Simple algorithm for evaluating SU(2) non-abelian geometric phase in the vicinity of molecular conical intersection of odd electron number

Core Contribution

The paper gives a closed-form, integration-free algorithm for computing the SU(2) non-abelian geometric phase (holonomy) along a discretized closed trajectory in the 5-dimensional branching space of a T²=−1 conical intersection, i.e. an odd-electron molecule with Kramers degeneracy. The key idea is that the Yang-monopole-like connection is abelian when restricted to any 3D S² subspace. The author inserts the north pole as an extra vertex and decomposes the S⁴-projected trajectory into a chain of spherical triangles. Each triangle contributes exp(−D̂σ/2) for the upper state or exp(−B̂σ/2) for the lower state. The area σ comes from the spherical SAS formula. The Berry connection factors D̂ and B̂ come from quaternion expressions of the longitudes. The total phase is a path-ordered product. The method needs no numerical integration of the connection and no explicit gauge transformations.

A second result relates the upper and lower state phases. Φ⁺ of a trajectory equals Φ⁻ of its inversion image on the opposite chart. The Wilson loop (the trace) is invariant under proper SO(5) transformations and swaps between states under improper ones. Trajectories with improper symmetry therefore have W⁺=W⁻. The examples are 5-simplex and 5-demicube trajectories. On the simplex, Wilson loop values split cleanly by permutation parity. On the demicube, the examples show path dependence and non-commutativity.

Methodological Rigor

The derivation is logically clean and builds on the author's earlier Yang-monopole paper. The abelianization on S² and the use of Stokes' theorem within charts, with a bridging segment λ to handle the chart transition, are carefully argued. The augmentation step relies on the pole-directed arcs contributing identity, which follows from A_θ=0. The parity result on the simplex is a nontrivial internal consistency check that the symmetry theorems predict correctly.

Weaknesses:

Potential Impact

The immediate audience is small: theorists working on geometric phases in molecular conical intersections with spin-orbit coupling, plus synthetic gauge field and Yang monopole communities (cold-atom work by Spielman et al.). The algorithm applies to any Hamiltonian of the form of Eq. 6, so it could serve holonomic quantum computing and the study of non-abelian Wilczek–Zee phases in quantum control. The paper suggests uses such as separating geometric forces from dynamical effects in non-adiabatic dynamics and designing molecular qubits. These are speculative at this stage. The Wilson loop symmetry relations (Eq. 55–56) are elegant and could guide interpretation, but they are of limited practical use without real-system data.

Timeliness & Relevance

Interest in non-abelian gauge structures, Yang monopoles and topological classification of non-adiabaticity in chemistry is rising. Spin-orbit-coupled non-adiabatic dynamics are a growing area, but the field is still bottlenecked by the lack of electronic structure tools for T²=−1 branching spaces. The paper addresses a computational gap that practitioners will only face once those tools mature, so adoption may lag.

Strengths

Limitations

Other Observations

The reproducibility is good for the algorithm since formulas are explicit. The examples are artificial, so the results are not surprising: they confirm theorems. The novelty is moderate, being a clever decomposition rather than a new paradigm. The impact will likely be modest, with citations from topological/gauge-theoretic chemical physics and perhaps synthetic gauge field work.