Cluster Algorithms¶

Metropolis flips one spin at a time. Near the critical temperature, this gets slow — the system needs exponentially many flips to decorrelate. Cluster algorithms fix this by flipping entire groups of spins at once.

The three algorithms¶

MetropolisWolffSwendsen-Wang

from mcising import Simulation, SimulationConfig, LatticeConfig, Algorithm

config = SimulationConfig(
    lattice=LatticeConfig(size=32),
    algorithm=Algorithm.METROPOLIS,
    temperatures=(2.269,),
    n_sweeps=1000,
)

results = Simulation(config).run()

Single-spin-flip with lookup tables. Best for general use, especially with J2, J3, or h couplings. Supports all lattice types and coupling combinations.

config = SimulationConfig(
    lattice=LatticeConfig(size=32),
    algorithm=Algorithm.WOLFF,
    temperatures=(2.269,),
    n_sweeps=1000,
)

results = Simulation(config).run()

Builds a single cluster from a random seed via DFS, then flips it. Dramatically reduces autocorrelation at Tc. One "sweep" = one cluster flip.

config = SimulationConfig(
    lattice=LatticeConfig(size=32),
    algorithm=Algorithm.SWENDSEN_WANG,
    temperatures=(2.269,),
    n_sweeps=1000,
)

results = Simulation(config).run()

Partitions the entire lattice into clusters via bond percolation (Union-Find), then independently flips each cluster with 50% probability. One "sweep" = one full partition + flip.

When to use which¶

Scenario	Best algorithm
General purpose, any coupling	Metropolis
Near Tc, J1-only	Wolff (fastest decorrelation)
Near Tc, many temperatures	Swendsen-Wang (good for parallel tempering)
J2 or J3 or h active	Metropolis (cluster algorithms require J2=J3=h=0)

Cluster algorithm constraints

Wolff and Swendsen-Wang require j2=0, j3=0, and h=0. This is a fundamental limitation of the bond-percolation approach — it only works for pure nearest-neighbor ferromagnets. If you need J2 or external field, use Metropolis.

Performance comparison¶

On a 32x32 square lattice at Tc=2.269, 10,000 sweeps:

Algorithm	Updates/sec
Metropolis	268M
Wolff	100M
Swendsen-Wang	48M

Metropolis has the highest raw throughput, but Wolff and Swendsen-Wang produce statistically independent samples much faster near Tc because each sweep decorrelates more effectively.

Cluster algorithms on any lattice¶

All three algorithms work on all 5 lattice types:

from mcising import LatticeType

# Wolff on triangular lattice
config = SimulationConfig(
    lattice=LatticeConfig(
        lattice_type=LatticeType.TRIANGULAR,
        size=32,
    ),
    algorithm=Algorithm.WOLFF,
    temperatures=(3.641,),
    n_sweeps=1000,
)