Ferromagnetic toroidal Ising (h = 0): Swendsen–Wang bond freezing with probability 1 − e^(−2βJ), build a same-spin cluster, flip it—fast mixing near T_c compared to single-spin Metropolis.
About this model
The Wolff (single-cluster) algorithm is a Swendsen–Wang–type cluster Monte Carlo move for the 2D ferromagnetic Ising model with periodic boundaries and h = 0. Parallel neighboring spins are frozen into the same cluster with probability p = 1 − e^(−2βJ); the whole cluster is then flipped, proposing a nonlocal update that bypasses critical slowing down of single-spin Metropolis dynamics. This page visualizes the lattice, highlights the last flipped cluster, and tracks |m|, energy per spin, and T/T_c (Onsager kT_c/J). It is a teaching sampler, not a finite-size scaling pipeline.
Who it's for: Students who already saw Metropolis Ising updates and want the standard cluster alternative emphasized near the Curie point.
Key terms
Wolff algorithm
Ising model
Cluster Monte Carlo
Swendsen–Wang
Critical slowing down
Onsager temperature
How it works
Single-cluster Wolff updates for the 2D ferromagnetic Ising model on a torus (h = 0): bonds between parallel spins freeze with probability 1 − e^(−2βJ), building a cluster that is then flipped—an O(1) rejection-free alternative to Metropolis near criticality.
Key equations
Bond-freezing probability p_f = 1 - exp(-2βJ) (here J = 1); grow a same-spin cluster and flip all its spins. At h = 0 this satisfies detailed balance for the Ising Boltzmann weight.
Frequently asked questions
Why is there no magnetic field h?
The elementary Wolff bond rule shown here is the standard zero-field ferromagnetic case. A non-zero field requires a modified cluster construction (ghost spins / generalized bonds); this page keeps the clean textbook presentation.
How does this relate to Swendsen–Wang?
Swendsen–Wang flips many clusters per sweep from the same bond configuration; Wolff grows one cluster from a random seed and flips it—often simpler to visualize while sharing the same bond probability.
Is one Wolff build equal to one “sweep”?
No. A sweep in Metropolis is O(N) spin trials; a Wolff step flips a random cluster whose mean size depends strongly on temperature. The sidebar counts Wolff builds and reports the last cluster size for intuition.