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Casilli N, Kaisar T, Colombo L, Ghosh S, Feng PXL, Cassella C. Parametric Frequency Divider Based Ising Machines. PHYSICAL REVIEW LETTERS 2024; 132:147301. [PMID: 38640363 DOI: 10.1103/physrevlett.132.147301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2023] [Accepted: 02/20/2024] [Indexed: 04/21/2024]
Abstract
We report on a new class of Ising machines (IMs) that rely on coupled parametric frequency dividers (PFDs) as macroscopic artificial spins. Unlike the IM counterparts based on subharmonic-injection locking (SHIL), PFD IMs do not require strong injected continuous-wave signals or applied dc voltages. Therefore, they show a significantly lower power consumption per spin compared to SHIL-based IMs, making it feasible to accurately solve large-scale combinatorial optimization problems that are hard or even impossible to solve by using the current von Neumann computing architectures. Furthermore, using high quality factor resonators in the PFD design makes PFD IMs able to exhibit a nanowatt-level power per spin. Also, it remarkably allows a speedup of the phase synchronization among the PFDs, resulting in shorter time to solution and lower energy to solution despite the resonators' longer relaxation time. As a proof of concept, a 4-node PFD IM has been demonstrated. This IM correctly solves a set of Max-Cut problems while consuming just 600 nanowatts per spin. This power consumption is 2 orders of magnitude lower than the power per spin of state-of-the-art SHIL-based IMs operating at the same frequency.
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Affiliation(s)
- Nicolas Casilli
- Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 02115, USA
| | - Tahmid Kaisar
- Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida 32611, USA
| | - Luca Colombo
- Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 02115, USA
| | - Siddhartha Ghosh
- Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 02115, USA
| | - Philip X-L Feng
- Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida 32611, USA
| | - Cristian Cassella
- Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 02115, USA
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Jiang K, Qiao J, Lan Y. Chaotic renormalization flow in the Potts model induced by long-range competition. Phys Rev E 2021; 103:062117. [PMID: 34271755 DOI: 10.1103/physreve.103.062117] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2021] [Accepted: 05/19/2021] [Indexed: 11/07/2022]
Abstract
A proper description of spin glass remains a hard subject in theoretical physics and is considered to be closely related to the emergence of chaos in the renormalization group (RG) flow. Previous efforts concentrate on models with either complicated or nonrealistic interactions in order to achieve this chaotic behavior. Here we find that the commonly used Potts model with long-range interaction could do the job nicely in a large parameter regime as long as the competition between the ferromagnetic and antiferromagnetic interaction is maintained. With this simplicity, the appearance of chaos is observed to sensitively depend on the detailed network structure: the parity of bond number in a branch of the basic RG substituting unit; chaos only emerges for even numbers of bonds. These surprising and universal findings may shed light on the study of spin glass.
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Affiliation(s)
- Kejin Jiang
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Jianyong Qiao
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Yueheng Lan
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China.,State Key Lab of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China
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3
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Çağlar T, Berker AN. Phase transitions between different spin-glass phases and between different chaoses in quenched random chiral systems. Phys Rev E 2017; 96:032103. [PMID: 29346860 DOI: 10.1103/physreve.96.032103] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2017] [Indexed: 11/07/2022]
Abstract
The left-right chiral and ferromagnetic-antiferromagnetic double-spin-glass clock model, with the crucially even number of states q=4 and in three dimensions d=3, has been studied by renormalization-group theory. We find, for the first time to our knowledge, four spin-glass phases, including conventional, chiral, and quadrupolar spin-glass phases, and phase transitions between spin-glass phases. The chaoses, in the different spin-glass phases and in the phase transitions of the spin-glass phases with the other spin-glass phases, with the non-spin-glass ordered phases, and with the disordered phase, are determined and quantified by Lyapunov exponents. It is seen that the chiral spin-glass phase is the most chaotic spin-glass phase. The calculated phase diagram is also otherwise very rich, including regular and temperature-inverted devil's staircases and reentrances.
