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Mu S, Gong J, Lemarié G. Kardar-Parisi-Zhang Physics in the Density Fluctuations of Localized Two-Dimensional Wave Packets. PHYSICAL REVIEW LETTERS 2024; 132:046301. [PMID: 38335351 DOI: 10.1103/physrevlett.132.046301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2023] [Revised: 09/08/2023] [Accepted: 12/14/2023] [Indexed: 02/12/2024]
Abstract
We identify the key features of Kardar-Parisi-Zhang (KPZ) universality class in the fluctuations of the wave density logarithm in a two-dimensional Anderson localized wave packet. In our numerical analysis, the fluctuations are found to exhibit an algebraic scaling with distance characterized by an exponent of 1/3, and a Tracy-Widom probability distribution of the fluctuations. Additionally, within a directed polymer picture of KPZ physics, we identify the dominant contribution of a directed path to the wave packet density and find that its transverse fluctuations are characterized by a roughness exponent 2/3. Leveraging on this connection with KPZ physics, we verify that an Anderson localized wave packet in 2D exhibits a stretched exponential correction to its well-known exponential localization.
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Affiliation(s)
- Sen Mu
- Department of Physics, National University of Singapore, Singapore 117542, Singapore
| | - Jiangbin Gong
- Department of Physics, National University of Singapore, Singapore 117542, Singapore
- Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore
- MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Unit, Singapore
- Joint School of National University of Singapore and Tianjin University, International Campus of Tianjin University, Binhai New City, Fuzhou 350207, China
| | - Gabriel Lemarié
- Department of Physics, National University of Singapore, Singapore 117542, Singapore
- Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore
- MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Unit, Singapore
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France
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Wiese KJ. Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2022; 85:086502. [PMID: 35943081 DOI: 10.1088/1361-6633/ac4648] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2021] [Accepted: 12/23/2021] [Indexed: 06/15/2023]
Abstract
Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.
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Affiliation(s)
- Kay Jörg Wiese
- Laboratoire de physique, Département de physique de l'ENS, École normale supérieure, UPMC Univ. Paris 06, CNRS, PSL Research University, 75005 Paris, France
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3
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Lemarié G. Glassy Properties of Anderson Localization: Pinning, Avalanches, and Chaos. PHYSICAL REVIEW LETTERS 2019; 122:030401. [PMID: 30735426 DOI: 10.1103/physrevlett.122.030401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2018] [Indexed: 06/09/2023]
Abstract
I present the results of extensive numerical simulations, which reveal the glassy properties of Anderson localization in dimension two at zero temperature: pinning, avalanches, and chaos. I first show that strong localization confines quantum transport along paths that are pinned by disorder but can change abruptly and suddenly (avalanches) when the energy is varied. I determine the roughness exponent ζ characterizing the transverse fluctuations of these paths and find that its value ζ=2/3 is the same as for the directed polymer problem. Finally, I characterize the chaos property, namely, the fragility of the conductance with respect to small perturbations in the disorder configuration. It is linked to interference effects and universal conductance fluctuations at weak disorder and more spin-glass-like behavior at strong disorder.
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Affiliation(s)
- G Lemarié
- Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, 31062 Toulouse, France
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Aamir MA, Karnatak P, Jayaraman A, Sai TP, Ramakrishnan TV, Sensarma R, Ghosh A. Marginally Self-Averaging One-Dimensional Localization in Bilayer Graphene. PHYSICAL REVIEW LETTERS 2018; 121:136806. [PMID: 30312065 DOI: 10.1103/physrevlett.121.136806] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2017] [Revised: 07/07/2018] [Indexed: 06/08/2023]
Abstract
The combination of a field-tunable band gap, topological edge states, and valleys in the band structure makes insulating bilayer graphene a unique localized system, where the scaling laws of dimensionless conductance g remain largely unexplored. Here we show that the relative fluctuations in lng with the varying chemical potential, in strongly insulating bilayer graphene (BLG), decay nearly logarithmically for a channel length up to L/ξ≈20, where ξ is the localization length. This "marginal" self-averaging, and the corresponding dependence of ⟨lng⟩ on L, suggests that transport in strongly gapped BLG occurs along strictly one-dimensional channels, where ξ≈0.5±0.1 μm was found to be much longer than that expected from the bulk band gap. Our experiment reveals a nontrivial localization mechanism in gapped BLG, governed by transport along robust edge modes.
