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Almeida STO, Nobre FD. Critical properties of short-range Ising spin glasses on a Wheatstone-bridge hierarchical lattice. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022102. [PMID: 26382339 DOI: 10.1103/physreve.92.022102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2015] [Indexed: 06/05/2023]
Abstract
An Ising spin-glass model with nearest-neighbor interactions, following a symmetric probability distribution, is investigated on a hierarchical lattice of the Wheatstone-bridge family characterized by a fractal dimension D≈3.58. The interaction distribution considered is a stretched exponential, which has been shown recently to be very close to the fixed-point coupling distribution, and such a model has been considered lately as a good approach for Ising spin glasses on a cubic lattice. An exact recursion procedure is implemented for calculating site magnetizations, mi=〈Si〉T, as well as correlations between pairs of nearest-neighbor spins, 〈SiSj〉T (〈〉T denote thermal averages), for a given set of interaction couplings on this lattice. From these local magnetizations and correlations, one can compute important physical quantities, such as the Edwards-Anderson order parameter, the internal energy, and the specific heat. Considering extrapolations to the thermodynamic limit for the order parameter, such as a finite-size scaling approach, it is possible to obtain directly the critical temperature and critical exponents. The transition between the spin-glass and paramagnetic phases is analyzed, and the associated critical exponents β and ν are estimated as β=0.82(5) and ν=2.50(4), which are in good agreement with the most recent results from extensive numerical simulations on a cubic lattice. Since these critical exponents were obtained from a fixed-point distribution, they are universal, i.e., valid for any coupling distribution considered.
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Affiliation(s)
- Sebastião T O Almeida
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro RJ, Brazil
- Colégio Técnico da Universidade Federal Rural do Rio de Janeiro, Seropédica, 23891-000 Rio de Janeiro RJ, Brazil
| | - Fernando D Nobre
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro RJ, Brazil
- National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro RJ, Brazil
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Almeida STO, Nobre FD. Fixed-point distributions of short-range Ising spin glasses on hierarchical lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032138. [PMID: 25871085 DOI: 10.1103/physreve.91.032138] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/18/2014] [Indexed: 06/04/2023]
Abstract
Fixed-point distributions for the couplings of Ising spin glasses with nearest-neighbor interactions on hierarchical lattices are investigated numerically. Hierarchical lattices within the Migdal-Kadanoff family with fractal dimensions in the range 2.58≤D≤7, as well as a lattice of the Wheatstone-Bridge family with fractal dimension D≈3.58 are considered. Three initial distributions for the couplings are analyzed, namely, the Gaussian, bimodal, and uniform ones. In all cases, after a few iterations of the renormalization-group procedure, the associated probability distributions approached universal fixed shapes. For hierarchical lattices of the Migdal-Kadanoff family, the fixed-point distributions were well fitted either by stretched exponentials, or by q-Gaussian distributions; both fittings recover the expected Gaussian limit as D→∞. In the case of the Wheatstone-Bridge lattice, the best fit was found by means of a stretched-exponential distribution.
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Affiliation(s)
- Sebastião T O Almeida
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
- Colégio Técnico da Universidade Federal Rural do Rio de Janeiro, Seropédica, 23891-000 Rio de Janeiro, RJ, Brazil
| | - Fernando D Nobre
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
- National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
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Salmon OR, Nobre FD. Spin-glass attractor on tridimensional hierarchical lattices in the presence of an external magnetic field. Phys Rev E 2009; 79:051122. [PMID: 19518431 DOI: 10.1103/physreve.79.051122] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2008] [Revised: 02/11/2009] [Indexed: 11/07/2022]
Abstract
A nearest-neighbor-interaction Ising spin glass, in the presence of an external magnetic field, is studied on different hierarchical lattices that approach the cubic lattice. The magnetic field is considered as uniform or random (following either a bimodal or a Gaussian probability distribution). In all cases, a spin-glass attractor is found, in the plane magnetic field versus temperature, associated with a low-temperature phase. The physical consequences of this attractor are discussed, in view of the present scenario of the spin-glass problem.
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Affiliation(s)
- Octavio R Salmon
- Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil.