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Affiliation(s)
- Tolga Çağlar
- Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, Istanbul 34956, Turkey
| | - A Nihat Berker
- Faculty of Engineering and Natural Sciences, Kadir Has University, Cibali, Istanbul 34083, Turkey.,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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4
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Çağlar T, Berker AN. Devil's staircase continuum in the chiral clock spin glass with competing ferromagnetic-antiferromagnetic and left-right chiral interactions. Phys Rev E 2017; 95:042125. [PMID: 28505813 DOI: 10.1103/physreve.95.042125] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2016] [Indexed: 11/07/2022]
Abstract
The chiral clock spin-glass model with q=5 states, with both competing ferromagnetic-antiferromagnetic and left-right chiral frustrations, is studied in d=3 spatial dimensions by renormalization-group theory. The global phase diagram is calculated in temperature, antiferromagnetic bond concentration p, random chirality strength, and right-chirality concentration c. The system has a ferromagnetic phase, a multitude of different chiral phases, a chiral spin-glass phase, and a critical (algebraically) ordered phase. The ferromagnetic and chiral phases accumulate at the disordered phase boundary and form a spectrum of devil's staircases, where different ordered phases characteristically intercede at all scales of phase-diagram space. Shallow and deep reentrances of the disordered phase, bordered by fragments of regular and temperature-inverted devil's staircases, are seen. The extremely rich phase diagrams are presented as continuously and qualitatively changing videos.
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Affiliation(s)
- Tolga Çağlar
- Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, Istanbul 34956, Turkey
| | - A Nihat Berker
- Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, Istanbul 34956, Turkey.,Faculty of Engineering and Natural Sciences, Kadir Has University, Cibali, Istanbul 34083, Turkey.,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Çağlar T, Berker AN. Chiral Potts spin glass in d=2 and 3 dimensions. Phys Rev E 2016; 94:032121. [PMID: 27739736 DOI: 10.1103/physreve.94.032121] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2016] [Indexed: 11/07/2022]
Abstract
The chiral spin-glass Potts system with q=3 states is studied in d=2 and 3 spatial dimensions by renormalization-group theory and the global phase diagrams are calculated in temperature, chirality concentration p, and chirality-breaking concentration c, with determination of phase chaos and phase-boundary chaos. In d=3, the system has ferromagnetic, left-chiral, right-chiral, chiral spin-glass, and disordered phases. The phase boundaries to the ferromagnetic, left- and right-chiral phases show, differently, an unusual, fibrous patchwork (microreentrances) of all four (ferromagnetic, left-chiral, right-chiral, chiral spin-glass) ordered phases, especially in the multicritical region. The chaotic behavior of the interactions, under scale change, are determined in the chiral spin-glass phase and on the boundary between the chiral spin-glass and disordered phases, showing Lyapunov exponents in magnitudes reversed from the usual ferromagnetic-antiferromagnetic spin-glass systems. At low temperatures, the boundaries of the left- and right-chiral phases become thresholded in p and c. In d=2, the chiral spin-glass Potts system does not have a spin-glass phase, consistently with the lower-critical dimension of ferromagnetic-antiferromagnetic spin glasses. The left- and right-chirally ordered phases show reentrance in chirality concentration p.
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Affiliation(s)
- Tolga Çağlar
- Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, Istanbul 34956, Turkey
| | - A Nihat Berker
- Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, Istanbul 34956, Turkey.,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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6
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Unraveling Quantum Annealers using Classical Hardness. Sci Rep 2015; 5:15324. [PMID: 26483257 PMCID: PMC4611884 DOI: 10.1038/srep15324] [Citation(s) in RCA: 54] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2015] [Accepted: 09/22/2015] [Indexed: 11/08/2022] Open
Abstract
Recent advances in quantum technology have led to the development and manufacturing of experimental programmable quantum annealing optimizers that contain hundreds of quantum bits. These optimizers, commonly referred to as 'D-Wave' chips, promise to solve practical optimization problems potentially faster than conventional 'classical' computers. Attempts to quantify the quantum nature of these chips have been met with both excitement and skepticism but have also brought up numerous fundamental questions pertaining to the distinguishability of experimental quantum annealers from their classical thermal counterparts. Inspired by recent results in spin-glass theory that recognize 'temperature chaos' as the underlying mechanism responsible for the computational intractability of hard optimization problems, we devise a general method to quantify the performance of quantum annealers on optimization problems suffering from varying degrees of temperature chaos: A superior performance of quantum annealers over classical algorithms on these may allude to the role that quantum effects play in providing speedup. We utilize our method to experimentally study the D-Wave Two chip on different temperature-chaotic problems and find, surprisingly, that its performance scales unfavorably as compared to several analogous classical algorithms. We detect, quantify and discuss several purely classical effects that possibly mask the quantum behavior of the chip.
