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Title: Conformal Invariance and Stochastic Loewner Evolution Processes in Two-Dimensional Ising Spin Glasses

Abstract

We present numerical evidence that the techniques of conformal field theory might be applicable to two-dimensional Ising spin glasses with Gaussian bond distributions. It is shown that certain domain wall distributions in one geometry can be related to that in a second geometry by a conformal transformation. We also present direct evidence that the domain walls are stochastic Loewner (SLE) processes with {kappa}{approx_equal}2.1. An argument is given that their fractal dimension d{sub f} is related to their interface energy exponent {theta} by d{sub f}-1=3/[4(3+{theta})], which is consistent with the commonly quoted values d{sub f}{approx_equal}1.27 and {theta}{approx_equal}-0.28.

Authors:
;  [1];  [2];  [3]
  1. School of Physics and Astronomy, University of Manchester, Manchester M13 9PL (United Kingdom)
  2. Institut fuer Theoretische Physik, Universitaet Goettingen, Friedrich-Hund-Platz 1, 37077, Goettingen (Germany)
  3. Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
Publication Date:
OSTI Identifier:
20861547
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review Letters; Journal Volume: 97; Journal Issue: 26; Other Information: DOI: 10.1103/PhysRevLett.97.267202; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 36 MATERIALS SCIENCE; CONFORMAL INVARIANCE; FRACTALS; GEOMETRY; MATHEMATICAL EVOLUTION; QUANTUM FIELD THEORY; SPIN GLASS STATE; STOCHASTIC PROCESSES; TRANSFORMATIONS; TWO-DIMENSIONAL CALCULATIONS

Citation Formats

Amoruso, C., Moore, M. A., Hartmann, A. K., and Hastings, M. B.. Conformal Invariance and Stochastic Loewner Evolution Processes in Two-Dimensional Ising Spin Glasses. United States: N. p., 2006. Web. doi:10.1103/PHYSREVLETT.97.267202.
Amoruso, C., Moore, M. A., Hartmann, A. K., & Hastings, M. B.. Conformal Invariance and Stochastic Loewner Evolution Processes in Two-Dimensional Ising Spin Glasses. United States. doi:10.1103/PHYSREVLETT.97.267202.
Amoruso, C., Moore, M. A., Hartmann, A. K., and Hastings, M. B.. Sun . "Conformal Invariance and Stochastic Loewner Evolution Processes in Two-Dimensional Ising Spin Glasses". United States. doi:10.1103/PHYSREVLETT.97.267202.
@article{osti_20861547,
title = {Conformal Invariance and Stochastic Loewner Evolution Processes in Two-Dimensional Ising Spin Glasses},
author = {Amoruso, C. and Moore, M. A. and Hartmann, A. K. and Hastings, M. B.},
abstractNote = {We present numerical evidence that the techniques of conformal field theory might be applicable to two-dimensional Ising spin glasses with Gaussian bond distributions. It is shown that certain domain wall distributions in one geometry can be related to that in a second geometry by a conformal transformation. We also present direct evidence that the domain walls are stochastic Loewner (SLE) processes with {kappa}{approx_equal}2.1. An argument is given that their fractal dimension d{sub f} is related to their interface energy exponent {theta} by d{sub f}-1=3/[4(3+{theta})], which is consistent with the commonly quoted values d{sub f}{approx_equal}1.27 and {theta}{approx_equal}-0.28.},
doi = {10.1103/PHYSREVLETT.97.267202},
journal = {Physical Review Letters},
number = 26,
volume = 97,
place = {United States},
year = {Sun Dec 31 00:00:00 EST 2006},
month = {Sun Dec 31 00:00:00 EST 2006}
}
  • Domain walls for spin glasses are believed to be scale invariant; a stronger symmetry, conformal invariance, has the potential to hold. The statistics of zero-temperature Ising spin glass domain walls in two dimensions are used to test the hypothesis that these domain walls are described by a Schramm-Loewner evolution SLE{sub {kappa}}. Multiple tests are consistent with SLE{sub {kappa}}, where {kappa}=2.32{+-}0.08. Both conformal invariance and the domain Markov property are tested. The latter does not hold in small systems, but detailed numerical evidence suggests that it holds in the continuum limit.
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  • The Schramm-Loewner evolution (SLE) is a powerful tool to describe fractal interfaces in 2D critical statistical systems, yet the application of SLE is well established for statistical systems described by quantum field theories satisfying only conformal invariance, the so-called minimal conformal field theories (CFTs). We consider interfaces in Z(N) spin models at their self-dual critical point for N=4 and N=5. These lattice models are described in the continuum limit by nonminimal CFTs where the role of a Z{sub N} symmetry, in addition to the conformal one, should be taken into account. We provide numerical results on the fractal dimension ofmore » the interfaces which are SLE candidates for nonminimal CFTs. Our results are in excellent agreement with some recent theoretical predictions.« less