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Affiliation(s)
- Md Ali Aamir
- Department of Physics, Indian Institute of Science, Bangalore 560 012, India
| | - Paritosh Karnatak
- Department of Physics, Indian Institute of Science, Bangalore 560 012, India
| | - Aditya Jayaraman
- Department of Physics, Indian Institute of Science, Bangalore 560 012, India
| | - T Phanindra Sai
- Department of Physics, Indian Institute of Science, Bangalore 560 012, India
| | - T V Ramakrishnan
- Department of Physics, Indian Institute of Science, Bangalore 560 012, India
| | - Rajdeep Sensarma
- Department of Theoretical Physics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Mumbai 400005, India
| | - Arindam Ghosh
- Department of Physics, Indian Institute of Science, Bangalore 560 012, India
- Centre for Nano Science and Engineering, Indian Institute of Science, Bangalore 560 012, India
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Corwin I, Ghosal P, Krajenbrink A, Le Doussal P, Tsai LC. Coulomb-Gas Electrostatics Controls Large Fluctuations of the Kardar-Parisi-Zhang Equation. PHYSICAL REVIEW LETTERS 2018; 121:060201. [PMID: 30141677 DOI: 10.1103/physrevlett.121.060201] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2018] [Indexed: 06/08/2023]
Abstract
We establish a large deviation principle for the Kardar-Parisi-Zhang (KPZ) equation, providing precise control over the left tail of the height distribution for narrow wedge initial condition. Our analysis exploits an exact connection between the KPZ one-point distribution and the Airy point process-an infinite particle Coulomb gas that arises at the spectral edge in random matrix theory. We develop the large deviation principle for the Airy point process and use it to compute, in a straightforward and assumption-free manner, the KPZ large deviation rate function in terms of an electrostatic problem (whose solution we evaluate). This method also applies to the half-space KPZ equation, showing that its rate function is half of the full-space rate function. In addition to these long-time estimates, we provide rigorous proof of finite-time tail bounds on the KPZ distribution, which demonstrate a crossover between exponential decay with exponent 3 (in the shallow left tail) to exponent 5/2 (in the deep left tail). The full-space KPZ rate function agrees with the one computed in Sasorov et al. [J. Stat. Mech. (2017) 063203JSMTC61742-546810.1088/1742-5468/aa73f8] via a WKB approximation analysis of a nonlocal, nonlinear integrodifferential equation generalizing Painlevé II which Amir et al. [Commun. Pure Appl. Math. 64, 466 (2011)CPMAMV0010-364010.1002/cpa.20347] related to the KPZ one-point distribution.
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Affiliation(s)
- Ivan Corwin
- Columbia University, Department of Mathematics 2990 Broadway, New York, New York 10027, USA
| | - Promit Ghosal
- Columbia University, Department of Statistics 1255 Amsterdam, New York, New York 10027, USA
| | - Alexandre Krajenbrink
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
| | - Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
| | - Li-Cheng Tsai
- Columbia University, Department of Mathematics 2990 Broadway, New York, New York 10027, USA
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Cao X, Rosso A, Bouchaud JP, Le Doussal P. Genuine localization transition in a long-range hopping model. Phys Rev E 2017; 95:062118. [PMID: 28709355 DOI: 10.1103/physreve.95.062118] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2016] [Indexed: 11/07/2022]
Abstract
We introduce and study a banded random matrix model describing sparse, long-range quantum hopping in one dimension. Using a series of analytic arguments, numerical simulations, and a mapping to a long-range epidemics model, we establish the phase diagram of the model. A genuine localization transition, with well defined mobility edges, appears as the hopping rate decreases slower than ℓ^{-2}, where ℓ is the distance. Correspondingly, the decay of the localized states evolves from a standard exponential shape to a stretched exponential and finally to a exp(-Cln^{κ}ℓ) behavior, with κ>1.
