Study note about Ising model

The main references of the study note are Monte Carlo simulation of classical Heisenberg model in three dimensions, Computational Quantum Physics and arXiv:0803.0217.

Ising Model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic ‘spins’ that can be in one of two states: +1 (up) or -1 (down). The spins are arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors.

Mathematical definition of Ising Model

Consider a set $\Lambda$ of lattice sites. For each lattice site $k\in\Lambda$, its spin state is represented by $\sigma_{k} = \{+1, -1\}$. The energy of a spin configuration $\sigma = \{\sigma_{k}\}_{k\in\Lambda}$ is given by the Hamiltonian function $H(\sigma)$:

$$H(\sigma) = -\sum_{<ij>} J_{ij}\sigma_{i}\sigma_{j} - \mu\sum_{j}h_{j}\sigma_{j}\tag{1}$$

where $\sigma_{i}$ and $\sigma_{j}$ are spin states (+1 or -1) of two adjacent sites $i, j \in \Lambda$, respectively. $J_{ij}$ is the interaction between site $i$ and site $j$. $J_{ij}>0$ indicates that the interaction is ferromagnetic, while $J_{ij}<0$ indicates that the interaction is antiferromagnetic. Quantity $\mu$ is the magnetic moment and $h_{j}$ is the external magnetic field interacting with site $j$. $h_{j}>0$ means the spin site $j$ desires to line up in the positive direction, while $h_{j}<0$ means the spin site $j$ desires to line up in the negative direction. The first sum is over pairs of adjacent spins (every pair is counted once). The notation $<ij>$ indicates that sites $i$ and $j$ are nearest neighbors.

The configuration probability is given by the Boltzmann distribution with inverse temperature $\beta = 1/(k_{B}T)\geq 0$:

$$P_{\beta}(\sigma) = \frac{e^{-\beta H(\sigma)}}{Z_{\beta}} = \frac{e^{-H(\sigma)/(k_{B}T)}}{Z_{\beta}}\tag{2}$$

where $k_{B}$ is the Boltzmann constant, $T$ is the temperature of the system, and $Z_{\beta} = \sum_{\sigma}e^{-\beta H(\sigma)}$ is the partition function. The configuration probabilities $P_{\beta }(\sigma)$ represent the probability that (in equilibrium) the system is in a state with configuration $\sigma$. Hence, $P_{\beta}(\sigma)\propto e^{-H(\sigma)/T}$ implies that lower energy states are exponentially more probable than higher energy states, especially at low temperature. As a result, the system tends to evolve over time towards configurations that minimize the total energy.

Simplication

Ignore the external magnetic field $h_{j}$, the corresponding Hamiltionian function of Ising Model becomes:

$$H(\sigma) = -\sum_{<ij>} J_{ij}\sigma_{i}\sigma_{j} \tag{3}$$

(Monte Carlo simulation of classical Heisenberg model in three dimensions) If the system is ferromagnetic, which means $J_{ij}>0$ holds for all adjacent sites in this system; it is obvious that configurations with spins to same direction are less energetic and therefore are more probable in the lower temperatures.

Monte-Carlo simulation of 2D Ising model

Here assumes $L = |\Lambda|$ is the number of sites on the square lattice. Since every spin site has two possible states: +1 (up) or -1 (down), there are $2^{L}$ different states that are possible. Hence, it is difficult to evaluate the Ising Model numerically with a large value of $L$, which motivates the reason for applying Monte Carlo Methods to simulate the Ising Model.

Metropolis-Hastings algorithm

In practice, Metropolis-Hastings algorithm with powerful versatility is the most commonly used Monte Carlo algorithm for the estimation of Ising Model. The main steps of Metropolis algorithm are:

1. Initialize a spin configuration $\sigma$ of $L$ spins.
2. Choose a random site $j$ on lattice, and then flip the spin of $j$ (Here just let $\sigma_{j} = -\sigma_{j}$). Finally record the spin configuration with flipped value of $\sigma_{j}$ as $\sigma^{'}$.
3. Caculate the change in energy $dE$ (Here $dE$ is just equivalent to $H(\sigma^{'}) - H(\sigma)$).
4. If $dE<0$, keep the flipped value. Otherwise, keep the flipped value with probability $e^{-\beta (H(\sigma^{'}) - H(\sigma))}$.
5. Repeat the step 2-4.

Implementation of the Ising Model simulations by Monte Carlo approach can be found in repository “Simulation”.