Lattice model (physics)

In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime.

Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice.

Currently, lattice models are quite popular in theoretical physics, for many reasons.

Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from perturbation theory.

The exact solution to many of these models (when they are solvable) includes the presence of solitons.

Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang–Baxter equation and quantum groups.

The solution of these models has given insights into the nature of phase transitions, magnetization and scaling behaviour, as well as insights into the nature of quantum field theory.

Physical lattice models frequently occur as an approximation to a continuum theory, either to give an ultraviolet cutoff to the theory to prevent divergences or to perform numerical computations.

An example of a continuum theory that is widely studied by lattice models is the QCD lattice model, a discretization of quantum chromodynamics.

However, digital physics considers nature fundamentally discrete at the Planck scale, which imposes upper limit to the density of information, aka Holographic principle.

Lattice models are also used to simulate the structure and dynamics of polymers.

The Ising model is given by the usual cubic lattice graph

is the edge set of nearest neighbours (the same letter is used for the energy functional but the different usages are distinguishable based on context).

The energy functional is The spin-variable space can often be described as a coset.

Generalising the XY model to higher dimensions gives the

We can define the partition function and there are no issues of convergence (like those which emerge in field theory) since the sum is finite.

In theory, this sum can be computed to obtain an expression which is dependent only on the parameters

In practice, this is often difficult due to non-linear interactions between sites.

Models with a closed-form expression for the partition function are known as exactly solvable.

Due to the difficulty of deriving exact solutions, in order to obtain analytic results we often must resort to mean field theory.

This mean field may be spatially varying, or global.

is replaced by the convex hull of the spin space

, that is, in the thermodynamic limit, the saddle point approximation tells us the integral is asymptotically dominated by the value at which

A simpler, but less mathematically rigorous approach which nevertheless sometimes gives correct results comes from linearising the theory about the mean field

then summing over configurations allows computation of the partition function.

dimensions provides insight into phase transitions.

This gives a spatially varying mean field

to bring the notation closer to field theory.

This allows the partition function to be written as a path integral where the free energy

is a Wick rotated version of the action in quantum field theory.