Ising model

• History of the Ising model

Ising Model

The Ising model is commonly defined over an ordered lattice. Each site of the lattice can adopt two states: either UP (S=+1) or DOWN (S=-1).

The energy of the system is the sum of pair interactions between nearest neighbors.

${\frac {U}{k_{B}T}}=-K\sum _{<ij>}S_{i}S_{j}$

where $<ij>$ indicates that the sum is done over nearest neighbors, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_i } indicates the state of the i-th site.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K } is called the Coupling constant.

to be continued:

• Ising in 1-d (exact solution)

• Usual lattices in 2d: Critical behavior

• Lattices in 3-d

• Ferromagnetic and antiferromagnetic couplings

• Frustration, etc

• Simulation procedures

• Theoretical methods