scholarly journals Hyperscaling Violation in Ising Spin Glasses

Entropy ◽  
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.

scholarly journals Non-perturbative effects in spin glasses

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
Michele Castellana ◽  
Giorgio Parisi
Keyword(s):  
Fixed Point ◽  
Ising Model ◽  
Spin Glasses ◽  
Numerical Study ◽  
Mean Field ◽  
Critical Dimension ◽  
Upper Critical Dimension ◽  
Ising Spin ◽  
Ferromagnetic Ising Model ◽  
The Mean

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.

Chaotic spin glasses: An upper critical dimension (invited)

1984 ◽  
Vol 55 (6) ◽  
pp. 1646-1648 ◽  
Author(s):  
Susan R. McKay ◽  
A. Nihat Berker
Keyword(s):  
Spin Glasses ◽  
Critical Dimension ◽  
Upper Critical Dimension

scholarly journals Lower critical dimension of Ising spin glasses

2001 ◽  
Vol 64 (18) ◽  
Author(s):  
A. K. Hartmann ◽  
A. P. Young
Keyword(s):  
Spin Glasses ◽  
Critical Dimension ◽  
Ising Spin

scholarly journals Scaling above the upper critical dimension in Ising models

1996 ◽  
Vol 54 (6) ◽  
pp. R3698-R3701 ◽  
Author(s):  
Giorgio Parisi ◽  
Juan J. Ruiz-Lorenzo
Keyword(s):  
Critical Dimension ◽  
Ising Models ◽  
Upper Critical Dimension

scholarly journals Finite-Size Effects in the φ4 Field and Lattice Theory Above the Upper Critical Dimension

1998 ◽  
Vol 09 (07) ◽  
pp. 1007-1019 ◽  
Author(s):  
X. S. Chen ◽  
V. Dohm
Keyword(s):  
Size Effects ◽  
Lattice Theory ◽  
Finite Size ◽  
Periodic Boundary Conditions ◽  
Critical Dimension ◽  
Spin Systems ◽  
Finite Size Effects ◽  
Upper Critical Dimension ◽  
Universal Properties ◽  
Binder Cumulant

We demonstrate that the standard O(n) symmetric φ4 field theory does not correctly describe the leading finite-size effects near the critical point of spin systems with periodic boundary conditions on a d-dimensional lattice with d>4. We show that these finite-size effects require a description in terms of a lattice Hamiltonian. For n →∞ and n=1, explicit results are given for the susceptibility and for the Binder cumulant. They imply that these quantities do not have the universal properties predicted previously and that recent analyses of Monte Carlo results for the five-dimensional Ising model are not conclusive.

Dynamic critical exponents in Ising spin glasses

1994 ◽  
Vol 49 (1) ◽  
pp. 728-731 ◽  
Author(s):  
L. Bernardi ◽  
I. A. Campbell
Keyword(s):  
Critical Exponents ◽  
Spin Glasses ◽  
Ising Spin
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