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Affiliation(s)
- Xiangyu Cao
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Alberto Rosso
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | | | - Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'École Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
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García-Mata I, Giraud O, Georgeot B, Martin J, Dubertrand R, Lemarié G. Scaling Theory of the Anderson Transition in Random Graphs: Ergodicity and Universality. PHYSICAL REVIEW LETTERS 2017; 118:166801. [PMID: 28474917 DOI: 10.1103/physrevlett.118.166801] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2016] [Indexed: 06/07/2023]
Abstract
We study the Anderson transition on a generic model of random graphs with a tunable branching parameter 1<K<2, through large scale numerical simulations and finite-size scaling analysis. We find that a single transition separates a localized phase from an unusual delocalized phase that is ergodic at large scales but strongly nonergodic at smaller scales. In the critical regime, multifractal wave functions are located on a few branches of the graph. Different scaling laws apply on both sides of the transition: a scaling with the linear size of the system on the localized side, and an unusual volumic scaling on the delocalized side. The critical scalings and exponents are independent of the branching parameter, which strongly supports the universality of our results.
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Affiliation(s)
- I García-Mata
- Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), CONICET-UNMdP, Funes 3350, B7602AYL Mar del Plata, Argentina
- Consejo Nacional de Investigaciones Científicas y Tecnológicas (CONICET), Godoy Cruz 2290 (C1425FQB) CABA, Argentina
| | - O Giraud
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - B Georgeot
- Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, 31062 Toulouse, France
| | - J Martin
- Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, Université de Liège, Bât. B15, B-4000 Liège, Belgium
| | - R Dubertrand
- Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, Université de Liège, Bât. B15, B-4000 Liège, Belgium
| | - G Lemarié
- Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, 31062 Toulouse, France
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De Luca A, Le Doussal P. Mutually avoiding paths in random media and largest eigenvalues of random matrices. Phys Rev E 2017; 95:030103. [PMID: 28415280 DOI: 10.1103/physreve.95.030103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2016] [Indexed: 06/07/2023]
Abstract
Recently, it was shown that the probability distribution function (PDF) of the free energy of a single continuum directed polymer (DP) in a random potential, equivalently to the height of a growing interface described by the Kardar-Parisi-Zhang (KPZ) equation, converges at large scale to the Tracy-Widom distribution. The latter describes the fluctuations of the largest eigenvalue of a random matrix, drawn from the Gaussian unitary ensemble (GUE), and the result holds for a DP with fixed end points, i.e., for the KPZ equation with droplet initial conditions. A more general conjecture can be put forward, relating the free energies of N>1 noncrossing continuum DP in a random potential, to the sum of the Nth largest eigenvalues of the GUE. Here, using replica methods, we provide an important test of this conjecture by calculating exactly the right tails of both PDFs and showing that they coincide for arbitrary N.
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Affiliation(s)
- Andrea De Luca
- Laboratoire de Physique Théorique et Modèles Statistiques (UMR CNRS 8626), Université Paris-Sud, Orsay, France
| | - Pierre Le Doussal
- Laboratoire de Physique Théorique de l'ENS, CNRS & Ecole Normale Supérieure de Paris, Paris, France
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9
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De Luca A, Le Doussal P. Crossing probability for directed polymers in random media. II. Exact tail of the distribution. Phys Rev E 2016; 93:032118. [PMID: 27078303 DOI: 10.1103/physreve.93.032118] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2015] [Indexed: 06/05/2023]
Abstract
We study the probability p ≡ p(η)(t) that two directed polymers in a given random potential η and with fixed and nearby endpoints do not cross until time t. This probability is itself a random variable (over samples η), which, as we show, acquires a very broad probability distribution at large time. In particular, the moments of p are found to be dominated by atypical samples where p is of order unity. Building on a formula established by us in a previous work using nested Bethe ansatz and Macdonald process methods, we obtain analytically the leading large time behavior of all moments p(m) ≃ γ(m)/t. From this, we extract the exact tail ∼ρ(p)/t of the probability distribution of the noncrossing probability at large time. The exact formula is compared to numerical simulations, with excellent agreement.
