Cluster Monte Carlo algorithm with a conserved order parameter

We propose a cluster simulation algorithm for statistical ensembles with fixed order parameter. We use the tethered ensemble, which features Helmholtz’s effective potential rather than Gibbs’s free energy and in which canonical averages are recovered with arbitrary accuracy. For the $D=2,3$ Ising model our method’s critical slowing down is comparable to that of canonical cluster algorithms. Yet, we can do more than merely reproduce canonical values. As an example, we obtain a competitive value for the 3D Ising anomalous dimension from the maxima of the effective potential.

Figure 1

(Color online) Computation of the effective potential ${\Omega }_N\left(\widehat{m},{\beta }_{\text{c}}\right)$ [Eq. (5)] for an $L=128$ Ising 3D lattice. Top: we simulate each point independently to measure ${⟨\widehat{h}⟩}_{\widehat{m},{\beta }_{\text{c}}}$ [Eq. (8)] (the lines are cubic splines and the errors are much smaller than the points). Bottom: this curve is then integrated to yield $p\left(\widehat{m}\right)$, which we show in a linear (◻) and in a logarithmic (○) scale. Inset: ${⟨\widehat{h}⟩}_{\widehat{m},{\beta }_{\text{c}}}$ for $\widehat{m}\simeq m+1/2>1/2$.

Figure 2

(Color online) Spin overlap [Eq. (12)] in a 2D critical $L=512$ Ising model at the central minimum (◻) and right maximum (○) of the effective potential [Eq. (5)].