Phase Transitions

←Previous

Homework assignment

In this assignment we will look at the Ising model and evaluate the spontaneous magnetisation of the spins in a square grid in 2 dimensions. To this end we will use the two methods discussed in lecture 8, namely Mean Field Approximation (MFA) and Monte Carlo simulation (MC) to evaulate the normalised spontaneous magnetisation M in dependence on the temperature T, which is given in units of jJ/k_B.

1. Mean field approximation

   As you may remember from lecture 8, in mean field approximation the normalised magnetisation M/M_max is given by the solution of an implicit equation, which we aim to solve in this first step. For this you need to
   • Implement a root finding algorithm in the file RootFinder.py. By default we assume the algorithm of choice is the Newton-Raphson method discussed in lecture 5, implemented as the method NewtonRaphson, see the lecture notes for details.
   • Implement f(x) and its first derivative in IsingModelAnalytical.py, encoding the class IsingMeanField. These two functions are made accessible through the __call__ method, with mode selecting whether the function (mode=0) or its dervaitve (mode=1) are returned.

2. Monte Carlo simulation

   In the MC solution, encoded in IsingModel.py, there are a number of classes to be made functional:
   • The Lattice-class:
     This class mainly carries a two-dimensional lattice with dim sites in each dimension:
     • In its initialisation, in the __init__ method, these sites are either filled randomly, through the method randomSpin() of the class Ising, or with a fixed default value given as the argument of the __init__ method.
     • The __call__ method returns the spin direction at position (lx, ly). Note: Employ periodic boundary conditions for the lattice.
     • The randomPosition method returns a random position x, y inside the lattice.
     • The setSpin method sets the spin at position lx, ly to the value spin.
   • The Ising-class:
     This class defines the interactions of the spins. Here you should fix the following methods:
     • randomSpin Return a random spin in the Ising model, i.e. eith +1 or -1.
     • modifySpin Modify the given spin such that it takes another than the original value.
     • calculateEnergy Calculate the contribution of a spin at the given position to the total energy. Note: You amy use the __call__ method of the Lattice class.
   • The Metropolis-class:
     Here, in each lattice sweep the method modifySpin is called dim² times. This method needs to be filled:
     • In modifySpin a single Metropolis step is performed for one spin: A spin position is selected at random and a new trial spin is generated, different from the original one. The energy difference is calculated and defines whether the trial spin is accepted or not. For details of how this happens, cf. Lecture 8.
   • The Observables-class:
     This class hosts the observables we would like to calculate and, in fact, also the scanning over the temperatures, in scanTemperatures. In each of the steps, the observables are calulcated in our case now it is just the normalised magnetisation M/M_max, realised in.
     • calculateMnorm. This method needs being fixed, see the lecture ntoes for details.