Problem Set #4

Ground Rules for Problem Sets

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>.
  2. Write a program to use the Metropolis algorithm to generate N vectors x_i and use them to evaluate <x^2>.

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:
  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.

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Copyright © 1997,1998 Richard Furnstahl and James Steele.