SU(2) Symmetry Examples¶

SU(2) symmetry represents full spin-rotation invariance. Unlike Abelian groups, each SU(2) charge carries an internal structure — a multiplet of 2j+1 magnetic states — so tensors with SU(2) symmetry work in the reduced tensor formalism: they store reduced tensor elements, and Clebsch–Gordan coefficients are handled automatically under the hood.

Common applications:

Heisenberg spin chains: Full SU(2) conservation for isotropic models
Hubbard model with SU(2) symmetry: Separate charge (U(1)) and spin (SU(2)) channels
Angular momentum coupling: General multi-particle spin states

SU(2) tensors automatically enforce the Wigner–Eckart theorem: a block labeled by charges (j1, j2, ...) exists only if the spins can be coupled to the total spin required by the tensor structure. Internally Nicole stores tensors in the reduced R-W-C representation; see the Yuzuha Protocol for details.

The SU(2) Group¶

group = SU2Group()

print(f"Group name:      {group.name}")
print(f"Neutral element: {group.neutral}   (spin-0 singlet)")
print(f"Is Abelian:      {group.is_abelian}\n")

# Irrep dimension: 2j+1 states per multiplet
for two_j in range(5):
    j = two_j / 2
    print(f"  2j={two_j}  →  j={j}  →  irrep_dim={group.irrep_dim(two_j)}")

Group name:      SU2
Neutral element: 0   (spin-0 singlet)
Is Abelian:      False

  2j=0  →  j=0.0  →  irrep_dim=1
  2j=1  →  j=0.5  →  irrep_dim=2
  2j=2  →  j=1.0  →  irrep_dim=3
  2j=3  →  j=1.5  →  irrep_dim=4
  2j=4  →  j=2.0  →  irrep_dim=5

Fusion Channels¶

Unlike Abelian groups where two charges fuse to a unique result, SU(2) spins can combine into multiple total spin channels:

# Spin-1/2 ⊗ spin-1/2 → singlet (j=0) or triplet (j=1)
print(f"1/2 ⊗ 1/2 → {group.fuse_channels(1, 1)}   (2j values: 0, 2)")

# Spin-1 ⊗ spin-1 → j=0, 1, or 2
print(f"1 ⊗ 1 →   {group.fuse_channels(2, 2)}  (2j values: 0, 2, 4)")

# Three spin-1/2 → j=1/2 or j=3/2
print(f"1/2⊗1/2⊗1/2 → {group.fuse_channels(1, 1, 1)}  (2j values: 1, 3)")

# SU(2) is self-dual
print(f"\nDual of 2j=3: {group.dual(3)}  (self-dual)")

1/2 ⊗ 1/2 → (0, 2)   (2j values: 0, 2)
1 ⊗ 1 →   (0, 2, 4)  (2j values: 0, 2, 4)
1/2⊗1/2⊗1/2 → (1, 3)  (2j values: 1, 3)

Dual of 2j=3: 3  (self-dual)

Creating an SU(2) Index¶

Each Sector in an SU(2) index stores a charge (= 2j) and dim (= number of multiplets). The total number of physical states differs from dim:

# Index with one singlet (dim=1), two doublets (dim=2), and one triplet (dim=1)
idx = Index(
    Direction.OUT,
    group,
    sectors=(
        Sector(charge=0, dim=1),  # 1 singlet  → 1×1 = 1 state
        Sector(charge=1, dim=2),  # 2 doublets → 2×2 = 4 states
        Sector(charge=2, dim=1),  # 1 triplet  → 1×3 = 3 states
    )
)

print(idx)
print(f"\ndim (multiplets): {idx.dim}")
print(f"num_states (physical states): {idx.num_states}")

  Index having 'SU2' with -

      Charge  Dims
         0     1
         1     2
         2     1

dim (multiplets): 4
num_states (physical states): 8

Creating a Random SU(2) Tensor¶

# Spin-1/2 index: one doublet sector
spin_half = Index(
    Direction.OUT,
    group,
    sectors=(Sector(charge=1, dim=1),),  # spin-1/2 multiplet
)

# Random 2-index SU(2) tensor (e.g. an operator in spin space)
T = Tensor.random([spin_half, spin_half.flip()], itags=["i", "j"], seed=42)
print(T)

  info:  2x { 1 x 1 }  having 'SU2',   Tensor,  { i*, j }
  data:  2-D float64 (8 B)    1 x 1 => 2 x 2  @ norm = 0.33669

     1.  1x1     |  2x2     [ 1 ; 1 ]   0.3367  {+}

The summary shows 'SU2' as the symmetry signature (rather than 'A' for Abelian tensors). When a block's reduced weight matrix is a single scalar, a sign indicator {+} or {-} is appended to flag its sign at a glance, without printing the full matrix.

