latticegpu.jl/docs/src/groups.md

Groups and Algebras

The module Groups contain generic data types to deal with group and algebra elements. Group elements g\in SU(N) are represented in some compact notation. For the case N=2 we use two complex numbers (Caley-Dickson representation, i.e. g=(z_1,z_2) with |z_1|^2 + |z_2|^2=1). For the case N=3 we only store two rows of the N\times N unitary matrix.

Group operations translate straightforwardly in matrix operations, except for SU(2), where we use

\forall g=(z_1, z_2)\in SU(2)\, \Longrightarrow\, g^{-1} =
(\overline z_1, -z_2)

and

\forall g=(z_1, z_2), g'=(w_1,w_2)\in SU(2)\, \Longrightarrow \, g g'= (z_1w_1 - \overline w_2 z_2, w_2z_1 + z_2\overline w_1)\,.

Group elements can be "casted" into arrays. The abstract type GMatrix represent generic N\\times N complex arrays.

Algebra elements X \in \mathfrak{su}(N) are traceless anti-hermitian matrices represented trough N^2-1 real numbers X^a. We have

  X = \sum_{a=1}^{N^2-1} X^a T^a

where T^a are a basis of the \mathfrak{su}(N) algebra with the conventions

  (T^a)^+ = -T^a \,\qquad
  {\rm Tr}(T^aT^b) = -\frac{1}{2}\delta_{ab}\,.

If we denote by \{\mathbf{e}_i\} the usual ortonormal basis of \mathbb R^N, the N^2-1 matrices T^a can always be written in the following way

1. symmetric matrices (N(N-1)/2)

   (T^a)_{ij} = \frac{\imath}{2}(\mathbf{e}_i\otimes\mathbf{e}_j +
   \mathbf{e}_j\otimes \mathbf{e}_i)\quad (1\le i < j \le N)
   

2. anti-symmetric matrices (N(N-1)/2)

   (T^a)_{ij} = \frac{1}{2}(\mathbf{e}_i\otimes\mathbf{e}_j -
   \mathbf{e}_j\otimes \mathbf{e}_i)\quad (1\le i < j \le N)
   

3. diagonal matrices (N-1). With l=1,...,N-1 and a=l+N(N-1)

   (T^a)_{ij} = \frac{\imath}{2}\sqrt{\frac{2}{l(l+1)}}
   \left(\sum_{j=1}^l\mathbf{e}_j\otimes\mathbf{e}_j -
     l\mathbf{e}_{l+1}\otimes \mathbf{e}_{l+1}\right)
   \quad (l=1,\dots,N-1;\, a=l+N(N-1))
   

For example in the case of \mathfrak{su}(2), and denoting the Pauli matrices by \sigma^a we have

  T^a = \frac{\imath}{2}\sigma^a \quad(a=1,2,3)\,,

while for \mathfrak{su}(3) we have T^a=\imath \lambda^a/2 (with \lambda^a the Gell-Mann matrices) up to a permutation.

Some examples

import Pkg # hide
Pkg.activate("/home/alberto/code/julia/LatticeGPU/") # hide
using LatticeGPU # hide

Here we just show some example codes playing with groups and elements. The objective is to get an idea on how group operations We can generate some random group elements.

# Generate random groups elements, 
# check they are actually from the grup
g = rand(SU2{Float64})
println("Are we in a group?: ", isgroup(g))
g = rand(SU3{Float64})
println("Are we in a group?: ", isgroup(g))

We can also do some simple operations

X = rand(SU3alg{Float64})
g1 = exp(X);
g2 = exp(X, -2.0); # e^{-2X}
g = g1*g1*g2;
println("Is this one? ", dev_one(g))

Types

Groups

The generic interface reads

Group

Concrete supported implementations are

SU2
SU3

Algebras

The generic interface reads

Algebra

With the following concrete implementations

SU2alg
SU3alg

GMatrix

The generic interface reads

GMatrix

With the following concrete implementations

M2x2
M3x3

Generic Group methods

inverse
dag
tr
dev_one
unitarize
isgroup
projalg

Generic Algebra methods

exp
expm
alg2mat