SU2Group¶
SU(2) non-Abelian symmetry group with spin-based quantum numbers.
dataclass
¶
Bases: UnitaryGroup
SU(2) symmetry group with integer-valued quantum numbers.
The SU(2) group represents rotational symmetry in quantum mechanics. Quantum numbers are non-negative integers representing twice the physical spin (2j convention): 0, 1, 2, 3, 4, ... corresponding to physical spins 0, ½, 1, 3/2, 2, ...
This convention avoids fractions while preserving all half-integer spins.
Fusion of two spins follows the triangular inequality: 2j1 ⊗ 2j2 → |2j1 - 2j2|, |2j1 - 2j2| + 2, ..., 2j1 + 2j2 (step size: 2)
Examples:
>>> # Spin-1/2 ⊗ spin-1/2 (1 ⊗ 1 in 2j notation) → 0, 2 (spin-0, spin-1)
>>> group.fuse_channels(1, 1)
(0, 2)
>>> # Spin-1 ⊗ spin-1 (2 ⊗ 2) → 0, 2, 4 (spin-0, spin-1, spin-2)
>>> group.fuse_channels(2, 2)
(0, 2, 4)
Attributes¶
Functions¶
Return dimension of spin-j irreducible representation.
For SU(2), the dimension of a spin-j representation is 2j+1.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
two_j
|
int
|
Quantum number (2j, where j is the physical spin). |
required |
Returns:
| Type | Description |
|---|---|
int
|
Dimension of the representation: 2j + 1. |
Return the dual representation.
For SU(2), all representations are self-dual (real), meaning the dual of a spin-j representation is spin-j itself.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
two_j
|
int
|
Quantum number (2j, where j is the physical spin). |
required |
Returns:
| Type | Description |
|---|---|
int
|
The same quantum number (2j). |
Fuse multiple spins, returning all possible total spin channels.
Returns all total spin values achievable by fusing the given spins, independent of the fusion tree structure. For SU(2), any value from the minimum to maximum in steps of 2 can be achieved by some choice of fusion tree.
In the 2j convention:
- For two spins: 2j1 ⊗ 2j2 → |2j1 - 2j2|, ..., 2j1 + 2j2 (step: 2)
- For n spins: 2j_min, 2j_min + 2, ..., 2j_max where:
- 2j_max = 2j1 + 2j2 + ... + 2jn (all aligned)
- 2j_min: largest value ≤ |2j_largest - sum_of_others| with matching integrality
Integrality constraint: All achievable total spins must be either all integer or all half-integer. In 2j notation, this means all returned values have the same parity (even for integer, odd for half-integer).
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
*two_js
|
int
|
One or more quantum numbers (2j values) to fuse. |
()
|
Returns:
| Type | Description |
|---|---|
Tuple[int, ...]
|
Tuple of all achievable total spin channels (2j values), sorted from smallest to largest. The actual fusion tree structure needed to achieve each value is determined by external packages. |
Examples:
Check if two quantum numbers are equal.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
a
|
int
|
Quantum numbers (2j values) to compare. |
required |
b
|
int
|
Quantum numbers (2j values) to compare. |
required |
Returns:
| Type | Description |
|---|---|
bool
|
True if the quantum numbers are equal. |
Validate that a charge is a valid SU(2) quantum number.
Valid SU(2) charges are non-negative integers representing 2j, where j is the physical spin: 0, 1, 2, 3, 4, ... corresponding to physical spins 0, ½, 1, 3/2, 2, ...
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
two_j
|
Any
|
Charge to validate (should be 2j as an integer). |
required |
Raises:
| Type | Description |
|---|---|
TypeError
|
If charge is not an integer. |
ValueError
|
If charge is negative. |
Examples:
>>> group = SU2Group()
>>> group.validate_charge(0) # Valid: spin-0 (2j=0)
>>> group.validate_charge(1) # Valid: spin-1/2 (2j=1)
>>> group.validate_charge(2) # Valid: spin-1 (2j=2)
>>> group.validate_charge(3) # Valid: spin-3/2 (2j=3)
>>> group.validate_charge(-1) # Raises ValueError
>>> group.validate_charge(1.5) # Raises TypeError
Description¶
Represents the SU(2) group of rotational symmetry. Unlike Abelian groups, SU(2) representations have an internal dimension (irrep dimension) and fuse into multiple channels simultaneously via Clebsch–Gordan decomposition.
