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Generic Client Code for Frustrated Spin Models

The frustmag client code is a generic Monte Carlo code for classical spin systems on frustrated lattices. It is highly modular and aims to allow for composability between different Hamiltonians (interactions) with different lattice geometries. This is enabled by a generic Bravais lattice class which allows for the iteration over either individual spins or lattice unit cells, and over nearest neighbors of each spin. Periodic and open boundary conditions are both supported. This way, both the interactions and most of the → update schemes can be implemented in a lattice-agnostic fashion. Type traits and template specialization are used to enable updates at compile time when applicable.

This code was used to study the classical Heisenberg model on the Kagome lattice using TK-SVM, the results of which are presented in chapter 9 of J. Greitemann's PhD thesis.

Building and Installation

CMake is used to build this code:

$ cd svm-order-params/frustmag
$ mkdir build && cd build
$ cmake ..
$ make -jN

Finally, using make install, the compiled executables can be copied to the bin directory at the location configured in CMAKE_INSTALL_PREFIX. This step is optional.

Refer to the top-level README for information on dependencies and cloning of submodules.

Build configuration

In addition to the build configuration options of the TK-SVM framework, this code allows for the selection of the model Hamiltonian and lattice geometry through the following build options:

Variable name Possible values
HAMILTONIAN heisenberg (default), ising
LATTICE chain, square, cubic, triangular, honeycomb, kagome (default), dice

To customize a variable, pass the appropriate flag on to CMake, e.g. the default configuration is explicitly stated as:

$ cmake -DHAMILTONIAN=heisenberg -DLATTICE=kagome ..

or use the interactive ccmake configurator.

Runtime parameters

This sections lists the runtime parameters which are defined by — and exclusive to — this client code. These parameters supplement those lists in the section → Runtime parameters of the top-level README.

Simulation runtime

Parameter name Default Description
total_sweeps 0 Number of MC steps per phase diagram point sampled
thermalization_sweeps 10000 Thermalization steps after each phase point change

Model parameters

Parameter name Default Description
total_sweeps 0 Number of MC steps per phase diagram point sampled
lattice.bravais.length required Linear size of the bravais lattice in units of unitcells
lattice.bravais.periodic true Boundary conditions on the simulation volume (PBC = true, OBC = false)

Update schemes

A number of different updates are implemented. Some of these are only possible in some Hamiltonians or lattice geometries. See J. Greitemann's PhD thesis, chapter 3, for detailed descriptions of each of the following updates.

Metropolis single spin flip (single_flip)

Randomly picks a number of single spins, corresponding to the total number of spins in the system, and attempts to update them with Metropolis probability. For Ising-type (Z2) spins, the spin is attempted to be flipped. For "O(3)" spins which are used in the Heisenberg model, by default a random spin is drawn uniformly from the unit sphere. In the latter case only, an additional parameter allows to instead draw the spin uniformly from a cone centered around the previous value of the spin, such that the original and tentatively updated spin span an angle of at most θ0.

Parameter name Default Description
update.single_flip.cos_theta_0 -1 cos(θ0)

Heat-bath algorithm (heatbath)

Randomly picks a number of single spins, corresponding to the total number of spins in the system, and updates them according to heat-bath probability. This update is currently only implemented for the Heisenberg model which constitutes a special case where the heat-bath algorithm can be made rejection-free. This update does not have any parameters associated with it.

Global transformation (global_trafo)

Applies the same transformation to all spins. For the Ising model, all spins are flipped; for the Heisenberg model, all spins are rotated by a random O(3) matrix. As both are symmetries of their respective Hamiltonians, this update is microcanonical and can thus always be accepted. Note that it does not help to reduce autocorrelation times of physical observables which preserve the same symmetry. It can however be beneficial in conjunction with TK-SVM, as these symmetries are not explicitly "taught" to the machine. This update does not have any parameters associated with it.

Overrelaxation update (overrelaxation)

The overrelaxation update is, too, a microcanonical update and can thus always be accepted. It can only be formulated for continuous spin degrees of freedom and is thus currently only available in the Heisenberg model. It randomly picks a number of single spins, corresponding to the total number of spins in the system, and updates them in a direction orthogonal to the local magnetization at that site such that the total energy remains unchanged. This can again help navigate the highly degenerate energy landscape of frustrated magnets. This update does not have any parameters associated with it.

Parallel tempering (parallel_tempering)

This code also implements a parallel tempering update using the facilities provided by the pt_adapter class. Refer to the section on Parallel Tempering in the top-level README file. The parallel_tempering update in this code merely initiates the PT update by calling negotiate_update. The callback to calculate the logarithmic weight is deferred to the Hamiltonian class. Both the Ising and Heisenberg model implement it.

The following parameters are used to control the frequency of the PT update. Only every pt.update_sweeps updates is a PT update actually initiated. By default, this never happens! As PT is a rather expensive proposition, the user must enable it explicitly by setting this parameter to a finite value. The second parameter, pt.query_freq determines how often within the duration of pt.update_sweeps each process should check whether it has been requested for an update. In computer systems with a fast interconnect, checking multiple times between PT updates may help reduce waiting times.

Parameter name Default Description
pt.update_sweeps ∞ Number of MC updates between PT updates
pt.query_freq 1 Number of PT queries in one PT update cycle

Tensorial kernel

The two options of the cluster parameter determine the choice of the spin cluster for the evaluation of the tensorial kernel. single corresponds to a single spin cluster while lattice uses the unitcells of the Bravais lattice as spin clusters.

Parameter name Default Description
rank required Rank of the monomial mapping
symmetrized 1 Eliminate redundant (symmetric) monomials (1) or not (0)
cluster lattice single or lattice