Assignment 2: Spin-Spin Correlation in the Ising Magnet

The purpose of this assignment will be to guide you through modifiying the ising_mc.c program I demonstrated in class so that it computes the spin-spin correlation function. You will plot this function for various temperatures.

Spin-spin correlation

Correlation between spins (or between a spin and itself) is the idea that the likelihood that the two spins have relative spin values that occur more frequently together than random chance would suggest. Consider two spins i and j with spin variables si and sj respectively. The expecations <si> and <sj> are not really interesting; they are just <s> since we compute that by averaging over all spins. The expectation <si*sj>, however, is interesting: if i and j are more likely to have the same spin value than not, this number is close to 1; if they are more likely to have opposite spin values, this number is close to -1. We define the correlation function C(d) such that

C(d) = <si*sj> - <s>*<s>

where d is the distance (along a row or column) between i and j.

A temperatures above the critical temperature, this correlation function has the property C(0) = 1 (since a spin is always positively correlated with itself) that dies out to 0 as d gets large. d cannot exceed L/2 for a finite lattice of side-length L with periodic boundaries. Its decay behavior mimics an exponential:

C(d) ~ exp(-d/xi)

where xi is called the "correlation length".

At temperatures below the critical temperature, C(d) no longer decays to 0 since we have large domains.

The Assignment

You are to modify ising_mc.c provided in this repository to compute the C(d) for lattices of size L= 20 and for temperatures of 2.3, 3, 5, and 10. You should generate a single plot in PNG format that plots C(d) vs d on [0:10] for each of these temperatures on the same axes.

What to do to the code

In main():

1. Declare arrays for sisj and sisjsum

   double * sisj, * sisjsum;
   

2. Allocate:

   sisj=(double*)malloc(L/2*sizeof(double));
   sisjsum=(double*)malloc(L/2*sizeof(double));
   

3. Sample:

• include sisj as an argument to samp()
• include a for loop from 0 to L/2 to update the sisjsum tally:

  for (i=0;i<L/2;i++) sisjsum[i]+=sisj[i];
  

4. Output:

   for (i=0;i<L/2;i++)
    fprintf(stdout,"%i  %.5lf\n",i,
      sisjsum[i]/nSamp-ssum*ssum/(nSamp*nSamp));
   

In samp()

1. Include double * sisj as a parameter declaration and d as a local integer

2. Before the loop over all spins, initialize sisj

   for (d=0;d<L/2;d++) sisj[d]=0.0;
   

3. Inside the loop over all spins, increment sisj by considering only eastern and southern neighbors a distance d away:

   for (d=0;d<L/2;d++) 
      sisj[d]+=F[i][j]*(F[(i+d)%L][j]+F[i][(j+d)%L]);
   

4. At the end of samp() divide all elements if sisj[] by 2*L*L

   for (d=0;d<L/2;d++) 
     sisj[d]/=(2*L*L);
   

Generating a plot

You must write a short python script to generate the required plot. A workflow you might want to consider might look like this:

1. Run the code for each temperature for at least a few thousand cycles, and copy and paste the numerical output into a unique file for each run; for instace C-T1.dat, C-T2.dat, etc.

2. Edit the python template my_plot.py to read each file into unique arrays:

d1,C1=np.loadtxt('C-T1.dat',unpack=True)
d2,C2=np.loadtxt('C-T2.dat',unpack=True)
(etc)

3. ...and for each, issue a plot command, e.g.,

ax.scatter(d1,C1,label='T=2.3')

Submitting

Just push this repository once you're happy with your plot.