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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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8
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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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9
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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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10
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Jörg T, Ricci-Tersenghi F. Entropic effects in the very low temperature regime of diluted Ising spin glasses with discrete couplings. PHYSICAL REVIEW LETTERS 2008; 100:177203. [PMID: 18518330 DOI: 10.1103/physrevlett.100.177203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2007] [Indexed: 05/26/2023]
Abstract
We study link-diluted +/-J Ising spin glass models on the hierarchical lattice and on a three-dimensional lattice close to the percolation threshold. We show that previously computed zero temperature fixed points are unstable with respect to temperature perturbations and do not belong to any critical line in the dilution-temperature plane. We discuss implications of the presence of such spurious unstable fixed points on the use of optimization algorithms, and we show how entropic effects should be taken into account to obtain the right physical behavior and critical points.
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Affiliation(s)
- Thomas Jörg
- LPTMS, Université de Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France
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11
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Monthus C, Garel T. Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:021132. [PMID: 18352012 DOI: 10.1103/physreve.77.021132] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2007] [Indexed: 05/26/2023]
Abstract
We study the wetting transition and the directed polymer delocalization transition on diamond hierarchical lattices. These two phase transitions with frozen disorder correspond to the critical points of quadratic renormalizations of the partition function. (These exact renormalizations on diamond lattices can also be considered as approximate Migdal-Kadanoff renormalizations for hypercubic lattices.) In terms of the rescaled partition function z=Z/Z(typ) , we find that the critical point corresponds to a fixed point distribution with a power-law tail P(c)(z) ~ Phi(ln z)/z(1+mu) as z-->+infinity [up to some subleading logarithmic correction Phi(ln z)], so that all moments z(n) with n>mu diverge. For the wetting transition, the first moment diverges z=+infinity (case 0<mu<1 ), and the critical temperature is strictly below the annealed temperature T(c)<T(ann). For the directed polymer case, the second moment diverges z(2)=+infinity (case 1<mu<2 ), and the critical temperature is strictly below the exactly known transition temperature T2 of the second moment. We then consider the correlation length exponent nu : the linearized renormalization around the fixed point distribution coincides with the transfer matrix describing a directed polymer on the Cayley tree, but the random weights determined by the fixed point distribution P(c)(z) are broadly distributed. This induces some changes in the traveling wave solutions with respect to the usual case of more narrow distributions.
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Affiliation(s)
- Cécile Monthus
- Service de Physique Théorique, CEA/DSM/SPhT, Unité de Recherche Associée au CNRS, Gif-sur-Yvette Cedex, France
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12
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Katzgraber HG, Krzakała F. Temperature and disorder chaos in three-dimensional Ising spin glasses. PHYSICAL REVIEW LETTERS 2007; 98:017201. [PMID: 17358500 DOI: 10.1103/physrevlett.98.017201] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2006] [Indexed: 05/14/2023]
Abstract
We study the effects of small temperature as well as disorder perturbations on the equilibrium state of three-dimensional Ising spin glasses via an alternate scaling ansatz. By using Monte Carlo simulations, we show that temperature and disorder perturbations yield chaotic changes in the equilibrium state and that temperature chaos is considerably harder to observe than disorder chaos.
