SC: superconductivity intertwined with topological order

The gist: This work develops a classification framework for (2+1)D gapped superconductors by focusing on the topological properties of superconducting vortices, leading to five classes: invertible SC, SC∗, twisted SC∗, SC, and primitive SC.

Classification Framework

The paper proposes an organizing principle for classifying all (2+1)-dimensional SC phases with gapped 1,...,(2n−1)-body spectra based on the topological properties of vortices. This classification distinguishes between phases where superconductivity and topological order simply coexist and those where they are irreducibly intertwined. The five classes are defined by the factorization properties of the elementary vortex sector, denoted as C1, into a product involving the deconfined sector (Cdym).

The classification scheme includes:

** Invertible SC:**

An invertible SC is characterized by its elementary vortex sector being C inv1, meaning it is an invertible phase with trivial Cdym. For odd n, it has 16n topologically inequivalent classes labeled by υ ∈ Z16n.

(Untwisted) SC∗:

An untwisted SC∗ is defined if its elementary vortex sector is factorized as C inv1 ⊗ Cdym. This suggests it is a stacking of an invertible SC and a neutral TO, implying the SC and TO are topologically decoupled.

(Twisted) SC∗:

A twisted SC∗ has an elementary vortex sector that takes the form V inv1 ⊗ Cdym. This twist implies that when there is a nontrivial v, the fusion result of 2n identical Abelian elementary vortices yields a deconfined anyon w, rather than the vacuum or an electron.

SC:

An SC is defined if its elementary vortex sector cannot be factorized as C inv1 ⊗ Cdym or V inv1 ⊗ Cdym. This defines a new class where the superconducting order and topological order are irreducibly intertwined.

Primitive SC and Constraints

A primitive SC is defined as an SC where all of its ph/(2ne) vortex sectors cannot be factorized as C invp ⊗ Cdym or V invp ⊗ Cdym. This implies that every elementary vortex is necessarily non-Abelian. A minimal example for illustration is the charge-4e SC described by U(2)4,0 Chern–Simons theory.

The classification imposes no-go results on SC phases:

1. The elementary vortex sector has quantum dimension at least √6.

2. It cannot satisfy Fibonacci fusion rules.

Furthermore, an SC cannot arise by disordering the relative phases of lower-charge superconducting condensates, suggesting it can be a primary order rather than a vestigial order. A necessary condition to construct a primitive SC is that it must be constructed by condensing (g, 1) from a parent UMTC.

Unified Construction and Anyon Condensation

A unified construction framework is developed based on anyon condensation [53, 64]. Every SC can be constructed via anyon condensation from a parent UMTC D stacked with an atomic insulator sRep(U(1)).

The recipe involves:

1. Searching for an Abelian anyon g with order r, where g r = 1 and r is the smallest positive integer.

2. Stacking D with an atomic insulator labeled by local objects k carrying electric charge ke.

3. Condensing the bosonic composite (g, k), which yields a charge-2ne SC with 2n = kr.

The deconfined anyons correspond to those in D having trivial monodromy with g, and the ph/(2ne) vortices correspond to those with monodromy M = e 2πi kp/2n for p = 1, ···, r−1.

Relation to Physical Mechanisms

The paper relates the classification to physical mechanisms such as vestigial order and anyon superconductivity.

** Vestigial Order:**

Vestigial order is defined as a resulting order produced by the proliferation of vortex–anti-vortex pairs Σ(i)Σ¯(j) for all 1 ≤ i < j ≤ k that restores the discrete G symmetry. The paper argues that a primitive SC cannot arise from vestigial order of lower-charge SC, and should be a genuine primary order.

Anyon Superconductivity:

Anyon superconductivity is termed the algebraic construction of SC from condensing fractionally charged anyons in a spinC-SET. This mechanism shows that every SC can be constructed from condensing a composite bosonic object (g, k) in a parent UMTC D stacked with sRep(U(1)).

The vortex sector structure provides diagnostics for when an SC phase can arise from an anyon superconductivity mechanism. For example, to obtain an SC from an Abelian FQH state, one inevitably needs “non-minimal” stacking of a non-Abelian TO.

Examples

Explicit constructions are provided for all classes of generalized 16-foldway invertible SC, (Twisted) SC∗ and anyon superconductivity, two SC series from SU(2)2n CS theory, charge-2ne primitive SC series from SU(2n)2n CS theory, charge-4e primitive SC series from Spin(4m+2)4 CS theory, and a charge-6e primitive SC from B´ezout’s identity.

The charge-6e primitive SC example from B´ezout’s identity is constructed by condensing [(0, 3), 3] in Sp(4)3. This state is described as the charge-6e primitive SC with smallest total quantum dimension.

The paper concludes that nonfactorizable vortex sector structure is a robust organizing principle for higher-charge topological superconductivity. It also suggests asking whether an analogous notion of SSB can be formulated for other symmetry-breaking phases with intrinsic topological order.

