monte-carlo-constrained-cpu

This solver uses the Constrained Monte Carlo algorithm [Phys. Rev. B 82, 054415 (2010)] to calculate equilibrium thermodynamics with classical (Rayleigh Jeans) statistics while constraining the order parameter to a given angle. The length of the order parameter is free to vary. Usually this solver is used in combination with the torque monitor to calculate free energy barriers.

The constraint angles are applied to the order parameter vector

\[\vec{N} = \sum_{i} \mu_{s,i} \left( \mathbb{T}_{i} \cdot \vec{S}_{i} \right)\]

where \(\mu_{s,i}\) and \(\mathbb{T}_{i}\) are the magnetic moment and the spin transform matrix of the \(i\)-th spins as defined for each material (see materials). The transformation matrix allows Constrained Monte Carlo to be used for example with ferrimagnets and antiferrimagnets by use of the Neel vector rather than the magnetisation.

This Monte Carlo solver moves two spins for every trial. We define One Monte Carlo step as one trial move of every spin on average. Therefore num_spins/2 trial moves of pairs of spins are made for each Monte Carlo step.

Trial spins are always chosen at random, both for the initial and second (compensation) spin in the pair. No consideration is given for whether the second spin is from the same material or unit cell position.

Note

The solver will periodically check the constraint is being correctly maintained. If it is not constrained correctly JAMS will quit with an error. This indicates there is either a problem with the input configuration or an unexpected bug in JAMS.

Required settings

max_steps

Maximum number of Monte Carlo steps to solve.

cmc_constraint_theta

Polar (from \(z\)-axis) constraint angle in degrees.

cmc_constraint_phi

Azimuthal (in \(xy\)-plane) constraint angle in degrees.

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).

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.