Critical slowing down in Monte Carlo (MC) simulations

Question

Here is what I understood from critical slowing down.

When we are near a phase transition, the autocorrelation time $\tau$ is very long.

Imagine we are doing MC simulations on a ferromagnetic near $T_c$ where $T_c$ is the temperature of phase transition. We are studying the observable $M$ which is the total magnetisation. The fact that $\tau$ is long physically mean that if I am in a microstate with magnetisation far from the average magnetization, it will take a long time (many steps) for the simulation to access to microstates that have magnetisation near the average magnetisation.

And, as the microstates that have magnetisation near the average magnetisation have a more statistical importance in the calculation, it means that we will have a very bad statistics if we don't sample long enough.

My questions:

• Am I right in this explanation?

• Why is $\tau$ very long near a phase transition? I am not sure to get it.

• Why are cluster algorithm better to treat such problems? This question has been asked here Critical slowing down in Monte-Carlo algorithm for classical 2D Ising Model but not really answered.

Answer 1

1) Near a second order phase transition the correlation length is very large, and there are fluctuations on all scales. This means that a local algorithm will have difficulty in sampling the space of relevant configurations efficiently. The mean magnetization may actually be more or less correct, but more complicated observables (higher moments of M, correlation functions, etc) are difficult to compute.

2) Consider the Landau theory of phase transitions. The free energy functional is $$ F = \int dx \left[ \kappa(\nabla M)^2 + a M^2 + b M^4 + \ldots \right] $$ and the correlation length (in the disordered phase) is $\xi \sim 1/\sqrt{a}$. At the phase transition $a\to 0$ and $\xi\to\infty$. Fluctuations modify the simple mean field scaling, and $\xi\sim 1/t^\nu$ where $t$ is the reduced temperature and $\nu$ is a critical exponent.

3) In order to study the relaxation time we have to make the Landau theory time dependent. This leads to a hydrodynamic theory known as model A. The equation of motion is $$ \partial_t M = -D \frac{\delta F}{\delta M} $$ which has eigenmodes of the form $\omega_k=iD(k^2+a)$. Usually the relaxation time is finite, but near the critical point $a\to 0$ and $\omega_k\sim ik^2$, so that modes with wave number $k\sim 1/\xi$ relax over a time $\tau \sim \xi^2$. Again, this is mean field and a more sophisticated analysis gives $\tau \sim \xi^z$ with a critical exponent z.

4) Cluster algorithms perform updates on all scales, and capture the physics in 2),3) better.