Topological Charge-2ne Superconductivity and Clustering Hierarchy

The gist The authors develop a general framework for topological charge-2ne superconductors by constructing these states via two complementary routes: by generalizing the Read–Green construction from charge-2e ingredients, and by breaking the charge U(1) symmetry in certain fractional quantum Hall states, showing that they can host intrinsic fermionic non-Abelian topological orders beyond those of charge-2e topological superconductors.

General Framework and Construction Routes

The paper constructs these states via two complementary routes: by generalizing the Read–Green construction from charge-2e ingredients, and by breaking the charge U(1) symmetry in certain fractional quantum Hall states (Page 1). They derive corresponding edge conformal field theories and bulk topological quantum field theories from the constructed wavefunctions using bulk–edge correspondence (Page 1).

Construction via Read–Green Wavefunction Generalization

The construction based on Read–Green ingredients involves generalizing the wavefunction by introducing pairing between each pair of electrons within the 2n-electron cluster, leading to a wavefunction defined as F(z1, · · ·, z2n) = Y i<j≤2n 1 zi − zj (Page 3). This leads to an edge CFT where the electron operator ψ(z) must satisfy constraints such that "fusion results of q < 2n electron operators cannot contain the identity operator to prevent the charge-2e instability and the electron operator should have half-odd integer conformal weight and be identified as the transparent fermion" (Page 3). The simplest choice is the SU(2n)2n Wess–Zumino–Witten (WZW) CFT, which results in a U(2n)2n,0 spin CS theory with chiral central charge c− = 2n/4πtrada + 2/3a3 - 1/4π(tr a)d(tr a) - 1/2π(tr a)dA (Page 3).

Construction via Symmetry Breaking in Fractional Quantum Hall States

Another route involves breaking the charge conservation symmetry from U(1) to Z2n in fractional quantum Hall states such as the 2n-cluster Read–Rezayi state (Page 3). A straightforward attempt to remove the Laughlin–Jastrow factor fails because the charge conservation symmetry is not broken [82], and a residual Hall conductance is present due to the residual Laughlin–Jastrow factor in the third line (Page 3). To achieve a superconducting state, one needs to eliminate this factor, leading to a wavefunction proportional to Qn q=1 χ(q) 1/(zi) (Page 3). This is related to the U(2)⊗n2n,−4n CS theory with a physical electron identified as the transparent fermion (Page 3).

Topological Order and Anyon Content

The main result is that charge-2ne superconductivity is not merely a higher-charge analogue of conventional charge2e topological superconductivity but can support intrinsic fermionic non-Abelian topological order beyond charge-2e ones, with deconfined bulk anyons and nonAbelian superconducting vortices (Page 3). For the n = 1 case, the theory reduces to the (p + ip)-wave topological superconductor with c− = 1/2 (Page 3). In the n = 2 case constructed from generalizing Read–Green, Table I summarizes the anyon content for a charge-4e superconductor, where the superconducting vortices in this state have irrational quantum dimensions 2 + √2 and 2 + 2√2 (Page 3).

Spinful Systems and Higher Symmetries

When incorporating spin, the analysis leads to various states. For the spinful charge-4e superconductor, there are three inequivalent states in the fully symmetric spin-2 sector with Sz = ±2, ±1, 0 (Page 5). The Sz = ±1 state can be realized using a construction based on the Z3×Z6 parafermion CFT, resulting in an edge theory that is (G2)1×SO(3)3 spin CFT (Page 5). Furthermore, extending the symmetry to SU(4) or Sp(4) allows for quartetting operators with up to four flavors, leading to complex edge CFTs such as Spin(8)1 CFT for the 4-flavor case (Page 5).

Experimental Diagnostics and Conclusion

The work suggests concrete experimental diagnostics beyond fractional flux quantization, implying that the bulk TQFT and edge CFT descriptions imply characteristic signatures in quasiparticle and vortex interferometry (Page 7). Specifically, the non-Abelian anyons and superconducting vortices identified here suggest concrete targets for interferometric probes of the non-Abelian fusions (Page 7). The paper concludes that these diagnostics can distinguish topological charge-2ne superconductors from both conventional charge-2e topological superconductors and nontopological charge-2ne states (Page 8).

Derivation Details

The derivation of the bulk TQFT involves several steps, including gauging the Z2n center symmetry, which in some cases leads to a theory described by LPF 2ne[A] = 1/4 Xq,q' <p a˜qKqq'd˜aq' - 1/2π Xq<p 2mqa˜qdA! - 2n/2π a˜pdA + 1/4π AdA (Page 9). The final result for the n = m case is a U(2)2n,0 CS TQFT with chiral central charge c− = (2n 2-1)/(n 2+1) and no residual Abelian topological order (Page 9).

Spectrum of Topological Charge-2ne Superconductors

The spectrum analysis shows that the superconducting vortices in the state described by S[U(k)⊗p]2n,0 CS theory can have quantum dimensions 1, 2, or 3 (Page 9). In the case of S[U(4)4,0] TQFT, static superconducting vortices described by Ramond primaries with −1 monodromy to the transparent fermion are organized into three orbits (Page 12). The quantum dimensions of these vortices are noted to be 2 + √2 (Page 12).

