辅导汇编、讲解Bitwise Multiplication汇编
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Theory
Ising model
The Ising model is used for modelling ferromagnetic and antiferromagnetic materials. This model represents a lattice occupied by atoms which can each have dipole moments or spins. It predicts a second order phase transition occurring at the Curie temperature for dimensions higher than 1. Phase transition is identified from ensemble properties and compared with the theoretical model which has been solved exactly for zero external field.
The magnetization per spin of the system is simply the average of the spins.
1/N M= 1/N ∑S_k
The total energy of a ferromagnetism is
E = -J ∑Si Sj ,
J is a constant and the sum is over all pairs of adjacent spins from assumption 3, the energy of two neighboring pairs is -E if they are parallel and +E if they are antiparallel.
Ferromagnetism and anti ferromagnetism
Ferromagnetic materials are strongly ordered and have net magnetization per site as ?1/1 under temperatures T < Tc. They are able to maintain spontaneous magnetization even under the absence of external fields. At temperatures above Tc, the tendency to stay ordered is disrupted due to competing effects from thermal motion. The ferromagnetic substance behaves like a paramagnetic substance at T>TC, showing no spontaneous magnetization. As far as the Ising Model goes, coupling parameter J > 0 for ferromagnetic substances and J < 0 for antiferromagnetic substances.
1.3 Hamiltonian
Each atom can adopt two states, corresponding to s = {?1, 1}, where s represents the spin and the spin interactions are dependent on the coupling parameter Jij. The lattice model has periodic boundary conditions and extends infinitely. This model is defined in the Canonical Ensemble (N, V,T).
Algoritm
An optimal value of 100 atoms was chosen to model this system with periodic boundary conditions. The lattice was represented by a 10 × 10 random matrix with each element being randomly assigned with the values -1 or 1. It is chosen as a 10 × 10 matrix so as to ensure that all the edges are periodic in nature.
All quantities are manipulated in normalized units. Temperature is normalized and kB s 1 unit. The coupling strength is taken as J = 1 unit. The whole procedure was done for different temperatures ranging from 1–5.
And then a series of measurements was made for J = 1k, T = 1to 5 K for B = {-1,1}.
Initialize the system randomly with spins, at a given Temperature. Set the value of the external field, in most cases h = 0. Make a random flip in the spin of some atom. Compute the Energy change arising from this, due to only the neighboring atoms. Ensure that the periodic boundary conditions are in place to
take care of edge effects. If ?E < 0, accept this configuration and continue this process If ?E > 0, accept this configuration with a probability of p = exp (? ?E/kB T ), else retain the old configuration. Once every m, iterations, sample the system for important ensemble properties. This sampling has to be done after discarding the edges. because they only represent the periodic boundary conditions. Now allow the system to equilibriate (typically takes_ n3 iterations)
Estimate the average properties, variance terms (Susceptibility and Cv )
Repeat this procedure at different temperatures.
In summarize:
we first choose a state randomly by using the random matrix. Then, use the subroutine, dU, to calculate the ΔU. In the subroutine, dU, we implement the Mean Field Approximation method: E = -Jn? , n : number of nearest neighbors (n=4 in 2_D) ? : average alignment of neighbors and periodic boundary conditions, which we wrap around the states at the boundary (edge). So that the right edge is immediately left of the left edge and the bottom edge is immediately above the top edge. At the end of the loop, we color the cell whose spin has flipped by using the subroutine, Color Cell.
Currie temperature:
If you like to find the Curie temperature of a ferro magnet, you have to set different T (temperature) values for different runs and compare their results. If the T we set is below the actual Curie temperature, you'll get the final picture is either totally red or totally blue. Then, you can try a higher T for the next run. Repeat this process until you get the best Curie temperature. Number of enough iterations to get the right result is depends on the domain size you define at the beginning of the run. Since the Ising model is based on random sampling. The curie temperature which I found was 2.6.
Magnetization and Susceptibility
In this section, we calculate the magnetization and susceptibility at different lattice sizes and temperatures.
5.2 Magnetization
Here we plot the magnetization. We see that for larger lattice sizes, the system has not reached equilibrium and the data is very noisy. Otherwise the plot matches what is known, i.e, full magnetization at low temperatures, disorder at higher temperatures, with a transition at the critical temperature. The transition is sharper at larger lattice sizes.
5.3 Susceptibility
Susceptibility is the second derivative of the energy and measures the extent to which the lattice will be
Magnetized. It is discontinuous at the critical temperature. We see that the peak gets sharper, and T/Tc
gets closer to unity with increasing lattice size. This means that larger lattices give better approximations
of Onsager’s Tc, and an inifite latice would have exact resemblance. The data looks very noisy at high
temperatures.
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