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The paper establishes that the elementary h/(2ne) vortex sector is a basic piece of topological data of the superconducting phase. It shows that this sector provides a sharp way to distinguish superconducting phases in which superconductivity and TO simply coexist from those in which they are irreducibly intertwined.

The classification organizes superconductors into invertible SC, untwisted and twisted SC∗, and SC phases. An invertible SC phase has each vortex sector composed of either two Abelian vortices related by fusing an electron, or a non-Abelian Ising vortex invariant upon such a fusion. An SC∗ or twisted SC∗ phase has the elementary vortex sector factorizes into an invertible superconductor vortex sector dressed by deconfined anyons, possibly with a twist. In contrast, an SC phase is defined by the failure of this factorization. In such a phase, the non-Abelian-ness of the elementary vortex is not an optional dressing by hidden topological order; it is intrinsic to the topologically ordered superconductor itself. This nonfactorization gives practical rank and quantum dimension diagnostics for identifying SC from its deconfined anyon and elementary vortex data.

The unified anyon condensation construction turns the characterization of superconducting vortices into a computable problem in the parent UMTC. The same framework also clarifies the relation to anyon superconductivity mechanism achieved by condensing fractionally charged anyons in a spinC-SET.

The paper illustrates that nonfactorizable vortex sector structure is not an isolated phenomenon, but a robust organizing principle for higher-charge topological superconductivity.

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The paper further identifies primitive SC, for which all nontrivial vortex sectors exhibit nonfactorization, not only the elementary one. Primitive SC are argued to be primary orders, not able to arise as vestigial orders, i.e., by disordering relative phases of lower-charge condensates. Examples include proposed series from SU(2n)2n and Spin(4m+2)4, as well as a charge-6e example by applying B´ezout’s identity.

The construction reproduces the generalized 16-fold way superconductors and SC∗ phases, and produces SC series from Chern–Simons theories such as SU(2)2n.

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The paper illustrates the framework with several classes of examples.

1. Bold header: Improved classification framework for topological superconductivity phases

This framework allows AI systems to precisely classify (2+1)D gapped superconductors based on their vortex properties, distinguishing between invertible SC, (Untwisted) SC∗, Twisted SC∗, and the central class SC.

2. Bold header: Novel diagnostics for identifying primitive SC phases

AI systems can now use specific criteria to identify if a charge-2ne superconducer is a primitive SC, such as checking if the elementary vortex sector cannot be factorized as Cinv1 ⊗ Cdym or Vinv1 ⊗ Cdym.

3. Bold header: Predictive power for topological order constraints

The framework provides no-go results that AI can use to constrain physical realizations, specifically noting that for SC phases, the elementary vortex sector has quantum dimension at least √6 and cannot satisfy Fibonacci fusion rules.

4. Bold header: Systematic construction via anyon condensation

AI systems can utilize the Unified construction from UMTC framework to systematically generate infinite families of examples by condensing simple currents, allowing them to construct SCs from a parent topological order through the recipe summarized in Section V.

5. Bold header: Linking superconductivity to anyon superconductivity mechanisms

The system can determine whether an SC arises from vestigial order or anyon superconductivity by examining the structure of its vortex sectors, as these mechanisms are constrained by which vortex sectors admit factorization.

6. Bold header: Analysis of topological hierarchy in charge series

AI systems can analyze the charge hierarchy derived from anyon condensation, understanding how condensing different simple currents generates infinite series of charge-kre SCs with increasing k, and how this process can sometimes sharpen an SC by eliminating Abelian vortices not mutually local to that algebra.

7. Bold header: Validation of non-Abelian vortex contributions in specific systems

By analyzing the fusion rules (e.g., Appendix A for SU(4)4), AI can verify the presence of non-Abelian vortices and determine their quantum dimensions, which is crucial for understanding their potential role in universal quantum computing.

Superconductivity and intrinsic topological order can intertwine in ways that go beyond simple coexistence. Here we develop a classification framework for (2+1)D gapped superconductors with condensate charge 2ne. By focusing on the topological properties of superconducting vortices, we precisely characterize this interplay. We obtain a classification with increasing levels of intertwinement, from the well known invertible superconductors without topological order, to untwisted and twisted SC, and finally to SC. Our main finding is the family SC, in which the elementary vortex carrying h/(2ne) flux cannot be factorized into an invertible superconductor vortex dressed by bulk anyons, and every elementary vortex is necessarily non-Abelian. For such phases, we also obtain no-go results: the elementary vortex sector has quantum dimension at least sqrt 6 and cannot satisfy Fibonacci fusion rules. We develop a systematic algebraic framework to obtain all these superconducting phases through anyon condensation, allowing us to construct several infinite families of examples and make connections to anyon superconductivity. Finally, we introduce primitive SC phases for which factorization of every nontrivial vortex sector is obstructed, and argue that they cannot arise by disordering the relative phases of lower-charge superconducting condensates. Our classification scheme based on vortex properties allows us to uncover new distinctions between topologically ordered superconductors and constrain how they can be realized.