Document Detail


Ground-state degeneracies of Ising spin glasses on diamond hierarchical lattices.
MedLine Citation:
PMID:  11970209     Owner:  NLM     Status:  PubMed-not-MEDLINE    
Abstract/OtherAbstract:
The total number of ground states for short-range Ising spin glasses, defined on diamond hierarchical lattices of fractal dimensions d=2, 3, 4, 5, and 2.58, is estimated by means of analytic calculations (three last hierarchy levels of the d=2 lattice) and numerical simulations (lower hierarchies for d=2 and all remaining cases). It is shown that in the case of continuous probability distributions for the couplings, the number of ground states is finite in the thermodynamic limit. However, for a bimodal probability distribution (+/-J with probabilities p and 1-p, respectively), the average number of ground states is maximum for a wide range of values of p around p=1 / 2 and depends on the total number of sites at hierarchy level n, Nn. In this case, for all lattices investigated, it is shown that the ground-state degeneracy behaves like exp[h(d)Nn], in the limit Nn large, where h(d) is a positive number which depends on the lattice fractal dimension. The probability of finding frustrated cells at a given hierarchy level n, Fn(p), is calculated analytically (three last hierarchy levels for d=2 and the last hierarchy of the d=3 lattice, with 0<or=p<or=1), as well as numerically (all other cases, with p=1 / 2). Except for d=2, in which case Fn(1 / 2) increases by decreasing the hierarchy level, all other dimensions investigated present an exponential decrease in Fn(1 / 2) for decreasing values of n. For d=2 our results refer to the paramagnetic phase, whereas for all other dimensions considered [which are greater than the lower critical dimension dl (dl approximately 2.5)], our results refer to the spin-glass phase at zero temperature; in the latter cases h(d) increases with the fractal dimension. For n>>1, only the last hierarchies contribute significantly to the ground-state degeneracy; such a dominant behavior becomes stronger for high fractal dimensions. The exponential increase of the number of ground states with the total number of sites is in agreement with the mean-field picture of spin glasses.
Authors:
E M Curado; F D Nobre; S Coutinho
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Publication Detail:
Type:  Journal Article    
Journal Detail:
Title:  Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics     Volume:  60     ISSN:  1063-651X     ISO Abbreviation:  Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics     Publication Date:  1999 Oct 
Date Detail:
Created Date:  2002-04-23     Completed Date:  2002-06-18     Revised Date:  2003-11-03    
Medline Journal Info:
Nlm Unique ID:  9887340     Medline TA:  Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics     Country:  United States    
Other Details:
Languages:  eng     Pagination:  3761-70     Citation Subset:  -    
Affiliation:
Centro Brasiliro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil.
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