New Approaches to Modern Physics/CONCEPTS


118093 - New Approaches to Modern Physics
WINTER 2005

Advanced Statistical Mechanics


[Course Summary][Links to Lecture Notes and Homework][Announcements] [For Lecturers]
[JOAN ADLER's lectures]

CONCEPTS

  1. We are discussing discrete lattices in various dimensions. The number of nearest neighbours will be indicated by z. Lattices I will mention later include:

    1. 2D - square, triangular (thanks Uri) and honeycomb (thanks Anastasia)

      what what what

    2. 3D - simple cubic lattice

      what

      You may view animations of many lattices here

    3. general hypercubic - has 2 times D neighbours in dimension D

    4. Cayley tree - also known as the Bethe lattice because the Bethe approximation is exact here. Each site except the surface sites have z neighbours, z =3,4,...... It is like an infinite dimension because in d dimensions the volume of a ``circle'' with radius r is proportional to rd while its surface is proportional to rd-1 . As dimension tends to infinity the surface is proportional to the volume, as for the Cayley tree. A Cayley tree of z=3 is shown here.

      what

    5. amorphous/random - an ideal random lattice has points whose coordinates are randomly located. In physical systems there is usually some sort of short range order.

  2. Universality - a simplistic definition which is adequate for us now is:
    different models fall in the same universality class if their critical exponents only depend on the geometric dimension and not on the lattice details. (Critical temperatures will always be lattice dependent.)

  3. Potts models - this a general family of models which includes both Ising models and percolation as special cases. The formal defintion can be seen here. The Ising model is the two state ( q=2) Potts model (think of spin up as one state and spin down as the other) and percolation is formally described to be the q approaches 1 limit. This is true in the formal sense that if one writes out expressions for the model in series expansions, taking the limit as q approaches 1 gives percolation results.

  4. Perhaps the easisst way to understand this is to define the 3 state model.Here there are three possible states or say, colors red green and blue.

  5. Although Potts models do not have a solution like the Onsager one, in two dimensions their critical points, exponents and transition order are known exactly. (First order transition for q greater than 4.)

  6. The critical points were calculated by Potts for the square lattice using the idea of duality. Duality is a concept from graph theory that means that each bond of the dual lattice intersects a bond of the original lattice once and once only and vice -versa. The square lattice is self-dual and the dual of the triangular is the honeycomb lattice.