This is a simple Matlab program to simulate the macroscopic properties of spin qubits in the Ising model. It uses a Monte Carlo simulation to estimate the state of the system as a function of temperature. This code is based off the analysis done by Tobin Fricke in 2006. A pdf copy of his report is saved in the directory labeled "papers" for reference. I expand on his work by analyzing qubits on a toroid, a Klein bottle, and a "double Klein bottle".
The toroidal topology will have periodic boundary conditions at all edges of the grid. The Klein bottle topology will have periodic boundary conditions on the sides of the grid, but anti-periodic boundary conditions on the top and bottom of the grid. Finally, the "double Klein bottle" will have anti-periodic boundary conditions on all sides of the grid.
The parameters for the simulation are listed in "lib/declare_parameters.m". They are:
- boltzmans_constant: The constant k, set to one.
- interaction_strength: The constant J, set to one.
- maximum_temperature: The maximum temperature tested, in Kelvins.
- number_of_random_temperatures_to_test: Number of data points in the Monte Carlo simulation.
- cells_on_side_of_grid: The length dimension of the grid.
- number_of_potential_spin_flips: Number of spin flips within each step of the simulation.
- parpool(4): If you have access to the Matlab parallel computing toolbox, you can set any number of cores that will be used for the simulation. If not, comment out this line.
The files "lib/TorIsing", "lib/KleinIsing", and "lib/TwistIsing" calculate the energy per site, the magnetization per site, and the temperature of the system for the toroid, Klein bottle, and "double Klein bottle" repsectively. Run "lib/montecarlo.m" to get plots for all of the topologies.
After running the simulations a few times, I have come to the conclusion that the three different topologies do not greatly influence the magnetization or energy curves that we see. I predict that the reason for this is the fact that the qubits are unaware of a boundary in any of the topologies due to symmetry. Thus, a qubit near the "edge" of any of these topologies should be indistinguishable from one in the "bulk" of the material.