Lattice model (physics)

The Ising model is given by the usual cubic lattice graph ${\displaystyle G=(\Lambda ,E)}$ where ${\displaystyle \Lambda }$ is an infinite cubic lattice in ${\displaystyle \mathbb {R} ^{d}}$ or a period ${\displaystyle n}$ cubic lattice in ${\displaystyle T^{d}}$, and ${\displaystyle E}$ 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 spin-variable space is ${\displaystyle S=\{+1,-1\}=\mathbb {Z} _{2}}$. The energy functional is

${\displaystyle E(\sigma )=-H\sum _{v\in \Lambda }\sigma (v)-J\sum _{\{v_{1},v_{2}\}\in E}\sigma (v_{1})\sigma (v_{2}).}$

The spin-variable space can often be described as a coset. For example, for the Potts model we have ${\displaystyle S=\mathbb {Z} _{n}}$. In the limit ${\displaystyle n\rightarrow \infty }$, we obtain the XY model which has ${\displaystyle S=SO(2)}$. Generalising the XY model to higher dimensions gives the ${\displaystyle n}$-vector model which has ${\displaystyle S=S^{n}=SO(n+1)/SO(n)}$.

We specialise to a lattice with a finite number of points, and a finite spin-variable space. This can be achieved by making the lattice periodic, with period ${\displaystyle n}$ in ${\displaystyle d}$ dimensions. Then the configuration space ${\displaystyle {\mathcal {C}}}$ is also finite. We can define the partition function

${\displaystyle Z=\sum _{\sigma \in {\mathcal {C}}}\exp(-\beta E(\sigma ))}$

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 ${\displaystyle \{g_{i}\}}$ and ${\displaystyle \beta }$. 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.

Examples of exactly solvable models are the periodic 1D Ising model, and the periodic 2D Ising model with vanishing external magnetic field, ${\displaystyle H=0,}$ but for dimension ${\displaystyle d>2}$, the Ising model remains unsolved.

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.

The configuration space ${\displaystyle {\mathcal {C}}}$ of functions ${\displaystyle \sigma }$ is replaced by the convex hull of the spin space ${\displaystyle S}$, when ${\displaystyle S}$ has a realisation in terms of a subset of ${\displaystyle \mathbb {R} ^{m}}$. We'll denote this by ${\displaystyle \langle {\mathcal {C}}\rangle }$. This arises as in going to the mean value of the field, we have ${\displaystyle \sigma \mapsto \langle \sigma \rangle$ :={\frac {1}{|\Lambda |}}\sum _{v\in \Lambda }\sigma (v)} .

As the number of lattice sites ${\displaystyle N=|\Lambda |\rightarrow \infty }$, the possible values of ${\displaystyle \langle \sigma \rangle }$ fill out the convex hull of ${\displaystyle S}$. By making a suitable approximation, the energy functional becomes a function of the mean field, that is, ${\displaystyle E(\sigma )\mapsto E(\langle \sigma \rangle ).}$ The partition function then becomes

${\displaystyle Z=\int _{\langle {\mathcal {C}}\rangle }d\langle \sigma \rangle e^{-\beta E(\langle \sigma \rangle )}\Omega (\langle \sigma \rangle )=:\int _{\langle {\mathcal {C}}\rangle }d\langle \sigma \rangle e^{-N\beta f(\langle \sigma \rangle )}.}$

As ${\displaystyle N\rightarrow \infty }$, that is, in the thermodynamic limit, the saddle point approximation tells us the integral is asymptotically dominated by the value at which ${\displaystyle f(\langle \sigma \rangle )}$ is minimised:

${\displaystyle Z\sim e^{-N\beta f(\langle \sigma \rangle _{0})}}$

where ${\displaystyle \langle \sigma \rangle _{0}}$ is the argument minimising ${\displaystyle f}$.

A simpler, but less mathematically rigorous approach which nevertheless sometimes gives correct results comes from linearising the theory about the mean field ${\displaystyle \langle \sigma \rangle }$. Writing configurations as ${\displaystyle \sigma (v)=\langle \sigma \rangle +\Delta \sigma (v)}$, truncating terms of ${\displaystyle {\mathcal {O}}(\Delta \sigma ^{2})}$ then summing over configurations allows computation of the partition function.

Such an approach to the periodic Ising model in ${\displaystyle d}$ dimensions provides insight into phase transitions.

Suppose the continuum limit of the lattice ${\displaystyle \Lambda }$ is ${\displaystyle \mathbb {R} ^{d}}$. Instead of averaging over all of ${\displaystyle \Lambda }$, we average over neighbourhoods of ${\displaystyle \mathbf {x} \in \mathbb {R} ^{d}}$. This gives a spatially varying mean field ${\displaystyle \langle \sigma \rangle$ :\mathbb {R} ^{d}\rightarrow \langle {\mathcal {C}}\rangle } . We relabel ${\displaystyle \langle \sigma \rangle }$ with ${\displaystyle \phi }$ to bring the notation closer to field theory. This allows the partition function to be written as a path integral

${\displaystyle Z=\int {\mathcal {D}}\phi e^{-\beta F[\phi ]}}$

where the free energy ${\displaystyle F[\phi ]}$ is a Wick rotated version of the action in quantum field theory.

• Ising model
• ANNNI model
• Potts model
• Chiral Potts model
• XY model
• Classical Heisenberg model
• n-vector model
• Vertex model
• Toda lattice

• Bond fluctuation model
• 2nd model

• QCD lattice model