Low energy excitations in a long prism geometry: computing the lower critical dimension of the Ising spin glass

TL;DR

This work introduces a rectangular prism (long-prism) geometry to study low-energy excitations in disordered spin systems and uses the scaling of the transverse correlation length $\xi(L)$ to extract the lower critical dimension $D_{lc}$ and the multifractal spectrum of correlations in a three-dimensional Ising spin glass at zero field. Through large-scale Monte Carlo simulations enhanced by Houdayer cluster moves and open boundary conditions along the long axis, the authors find $a_3=1.34(3)$ and $D_{lc}=2.49(3)$, in strong agreement with Mean Field Theory predictions ($a_3=4/3$, $D_{lc}=5/2$) and slightly above Droplet-model expectations. The multifractal analysis reveals a nontrivial spectrum, indicating that simple one-dimensional toy models require refinement to capture the prism’s full behavior and suggesting a route to discriminate between competing spin-glass theories. The approach provides a practical, scalable framework to study soft excitations in spin glasses and could be extended to other disordered systems and operator classes.

Abstract

We propose a general method for studying systems that display excitations with arbitrarily low energy in their low-temperature phase. We argue that in a rectangular right prism geometry, with longitudinal size much larger than the transverse size, correlations decay exponentially (at all temperatures) along the longitudinal dimension, but the scaling of the correlation length with the transverse size carries crucial information from which the lower critical dimension can be inferred. The method is applied in the particularly demanding context of Ising spin glasses at zero magnetic field. The lower critical dimension and the multifractal spectrum for the correlation function are computed from large-scale numerical simulations. Several technical novelties (such as the unexpectedly crucial performance of Houdayer's cluster method or the convenience of using open - rather than periodic - boundary conditions) allow us to study three-dimensional prisms with transverse dimensions up to $L=24$ and effectively infinite longitudinal dimensions down to low temperatures. The value that we find for the lower critical dimension turns out to be in agreement with expectations from both the Replica Symmetry Breaking theory and the Droplet model for spin glasses. We argue that our novel setting holds promise in clarifying which of the two competing theories more accurately describes three-dimensional spin glasses.

Figures (6)

• Figure 1: The eight largest clusters for a typical configuration on a lattice $8\times8\times512$ with periodic boundary conditions at $T=0.7$ are depicted at $y=4$. There is no percolation through the lattice along the Z direction. The Z scale has been reduced by a factor of 10, in order to improve visibility.
• Figure 2: A comparison of the correlation functions $C^{(1)}(r)$ obtained on a lattice $16\times16\times512$ with PBC and on a lattice $16\times16\times48$ with OBC shows full compatibility within the errors. Inset:$\xi_{n=1}$ as a function of the minimal distance considered $r_\text{min}$ in the analysis supplemental comes out compatible with both boundary conditions.
• Figure 3: The growth of the longitudinal correlation length $\xi_{n=1}$ as a function of the transverse size $L$ for $T=0.7\simeq 0.635 T_\text{c}$ and $T=1.1019 \simeq T_\text{c}$. The solid lines are fits to $\xi_{n=1}(L)\!=\!B L^{4/3}+c$ for $T\!=\!0.7$ [$B\!=\!1.046(5)$, $c\!=\!2.01(8)$, $\chi^2/\text{DoF}\!=\!1.34/3$], and $\xi_{n=1}(L)\!=\!AL$ for $T\!=\!1.1019$ [$A\!=\!0.977(2)$, $\chi^2/\text{DoF}\!=\!3.82/4$]. In both fits, the smallest system size $L\!=\!4$ is excluded.
• Figure 4: Ratio of correlation functions $C^{(n)}/[C^{(1)}]^n$ versus $C^{(1)}$, see Eq. \ref{['eq:eq-C3D']}, as computed for $L=16$ at $T=0.7$ and $T=1.1019$. The slope in logarithmic scale is $\tau_n-n$ where $\tau_n$ is the so-called multifractal spectrum ($r$ increases from right to left: we are interested in the limit $C^{(1)}\to 0$, hence $r\to\infty$). The negative slopes indicate that $\tau_n<n$ for $n>1$, hence $C^{(n)}\gg [C^{(1)}]^n$ at long distances (multifractal scaling).
• Figure 5: Scaling of the logarithm of the (normalized) correlation function $C^{(1)}(r)$, see Eq. \ref{['EMeq:K-def']}. Data for systems of size $L=4,6,8,12,16$, and $24$ at temperature $T=0.7$. The black line is the best fit to $K(r)=(r/\xi_{n=1})+d$ for the $L=16$ data in the range $0.05 < r/\xi_{n=1} < 1.1$. Inset: zoom in near the origin. The dotted line is the best fit to $K(r)=A(r/\xi_{n=1})^a$ for the $L=16$ data in the range $r/\xi_{n=1} < 0.15$ (with exponent $a\approx 1.23$) and interpolates well data for all $L$ values.