Cluster representations and the Wolff algorithm in arbitrary external fields

We introduce a natural way to extend celebrated spin-cluster Monte Carlo algorithms for fast thermal lattice simulations at criticality, such as the Wolff algorithm, to systems in arbitrary fields, be they linear magnetic vector fields or nonlinear anisotropic ones. By generalizing the “ghost spin” representation to one with a “ghost transformation,” global invariance to spin symmetry transformations is restored at the cost of an extra degree of freedom which lives in the space of symmetry transformations. The ordinary cluster-building process can then be run on the representation. We show that this extension preserves the scaling of accelerated dynamics in the absence of a field for Ising, Potts, and $\mathrm{O}\left(n\right)$ models and demonstrate the method's use in modeling the presence of novel nonlinear fields. We also provide a c++ library for the method's convenient implementation for arbitrary models.

Figure 1

The scaled autocorrelation time of the energy $\mathcal{H}$ for the Wolff algorithm on a $32\times 32\times 32$ XY model at its critical temperature as a function of applied vector field magnitude $\left|H\right|$. The red points correspond to reflections sampled uniformly, whereas the green points represent reflections sampled as described in Sec. 4b.

Figure 2

Scaling collapse of autocorrelation times $\tau$ for the energy $\mathcal{H}$ scaled by the average cluster size as a function of an external field for various models of Table 1. Critical exponents are model dependent. The colored lines and points depict values as measured by the extended algorithm. The solid black lines show a plot proportional to $h^{-z\nu /\beta \delta }$ for each model. The dynamic exponents $z$ are roughly measured as two-dimensional (2D) Ising: 0.23(5); 3D Ising: 0.28(5); 2D three-state Potts: 0.55(5); 2D four-state Potts: 0.94(5); 3D O(2): 0.17(5); 3D O(3): 0.13(5). $\mathrm{O}\left(n\right)$ models use the distribution of transformations described in Sec. 4b. The curves stop collapsing at high fields when the correlation length falls to near the lattice spacing; here noncluster algorithms can be efficiently used.

Figure 3

Collapses of rescaled average Wolff cluster size ${\langle s\rangle }_{\text{1c}}{L}^{-\gamma /\nu }$ as a function of field scaling variable $h{L}^{\beta \delta /\nu }$ for a variety of models. Critical exponents $\gamma ,\nu ,\beta$, and $\delta$ are model dependant. The colored lines and points depict values as measured by the extended algorithm. The solid black lines show a plot of $g(0,x)\propto x^{2/\delta }$ for each model.

Figure 4

Susceptibilities as a function of system size for a 2D O(2) model at $T=0.7$ and with (top) fourfold symmetric and (bottom) sixfold symmetric perturbing fields. The different field strengths are shown in different colors.