Ising model

In statistical mechanics, the Ising model is a simplified representation of the interaction between individual magnetic moments in a ferromagnetic substance, for example, a refrigerator magnet. These moments, called spins, are presumed to point either "up" ($+1$) or "down" ($-1$) and interact with other spins depending on their relative position.

The behavior of such a magnetic material is temperature-dependent:

1. At high temperature, thermal excitations cause the spins to readily change orientation and there is little organization in the system.
2. As the temperature decreases, lower-energy states are favored. These are attained when neighboring spins agree, causing small aligned domains to form.
3. Once this alignment translates to a substantial part of the system, the individual moments add up to an overall magnetic field.

Ising Hamiltonian

The energy of an Ising system is described by the Hamiltonian

$$ \mathcal{H} = -\sum_{\langle i,j\rangle} J \sigma_i\sigma_j, $$

where $\sigma_i \in \{\pm1\}$ are the spin variables, the sum is over interacting, or neighboring, spin pairs, and $J>0$ is an interaction constant that describes how strongly neighboring spins interact. For each pair of spins which are aligned, there is a contribution of $-J$ to the overall energy.

Note

The Ising Model has no disorder in the interaction. At low temperature, all spins have a tendency to align with their respective neighbors. The lowest-energy state, or ground state, is attained when all spin variables have the same value; for example, they are either all $+1$ or all $-1$.

Dynamics of the system

The probability of finding a statistical-mechanical system in a specific state is given by the Boltzmann distribution. It depends on the energy of this state (compared to that of all other possible states) and the temperature

$$ p(\vec{\sigma}) = \frac{e^{-\mathcal{H}(\vec\sigma)/k_\mathrm{B}T}}{\sum_{\vec{s}}e^{-\mathcal{H}(\vec{s})/k_\mathrm{B}T}} \propto e^{-\mathcal{H}(\vec\sigma)/k_\mathrm{B}T},$$

where the normalizing sum in the denominator is over all of the states (the partition function), $k_\mathrm{B}$ is the Boltzmann constant (typically simplified to $k_\mathrm{B}=1$ in theoretical models), and $T$ is the current temperature (often expressed as the inverse temperature $\beta = 1/T$).

Note

For the purpose of optimization, the key detail to note is that for $T\to 0$ ($\beta\to\infty$), the lowest-energy state dominates the sum and the chance of finding the system in this ground state tends to $p(\vec\sigma_{\mathrm{GS}})\to 1$.

Disordered Ising systems

While the (ferromagnetic) Ising model offers some interesting dynamics, the system becomes more complex if the interaction constants $J$ are allowed to be bond-dependent, such as using a different constant for each set of interacting spins.

$$ \mathcal{H} = -\sum_{\langle i,j\rangle} J_{ij} \sigma_i\sigma_j. $$

These are know as Ising Spin Glasses, and exhibit the following behavior:

• A term with $J_{ij}<0$ represents anti-ferromagnetic interaction, that is, spins which favor anti-alignment (alternating $+1$ and $-1$ spin values).

• A selection of $J_{ij}$'s with different signs (and possibly magnitudes), along with the interaction graph, can lead to competing interactions; a situation where no spin-variable assignment satisfies all the bonds (known as frustration).

• When finding the ground state, the choice of how to assign variables with competing interactions can have a ripple effect across the system by changing how adjacent spins must be arranged, in turn impacting their respective neighbors.

As a result, finding the ground state is much more challenging in these complex systems.