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Hartmann AK, Young AP. Metastate analysis of the ground states of two-dimensional Ising spin glasses. Phys Rev E 2023; 108:024142. [PMID: 37723747 DOI: 10.1103/physreve.108.024142] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2023] [Accepted: 06/16/2023] [Indexed: 09/20/2023]
Abstract
Using an efficient polynomial-time ground-state algorithm we investigate the Ising spin glass state at zero temperature in two dimensions. For large sizes, we show that the spin state in a central region is independent of the interactions far away, indicating a "single-state" picture, presumably the droplet model. Surprisingly, a single power law describes corrections to this result down to the smallest sizes studied.
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Affiliation(s)
- A K Hartmann
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany
| | - A P Young
- Department of Physics, University of California, Santa Cruz, California 95064, USA
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2
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Newman CM, Read N, Stein DL. Proof of Single-Replica Equivalence in Short-Range Spin Glasses. PHYSICAL REVIEW LETTERS 2023; 130:077102. [PMID: 36867816 DOI: 10.1103/physrevlett.130.077102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2022] [Accepted: 01/19/2023] [Indexed: 06/18/2023]
Abstract
We consider short-range Ising spin glasses in equilibrium at infinite system size, and prove that, for fixed bond realization and a given Gibbs state drawn from a suitable metastate, each translation and locally invariant function (for example, self-overlaps) of a single pure state in the decomposition of the Gibbs state takes the same value for all the pure states in that Gibbs state. We describe several significant applications to spin glasses.
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Affiliation(s)
- C M Newman
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
- NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
| | - N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
- Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA
| | - D L Stein
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
- NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
- Department of Physics, New York University, New York, New York 10012, USA
- NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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3
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Read N. Complexity as information in spin-glass Gibbs states and metastates: Upper bounds at nonzero temperature and long-range models. Phys Rev E 2022; 105:054134. [PMID: 35706314 DOI: 10.1103/physreve.105.054134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2022] [Accepted: 03/24/2022] [Indexed: 06/15/2023]
Abstract
In classical finite-range spin systems, especially those with disorder such as spin glasses, a low-temperature Gibbs state may be a mixture of a number of pure or ordered states; the complexity of the Gibbs state has been defined in the past roughly as the logarithm of this number, assuming the question is meaningful in a finite system. As nontrivial pure-state structure is lost in finite size, in a recent paper [Phys. Rev. E 101, 042114 (2020)2470-004510.1103/PhysRevE.101.042114] Höller and the author introduced a definition of the complexity of an infinite-size Gibbs state as the mutual information between the pure state and the spin configuration in a finite region, and applied this also within a metastate construction. (A metastate is a probability distribution on Gibbs states.) They found an upper bound on the complexity for models of Ising spins in which each spin interacts with only a finite number of others, in terms of the surface area of the region, for all T≥0. In the present paper, the complexity of a metastate is defined likewise in terms of the mutual information between the Gibbs state and the spin configuration. Upper bounds are found for each of these complexities for general finite-range (i.e., short- or long-range, in a sense we define) mixed p-spin interactions of discrete or continuous spins (such as m-vector models), but only for T>0. For short-range models, the bound reduces to the surface area. For long-range interactions, the definition of a Gibbs state has to be modified, and for these models we also prove that the states obtained within the metastate constructions are Gibbs states under the modified definition. All results are valid for a large class of disorder distributions.
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Affiliation(s)
- N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA and Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA
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4
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Newman CM, Stein DL. Ground-state stability and the nature of the spin glass phase. Phys Rev E 2022; 105:044132. [PMID: 35590620 DOI: 10.1103/physreve.105.044132] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2021] [Accepted: 02/23/2022] [Indexed: 06/15/2023]
Abstract
We propose an approach toward understanding the spin glass phase at zero and low temperature by studying the stability of a spin glass ground state against perturbations of a single coupling. After reviewing the concepts of flexibility, critical droplet, and related quantities for both finite- and infinite-volume ground states, we study some of their properties and review three models in which these quantities are partially or fully understood. We also review a recent result showing the connection between our approach and that of disorder chaos. We then view four proposed scenarios for the low-temperature spin glass phase-replica symmetry breaking, scaling-droplet, TNT, and chaotic pairs-through the lens of the predictions of each scenario for the lowest-energy large-lengthscale excitations above the ground state. Using a new concept called σ-criticality, which quantifies the sensitivity of ground states to single-bond coupling variations, we show that each of these four pictures can be identified with different critical droplet geometries and energies. We also investigate necessary and sufficient conditions for the existence of multiple incongruent ground states.
