# Problem Set #4

## Ground Rules for Problem Sets

• Due on March 12.
• Will be handed back for corrections or revisions until we're happy.
• Answers to articulation problems are limited to three reasonable length sentences. You should imagine answering a question from a graduate student at a colloquium on the current topic. This is hard!
• For any of the problems you can use any math tool such as maple or mathematica. Provide a printout of the session.

## Problem Set 4

### Articulation Questions

1. Why use a path integral formulation to make a simulation of QCD?
2. Why is lattice QCD formulated in Euclidean space instead of Minkowski space?
3. Why do lattice gauge theories use link variables instead of the gauge fields themselves?
4. What is the difference between a first-order and a second-order phase transition?
5. Why are finite temperature lattice simulations easy but finite density simulations hard?

### Project A: Monte Carlo Integration and Metropolis Algorithm

In this project, you will write programs that perform first "naive" Monte Carlo integration (as discussed in class) and then use the Metropolis algorithm (see handout). The ratio of integrals to be used for testing is
<x^2> = [\int d^{D}x x^2 e^{-|x}|}] / [\int d^{D}x e^{-|x|}] = D(D+1)
where D is the number of dimensions and x is a vector in D-dimensional space.

There are three parameters we will consider in evaluating <x^2>: the number of dimensions D, the size of the cutoff L (i.e., in each dimension integrate from -L to +L), and the number of points N used (that is, the number of times the integrands are evaluated).

1. Write a program to perform naive Monte Carlo integration and evaluate <x^2>.
• Discuss (with graphs) how the accuracy of the Monte Carlo result depends on L, N, and D. Consider D = 1,5,10,20. What is a good choice for L?
2. Write a program to use the Metropolis algorithm to generate N vectors x_i and use them to evaluate <x^2>.
• Choose a random starting size inside the box of size L (use your good choice'' for L from part 1).
• Generate new vectors (or candidates for new vectors) by making a step of magnitude \delta in a random direction. What is a good size for \delta?
• Compare the accuracy with N of the Metropolis result to naive Monte Carlo integration for D = 1,5,10,20.

### Project B: Ising Model in Two Dimensions

The Ising model is a simple model of a magnetic system. It consists of a finite lattice in D dimensions with spins on each lattice site i with spin-projection S_{i} = +/- 1. The interaction energy at zero external field between two adjacent spins at sites i and j is E_{ij} = -2 J S_{i} S_{j}. That is, the interaction energy is either -2J or +2J. To find the total energy E, sum E_{ij} over each pair of nearest neighbors (be careful not to double count!).

1. Write a program to simulate the Ising model in zero field for two dimensions. Use the Metropolis algorithm with a Boltzmann weighting factor e^{-\beta E} to select configurations. To generate a new configuration:
• Step through each lattice site (rather than simply picking a new site at random).
• Decide whether or not to flip the spin on that site by the Metropolis algorithm.
• When each site has been considered, save the new configuration or use it to evaluate the contribution to the average spin.
2. Take D=2 and use a 30 x 30 lattice (you can use larger if you wish!) with periodic boundary conditions. Work in units with J=1.
3. Measure the average spin <S(\beta)> as a function of \beta and locate the critical point \beta_c. How does the equilibration time depend on \beta?
4. Compare your results to the mean-field result. Comment on the similarities and differences.