Tensor Contraction¶

All standard operations work identically for SU(2) tensors. Contraction automatically handles the Clebsch–Gordan algebra:

# Spin-1 index with one triplet
spin_one = Index(
    Direction.OUT,
    group,
    sectors=(Sector(charge=2, dim=1),),
)

A = Tensor.random([spin_one, spin_one.flip()], itags=["i", "mid"], seed=1)
B = Tensor.random([spin_one, spin_one.flip()], itags=["mid", "j"], seed=2)

result = contract(A, B)
print(f"A:\n{A}\n")
print(f"A @ B:\n{result}")

A:

  info:  2x { 1 x 1 }  having 'SU2',   Tensor,  { i*, mid }
  data:  2-D float64 (8 B)    1 x 1 => 3 x 3  @ norm = 0.661352

     1.  1x1     |  3x3     [ 2 ; 2 ]   0.6614  {+}

A @ B:

  info:  2x { 1 x 1 }  having 'SU2',   Tensor,  { i*, j }
  data:  2-D float64 (8 B)    1 x 1 => 3 x 3  @ norm = 0.149791

     1.  1x1     |  3x3     [ 2 ; 2 ]  0.08648  {+}

SVD Decomposition¶

# Multi-sector index with spin-0 and spin-1 sectors
idx = Index(
    Direction.OUT,
    group,
    sectors=(
        Sector(charge=0, dim=2),  # 2 singlets
        Sector(charge=2, dim=1),  # 1 triplet
    )
)

T = Tensor.random([idx, idx.flip()], itags=["i", "j"], seed=10)
U, S, Vh = decomp(T, axes=0, mode="SVD")

print(f"T:\n{T}\n")
print(f"U:\n{U}\n")
print(f"S:\n{S}")

T:

  info:  2x { 2 x 1 }  having 'SU2',   Tensor,  { i*, j }
  data:  2-D float64 (40 B)    3 x 3 => 5 x 5  @ norm = 1.95727

     1.  2x2     |  1x1     [ 0 ; 0 ]    32 B   {+}
     2.  1x1     |  3x3     [ 2 ; 2 ]   0.9198  {+}

U:

  info:  2x { 2 x 1 }  having 'SU2',   Tensor,  { i*, _bond_L* }
  data:  2-D float64 (40 B)    3 x 3 => 5 x 5  @ norm = 2.23607

     1.  2x2     |  1x1     [ 0 ; 0 ]    32 B   {+}
     2.  1x1     |  3x3     [ 2 ; 2 ]       1.  {+}

S:

  info:  2x { 2 x 1 }  having 'SU2',  Diagonal,  { _bond_L, _bond_R }
  data:  2-D float64 (40 B)    3 x 3 => 5 x 5  @ norm = 2.35012

     1.  2x2     |  1x1     [ 0 ; 0 ]    32 B   {+}
     2.  1x1     |  3x3     [ 2 ; 2 ]   0.9198  {+}

Loading a Spin SU(2) Space¶

# Spin-1/2 system with full SU(2) symmetry
Spc, Op = load_space("Spin", "SU2", {"J": 0.5})

print(f"Physical space dimension: {Spc.dim}")
print(f"Physical states: {Spc.num_states}")
print(f"Available operators: {list(Op.keys())}\n")

# The spin operator S is a rank-1 spherical tensor
# It stores the reduced tensor element (Wigner-Eckart theorem)
print(f"S operator (reduced tensor element):\n{Op['S']}")

Physical space dimension: 1
Physical states: 2
Available operators: ['S', 'vac']

S operator (reduced tensor element):

  info:  3x { 1 x 1 }  having 'SU2',  Operator,  { _init_, _init_*, _aux_* }
  data:  3-D float64 (8 B)    1 x 1 x 1 => 2 x 2 x 3  @ norm = 1.22474

     1.  1x1x1   |  2x2x3   [ 1 ; 1 ; 2 ]    0.866  {+}

With SU(2) symmetry, there is only one independent spin operator S (the spherical tensor), compared to three separate operators (Sp, Sm, Sz) needed for the explicit U(1) case. Nicole uses the Wigner–Eckart theorem to reconstruct full tensor elements on the fly.