monte-carlo-metropolis-cpu

This solver uses the Metropolis Monte Carlo algorithm to calculate equilibrium thermodynamics with classical (Rayleigh Jeans) statistics.

One Monte Carlo step is defined as a trial move having been attempted for every spin on average. On average is because spins can be selected at random so there is no guarantee that every spin has a trial move, only that num_spins trial moves occur.

Required settings

max_steps

Maximum number of Monte Carlo steps to solve.

Optional settings

min_steps = 0

Minimum number of Monte Carlo steps to solve (in case a monitor can stop the solver due to a convergence criterion).

use_random_spin_order = true

Toggle whether spins are selected randomly for trial moves or if the spins are selected in-order. On some systems/compilers the in-order selection has been seen to be 4x faster. Both methods should converge to the correct thermodynamic solution given enough steps.

use_total_energy = false

Toggle whether energy difference are calculated per spin or using the total energy. Setting use_total_energy to true will be much slower. This setting exists primarily to test that the Hamiltonians give the same result for the total energy and one spin energy calculations.

output_write_steps = 1000

Number of Monte Carlo steps between outputting trial move statistics to the terminal.

Trial Moves

Different types of trial spin moves can be used. Selecting different types or combination of types can give a much faster convergence to equilibrium. The total move fraction should add to 1 (JAMS will normalise anyway).

The move to use for a given Monte Carlo step is chosen randomly but the same move is used for every trial move within one step.

Statistics about how many moves were accepted of each type are printed to the terminal every output_write_steps steps.

move_fraction_uniform = 0.0

Fraction between 0 and 1 of trial moves which move a spin to a uniform random angle on the sphere.

\[(S_x, S_y, S_z) \rightarrow (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta) \quad \mathrm{where}\quad \theta\sim[0,\pi],\phi\sim[0,2\pi)\]

move_fraction_angle = 1.0

Fraction between 0 and 1 of trial moves which move a spin by a limited angle. The size of the angle is controlled by move_angle_sigma.

move_angle_sigma = 0.5

The size of \(\sigma\) in move_fraction_angle.

\[(S_x, S_y, S_z) \rightarrow (S_x, S_y, S_z) + \sigma(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta) \quad \mathrm{where}\quad \theta\sim[0,\pi],\phi\sim[0,2\pi)\]

move_fraction_reflection = 0.0

Fraction between 0 and 1 of trial moves reflect a spin.

\[(S_x, S_y, S_z) \rightarrow (-S_x, -S_y, -S_z)\]

Warning

This trial move is non-ergodic for Heisenberg spins and must be used in combination with other types of trial move.

Preconditioners

Preconditioners do some initial work before the main Monte Carlo loop to try and get to a reasonable starting configuration (close to a global energy minimum). This is a short thermalisation (500 Monte Carlo steps) followed by some algorithm.

Currently where is only one preconditioner (SystematicPreconditioner) which does a quick search of angles for the total magnetisation to see which orientation gives the lowest energy.

preconditioner_theta = 5.0

preconditioner_phi = 5.0

The \(\Delta\theta,\Delta\phi\) to search for the minimum energy over.

A data file seedname_mc_pre.tsv will be written with the columns:

1. theta (deg)

2. phi (deg)

3. energy (J)