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Affiliation(s)
- C M Newman
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
| | - D L Stein
- Department of Physics and Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA; NYU-ECNU Institutes of Physics and Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China; and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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5
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Jensen S, Read N, Young AP. Nontrivial maturation metastate-average state in a one-dimensional long-range Ising spin glass: Above and below the upper critical range. Phys Rev E 2021; 104:034105. [PMID: 34654158 DOI: 10.1103/physreve.104.034105] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/25/2021] [Accepted: 08/13/2021] [Indexed: 11/07/2022]
Abstract
Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature T_{c} for a one-dimensional Ising spin-glass model with diluted long-range interactions. In this model, the probability P_{ij} that an edge {i,j} has nonvanishing interaction falls as a power law with chord distance, P_{ij}∝1/R_{ij}^{2σ}, and we study a range of values of σ with 1/2<σ<1. We consider a correlation function C_{4}(r,t). A dynamic correlation length that shows power-law growth with time ξ(t)∝t^{1/z} can be identified in the data and, for large time t, C_{4}(r,t) decays as a power law r^{-α_{d}} with distance r when r≪ξ(t). The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent α_{d} differentiates between a trivial MMAS (α_{d}=0), as expected in the droplet picture of spin glasses, and a nontrivial MMAS (α_{d}≠0), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero α_{d} even in the regime σ>2/3 which corresponds to short-range systems below six dimensions. For σ<2/3, the decay exponent α_{d} follows the RSB prediction for the decay exponent α_{s}=3-4σ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches α_{d}=1-σ in the regime 2/3<σ<1; however, it deviates from both lines in the vicinity of σ=2/3.
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Affiliation(s)
- S Jensen
- Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
| | - N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA.,Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA
| | - A P Young
- Physics Department, University of California, Santa Cruz, Santa Cruz, California 95064, USA
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6
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Moore MA. Droplet-scaling versus replica symmetry breaking debate in spin glasses revisited. Phys Rev E 2021; 103:062111. [PMID: 34271696 DOI: 10.1103/physreve.103.062111] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2021] [Accepted: 05/21/2021] [Indexed: 11/07/2022]
Abstract
Simulational studies of spin glasses since the early 2010s have focused on the so-called replicon exponent α as a means of determining whether the low-temperature phase of spin glasses is described by the replica symmetry breaking picture of Parisi or by the droplet-scaling picture. On the latter picture, it should be zero, but we shall argue that it will only be zero for systems of linear dimension L>L^{*}. The crossover length L^{*} may be of the order of hundreds of lattice spacings in three dimensions and approach infinity in six dimensions. We use the droplet-scaling picture to show that the apparent nonzero value of α when L<L^{*} should be 2θ, where θ is the domain wall energy scaling exponent. This formula is in reasonable agreement with the reported values of α.
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Affiliation(s)
- M A Moore
- Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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7
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Höller J, Read N. One-step replica-symmetry-breaking phase below the de Almeida-Thouless line in low-dimensional spin glasses. Phys Rev E 2020; 101:042114. [PMID: 32422847 DOI: 10.1103/physreve.101.042114] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2019] [Accepted: 03/16/2020] [Indexed: 06/11/2023]
Abstract
The de Almeida-Thouless (AT) line is the phase boundary in the temperature-magnetic field plane of an Ising spin glass at which a continuous (i.e., second-order) transition from a paramagnet to a replica-symmetry-breaking (RSB) phase occurs, according to mean-field theory. Here, using field-theoretic perturbative renormalization group methods on the Bray-Roberts reduced Landau-Ginzburg-type theory for a short-range Ising spin glass in space of dimension d, we show that at nonzero magnetic field the nature of the corresponding transition is modified as follows: (a) For d-6 small and positive, with increasing field on the AT line, first, the ordered phase just below the transition becomes the so-called one-step RSB, instead of the full RSB that occurs in mean-field theory; the transition on the AT line remains continuous with a diverging correlation length. Then at a higher field, a tricritical point separates the latter transition from a quasi-first-order one, that is one at which the correlation length does not diverge, and there is a jump in part of the order parameter, but no latent heat. The location of the tricritical point tends to zero as d→6^{+}. (b) For d≤6, we argue that the quasi-first-order transition could persist down to arbitrarily small nonzero fields, with a transition to full RSB still expected at lower temperature. Whenever the quasi-first-order transition occurs, it is at a higher temperature than the AT transition would be for the same field, preempting it as the temperature is lowered. These results may explain the reported absence of a diverging correlation length in the presence of a magnetic field in low-dimensional spin glasses in some simulations and in high-temperature series expansions. We also draw attention to the similarity of the "dynamically frozen" state, which occurs at temperatures just above the quasi-first-order transition, and the "metastate-average state" of the one-step RSB phase, and discuss the issue of the number of pure states in either.