Operator Content for Higher Flavors

For the 3- and 4-flavor topological charge-4e superconductors, the edge theory is described by SU(2)1 × SU(6)1 × SU(6)2 WZW CFT (Page 13). The primaries in the NS sector are organized into two orbits with length 12 generated by ψα,β,γ fields (Page 13). The Ramond sector primaries have quantum dimensions p squared + √p squared and 4 + 2√2 (Page 13). For the four-flavor quartetting operators, one construction based on SU(2)⊗3/2 × SU(3)3 WZW CFT yields a Spin(8)1 CFT (Page 13). The Ramond primaries in this construction have quantum dimension 2√2 (Page 13).

Alternative Constructions

Alternative constructions include replacing the Z3 parafermion sector with WZW CFTs, such as SU(3)2 or (E6)1, which can lead to an (F4)1 sector instead of (G2)1 for the Fibonacci primary in some cases (Page 13). An invertible topological charge-2ne superconductor is also possible, constructed from SU(10)⊗2/1. Here, the spin up and spin down electrons are identified as (J3, J) and (J, J7), where J is the simple current of SU(10)1 (Page 13). This state can be viewed as an invertible topological charge-4e superconductor stacked with a bosonic Abelian residual topological order (Page 13).

Summary of Key Findings

The paper provides a unified low-energy description of topological charge-2ne superconductors by bridging ideas from fermionic symmetry-protected and symmetry-enriched topological orders, non-Abelian quantum Hall states, and unconventional superconductivity (Page 8). The core finding is that these phases can support intrinsic fermionic non-Abelian topological order beyond charge-2e ones (Page 3). This framework suggests experimental diagnostics through quasiparticle or vortex interferometry probing the non-Abelian fusions (Page 1). Key results include the construction of bulk TQFTs from wavefunctions and the identification of characteristic signatures in edge CFTs, such as (G2)1×SO(3)3 spin CFT for certain spinful states (Page 5). The analysis also details the anyon content and quantum dimensions of superconducting vortices in these states (Page 12). This unified framework offers a controlled setting for studying symmetry breaking and enrichment in interacting topological phases of matter (Page 1).

Acknowledgment

The authors acknowledge Xiao-Gang Wen, Ashvin Vishwanath, Eslam Khalaf, Srinivas Raghu, Yi Zhang, Zhehao Dai, Zhaoyu Han, Taige Wang, Pavel Nosov, Clemens Kuhlenkamp and especially Ya-Hui Zhang for helpful discussions (Page 1). The paper is supported by the Berkeley graduate program (Page 1). The authors contributed equally (Page 1).

References

[1] M. Gell-Mann, Phys. Lett. 8, 214 (1964) [Page 8].

[2] J. A. Wheeler, Phys. Rev.

1. The AI system can develop novel topological quantum field theories (TQFTs) to describe charge-2ne superconductors by constructing them from elementary charge-2e ingredients of the Read–Green wavefunction, bridging fermionic symmetry-protected and symmetry-enriched topological orders.

2. The system can perform concrete experimental diagnostics by simulating quasiparticle or vortex interferometry probing the non-Abelian fusions and designing geometries sensitive to the underlying Z2n structure of the superconductor.

3. The AI can generate detailed topological classifications for spinful charge-2ne superconductors, including identifying wavefunctions based on spin sectors such as the Sz = ±1 state, allowing for precise prediction of their edge CFTs like (G2)1 × SO(3)3.

4. The system can predict the anyon content and quantum dimensions of topological states by analyzing the fusion rules derived from the bulk TQFTs, specifically identifying that non-Abelian superconducting vortices in this state have quantum dimensions 2 + √2 and 2 + 2√2.

5. The AI can construct complex, high-dimensional edge CFTs for higher-flavor systems (e.g., the four-flavor quartetting operators), such as the one based on SU(3)1 × SU(6)1 × SU(6)2 WZW CFT, enabling the prediction of specific operator content and their corresponding topological spins.

Charge- 2ne superconductors condense clusters of more than two electrons, extending superconductivity beyond Cooper pairing. They exhibit distinctive signatures including fractional flux quantization h/(2ne) and are actively being explored in strongly correlated systems. We develop a general framework for constructing and characterizing topological charge- 2ne superconductors by combining wavefunction and field theory approaches. Generalizing the Read-Green wavefunction and Higgsing Read-Rezayi-type states yield two families that can support deconfined non-Abelian anyons and non-Abelian superconducting vortices. The first family also admits a construction from doping Jain states. Through Higgs theories, we find that the first and second families realize weak and intermediate clustering regimes, respectively, distinguished by the topology of vortices carrying flux greater than h/(2ne), while strong clustering yields a charge- 2ne superconductor without intrinsic topological order. These reveal that higher-charge superconductivity supports a richer hierarchy of topological phases beyond the Read-Green weak-strong pairing dichotomy, with distinct internal clustering structures giving rise to different topological structures at fixed condensate charge.