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Sasaki M, Hukushima K, Yoshino H, Takayama H. Temperature chaos and bond chaos in Edwards-Anderson Ising spin glasses: domain-wall free-energy measurements. PHYSICAL REVIEW LETTERS 2005; 95:267203. [PMID: 16486393 DOI: 10.1103/physrevlett.95.267203] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/05/2004] [Indexed: 05/06/2023]
Abstract
Domain-wall free energy sigmaF, entropy sigmaS, and the correlation function C(temp) of sigmaF are measured independently in the four-dimensional +/-J Edwards-Anderson (EA) Ising spin glass. The stiffness exponent theta, the fractal dimension of domain walls d(s), and the chaos exponent zeta are extracted from the finite-size scaling analysis of sigmaF, sigmaS, and C(temp), respectively, well inside the spin-glass phase. The three exponents are confirmed to satisfy the scaling relation zeta = d(s)/2 - theta derived by the droplet theory within our numerical accuracy. We also study bond chaos induced by random variation of bonds, and find that the bond and temperature perturbations yield the universal chaos effects described by a common scaling function and the chaos exponent. These results strongly support the appropriateness of the droplet theory for the description of chaos effect in the EA Ising spin glasses.
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Affiliation(s)
- M Sasaki
- Institute for Solid State Physics, University of Tokyo, Kashiwa, Japan
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Sasaki M, Martin OC. Temperature chaos, rejuvenation, and memory in Migdal-Kadanoff spin glasses. PHYSICAL REVIEW LETTERS 2003; 91:097201. [PMID: 14525205 DOI: 10.1103/physrevlett.91.097201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2003] [Indexed: 05/24/2023]
Abstract
We use simulations within the Migdal-Kadanoff approach to probe the scales relevant for rejuvenation and memory in Ising spin glasses. First we investigate scaling laws for domain wall free energies and extract the chaos overlap length l(T,T'). Then we perform out of equilibrium simulations that follow experimental protocols. We find that (1) a rejuvenation signal arises at a length scale significantly smaller than l(T,T'), and (2) memory survives even if equilibration goes out to length scales larger than l(T,T'). Theoretical justifications of these phenomena are then considered.
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Affiliation(s)
- M Sasaki
- Institute for Solid State Physics, University of Tokyo, Kashiwa-no-ha 5-1-5, Kashiwa 277-8581, Japan
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Aspelmeier T, Bray AJ, Moore MA. Why temperature chaos in spin glasses is hard to observe. PHYSICAL REVIEW LETTERS 2002; 89:197202. [PMID: 12443143 DOI: 10.1103/physrevlett.89.197202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2002] [Indexed: 05/24/2023]
Abstract
The overlap length of a three-dimensional Ising spin glass on a cubic lattice with Gaussian interactions has been estimated numerically by transfer matrix methods and within a Migdal-Kadanoff renormalization group scheme. We find that the overlap length is large, explaining why it has been difficult to observe spin glass chaos in numerical simulations and experiment.
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Affiliation(s)
- T Aspelmeier
- Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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Sales M, Yoshino H. Fragility of the free-energy landscape of a directed polymer in random media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:066131. [PMID: 12188807 DOI: 10.1103/physreve.65.066131] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2002] [Indexed: 05/23/2023]
Abstract
We examine the sensitiveness of the free-energy landscape of a directed polymer in random media with respect to various kinds of infinitesimally weak perturbation including the intriguing case of temperature chaos. To this end, we combine the replica Bethe Ansatz approach outlined by Sales and Yoshino (e-print cond-mat/0112384), the mapping to a modified Sinai model, and numerically exact calculations by the transfer-matrix method. Our results imply that for all the perturbations under study there is a slow crossover from a weakly perturbed regime, where rare events take place, to a strongly perturbed regime at larger length scales beyond the so-called overlap length, where typical events take place leading to chaos, i.e., a complete reshuffling of the free-energy landscape. Within the replica space, the evidence for chaos is found in the factorization of the replicated partition function induced by infinitesimal perturbations. This is the reflex of explicit replica-symmetry breaking.