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Affiliation(s)
- Andrea De Luca
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Pierre Le Doussal
- Laboratoire de Physique Théorique de l'ENS, CNRS and Ecole Normale Supérieure de Paris, Paris, France
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10
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De Luca A, Le Doussal P. Crossing probability for directed polymers in random media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:040102. [PMID: 26565146 DOI: 10.1103/physreve.92.040102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2015] [Indexed: 06/05/2023]
Abstract
We study the probability that two directed polymers in the same random potential do not intersect. We use the replica method to map the problem onto the attractive Lieb-Liniger model with generalized statistics between particles. Employing both the nested Bethe ansatz and known formula from MacDonald processes, we obtain analytical expressions for the first few moments of this probability and compare them to a numerical simulation of a discrete model at high temperature. From these observations, several large time properties of the noncrossing probabilities are conjectured. Extensions of our formalism to more general observables are discussed.
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Affiliation(s)
- Andrea De Luca
- Laboratoire de Physique Théorique et Modèles Statistiques (UMR CNRS 8626), Université Paris-Sud, Orsay, France
| | - Pierre Le Doussal
- Laboratoire de Physique Théorique de l'ENS, CNRS and Ecole Normale Supérieure de Paris, Paris, France
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Somoza AM, Ortuño M, Gasparian V, Pino M. Green functions of interacting systems in the strongly localized regime. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:335503. [PMID: 26235315 DOI: 10.1088/0953-8984/27/33/335503] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We have developed an approach to calculate the single-particle Green function of a one-dimensional many-body system in the strongly localized limit at zero temperature. Our approach sums the contributions of all possible forward scattering paths in configuration space. We demonstrate that for fermions and nearest neighbors interactions the Green function factorizes at every link connecting two sites with the same occupation. As a consequence, the conductance distribution function for interacting systems is log-normal, in the same universality class as non-interacting systems. We have developed a numerical procedure to calculate the ground state and the Green function, generating all possible paths in configuration space. Our results agree with results obtained by exact diagonalization of small systems in the limit of large disorder.
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Affiliation(s)
- A M Somoza
- Departamento de Física-CIOyN, Universidad de Murcia, Murcia 30.071, Spain
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12
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Multiple virtual tunneling of Dirac fermions in granular graphene. Sci Rep 2013; 3:3404. [PMID: 24296942 PMCID: PMC3847705 DOI: 10.1038/srep03404] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2013] [Accepted: 11/08/2013] [Indexed: 12/03/2022] Open
Abstract
Graphene charge carriers behave as massless Dirac fermions, opening the exciting possibility to observe long-range virtual tunneling of electrons in a solid. In granular metals, electron hops arising from series of virtual transitions are predicted to yield observable currents at low-enough temperatures, but to date experimental evidence is lacking. We report on electron transport in granular graphene films self-assembled by hydrogenation of suspended graphene. While the log-conductance shows a characteristic T−1/2 temperature dependence, cooling the samples below 10 K drives a triple crossover: a slope break in log-conductance, simultaneous to a substantial increase in magneto-conductance and onset of large mesoscopic conductance fluctuations. These phenomena are signatures of virtual transitions of electrons between distant localized states, and conductance statistics reveal that the high crossover-temperature is due to the Dirac nature of granular graphene charge carriers.