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Affiliation(s)
- J Höller
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
| | - N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
- Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA
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8
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Read N. Triviality of the ground-state metastate in long-range Ising spin glasses in one dimension. Phys Rev E 2018; 97:012134. [PMID: 29448455 DOI: 10.1103/physreve.97.012134] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2017] [Indexed: 11/07/2022]
Abstract
We consider the one-dimensional model of a spin glass with independent Gaussian-distributed random interactions, which have mean zero and variance 1/|i-j|^{2σ}, between the spins at sites i and j for all i≠j. It is known that, for σ>1, there is no phase transition at any nonzero temperature in this model. We prove rigorously that, for σ>3/2, any translation-covariant Newman-Stein metastate for the ground states (i.e., the frequencies with which distinct ground states are observed in finite-size samples in the limit of infinite size, for given disorder) is trivial and unique. In other words, for given disorder and asymptotically at large sizes, the same ground state, or its global spin flip, is obtained (almost) always. The proof consists of two parts: One is a theorem (based on one by Newman and Stein for short-range two-dimensional models), valid for all σ>1, that establishes triviality under a convergence hypothesis on something similar to the energies of domain walls and the other (based on older results for the one-dimensional model) establishes that the hypothesis is true for σ>3/2. In addition, we derive heuristic scaling arguments and rigorous exponent inequalities which tend to support the validity of the hypothesis under broader conditions. The constructions of various metastates are extended to all values σ>1/2. Triviality of the metastate in bond-diluted power-law models for σ>1 is proved directly.
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Affiliation(s)
- N Read
- Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA
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9
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Wang W, Moore MA, Katzgraber HG. Fractal Dimension of Interfaces in Edwards-Anderson and Long-range Ising Spin Glasses: Determining the Applicability of Different Theoretical Descriptions. PHYSICAL REVIEW LETTERS 2017; 119:100602. [PMID: 28949153 DOI: 10.1103/physrevlett.119.100602] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2017] [Indexed: 06/07/2023]
Abstract
The fractal dimension of excitations in glassy systems gives information on the critical dimension at which the droplet picture of spin glasses changes to a description based on replica symmetry breaking where the interfaces are space filling. Here, the fractal dimension of domain-wall interfaces is studied using the strong-disorder renormalization group method pioneered by Monthus [Fractals 23, 1550042 (2015)FRACEG0218-348X10.1142/S0218348X15500425] both for the Edwards-Anderson spin-glass model in up to 8 space dimensions, as well as for the one-dimensional long-ranged Ising spin-glass with power-law interactions. Analyzing the fractal dimension of domain walls, we find that replica symmetry is broken in high-enough space dimensions. Because our results for high-dimensional hypercubic lattices are limited by their small size, we have also studied the behavior of the one-dimensional long-range Ising spin-glass with power-law interactions. For the regime where the power of the decay of the spin-spin interactions with their separation distance corresponds to 6 and higher effective space dimensions, we find again the broken replica symmetry result of space filling excitations. This is not the case for smaller effective space dimensions. These results show that the dimensionality of the spin glass determines which theoretical description is appropriate. Our results will also be of relevance to the Gardner transition of structural glasses.
