XY Model X64

XY Model is a very useful, Java based application designed to implement a Monte Carlo simulation of the planar ferromagnet or XY Model of spins on a lattice. The simulation returns the configuration of spins with the option of showing the vortices. The default system is a lattice of linear dimension L = 20 at temperature T = 0.89.

In the lattice model of 2-D Ising or XY model spins are placed on a two-dimensional square lattice. The spins have a preferred direction, and they can point to only two orientations, parallel or anti-parallel to each other. Each spin has a random strength, which can be positive or negative. The spin orientation is thus now represented by 2-D vectors with a two-component vector in each point of the lattice. Therefore the lattice consists of 2^N discrete points, where N is the number of spins. The lattice has periodic boundary conditions, i.e. spins at the lattice boundaries are equivalent to those inside.
Each spin is a member of one of two possible states called “up” and “down”, and their spin vector is then a 2-D vector:
[up]=[(1,0),(0,1)]: “2”
[down]=[(0,1),(1,0)]: “-1”
Vortices are defined as regions with vortex lines where the sign of the spins is opposite to their neighbours (as opposed to antiferromagnetic ordering where spins are aligned with their neighbours). A vortex is formed when a spin goes from a favourable orientation to an unfavourable one, i.e. a spin flips its orientation from “up” to “down”.
The XY Model is solved by using the Metropolis algorithm in a high temperature limit. In a high temperature limit the spins are supposed to become random, but the probability distribution is equal for up and down spins.
The high temperature limit is useful to obtain long-range order on a lattice because the thermal energy W is bigger than the ferromagnetic energy gap, which is proportional to the temperature.
By comparing the energy difference between the spin configurations with and without vortex lines, one sees that the system is in a long-range ordered state for TTc the system is disordered with no vortices. The actual value of Tc depends on the strength and range of the interaction.
The temperatures T are specified with an integer N, which is the number of spins in the lattice, where, N=20 in the example below.

A:

Below I’ll just copy-paste the code from the source
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The XY model is a classical spin system with infinitely many degrees of freedom (spin configurations) on a lattice. The goal of the simulation is to find the Gibbs distribution of the lattice, the Boltzmann distribution of the lattice, or a slightly shifted Gibbs distribution for which the vortices are frozen to the ground state. The motivation for the simulation is to find the Gibbs distribution of the two-dimensional XY model, which is a quantum system, and thus has no classical limit in the limit of zero spin quantum numbers. However, this quantum Monte Carlo simulation can be made classically approximate by modifying the QMC algorithm to use a fixed spin value.
Version 2.0.1 of the XY model uses the Lattice Boltzmann algorithm to find the Gibbs distribution of the model. The multi-grid algorithm is used for finding the ground state vortex structure, and the fact that there are no external field terms allows the Monte Carlo simulation to easily capture the vortex structure.
The multigrid algorithm was inspired by the algorithm for simulating the transverse field Ising model published in [@PanagiotidesTISG]. A sequence of small perpendicular magnetic fields is applied to the lattice, as described in [@Smith1994]. The initial field is a small initial one, proportional to the temperature (so that it never leaves the spin-up state). A sequence of successively larger fields can be generated by feeding the sequence into the Ising model with a multiple of the initial field. This approach gives two advantages: the first is the ability to obtain the vortex structure, and the second is the approximate fixing of the spins to the ground state (because of the missing external field terms) thus reducing the number of sample points to the ground state. The algorithm is equivalent to a Brownian motion with long-ranged drift of a particle with a small mass moving under gravity on a suitably chosen potential.
The lattice is decomposed into small cells and then the system is run on each cell in sequence. This results in an energy functional: $$\mathcal{E} = \sum_{c} \left[ \mathcal{E}_0(c) + \sum_{j} \mathcal{E}_s(c, j) \right].$$
The Gibbs distribution is found by adding energy to the system: $$e^{ -\beta \mathcal{E}} \equiv e^{ -\beta \mathcal{E}_0

The XY Model is a Monte Carlo simulation of a 2-dimensional ferromagnet that has been widely studied by physicists because of its simplicity and resemblance to the real world.
The system consists of Ising spins, with spin values S = 0 or 1, on a square lattice. The Hamiltonian for the system consists of the sum of the interactions between nearest neighbour spins. These interactions favour the alignment of spins in the same direction.
The mean field approximation is then used to predict the overall magnetization of the system.
The system is evolved using a stochastic Monte Carlo simulation. That is, each spin is selected at random and the spin is flipped. The simulation returns a configuration of spins and each spin’s orientation is recorded.
Each spin configuration is sampled once at the start of the simulation and is evolved by flipping a certain number of spins for a set period of time. (A set number of spins per unit time or simulation time step). The number of spin flips determines the accuracy of the simulation. As the number of spins increases, so does the precision of the simulation. With this in mind, the parameter used in this program is the flip rate k.
The XY Model is the most popular test of the speed of the simulation and the precision of the measurement. The measurement can be either a histogram of the spins or the number of vortices.

*This is a sample of the vortices. As the program evolves, vortices will move around the lattice.

A vortex is a location where two spins are pointing in the same direction. In the real world, the vortices point out in pairs. The actual position of a vortex will be a point on a lattice. On a lattice, the vortices are intersections of lines of spins.

A vortex pair can be an object worth studying. Suppose for example that you had an infinite lattice with periodic boundary conditions. Then, a vortex pair would be an infinite strand of spins pointing in the same direction, with the pair running parallel to the lattice.
For example, at the top right of the lattice, there is a vortex pair. This pair comes from the left across the right edge of the lattice and then across the bottom edge of the lattice. On the bottom, there is another vortex pair. This time, the two vortices pass along the top and left edge of the lattice before passing across the bottom edge to the right. For a therm

Minimum:
OS: Windows 7, 8, 8.1, 10
Processor: Intel Core i3-2350M 2.3 GHz, Intel Core i5-2400 2.4 GHz or AMD A10-5700 2.2 GHz
Memory: 4 GB RAM
Graphics: NVIDIA GeForce 8400, AMD Radeon HD 7570, Intel HD 4000
DirectX: Version 11
Hard Drive: 30 GB available space
Sound Card: DirectX-compatible sound card with output capabilities: Stereo

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