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arXiv:cond-mat/0611433 · Physical Review B

Possible description of domain walls in two-dimensional spin glasses by stochastic Loewner evolutions

Denis Bernard, Pierre Le Doussal, A. Alan Middleton

Published Jul 10, 2007• 66 citations
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Abstract

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 ${\mathrm{SLE}}_{\ensuremath{\kappa}}$. Multiple tests are consistent with ${\mathrm{SLE}}_{\ensuremath{\kappa}}$, where $\ensuremath{\kappa}=2.32\ifmmode\pm\else\textpm\fi{}0.08$. 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.

DOI
10.1103/physrevb.76.020403

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Comment on "Conformal invariance and stochastic Loewner evolution processes in two-dimensional Ising spin glasses"

Ronald Fisch

By combining the scaling relation of Amoruso {\it et al.}, PRL {\bf 97}, 267202 (2006) with standard droplet model assumptions, a value $θ= (\sqrt{6} - 3) / 2$ is obtained. This conjecture is reasonably consistent with the best existing numerical calculations, and may be exact.

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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\ensuremathκ{\mathrm{SLE}}_{\ensuremath{\kappa}}SLE\ensuremathκ​. Multiple tests are consistent with SLE\ensuremathκ{\mathrm{SLE}}_{\ensuremath{\kappa}}SLE\ensuremathκ​, where \ensuremathκ=2.32\ifmmode±\else\textpm\fi0.08\ensuremath{\kappa}=2.32\ifmmode\pm\else\textpm\fi{}0.08\ensuremathκ=2.32\ifmmode±\else\textpm\fi0.08. 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.
The concept of Schramm-Loewner evolution provides a unified description of domain boundaries of many lattice spin systems in two dimensions, possibly even including systems with quenched disorder. Here, we study domain walls in the random-field Ising model. Although, in two dimensions, this system does not show an ordering transition to a ferromagnetic state, in the presence of a uniform external field spin domains percolate beyond a critical field strength. Using exact ground-state calculations for very large systems, we examine ground-state domain walls near this percolation transition finding strong evidence that they are conformally invariant and satisfy the domain Markov property, implying compatibility with Schramm-Loewner evolution (SLE κ ) with parameter κ = 6. These results might pave the way for new field-theoretic treatments of systems with quenched disorder.
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 approximately 2.1. An argument is given that their fractal dimension d(f) is related to their interface energy exponent theta by d(f) - 1 = 3/[4(3 + theta)], which is consistent with the commonly quoted values d(f) approximately 1.27 and theta approximately -0.28.
By combining the scaling relation of Amoruso {\it et al.}, PRL {\bf 97}, 267202 (2006) with standard droplet model assumptions, a value θ=(6−3)/2θ= (\sqrt{6} - 3) / 2θ=(6​−3)/2 is obtained. This conjecture is reasonably consistent with the best existing numerical calculations, and may be exact.
The distribution of energy and free energy of domain walls near the spin-glass transition is studied numerically in the three-dimensional Ising spin-glass model with discrete $\ifmmode\pm\else\textpm\fi{}J$ interactions. Temperature dependence of the width of the free-energy distribution clearly shows a phase transition to a low-temperature phase with long-range spin correlations at $\frac{k{T}_{g}}{J}\ensuremath{\simeq}1.2$. Simple scaling analysis is consistent with previous studies of this system.
We study domain-wall excitations in two-dimensional random-bond Ising spin systems on a square lattice with side length LLL, subject to two different continuous disorder distributions. In both cases an adjustable parameter allows tuning of the disorder so as to yield a transition from a spin-glass ordered ground state to a ferromagnetic ground state. We formulate an auxiliary graph-theoretical problem in which domain walls are given by undirected shortest paths with possibly negative distances. Due to the details of the mapping, standard shortest-path algorithms cannot be applied. To solve such shortest-path problems we have to apply minimum-weight perfect-matching algorithms. We first locate the critical values of the disorder parameters, where the ferromagnet to spin-glass transition occurs for the two types of the disorder. For certain values of the disorder parameters close to the respective critical point, we investigate the system size dependence of the width of the average domain-wall energy $(\ensuremath{\sim}{L}^{\ensuremath{\theta}})$ and the average domain-wall length $(\ensuremath{\sim}{L}^{{d}_{f}})$. Performing a finite-size scaling analysis for systems with a side length up to L=512L=512L=512, we find that both exponents remain constant in the spin-glass phase, i.e., $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.28$ and ${d}_{f}\ensuremath{\approx}1.275$. This is consistent with conformal field theory, where it seems to be possible to relate the exponents from the analysis of stochastic Loewner evolutions via ${d}_{f}\ensuremath{-}1=3/[4(3+\ensuremath{\theta})]$. Finally, we characterize the transition in terms of ferromagnetic clusters of spins that form, as one proceeds from spin-glass ordered to ferromagnetic ground states.
We study the time evolution of quantum one-dimensional gapless systems evolving from initial states with a domain wall.We generalize the path integral imaginary time approach that together with boundary conformal field theory allows us to derive the time and space dependence of general correlation functions.The latter are explicitly obtained for the Ising universality class, and the typical behavior of one-and two-point functions is derived for the general case.Possible connections with the stochastic Loewner evolution are discussed and explicit results for one-point time dependent averages are obtained for generic κ for boundary conditions corresponding to stochastic Loewner evolution.We use this set of results to predict the time evolution of the entanglement entropy and obtain the universal constant shift due to the presence of a domain wall in the initial state.