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Choe DH, Chang KJ. Effect of dimensionality on the localization behavior in hydrogenated graphene systems. NANO LETTERS 2012; 12:5175-5180. [PMID: 22963004 DOI: 10.1021/nl302207p] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Recently, several experiments have shown that graphene exhibits a metal-to-insulator transition by hydrogenation. Here we theoretically study the transport properties of hydrogenated graphene and graphene nanoribbons (GNRs), focusing on the conductance fluctuation behavior in the localized regime. Using a simple model for the conductance distribution in the quasi-localized regime where the conventional theory fails, we derive the modified single parameter scaling (SPS) relations for quasi-one-dimensional (Q1D) GNRs as well as two-dimensional (2D) graphene. We show that, as the dimensional crossover occurs from 2D to Q1D, the shape of the conductance distribution evolves from a positively skewed distribution to a log-normal distribution. We predict that GNRs with widths much larger than the localization lengths do not behave as a Q1D system. Our results provide fundamental insights into the dimensionality change not only in graphene, but also in general mesoscopic systems in the localized regime.
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Affiliation(s)
- Duk-Hyun Choe
- Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea
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14
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Fridman M, Pugatch R, Nixon M, Friesem AA, Davidson N. Measuring maximal eigenvalue distribution of Wishart random matrices with coupled lasers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:020101. [PMID: 22463135 DOI: 10.1103/physreve.85.020101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/16/2011] [Indexed: 05/31/2023]
Abstract
We determined the probability distribution of the combined output power from 25 coupled fiber lasers and show that it agrees well with the Tracy-Widom and Majumdar-Vergassola distributions of the largest eigenvalue of Wishart random matrices with no fitting parameters. This was achieved with 500,000 measurements of the combined output power from the fiber lasers, that continuously changes with variations of the fiber lasers lengths. We show experimentally that for small deviations of the combined output power over its mean value the Tracy-Widom distribution is correct, while for large deviations the Majumdar-Vergassola distribution is correct.
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Affiliation(s)
- Moti Fridman
- Weizmann Institute of Science, Department of Physics of Complex Systems, Rehovot 76100, Israel
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15
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Castellana M, Decelle A, Zarinelli E. Extreme value statistics distributions in spin glasses. PHYSICAL REVIEW LETTERS 2011; 107:275701. [PMID: 22243317 DOI: 10.1103/physrevlett.107.275701] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2011] [Indexed: 05/31/2023]
Abstract
We study the probability distribution of the pseudocritical temperature in a mean-field and in a short-range spin-glass model: the Sherrington-Kirkpatrick and the Edwards-Anderson (EA) model. In both cases, we put in evidence the underlying connection between the fluctuations of the pseudocritical point and the extreme value statistics of random variables. For the Sherrington-Kirkpatrick model, both with Gaussian and binary couplings, the distribution of the pseudocritical temperature is found to be the Tracy-Widom distribution. For the EA model, the distribution is found to be the Gumbel distribution. Being the EA model representative of uniaxial magnetic materials with quenched disorder like Fe(0.5)Mn)0.5)TiO(3) or Eu(0.5)Ba(0.5)MnO(3), its pseudocritical point distribution should be a priori experimentally accessible.
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Dobrinevski A, Le Doussal P, Wiese KJ. Interference in disordered systems: a particle in a complex random landscape. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061116. [PMID: 21797311 DOI: 10.1103/physreve.83.061116] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2011] [Indexed: 05/31/2023]
Abstract
We consider a particle in one dimension submitted to amplitude and phase disorder. It can be mapped onto the complex Burgers equation, and provides a toy model for problems with interplay of interferences and disorder, such as the Nguyen-Spivak-Shklovskii model of hopping conductivity in disordered insulators and the Chalker-Coddington model for the (spin) quantum Hall effect. We also propose a direct realization in an experiment with cold atoms. The model has three distinct phases: (I) a high-temperature or weak disorder phase, (II) a pinned phase for strong amplitude disorder, and (III) a diffusive phase for strong phase disorder, but weak amplitude disorder. We compute analytically the renormalized disorder correlator, equivalent to the Burgers velocity-velocity correlator at long times. In phase III, it assumes a universal form. For strong phase disorder, interference leads to a logarithmic singularity, related to zeros of the partition sum, or poles of the complex Burgers velocity field. These results are valuable in the search for the adequate field theory for higher-dimensional systems.
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Affiliation(s)
- Alexander Dobrinevski
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, Paris, France.
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