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Affiliation(s)
- Wenlong Wang
- Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
| | - M A Moore
- School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
| | - Helmut G Katzgraber
- Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
- 1QB Information Technologies (1QBit), Vancouver, British Columbia, Canada V6B 4W4
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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10
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Billoire A, Fernandez LA, Maiorano A, Marinari E, Martin-Mayor V, Moreno-Gordo J, Parisi G, Ricci-Tersenghi F, Ruiz-Lorenzo JJ. Numerical Construction of the Aizenman-Wehr Metastate. PHYSICAL REVIEW LETTERS 2017; 119:037203. [PMID: 28777596 DOI: 10.1103/physrevlett.119.037203] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2017] [Indexed: 06/07/2023]
Abstract
Chaotic size dependence makes it extremely difficult to take the thermodynamic limit in disordered systems. Instead, the metastate, which is a distribution over thermodynamic states, might have a smooth limit. So far, studies of the metastate have been mostly mathematical. We present a numerical construction of the metastate for the d=3 Ising spin glass. We work in equilibrium, below the critical temperature. Leveraging recent rigorous results, our numerical analysis gives evidence for a dispersed metastate, supported on many thermodynamic states.
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Affiliation(s)
- A Billoire
- Institute de Physique Théorique, CEA Saclay and CNRS, 91191 Gif-sur-Yvette, France
| | - L A Fernandez
- Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
| | - A Maiorano
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
- Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Rome, Italy
| | - E Marinari
- Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Rome, Italy
- Nanotec, Consiglio Nazionale delle Ricerche, I-00185 Rome, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, I-00185 Rome, Italy
| | - V Martin-Mayor
- Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
| | - J Moreno-Gordo
- Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
| | - G Parisi
- Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Rome, Italy
- Nanotec, Consiglio Nazionale delle Ricerche, I-00185 Rome, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, I-00185 Rome, Italy
| | - F Ricci-Tersenghi
- Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Rome, Italy
- Nanotec, Consiglio Nazionale delle Ricerche, I-00185 Rome, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, I-00185 Rome, Italy
| | - J J Ruiz-Lorenzo
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
- Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, 06071 Badajoz, Spain
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11
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Wang W. Numerical simulations of Ising spin glasses with free boundary conditions: The role of droplet excitations and domain walls. Phys Rev E 2017; 95:032143. [PMID: 28415332 DOI: 10.1103/physreve.95.032143] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2016] [Indexed: 06/07/2023]
Abstract
The relative importance of the contributions of droplet excitations and domain walls on the ordering of short-range Edwards-Anderson spin glasses in three and four dimensions is studied. We compare the spin overlap distribution functions of periodic and free boundary conditions using population annealing Monte Carlo. For system sizes up to about 1000 spins, spin glasses show nontrivial spin overlap distributions. Periodic boundary conditions may trap diffusive domain walls which can contribute to small spin overlaps, and the other contribution is the existence of low-energy droplet excitations within the system. We use free boundary conditions to minimize domain-wall effects, and show that low-energy droplet excitations are the major contribution to small overlaps in numerical simulations. Free boundary conditions has stronger finite-size effects, and is likely to have the same thermodynamic limit with periodic boundary conditions.
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Affiliation(s)
- Wenlong Wang
- Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
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12
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Arguin LP, Newman CM, Stein DL. Thermodynamic Identities and Symmetry Breaking in Short-Range Spin Glasses. PHYSICAL REVIEW LETTERS 2015; 115:187202. [PMID: 26565493 DOI: 10.1103/physrevlett.115.187202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2015] [Indexed: 06/05/2023]
Abstract
We present a technique to generate relations connecting pure state weights, overlaps, and correlation functions in short-range spin glasses. These are obtained directly from the unperturbed Hamiltonian and hold for general coupling distributions. All are satisfied in phases with simple thermodynamic structure, such as the droplet-scaling and chaotic pairs pictures. If instead nontrivial mixed-state pictures hold, the relations suggest that replica symmetry is broken as described by a Derrida-Ruelle cascade, with pure state weights distributed as a Poisson-Dirichlet process.
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Affiliation(s)
- L-P Arguin
- Department of Mathematics, City University of New York, Baruch College and Graduate Center, New York, New York 10010, USA
| | - C M Newman
- Courant Institute of Mathematical Sciences, New York, New York 10012, USA and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
| | - D L Stein
- Department of Physics and Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA and NYU-ECNU Institutes of Physics and Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
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