Monte Carlo Simulations of Spins
Introduction
The spinmc package is one of the applications of the ALPS project. It provides a a generic implementation of local and cluster updates for classical spin systems. The applications supports the following models on arbitrary lattices:
- Ising models
- XY models
- Heisenberg models
- 3-, 4- and 10-state Potts models
The application can easily be extended to additional q states Potts models and $O(N)$ models by editing the file mc/spins/spinmc_factory.C in a straightforward manner.
Running a simulation
is discussed in the tutorial.
Input parameters
In addition to the general input parameters of the ALPS scheduler library the spinmc application takes the following input parameters:
| Name | Default | Description |
|---|---|---|
| LATTICE_LIBRARY | lattices.xml | path to a file containing lattice descriptions |
| LATTICE | name of the lattice | |
| MODEL | either Ising, XY, Heisenberg or Potts | |
| q | the number of different states in a Potts model | |
| UPDATE | the update type, either local or cluster | |
| ERROR_VARIABLE | the name of an observable whose error you would like ALPS to monitor (must be used with ERROR_LIMIT) | |
| ERROR_LIMIT | once ERROR_VARIABLE’s absolute error is less than this amount, ALPS will stop the task (must be used with ERROR_VARIABLE) | |
| T | the temperature | |
| J | the default coupling constant | |
| J# | J | the coupling constant on a bond with type # (#=0,1,…). |
| D | onsite single-ion anisotropy coupling constants (one for each spin component in a list, e.g. D=“0.0 0.0 10.0”) | |
| CONVENTION | classical | specifies whether the classical or quantum conventions are used (see below) |
| S | 1 if CONVENTION=classical 1/2 if CONVENTION=quantum | the default spin size |
| S# | S | the spin size on a site with type # (#=0,1,…). |
| $g$ | 1 | the Landee $g$-factor, used for suscpetibility measurements |
| h | 0 | external magnetic field (only with local update) |
In addition, the lattice description can require further parameters (e.g. L or W) as specified in the lattice description file. Note: while the classical Monte carlo program uses XML lattice description it does not use XML model descriptions. The model is instead specifiec by the parameters in the table above.
Local versus cluster updates
Cluster updates should be used as long as there is no magnetic field applied and the spin system is not frustrated. Otherwise local updates are preferred.
Quantum versus classical conventions
Quantum and classical spin models often use different conventions for the coupling constants, and the CONVENTION parameter allows to choose between the two.
- classical convention is to have positive signs denote ferromagnetic coupling. The coupling strengths are multiplied by $S^2$ if a parameter S is specified.
- quantum convention is to have positive signs denote anti-ferromagnetic coupling. The coupling strengths are multiplied by S(S+1) where S defaults to 1/2.
Measurements
The following observables are measured by the spinmc application:
| Name | Description |
|---|---|
| Energy | the total energy of the system |
| Energy Density | the energy density (energy per site) of the system |
| Specific Heat | the specific heat per site of the system |
| Magnetization | the z-component of the magnetization |
| |Magnetization| | absolute value of the z-component of the magnetization |
| Magnetization^2 | square of the z-component of the magnetization |
| Magnetization along Field | the component of the magnetization along the external magnetic field |
| Staggered Magnetization | the z-component of the staggered magnetization (only on bipartite lattices) |
| Staggered Magnetization^2 | square of the z-component of the staggered magnetization (only on bipartite lattices) |
| Susceptibility | the uniform susceptibility, includes a factor of $g^2$ |
| Cluster size | the mean cluster size as a fraction of the lattice volume (only for cluster updates) |
Note: To evaluate the specific heat the evaluation program spinmc_evaluate has to be run on the task files (*task*.xml).
Contributors
The following persons have contributed to the classical spins application:
- Mathias Koerner
- Matthias Troyer
- Synge Todo
- Fabian Stoeckli