Introduction

The Monte Carlo loop

The mc_generic class is an implementation of the Monte Carlo loop. Its goal is to propose and then accept or reject changes to a configuration according to this loop:

As shown in the figure, after a first initialization, the loop starts by proposing an update. In the following, we generically refer to this proposal as a proposed move. The move proposes a modification of the state of the system, which we call the configuration of the system. After having computed the transition probabilities between this proposed configuration and the old one, as well as their probability density, we compute an acceptance probability for the move. Based on this probability, the move is either accepted or rejected. If it is rejected, nothing happens and we remain in the same configuration. If it is accepted, the configuration is updated.

This procedure is the heart of the Monte Carlo algorithm and is repeated at every Monte Carlo step (meaning one loop). Measurements are not made at every step, to allow for some decorrelation between measured configurations. Thus, measurements are made every \(L\) steps. We say that these \(L\) steps form a cycle and \(L\) is the length of a cycle.

At the very beginning of the simulation, one usually allows for \(W\) warmup (thermalization) cycles. This means that there will be no measurements during these first \(W\) cycles. After that, we define \(N\), the number of cycles that will be done until the end of the simulation.

At the end of the simulation, the code will have done:

• \(N\) measurements

• \(N + W\) cycles

• \((N + W) \times L\) steps

C++ variable names

In the C++ examples, these variables will be called:

• n_cycles \(= N\)

• length_cycle \(= L\)

• n_warmup_cycle \(= W\)

You will also have to use these names if you will construct an mc_generic instance from a dictonary (see full documentation/manual/triqs below).

Monte Carlo loop and connection with moves and measures

We will cover this in more details, but let us already mention here that the mc_generic class only implements the Monte Carlo loop. It doesn’t need (and in fact doesn’t) know anything about what the configuration is or what the moves and measurements really do. All it does, is to use external classes which take care of making the moves. It just expects back a Metropolis ratio so that it can decide wether the move should be accepted or rejected. Once this choice is made, it tells the external class which again does we is needed if the move is accepted or rejected. The same is true for measurements which are external classes called by the loop. This will become clearer with an example in the following section.

Note

Above, we described the Metropolis algorithm. A different accept/reject scheme could be used but the mechanism remains the same.