Geometry of Cooper Pairs in Momentum Space: Monopole and Spinor Superconductivity

When the geometry of electronic wave functions reshapes superconductivity -- Conventional superconductivity is described by Cooper pairs characterized by relatively familiar angular-momentum symmetries. In topological quantum materials, however, the electronic states themselves can possess nontrivial momentum-space geometry. Pairing electrons living on such states can force the superconducting order parameter into forms that have no direct counterpart in ordinary superconductors.

    Our work develops this idea through monopole superconductivity, topological nodal pairing, spinor pairing, and phase-sensitive probes of unconventional order.

Scheme of Josephson phase sensitive detection of the monopole charge of Cooper pairs.

Monopole superconductivity

Topology of the normal-state electronic wave functions can enforce topology of the superconducting order itself. The work Topological Nodal Cooper Pairing in Doped Weyl Metals, in collaboration with F. D. M. Haldane, showed that pairing between Fermi surfaces carrying nonzero Berry flux acquires a topological obstruction in momentum space. The superconducting gap cannot be globally represented as an ordinary scalar function over Fermi surface in momentum space; instead, its structure is governed by monopole harmonics.

This geometric structure also extends beyond Cooper pairing. With Eric Bobrow and Canon Sun, we developed monopole charge-density-wave order in Weyl semimetal systems, demonstrating that particle-hole condensates can inherit analogous momentum-space topology. This broadened the research program from a theory of one exotic superconductor to a more general theory of many-body order parameters defined over topologically nontrivial electronic states.

How can monopole superconductivity be detected?

A major challenging question in monopole ordering was the experimental identification of monopole pairing order. In Designing Phase Sensitive Probes of Monopole Superconducting Order (Physical Review Research, 2024), our group developed combined topology and symmetry-based Josephson experimental proposal to distinguish monopole superconductivity through two sets of phase-sensitive measurements.

The work pushes the monopole superconductivity from academic interest toward experimentally falsifiable detection signatures of unconventional superconducting order.

Spinor pairing: when the order parameter transforms like a quantum spinor

A further development is Spinor Pairing Order Enforced by Berry Phase (Physical Review Research – Letter, 2026). Here Berry-phase structure enforces an even more unusual order parameter whose transformation properties are spinor rather than those of a conventional scalar superconducting gap.

Together, these works ask a general question:

Which fundamentally novel many-body quantum states can we have when quantum geometry is built into the underlying electronic states?

Representative works

Topological Nodal Cooper Pairing in Doped Weyl Metals

Yi Li, F. D. M. Haldane

Physical Review Letters 120, 067003 (2018)

Monopole Charge Density Wave States in Weyl Semimetals

Eric Bobrow, Canon Sun, Yi Li

Physical Review Research 2, 012078(R) (2020)

Designing Phase Sensitive Probes of Monopole Superconducting Order

Grayson R. Frazier, Junjia Zhang, Junyi Zhang, Xinyu Sun, Yi Li

Physical Review Research 6, 043189 (2024)

Spinor Pairing Order Enforced by Berry Phase

Yi Li, Grayson Frazier

Physical Review Research – Letter 8, L022037 (2026)