Symmetry Groups Overview¶

Nicole's symmetry system enables efficient block-sparse tensor operations by exploiting conserved quantum numbers.

Available Groups¶

Abelian Groups¶

U1Group: Continuous U(1) symmetry with integer charges
Z2Group: Binary Z(2) symmetry with charges 0 or 1
ProductGroup: Combination of Abelian groups (e.g. U(1) × Z(2))

Non-Abelian Groups¶

SU2Group: SU(2) rotational symmetry with spin-based quantum numbers
ProductGroup: Combination of Abelian groups with SU2Group as the last component (e.g. U(1) × SU(2))

Core Concepts¶

Charges¶

Charges are quantum numbers that label symmetry sectors:

U(1): Integers (..., -2, -1, 0, 1, 2, ...)
Z(2): Binary (0, 1)
SU(2): Non-negative integers using the 2j convention (0, 1, 2, 3, ...) for physical spins 0, ½, 1, 3/2, ...
ProductGroup: Tuples of component charges, e.g. (n, 2j) for U(1) × SU(2)

Charge Operations¶

All symmetry groups support:

neutral: Identity element (0 for U1, Z2, and SU2)
dual(q): Dual/contragredient representation; self-dual for SU(2)
equal(a, b): Test equality
is_abelian: True for U1/Z2/Abelian ProductGroup; False for SU2Group and ProductGroup containing SU2
irrep_dim(q): Dimension of irrep for charge q; always 1 for Abelian groups; 2j+1 for SU(2)

Charge Fusion¶

Charge fusion depends on whether the group is Abelian:

Abelian groups (U1, Z2, Abelian ProductGroup): fuse_unique(*qs) combines any number of charges into a single result (addition for U1, XOR for Z2).
Non-Abelian groups (SU2, ProductGroup with SU2): two charges can fuse into multiple result channels. Use fuse_channels instead:

from nicole import U1Group, SU2Group

# Abelian: always a unique result
u1 = U1Group()
u1.fuse_unique(2, 3)          # 5

# Non-Abelian: multiple channels
su2 = SU2Group()
su2.fuse_channels(1, 1)       # (0, 2)  — spin-0 or spin-1
su2.fuse_channels(2, 2)       # (0, 2, 4)  — spin-0, spin-1, or spin-2

Charge Conservation¶

Tensor blocks must satisfy:

∑(OUT charges) - ∑(IN charges) = neutral element

This is enforced automatically throughout Nicole.

Physical Examples¶

U(1) Applications¶

Particle number conservation
Magnetization (Sz) conservation
Electric charge

Z(2) Applications¶

Fermion parity (even/odd)
Spatial inversion symmetry
Time-reversal (in some contexts)

SU(2) Applications¶

Isotropic spin chains (Heisenberg model with full spin-rotation invariance)
Angular momentum conservation in multi-particle systems
Hubbard model with spin SU(2) symmetry (U1 × SU2)

SU(2) tensors use the Wigner–Eckart decomposition internally; see the Yuzuha Protocol for the R-W-C structure and compression rationale.

ProductGroup Applications¶

Particle number + spin (U1 × U1)
Particle number + parity (U1 × Z2)
Particle number + full spin rotation (U1 × SU2)
Parity + full spin rotation (Z2 × SU2)

Notes¶

Symmetry groups are immutable and lightweight. Groups are typically shared across many indices and tensors.