In
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
and
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Bethe lattice (also called a regular tree) is an infinite
connected cycle-free graph where all vertices have the same number of neighbors. The Bethe lattice was introduced into the physics literature by
Hans Bethe
Hans Albrecht Bethe (; July 2, 1906 – March 6, 2005) was a German-American theoretical physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics, and solid-state physics, and who won the 1967 Nobel ...
in 1935. In such a graph, each node is connected to ''z'' neighbors; the number ''z'' is called either the
coordination number
In chemistry, crystallography, and materials science, the coordination number, also called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it. The ion/molecule/atom surrounding the central i ...
or the
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
, depending on the field.
Due to its distinctive topological structure, the statistical mechanics of
lattice models
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 co ...
on this graph are often easier to solve than on other lattices. The solutions are related to the often used
Bethe approximation for these systems.
Basic Properties
When working with the Bethe lattice, it is often convenient to mark a given vertex as the root, to be used as a reference point when considering local properties of the graph.
Sizes of Layers
Once a vertex is marked as the root, we can group the other vertices into layers based on their distance from the root. The number of vertices at a distance
from the root is
, as each vertex other than the root is adjacent to
vertices at a distance one greater from the root, and the root is adjacent to
vertices at a distance 1.
In statistical mechanics
The Bethe lattice is of interest in statistical mechanics mainly because lattice models on the Bethe lattice are often easier to solve than on other lattices, such as the
two-dimensional square lattice. This is because the lack of cycles removes some of the more complicated interactions. While the Bethe lattice does not as closely approximate the interactions in physical materials as other lattices, it can still provide useful insight.
Exact solutions to the Ising model
The
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
is a mathematical model of
ferromagnetism
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
, in which the magnetic properties of a material are represented by a "spin" at each node in the lattice, which is either +1 or -1. The model is also equipped with a constant
representing the strength of the interaction between adjacent nodes, and a constant
representing an external magnetic field.
The Ising model on the Bethe lattice is defined by the partition function
Magnetization
In order to compute the local magnetization, we can break the lattice up into several identical parts by removing a vertex. This gives us a recurrence relation which allows us to compute the magnetization of a Cayley tree with ''n'' shells (the finite analog to the Bethe lattice) as
where
and the values of
satisfy the recurrence relation
In the
case when the system is ferromagnetic, the above sequence converges, so we may take the limit to evaluate the magnetization on the Bethe lattice. We get
where ''x'' is a solution to
.
There are either 1 or 3 solutions to this equation. In the case where there are 3, the sequence
will converge to the smallest when
and the largest when
.
Free energy
The free energy ''f'' at each site of the lattice in the Ising Model is given by