SU(3)¶

Defining representation \(\bf{3}\) of su(3) algebra

class groups.su3.SU3_DEFINING(p=1, q=0, dtype=torch.complex128, device='cpu')[source]¶

Parameters:

p (int) – (p,q) labels of the highest weight state of su(3) representation. For defining representation p=1, q=0.
q (int) –
dtype (torch.dtype) – data type of matrix representation of operators
device (int) – device on which the torch.tensor objects are stored

Build the defining representation \(\bf{3}\) of su(3) Lie algebra using the Cartan-Weyl (C-W) basis. In terms of the standard Gell-Mann matrices \(\lambda\), the C-W basis is:

\[\begin{split}\begin{align*} T^\pm &= \frac{1}{2} (\lambda_1 \pm i\lambda_2) = (F_1 \pm iF_2)\\ T^z &= \frac{1}{2} \lambda_3 = F_3\\ V^\pm &= \frac{1}{2} (\lambda_4 \pm i\lambda_5) = (F_4 \pm iF_5)\\ U^\pm &= \frac{1}{2} (\lambda_6 \pm i\lambda_7) = (F_6 \pm iF_7)\\ Y &= \frac{1}{\sqrt{3}} \lambda_8 = \frac{2}{\sqrt{3}} F_8 \end{align*}\end{split}\]

The quadratic Casimir operator of su(3) can be expressed in terms of the C-W basis, defined as follow.

\[\begin{split}\begin{align*} C_1 = \sum_{k}{F_k F_k} &= \frac{1}{2} (T^+ T^- + T^- T^+ + V^+ V^- + V^- V^+ + U^+ U^- + U^- U^+) \\ &+ T^z T^z + \frac{3}{4} Y Y \end{align*}\end{split}\]

C1()[source]¶

Returns:: The quadratic Casimir of su(3) as rank-4 for tensor
Return type:: torch.tensor

C2()[source]¶

Returns:: The cubic Casimir of su(3) as rank-6 for tensor
Return type:: torch.tensor

Cartan_Weyl()[source]¶

Returns:: vector of generators forming Cartan-Weyl basis ordered as [T^+, T^-, T^z, V^+, V^-, U^+, U^-, Y].
Return type:: torch.tensor

Returns a rank-3 tensor with first index running over generators.

I()[source]¶

Returns:: Identity operator of irrep
Return type:: torch.tensor

J_Gell_Mann()[source]¶

Returns:: \(\vec{\lambda}\) vector of Gell-Mann matrices
Return type:: torch.tensor

Returns a rank-3 tensor with first index running over generators.

TM()[source]¶

Returns:: \(T^-\) operator of irrep
Return type:: torch.tensor

TP()[source]¶

Returns:: \(T^+\) operator of irrep
Return type:: torch.tensor

TZ()[source]¶

Returns:: \(T^z\) operator of irrep
Return type:: torch.tensor

UM()[source]¶

Returns:: \(U^-\) operator of irrep
Return type:: torch.tensor

UP()[source]¶

Returns:: \(U^+\) operator of irrep
Return type:: torch.tensor

VM()[source]¶

Returns:: \(V^-\) operator of irrep
Return type:: torch.tensor

VP()[source]¶

Returns:: \(V^+\) operator of irrep
Return type:: torch.tensor

Y()[source]¶

Returns:: \(Y\) operator of irrep
Return type:: torch.tensor

SU(3) with explicit U(1)xU(1) structure¶

Defining representation \(\bf{3}\) of su(3) algebra in block-sparse form leveraging abelian U(1)xU(1) subgroup.

class groups.su3_abelian.SU3_DEFINING_U1xU1(settings, p=1, q=0)[source]¶

Parameters:

p (int) – (p,q) labels of the highest weight state of su(3) representation. For defining representation p=1, q=0.
q (int) –
settings (NamedTuple or SimpleNamespace (TODO link to definition)) – YAST configuration

Build the defining representation \(\bf{3}\) of su(3) Lie algebra using the Cartan-Weyl basis. In terms of the standard Gell-Mann matrices \(\lambda\), the C-W basis is:

\[\begin{split}\begin{align*} T^\pm &= \frac{1}{2} (\lambda_1 \pm i\lambda_2) = (F_1 \pm iF_2)\\ T^z &= \frac{1}{2} \lambda_3 = F_3\\ V^\pm &= \frac{1}{2} (\lambda_4 \pm i\lambda_5) = (F_4 \pm iF_5)\\ U^\pm &= \frac{1}{2} (\lambda_6 \pm i\lambda_7) = (F_6 \pm iF_7)\\ Y &= \frac{1}{\sqrt{3}} \lambda_8 = \frac{2}{\sqrt{3}} F_8 \end{align*}\end{split}\]

The U(1)xU(1) charges for states spanning \(\mathbf{3}=(1,0)\) irrep can be assigned as (rescaled) eigenvalues of diagonal \(T^z\) and Y operators:

( 1,  1)
(-1,  1)
( 0, -2)

The signature convention \(O = \sum_{ij} O_{ij}|i\rangle\langle j|\) is -1 for index i (\(|ket\rangle\)) and +1 for index j (\(\langle bra|\)).

The quadratic Casimir operator of su(3) can be expressed in terms of the C-W basis, defined as follow.

\[\begin{split}\begin{align*} C_1 = \sum_{k}{F_k F_k} &= \frac{1}{2} (T^+ T^- + T^- T^+ + V^+ V^- + V^- V^+ + U^+ U^- + U^- U^+) \\ &+ T^z T^z + \frac{3}{4} Y Y \end{align*}\end{split}\]

C1()[source]¶