How does superconductivity intertwine with topological order in SC-sharp phases?

Vortex factorization defines the degree of intertwinement

The paper's answer is a classification, not a single material prediction. It treats a superconductor as a spontaneous breaking of charge U(1) down to a residual Z_2n symmetry containing fermion parity, then asks how the resulting vortices relate to the deconfined bulk anyons. The organizing test is whether the elementary vortex sector, the set of topologically distinct vortices carrying one flux quantum h/(2ne), can be written as a product of an invertible-superconductor vortex sector and the dynamical anyon sector.

That test produces a ladder. Invertible superconductors have a trivial dynamical anyon sector. Untwisted SC* factorizes as C_inv_1 times C_dym, matching an invertible superconductor stacked with a neutral topological order. Twisted SC* factorizes as V_inv_1 times C_dym, where an Abelian object v satisfies v^2n = w, a nontrivial element of C_dym. SC-sharp is the case where neither factorization works, which is what the authors mean by irreducible intertwinement.

Why SC-sharp vortices must be non-Abelian

The central finding is the SC-sharp family itself. In these phases the elementary vortex carrying h/(2ne) flux cannot be decomposed into an ordinary invertible-superconductor vortex plus bulk anyons, and every elementary vortex is necessarily non-Abelian. The authors also derive no-go results for this sector: its quantum dimension is at least sqrt(6), and it cannot satisfy Fibonacci fusion rules.

The quantum-dimension bound is the sharpest reader-facing discriminator. A sector with quantum dimension at least sqrt(6) is not a collection of Abelian vortices, and it is not the Fibonacci case either. The paper notes that anyon permutation by the residual Z_2n symmetry is sufficient but not necessary for the nonfactorization, so SC-sharp is broader than a symmetry-permutation story.

Anyon condensation generates every phase in the ladder

The construction is algebraic. Start with a parent unitary modular tensor category D stacked with an atomic insulator, which supplies integer-charged local excitations labeled by charge ke. Find an Abelian anyon g of order r, then condense the bosonic composite (g, k); this yields a charge-2ne superconductor with 2n = kr. Deconfined anyons are those with trivial monodromy with respect to g, while the ph/(2ne) vortices are those with monodromy exp(2 pi i kp/(2n)).

The authors argue this recipe is exhaustive: every (2+1)-dimensional superconductor in their setting can be obtained by condensing a single self-bosonic anyon from a parent topological order. That claim is what lets a vortex-based classification double as a generative framework, and it connects the classification to anyon superconductivity in the stack-and-condense picture.

The SU(2)_2n series shows an even-odd split

The clearest worked family condenses (j = n, n) in SU(2)_2n stacked with an atomic insulator. For n > 1 this produces charge-2ne SC-sharp phases, described by a U(2)_{2n,0} Chern-Simons theory. The rank bookkeeping differs by parity of n, and that difference is the evidence for the SC-sharp label.

For even n, the j = n/2 anyon is fixed by the Z_2 one-form symmetry and falls into the dynamical anyon sector, splitting into two deconfined anyons; the dynamical sector has rank (n+4)/2 while the elementary vortex sector has rank n. For odd n, the condensed current is identified with the physical fermion rather than the vacuum, so j = n/2 does not split; it binds a Majorana zero mode and returns to itself upon fusing an electron, similar to the vortex of a charge-2e (p+ip)-wave superconductor. The dynamical and elementary vortex sectors then have ranks (n+1)/2 and n. In neither parity case does the elementary vortex sector have the rank relation expected of a factorizable phase.

Twisted SC* needs non-Abelian anyons or higher charge

The paper separates twisted SC* from untwisted SC* with explicit constructions. A charge-2e superconductor obtained by condensing a Cooper pair in a Laughlin-1/3 state is untwisted: its elementary vortex sector factorizes as C_inv_1 times C_dym, and its vortices are Abelian. A genuine twisted SC* requires either non-Abelian deconfined anyons or a condensate charge greater than 2e.

The charge-4e example makes the higher-charge route concrete. Condensing (g^8, 2) in U(1)_16 yields a charge-4e superconductor with U(1)_4 topological order; every choice of elementary vortex v satisfies v^4 = g^4, a nontrivial anyon, so the sector is twisted. For charge 2e, condensing [(psi, g^4), 1] in Ising times U(1)_8 gives a twisted SC* whose elementary vortex fuses with itself to (1, g^2), which cannot be trivialized by attaching an Abelian anyon. The authors describe this as stacking a trivial charge-2e superconductor on a Pfaffian state, and note that only a neutral topological order necessarily gives untwisted SC*.

Stacking topological order can collapse invertible distinctions

One result changes how readers should compare superconductors by charge alone. A trivial charge-8e superconductor stacked with a neutral U(1)_8 topological order becomes an SC*. Through relabeling V' = g times V and g' = g^3 times f, the same state can be viewed as a nontrivial charge-8e invertible superconductor in class upsilon = 8 of the Z_16 classification, whose elementary vortex has topological spin 1/16, stacked with a neutral U(1)_8 topological order.

So two inequivalent charge-8e invertible superconductors become topologically equivalent charge-8e SC* once the same U(1)_8 topological order is stacked. The condensate charge and the invertible classification do not by themselves fix the topological type; the vortex sector does.

Primitive SC-sharp constrains vestigial-order mechanisms

The authors introduce primitive SC-sharp phases, in which every nontrivial vortex sector obstructs factorization, not just the elementary one. They argue these cannot arise by disordering the relative phases of lower-charge superconducting condensates. The argument uses the smallest r dividing 2n for which the rh/(2ne) vortex sector factorizes: it indicates the coexisting topological order is enriched only by the Z_r subgroup of the residual Z_2n symmetry, suggesting the state can be viewed as bound charge-ke clusters with k = 2n/r.

The consequence is a mechanism constraint. A state that would have to come from partially melting a multicomponent superconductor with condensate charge greater than ke is excluded. In the extreme case k = 1, the SC-sharp phase cannot arise as a quantum vestigial order at all and could only be a primary order, which is why the authors call it primitive. The paper constructs primitive series from SU(2n)_2n, Spin(4m+2)_4, and a charge-6e example using Bezout's identity.

What the classification changes for comparing superconductors

The practical priority is to classify by the elementary vortex sector rather than by condensate charge or by whether topological order is present. Two superconductors with the same 2ne charge can sit at different rungs: untwisted SC*, twisted SC*, or SC-sharp. The factorization test, the quantum-dimension bound of at least sqrt(6), and the exclusion of Fibonacci fusion rules give concrete checks that separate SC-sharp from the SC* cases.

The scope is gapped (2+1)-dimensional superconductors with condensate charge 2ne and gapped 1 through (2n-1)-body spectra, with Goldstone modes gapped by a weak symmetry-breaking field. Within that scope, the paper's distinctive claim is that SC-sharp phases are not merely superconductors stacked with topological order: their elementary vortices are irreducibly non-Abelian, and primitive SC-sharp phases additionally rule out a vestigial-order origin.

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