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SU(2) Protocol

This page describes the internal structure of SU(2)-symmetric tensors in Nicole and the mathematical framework behind their implementation. Nicole follows the Yuzuha Protocol — a specification for non-Abelian tensor algebra — and uses the Yuzuha SU(2) recoupling engine as its computational backend.

Prerequisites

This page assumes familiarity with Core Concepts, particularly the section on non-Abelian groups and irrep dimension.

The Wigner–Eckart Theorem

At the heart of SU(2)-symmetric tensor networks is the Wigner–Eckart theorem. For a tensor operator \(\hat{T}\) with definite symmetry properties, the theorem states that all matrix elements within a given charge block factorize as:

\[ \langle j', m' \| \hat{T}^{(k)}_q \| j, m \rangle = \langle j' \| \hat{T}^{(k)} \| j \rangle \cdot \langle j, m; k, q \mid j', m' \rangle \]

where:

  • \(\langle j' \| \hat{T}^{(k)} \| j \rangle\) is the reduced matrix element (RME) — a single number depending only on the spin labels \(j', k, j\), not on the magnetic quantum numbers \(m', q, m\)
  • \(\langle j, m; k, q \mid j', m' \rangle\) is the Clebsch–Gordan (CG) coefficient — a universal recoupling factor

This factorization extends to arbitrary \(n\)th-order tensors: a block of an \(n\)th-order tensor labeled by spins \((j_1, j_2, \ldots, j_n)\) has \(\prod_\ell (2j_\ell + 1)\) physical components, but all of them are determined by a small number of reduced tensor elements and the corresponding CG recoupling tensor.

j₁ j₂ ⋯ jₙ R̃ α C m₁ m₂ ⋯ mₙ

Tensor \(\tilde R\) (left) is the reduced tensor, with one index per tensor position running over the multiplet space (size Sector.dim). Tensor \(C\) (right) is the CG basis tensor, with one index per position running over the \(2j_\ell+1\) magnetic substates. The connecting index \(α\) is the outer multiplicity (OM) index, enumerating the independent fusion-tree configurations consistent with the charge labels \((j_1, \ldots, j_n)\).

Nicole exploits this structure to store tensors in reduced form, gaining both memory compression and computational acceleration proportional to \(\prod_\ell (2j_\ell + 1)\).

Outer Multiplicity

The outer multiplicity (OM) counts how many linearly independent ways a given set of spins \((j_1, j_2, \ldots, j_n)\) can be fused to a definite total spin \(J\). Each independent configuration corresponds to a distinct fusion tree — a choice of intermediate coupling at each step of the sequential spin addition. The OM index \(\alpha\) runs over these configurations and is the key internal index in the R-W-C decomposition.

Concept Dimension Scales with
Magnetic substates \(2j+1\) per index Spin magnitude — fixed by physics
Outer multiplicity (OM) \(\text{dim}(\alpha)\) Fusion-tree configurations

For a fixed charge block \((j_1, j_2, \ldots, j_n)\), the OM is a combinatorial number determined entirely by the spin labels. It counts the number of valid intermediate spin values in the left-associative fusion tree and generally grows with both the tensor order \(n\) and the spin magnitudes \(j_\ell\). For SU(2) tensors:

  • A 2nd-order block \((j_1, j_2)\) has OM \(= 1\) — there is no coupling, just an identity
  • A 3rd-order block \((j_1, j_2, j_3)\) has OM \(= 1\) — the block is a single CG coefficient
  • OM \(> 1\) first appears at 4th order, where multiple intermediate spin paths may be valid
  • For larger spin magnitudes, more intermediate values are compatible, further increasing the OM dimension

By operating entirely in the reduced (OM) space, Nicole avoids ever constructing the full \(\prod_\ell (2j_\ell+1)\)-dimensional physical block. All contractions, decompositions, and manipulations are carried out using X-symbols and R-symbols computed by Yuzuha.

Nicole's R-W-C Decomposition

A block labeled by charge \((j_1, \ldots, j_n)\) lives in an OM space of dimension \(\text{dim}(\alpha)\), but the actual physical content of that block may only occupy a small subspace of it. This is particularly valuable when the OM space is large (high tensor order or large spins): rather than storing a full vector across the entire OM space, Nicole introduces a weight matrix \(W\) with shape \((\text{dim}(r),\, \text{dim}(\alpha))\) where \(\text{dim}(r) \ll \text{dim}(\alpha)\). Each row of \(W\) is a vector in the OM space, and \(\text{dim}(r)\) tracks how many independent directions the physical content spans. This is the key motivation for the R-W-C structure.

