Efficient Cluster Algorithm for Spin Glasses in Any Space Dimension

Zheng Zhu, Andrew J. Ochoa, and Helmut G. Katzgraber

Phys. Rev. Lett. 115, 077201 – Published 14 August 2015

Abstract

Spin systems with frustration and disorder are notoriously difficult to study, both analytically and numerically. While the simulation of ferromagnetic statistical mechanical models benefits greatly from cluster algorithms, these accelerated dynamics methods remain elusive for generic spin-glass-like systems. Here, we present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by at least one order of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits of the isoenergetic cluster moves in two and three space dimensions, as well as the nonplanar chimera topology found in the D-Wave Inc. quantum annealing machine.

Received 22 January 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.077201

Authors & Affiliations

Zheng Zhu¹, Andrew J. Ochoa¹, and Helmut G. Katzgraber^1,2,3

¹Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
²Materials Science and Engineering Program, Texas A&M University, College Station, Texas 77843, USA
³Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

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Vol. 115, Iss. 7 — 14 August 2015

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Figure 1
(Top panel) Fraction of spins $p$ of potential cluster sites as a function of temperature $T$ for different system sizes $N$ in two space dimensions (2D). The horizontal line represents the percolation threshold of a two-dimensional square lattice, i.e., $p_{c} \approx 0.592$ [15]. Because $p \to 0.5$ for $T \to \infty$ , for all $T$ , clusters do not percolate, which is why the HCA is efficient in two-dimensional planar geometries. (Center panel) $p$ as a function of temperature $T$ for different system sizes $N$ on the chimera topology. The horizontal line represents the percolation threshold of the nonplanar chimera topology, namely, $p_{c} \approx 0.387$ , computed here using the approach developed in Ref. [16] (see the Supplemental Material [17]). For $T ≳ J = 1$ clusters percolate and cluster updates provide no gain. (Bottom panel) $p$ as a function of temperature $T$ for different system sizes $N$ in three space dimensions (3D). The horizontal line represents the percolation threshold of the three-dimensional cubic lattice ( $p_{c} \approx 0.311$ [18]). For $T ≳ J = 1$ clusters percolate. In all panels, error bars are computed via a jackknife analysis over configurations and are smaller than the symbols.
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Figure 2
(Top panel) $Δ$ [Eq. (2)] as a function of simulation time $t = 2^{b}$ measured in Monte Carlo sweeps in two space dimensions (2D) for $N = 1024$ and $T = 0.212$ . Simulations using vanilla PT thermalize at least $2^{25}$ Monte Carlo sweeps, whereas with the addition of ICMs thermalization is reduced to approximately $2^{16}$ Monte Carlo sweeps. This means approximately 2 orders of magnitude improvement. (Center panel) $Δ$ as a function of simulation time $t = 2^{b}$ measured in Monte Carlo sweeps for an Ising spin glass on chimera with $N = 1152$ spins at $T = 0.212$ . Simulations using PT thermalize at approximately $2^{25}$ Monte Carlo sweeps, whereas the addition of ICMs reduces thermalization to $2^{18}$ Monte Carlo sweeps. Again, approximately 2 orders of magnitude speedup. (Bottom panel) $Δ$ as a function of simulation time $t = 2^{b}$ measured in Monte Carlo sweeps in three space dimensions (3D) for $N = 1728$ and $T = 0.42 \sim 0.43 T_{c}$ . Using standard PT, the system thermalizes approximately after $2^{23}$ Monte Carlo sweeps. This time is reduced to $\sim 2^{20}$ Monte Carlo sweeps when ICMs are added. In all panels, error bars are computed via a jackknife analysis over configurations.
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Figure 3
Ratio between the approximate average thermalization time of PT and $PT + ICM$ for different topologies at the lowest simulation temperature (see Table 1) as a function of system size $N$ . In all cases the speedup increases with increasing system size. Note that thermalization times have been determined by eye.
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