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Affiliation(s)
- Marta Sales
- Departament de Física Fonamental, Facultat de Física, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain
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17
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Hukushima K. Domain-wall free energy of spin-glass models: numerical method and boundary conditions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:3606-13. [PMID: 11970194 DOI: 10.1103/physreve.60.3606] [Citation(s) in RCA: 23] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/1999] [Indexed: 04/18/2023]
Abstract
An efficient Monte Carlo method is extended to evaluate directly domain-wall free energy for randomly frustrated spin systems. Using the method, critical phenomena of spin-glass phase transition are investigated in the 4d+/-J Ising model under the replica boundary condition. Our values of the critical temperature and exponent, obtained by finite-size scaling, are in good agreement with those of the standard Monte Carlo and the series expansion studies. In addition, two exponents, the stiffness exponent and the fractal dimension of the domain wall, which characterize the ordered phase, are obtained. The latter value is larger than d-1, indicating that the domain wall is really rough in the 4d Ising spin-glass phase.
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Affiliation(s)
- K Hukushima
- Institute for Solid State Physics, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106-8666, Japan
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18
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Curado EM, Nobre FD, Coutinho S. Ground-state degeneracies of Ising spin glasses on diamond hierarchical lattices. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:3761-70. [PMID: 11970209 DOI: 10.1103/physreve.60.3761] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/21/1998] [Revised: 05/10/1999] [Indexed: 11/07/2022]
Abstract
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.
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Affiliation(s)
- E M Curado
- Centro Brasiliro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
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Azcoiti V, Follana E, Ritort F. Static chaos in spin glasses: the case of quenched disorder perturbations. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/28/14/008] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Mattsson J, Jonsson T, Nordblad P, Ito A. No phase transition in a magnetic field in the ising spin glass Fe0.5Mn0.5TiO3. PHYSICAL REVIEW LETTERS 1995; 74:4305-4308. [PMID: 10058467 DOI: 10.1103/physrevlett.74.4305] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Djurberg C, Granberg P, Mattsson J, Nordblad P. Overlap length in spin glasses imposed by magnetic field perturbations. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:15775-15778. [PMID: 9975943 DOI: 10.1103/physrevb.50.15775] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Ritort F. Static chaos and scaling behavior in the spin-glass phase. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:6844-6853. [PMID: 9974640 DOI: 10.1103/physrevb.50.6844] [Citation(s) in RCA: 26] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Li MS, Cieplak M. Renormalization-group study of the coupled XY-Ising models. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:955-964. [PMID: 9975762 DOI: 10.1103/physrevb.50.955] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Andersson J, Mattsson J, Svedlindh P. Monte Carlo studies of Ising spin-glass systems: Investigations of the overlap length in spin glasses. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 49:1120-1126. [PMID: 10010417 DOI: 10.1103/physrevb.49.1120] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Schins AG, Dons EM, Arts AF, Vincent E, Leylekian L, Hammann J. Aging in two-dimensional Ising spin glasses. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:16524-16532. [PMID: 10008235 DOI: 10.1103/physrevb.48.16524] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Andersson JO, Mattsson J, Nordblad P. Overlap length in a Cu-Mn spin glass probed by ac susceptibility. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:13977-13980. [PMID: 10007800 DOI: 10.1103/physrevb.48.13977] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Mattsson J, Djurberg C, Nordblad P, Hoines L, Stubi R, Cowen JA. Overlap length in two-dimensional Cu(Mn) spin-glass films. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:14626-14629. [PMID: 10005832 DOI: 10.1103/physrevb.47.14626] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Schins AG, Arts AF. Domain growth by aging in nonequilibrium two-dimensional random Ising systems. PHYSICAL REVIEW LETTERS 1993; 70:2340-2343. [PMID: 10053536 DOI: 10.1103/physrevlett.70.2340] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Cieplak M, Li MS, Banavar JR. Universality and chaos in XY spin glasses. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:5022-5026. [PMID: 10006663 DOI: 10.1103/physrevb.47.5022] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Svedlindh P, Gunnarsson K, Andersson JO, Ito A. Time-dependent ac susceptibility in spin glasses. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:13867-13873. [PMID: 10003450 DOI: 10.1103/physrevb.46.13867] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Andersson J, Mattsson J, Svedlindh P. Monte Carlo studies of Ising spin-glass systems: Aging behavior and crossover between equilibrium and nonequilibrium dynamics. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:8297-8304. [PMID: 10002591 DOI: 10.1103/physrevb.46.8297] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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