Charge Convention: 2j Integers¶
Charges are non-negative integers representing twice the physical spin (the 2j convention):
| Charge (2j) | Physical spin j | Irrep dim (2j+1) |
|---|---|---|
| 0 | 0 | 1 (singlet) |
| 1 | 1/2 | 2 (doublet) |
| 2 | 1 | 3 (triplet) |
| 3 | 3/2 | 4 (quartet) |
| 4 | 2 | 5 (quintet) |
This convention keeps all quantum numbers as plain integers, avoiding fractions for half-integer spins.
Irrep Dimension¶
Every SU(2) charge carries an irrep dimension equal to 2j + 1. This is the number of magnetic quantum number states (m = -j, ..., +j) within each multiplet. Nicole works in the reduced tensor formalism: each block in a tensor stores the reduced tensor element, and the full tensor elements are reconstructed by contracting with Clebsch–Gordan coefficients stored in Bridge intertwiners.
Fusion Channels¶
SU(2) spins fuse via the triangular inequality — two spins j1 and j2 can combine into any total spin from |j1 − j2| to j1 + j2 in integer steps. In 2j notation:
from nicole import SU2Group
group = SU2Group()
# Spin-1/2 ⊗ spin-1/2 → spin-0 or spin-1
group.fuse_channels(1, 1) # (0, 2)
# Spin-1 ⊗ spin-1 → spin-0, spin-1, or spin-2
group.fuse_channels(2, 2) # (0, 2, 4)
# Three spin-1/2 → spin-1/2 or spin-3/2
group.fuse_channels(1, 1, 1) # (1, 3)
Self-Duality¶
All SU(2) representations are self-dual — the contragredient of a spin-j representation is again spin-j:
group.dual(0) # 0 (singlet is its own dual)
group.dual(1) # 1 (doublet is its own dual)
group.dual(2) # 2 (triplet is its own dual)
Physical Applications¶
- Spin-S systems: Arbitrary spin magnitude, e.g. spin-½ chains, spin-1 Haldane chains
- Angular momentum conservation: Total angular momentum of a multi-particle system
- Wigner–Eckart theorem: Operator matrix elements factorize into CG coefficients and a single reduced matrix element
Using with Product Group¶
SU2Group can be combined with Abelian groups via ProductGroup. The SU(2) factor must always be the last component:
from nicole import ProductGroup, U1Group, Z2Group, SU2Group
# U(1) × SU(2): particle number + full spin rotation (e.g. Hubbard model)
group = ProductGroup([U1Group(), SU2Group()])
print(group.name) # "U1×SU2"
print(group.neutral) # (0, 0)
print(group.is_abelian) # False
# Charges are tuples (n, 2j)
# e.g. (2, 1) means 2 particles above, spin-1/2 multiplet
Index and Sector Semantics¶
For SU(2) tensors, Sector(charge=2j, dim=n) means n independent spin-j multiplets. The total number of physical states in that sector is n × (2j + 1), accessible via Index.num_states:
from nicole import Index, Sector, Direction, SU2Group
group = SU2Group()
idx = Index(
Direction.OUT,
group,
sectors=(
Sector(charge=0, dim=1), # 1 singlet → 1 physical state
Sector(charge=1, dim=2), # 2 doublets → 4 physical states
Sector(charge=2, dim=1), # 1 triplet → 3 physical states
)
)
print(idx.dim) # 4 (total multiplets: 1 + 2 + 1)
print(idx.num_states) # 8 (total states: 1×1 + 2×2 + 1×3)
See Also¶
- Overview: Symmetry system introduction
- Bridge: CG intertwiner data attached to each SU(2) tensor block
- U1Group: Abelian U(1) symmetry
- Z2Group: Abelian Z(2) symmetry
- ProductGroup: Combine SU(2) with Abelian groups
- Examples: SU(2)
Notes¶
- SU(2) tensors carry intertwiners (yuzuha
Bridgeobjects) attached to each data block; these encode the Clebsch–Gordan structure in the R-W-C decomposition and are managed automatically by Nicole. See the Yuzuha Protocol for a full explanation. - All standard operations (
contract,trace,decomp,permute,conj, etc.) handle SU(2) transparently. is_abelianreturnsFalseforSU2Groupand for anyProductGroupcontaining it.