this class, have been intensively studied. Here, numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlation length are presented for dimensions 3, 4, 5, and 7. Hyperscaling is clearly violated in dimensions 3 and 4, as well as above the upper critical dimension D = 6 . Estimates are obtained for the “violation of hyperscaling exponent” values in the various models.

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Spin glasses: recent advances in mean-field theory

Canadian Journal of Physics ◽

10.1139/p87-199 ◽

1987 ◽

Vol 65 (10) ◽

pp. 1245-1250 ◽

Cited By ~ 2

Author(s):

B. W. Southern

Keyword(s):

Field Theory ◽

Spin Glasses ◽

Glass Phase ◽

Mean Field Theory ◽

Mean Field ◽

Three Dimensional ◽

Spin Glass Phase ◽

Ising Spin ◽

Recent Advances ◽

The Mean

A survey of recent advances in the mean-field theory of Ising spin glasses is presented. The physical picture of the spin-glass phase predicted by this theory is described, and its relationship to real three-dimensional systems is discussed.

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Power-law bounds for critical long-range percolation below the upper-critical dimension

Probability Theory and Related Fields ◽

10.1007/s00440-021-01043-7 ◽

2021 ◽

Author(s):

Tom Hutchcroft

Keyword(s):

Long Range ◽

Power Law ◽

Upper Bound ◽

Mean Field ◽

Maximum Size ◽

Finite Graph ◽

Critical Dimension ◽

Point Function ◽

Upper Critical Dimension ◽

Infinite Cluster

AbstractWe study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

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