For every block of an SU(2)-symmetric tensor, Nicole stores the physical content as a three-factor product:

j₁ j₂ ⋯ jₙ R r W α C m₁ m₂ ⋯ mₙ

R — Reduced Tensor

R is the reduced tensor, stored in Tensor.data[key]. Its shape is over the multiplet space — the size along each dimension equals the number of independent multiplets for the corresponding Index (i.e. Sector.dim, the outer multiplicity per index). R contains the physically meaningful content of the tensor: the reduced tensor elements themselves.

For a 2nd-order operator block (j, j), R is a scalar (or a small matrix if multiple multiplets are present). For a 3rd-order isometry or operator, R is a matrix over the multiplet dimensions of the free indices.

C — CG Basis

C is the Clebsch–Gordan basis tensor, computed on demand by the Yuzuha recoupling engine from the CGSpec stored in the Bridge. It has one edge per external Index (running over the \(2j+1\) magnetic quantum numbers) plus one axis for the outer multiplicity (OM) space. C encodes all fusion tree configurations that are consistent with the charge labels and total spin, in a canonical orthonormal basis.

C is delegated to a dedicated Yuzuha CG engine via the CGSpec stored in the Bridge.

W — Weight Matrix

W is the weight matrix, stored as Bridge.weights with shape \((\text{dim}(r),\, \text{dim}(\alpha))\):

  • om_dimension \(= \text{dim}(\alpha)\): the total number of linearly independent canonical basis states in the OM space — i.e. the number of distinct left-associative fusion-tree configurations for the given charge labels. This is computed by Yuzuha from the CGSpec.
  • num_components \(= \text{dim}(r)\): the number of independent directions the physical content spans in the OM space. This starts at 1 and can grow when tensors with overlapping OM structure are added or multiplied together.

Each row of W is a vector in the OM space, representing the coefficient of one component of the physical content projected onto the canonical CG basis.

Recoupling: X-Symbols and R-Symbols

When two SU(2) tensors are contracted or permuted, their Bridge objects must be combined:

  • X-symbol (compute_xsymbol): the recoupling coefficient needed when contracting two tensors along a shared bond. It transforms the product of two OM spaces into the OM space of the result.
  • R-symbol (compute_rsymbol): the recoupling coefficient needed when permuting axes of a tensor. It transforms the OM space under a reordering of the fusion tree.

Both symbols are computed by the Yuzuha engine and are cached in its database for efficiency. Nicole calls these internally — users never invoke X- or R-symbols directly.

Nicole's Implementation

Nicole stores the R-W-C structure transparently through two tensor attributes:

Attribute Content
Tensor.data[key] R — reduced tensor (PyTorch tensor, multiplet-space shape)
Tensor.intw[key] Bridge(cgspec, weights) — contains W and the CGSpec for C

The Bridge class (from nicole.symmetry.delegate) is Nicole's adapter to the Yuzuha engine:

from nicole.symmetry.delegate import Bridge

bridge = T.intw[(1, 1)]          # Bridge for block with charges (j=1/2, j=1/2)
print(bridge.om_dimension)       # OM dimension from yuzuha CGSpec
print(bridge.num_components)     # Number of rows in W
print(bridge.weights.shape)      # (num_components, om_dimension)

All operations in Nicole that involve SU(2) tensors (contract, trace, decomp, permute, conj, etc.) automatically update both R and the Bridge in a consistent way, using X- and R-symbols from Yuzuha. The user never needs to handle intertwiners directly.

Summary

Symbol Name Where stored Role
R Reduced tensor Tensor.data[key] Physically meaningful content; reduced tensor elements
W Weight matrix Bridge.weights Expansion of physical content in the OM canonical basis
C CG basis Bridge.cgspec Universal recoupling factor

The compression gain of SU(2)-symmetric tensors over explicit Abelian (U(1)-resolved) tensors comes entirely from replacing the full magnetic-substate block with the compact R-W-C representation. A block with spin labels \((j_1, \ldots, j_n)\) has \(\prod_\ell (2j_\ell + 1)\) physical components, but is represented by a single reduced tensor element (times the small W and C factors). The gain therefore grows rapidly with both the spin magnitudes and the tensor order.

See Also

References

[1] A. Weichselbaum, Non-abelian symmetries in tensor networks: A quantum symmetry space approach, Ann. Phys. 327, 2972 (2012).

[2] A. Weichselbaum, X-symbols for non-Abelian symmetries in tensor networks, Phys. Rev. Res. 2, 023385 (2020).

[3] S. Singh and G. Vidal, Tensor network states and algorithms in the presence of a global SU(2) symmetry, Phys. Rev. B 86, 195114 (2012).

[4] P. Schmoll, S. Singh, M. Rizzi, and R. Orús, A programming guide for tensor networks with global SU(2) symmetry, Annals of Physics 419, 168232 (2020).