The Ising Model
The Ising Model
The Ising model is one of the simplest models of an interacting system. In this model the particles or spins sit on a lattice of identical sites and each particle interacts with its nearest neighbors only. An exact expression for the partition function of this system can be calculated if the spins are on a one or two dimensional lattice. Calculating the exact parition function for the two dimensional Ising model would be well beyond the scope of most undergraduate physics/maths courses.
Syllabus Aims
- You should be able to explain how the spins interact in a closed 1D-Ising model.
- You should be able to write out the Hamiltonian for a closed 1D-Ising model.
- You should be able to write a python function to calculate the Hamiltonian for a 1D-Ising model.
- You should be able to write a program to calculate the ensemble average of the energy and the free energy as a function of magnetisation for a 1D-Ising model by brute force.
- You should be able to use the Perron Frobenius theorem to show that $\lim_{n\rightarrow \infty} \textrm{Tr}(\mathbf{A})^n = \lambda_1^n$ where $\mathbf{A}$ is any square matrix with all positive elements and $\lambda_1$ is its top eigenvalue.
- You should be able to write out the summations that have to be evaluated in order to evaluate the partition function for closed and open 1D-Ising models. You should then be able to use the transfer matrix approach to rewrite these canonical partition function summations as powers of a matrix. By doing so you should thus be able to find an exact expression for canonical partition function for this model.
- You should be able to use the Perron-Frobenius theorem to evaluate the partition function for the Ising models in the thermodynamic limit.
Description and link | Module | Author | ||
| A video that explains how the partition function for a one dimensional system of Ising spins can be calculated exactly. | AMA4004 | G. Tribello |
Description and link | Module | Author | ||
| Problems on using transfer matrices to calculate partition functions and ensemble averages for interacting systems | AMA4004 | G. Tribello | ||
| An app which you can use to simulate the one dimensional ising model | AMA4004 | G. Tribello |
Description and link | Module | Author | ||
| An exercise that will teach you how to write functions that evaluate Hamiltonians for various lattice systems. | AMA4004 | G. Tribello | ||
| An exercise that will teach you how to evaluate a weighted and unweighted histograms. | AMA4004 | G. Tribello | ||
| An exericse that will teach you how to evaluate correlation functions between spins. | AMA4004 | G. Tribello |
Contact Details
School of Mathematics and Physics,
Queen's University Belfast,
Belfast,
BT7 1NN
Email: g.tribello@qub.ac.uk
Website: mywebsite