Localization and Critical Dynamics in a Vector Spin Glass Chain

Pimentel, I. R.; Stinchcombe, R B

doi:10.1209/0295-5075/6/8/009

Cited by 9 publications

(23 citation statements)

References 9 publications

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“…and then employ numerical scaling based on a method developed by Pimentel and Stinchcombe [20] to treat the equation of motion of a 1D Mattis-transformed Edwards-Anderson Heisenberg spin glass. We write Eq.…”

Section: A Scaling and Localization In The Steady Statementioning

confidence: 99%

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Disordered asymmetric simple exclusion process: Mean-field treatment

Harris

Stinchcombe

2004

Phys. Rev. E

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We provide two complementary approaches to the treatment of disorder in a fundamental nonequilibrium model, the asymmetric simple exclusion process. First, a mean-field steady-state mapping is generalized to the disordered case, where it provides a mapping of probability distributions and demonstrates how disorder results in a new flat regime in the steady-state current-density plot for periodic boundary conditions. This effect was earlier observed by Phys. Rev. E 58, 1911 (1998)] but we provide a treatment for more general distributions of disorder, including both numerical results and analytic expressions for the width 2 Delta(C) of the flat section. We then apply an argument based on moving shock fronts [Europhys. Lett. 48, 257 (1999)]] to show how this leads to an increase in the high-current region of the phase diagram for open boundary conditions. Second, we show how equivalent results can be obtained easily by taking the continuum limit of the problem and then using a disordered version of the well-known Cole-Hopf mapping to linearize the equation. Within this approach we show that adding disorder induces a localization transformation (verified by numerical scaling), and Delta(C) maps to an inverse localization length, helping to give a physical interpretation to the problem.

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Section: A Scaling and Localization In The Steady Statementioning

confidence: 99%

“…Equation (46) is of just the form considered in [20] and can be exactly scaled by b = 2 decimation to give…”

Section: A Scaling and Localization In The Steady Statementioning

confidence: 99%

Disordered asymmetric simple exclusion process: Mean-field treatment

Harris

Stinchcombe

2004

Phys. Rev. E

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“…For p =1/2 in one dimension, it was predicted analytically 15,16 and verified by numerical calculations [20][21][22] that z =3/2. Still in one dimension, but at general p, Eq.…”

Section: Heisenberg-mattis Spin Glasses: Scaling In One Dimensionmentioning

confidence: 75%

“…This comes via the identification of the scaling properties of the low-energy excitations of the HSMG chain with those of the classical Heisenberg model in one dimension 15,16,20,21 plus the general applicability of the FP equation in the continuum. 1 It is expected that treatments along these lines can be devised, with similar degree of success, for other physical systems in whose description Lyapunov exponents play a prominent role.…”

Section: Discussionmentioning

confidence: 99%

“…As a model system to apply the ideas developed here, we consider spin waves ͑magnons͒ in a simplified spin glass model, the so-called Heisenberg-Mattis spin glass ͑HMSG͒. 10 In one dimension, the dynamics of HMSGs turns out to exhibit many nontrivial features, [11][12][13][14][15][16][17][18] including dynamic exponents, 15,16,18 different from the standard hydrodynamic predictions. 19 In one dimension, all eigenstates are localized for any amount of disorder, though as energy → 0, the localization length ͑͒ diverges continuously.…”

Section: Introductionmentioning

confidence: 99%

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Distribution of local Lyapunov exponents in spin-glass dynamics

Queiroz

Stinchcombe

2007

Phys. Rev. B

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We investigate the statistical properties of local Lyapunov exponents which characterize magnon localization in the $d=1$ Heisenberg-Mattis spin glass (HMSG) at zero temperature, by means of a connection to a suitable version of the Fokker-Planck (F-P) equation. We consider the local Lyapunov exponents (LLE), in particular the case of {\em instantaneous} LLE. We establish a connection between the transfer-matrix recursion relation for the problem, and an F-P equation governing the evolution of the probability distribution of the instantaneous LLE. The closed-form (stationary) solutions to the F-P equation are in excellent accord with numerical simulations, for both the unmagnetized and magnetized versions of the HMSG. Scaling properties for non-stationary conditions are derived from the F-P equation in a special limit (in which diffusive effects tend to vanish), and also shown to provide a close description to the corresponding numerical-simulation data.Comment: Final version, to be published in Phys. Rev.

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