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Nobre FD, Curado EMF. Ground-state entropies of the Potts antiferromagnet on diamond hierarchical lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:036107. [PMID: 12366184 DOI: 10.1103/physreve.66.036107] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2002] [Indexed: 05/23/2023]
Abstract
The ground-state degeneracies of the q-state Potts antiferromagnet on general diamond hierarchical lattices are computed, for q> or =3, by means of two distinct methods. The first method, denominated the recursive approach, is based on exact recursion relations for the total number of ground states, leading to the exact ground-state entropy in the thermodynamic limit. The second method, called the factorization approach, consists in a simple approximation, where the total number of ground states is factorized as a product of the number of ground states at each hierarchy level. The factorization approach appears to be a poor approximation for small values of q, but its accuracy improves substantially as q increases, and it becomes exact in the limit q--> infinity. In spite of the fact that such a model presents no frustration, a residual entropy at zero temperature is found for all q> or =3. Similarly to what happens on Bravais lattices, the residual entropy approaches its maximum allowed value, ln q, as q increases.
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Affiliation(s)
- Fernando D Nobre
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil.
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Nobre FD. Phase diagram of the two-dimensional +/-J Ising spin glass. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:046108. [PMID: 11690091 DOI: 10.1103/physreve.64.046108] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2001] [Indexed: 05/23/2023]
Abstract
The +/-J Ising spin glass [probabilities p and (1-p) associated with ferromagnetic and antiferromagnetic couplings, respectively] is studied by applying a real-space renormalization-group technique on a hierarchical lattice that approaches the square lattice. Within such a procedure, there is no spin-glass phase and only two finite-temperature phases are found, namely, the paramagnetic and ferromagnetic ones. In spite of a reasonably small computational effort, an accurate paramagnetic-ferromagnetic boundary is presented: the estimate for the slope at p=1 is in very good agreement with the well-known exact result, whereas the coordinates of the Nishimori point are determined within a high precision. Below the Nishimori point, such a boundary is not strictly vertical-contrary to the usual belief-in such a way that a small reentrance is found at low temperatures.
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Affiliation(s)
- F D Nobre
- Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Campus Universitário, Caixa Postal 1641 59072-970 Natal, RN, Brazil.
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Curado EM, Nobre FD. Lower bounds for the ground-state degeneracies of frustrated systems on fractal lattices. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:8814-6. [PMID: 11138190 DOI: 10.1103/physreve.62.8814] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2000] [Indexed: 11/07/2022]
Abstract
The total number of ground states for nearest-neighbor-interaction Ising systems with frustrations, defined on hierarchical lattices, is investigated. A simple method is presented, which allows one to factorize the ground-state degeneracy, at a given hierarchy level n, in terms of contributions due to all hierarchy levels. Such a method may yield the exact ground-state degeneracy of uniformly frustrated systems, whereas it works as an approximation for randomly frustrated models. In the latter cases, it is demonstrated that such an approximation yields lower-bound estimates for the ground-state degeneracies.
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Affiliation(s)
- EM Curado
- Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
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Nobre FD, Curado EM. Calculation of ground-state entropies of highly frustrated systems on fractal lattices. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:4597-605. [PMID: 11088998 DOI: 10.1103/physreve.62.4597] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2000] [Indexed: 11/07/2022]
Abstract
The extensive ground-state entropy of frustrated systems on fractal lattices is investigated. Two methods of calculation are proposed, namely, recursive and factorization approaches. In the recursive approach the calculation is based on exact recursion relations for the total number of ground states. The latter procedure, which is in principle an approximation, is proposed as an alternative for dealing with complicated systems (for cases where the recursive approach may become impracticable), such as randomly frustrated models; it consists of factorizing the total number of ground states in terms of the number of ground states at each hierarchy level. Some examples of antiferromagnetic Ising models on different fractal lattices are considered, for which both procedures are applied. It is shown that the factorization approach may lead, in some cases, to the exact ground-state entropy, whereas in other cases it yields very accurate (although slightly lower) estimates.
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Affiliation(s)
- FD Nobre
- Departamento de Fisica Teorica e Experimental, Universidade Federal do Rio Grande do Norte, Campus Universitario, Caixa Postal 1641, 59072-970 Natal, Rio Grande do Norte, Brazil
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