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

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

Figure 1

(Color online) Plots of $P(\varphi ,R)-P_2\left(\varphi \right)$, where $P(\varphi ,R)$ is the probability for the domain wall to pass to the left of a point with polar coordinates $(R,\varphi )$ (see inset). The magnitude of statistical errors (not shown) is consistent with the apparent fluctuations of the data lines. The predicted $P_2\left(\varphi \right)=[\varphi -\frac{1}{2}\sin \left(2\varphi \right)]∕\pi$ for $\kappa =2$ is subtracted to display small variations clearly. The data from FPA paths agree with SLE predictions for $\kappa$ in the range $2.24<\kappa <2.40$, while LPA paths give $\kappa \approx 2.85\left(10\right)$. Inset: A domain wall of length $S=9$ in a sample with $L=4$ rows and $W=6$ columns.

Figure 2

(Color online) Plot of an effective diffusion constant ${\kappa }_{\mathrm{eff}}=\overline{{\xi }^2\left(2t\right)-{\xi }^2\left(t\right)}∕t$, for $W⩾4L$. Lines indicate $\kappa =2.24$, 2.32, 2.40, 2.85, and 3.00. The range $2.24<\kappa <2.40$ fits the data for curves with FAP BCs, while $2.85<\kappa <3.00$ describes the diffusion measured from a constrained domain wall end. Inset: Part of a sample conversion of a domain wall in the 2D ISG to a sequence $\xi \left(t_i\right)$, $i=1\dots S$. The left curve is the initial domain wall with $\xi \left(0\right)=0$, while the red (lighter) curve is the remainder after 500 applications of the dipolar map, giving $\xi (t_{500}\approx 7239.4)\approx 101.5$.

Figure 3

(Color online) Plot of $C\left(x\right)$, cumulative probability of ranked values for $r({\gamma }_1,{\gamma }_2)$, as defined in the text, with $W=L=8$. Large deviations from $r=1$, as clearly seen for RLERW, indicate a failure of the domain Markov property.