Theory of Topologically Ordered Superfluids in 2+1 Dimensions

arXiv:2610.10668 · cond-mat.str-el, cond-mat.supr-con, hep-th · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Theory of Topologically Ordered Superfluids in 2+1 Dimensions".

Mira: The gist:

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So we’re looking at this paper today, "Theory of Topologically Ordered Superfluids in two plusone Dimensions," and it's about how to characterize these exotic phases where you have these deconfined anyons but they still have that gapless mode because of the Goldstone modes <ref:2610.10668#pg1,Theory of Topologically Ordered Superfluids in 2+1 Dimensions>.

Mira: Exactly. It sets up a general algebraic framework for these two-plus-one dimensional bosonic systems that spontaneously break a global U(1c) charge symmetry <ref:2610.10668#pg1>. It’s essentially building a systematic way to look at the topological data of these TOSFs, which is really about how the system organizes itself topologically.

Lev: From an error correction standpoint, that mapping idea—going from the gapless TOSF phase C to a gapped topological order D by gauging—that sounds like a very powerful tool for understanding what's actually going on at the microscopic level.

Kai: Right, so the core idea here is this two-way mapping between the TOSF C and a dual U(one)dual-enriched TO D, where gauging U(1c) maps C to a fully gapped TO D that has some extra structure from a dual symmetry <ref:2610.10668#pg1,a dual U(1)dual>.

Mira: That enrichment is determined by just two things: the anyon generated by the two pi symmetry flux, which they call the vison, and the Hall conductance sigma xy <ref:2610.10668#pg1>.

Lev: If we’re talking about real hardware, knowing that sigma xy constrains the vison's topological spin theta = e i pi sigma xy is a very concrete piece of information we can actually measure or simulate.

Kai: And if you have a U(one)dual-enriched TO D where the Hall conductance vanishes, that implies the vison in D has bosonic self-statistics, which is a specific constraint on the dual theory <ref:2610.10668#pg1>.

Mira: The two-way mapping means that all the topological data of that TOSF C—like how many anyons are there, what’s the condensate charge—can be systematically encoded by looking at this dual TO D.

Lev: That systematic encoding sounds like something useful for designing experiments because it gives us a clear target structure to aim for when we build things.

Kai: And they show that superfluid vortices in the original TOSF C share the same topological properties as the flux defects from a residual discrete symmetry in that gapped anyon sector.

Mira: That means the vortex structure isn't completely separate from the anyon content; it’s intrinsically tied to those discrete symmetries left over after you gauge down to a simpler system.

Lev: So, if we want to study vortices, we don't have to treat them in isolation; we can map their properties onto something that is easier to handle in the gapped sector of the dual theory.

Title and authors: Kai: They also provided a field-theoretic formulation, treating Abelian and non-Abelian TOs separately. For a generic Abelian one, they use a Lagrangian involving X N terms and coupling to the background one-form gauge field A <ref:2610.10668#pg1>.

Mira: And when you gauge that dual U(one)dual symmetry, the Lagrangian changes into something like LCA which adds a term involving alpha and the Hall conductance sigma xy <ref:2610.10668#pg1,dual U(1)dual symmetry>.

Lev: That’s where I start thinking about implementation; how do we translate those field equations into something that runs on actual quantum hardware? Does it map to any standard Hamiltonian we can actually program?

Kai: They give an example with a charge-four TOSF featuring non-Abelian vortices, and they use a parton construction where the boson splits into two parts, phi and chi, which breaks the U(1c) down to its Z four subgroup.

Mira: This specific example shows how you can construct something with non-Abelian vortices that fits this framework, identifying the elementary vortex of C with a two pi/four flux of A in D/A.

Lev: That parton construction is interesting because it gives us a way to think about the fundamental excitations without just looking at the bulk topological order. It connects the microscopic building blocks to the macroscopic topological features.

Kai: They also touched on a phase transition prediction, suggesting that as the chemical potential mu changes sign from negative to positive, you naturally transition out of this TOSF phase into a gapped phase where all condensates vanish, meaning = zero <ref:2610.10668#pg2>.

Mira: That transition is mapped onto an Abelian-anyon proliferation transition in the dual theory D. It suggests that the gapless nature of the TOSF is tied to some kind of condensation that dissolves when you cross this critical point.

Lev: If we are trying to build a robust quantum computer, knowing exactly where that phase boundary is—where it goes from condensed to gapped—that helps us define our operating regime.

Kai: They also showed how stacking the TOSF C with a BIQH state of Hall conductance p not equal to zero induces a new dual TO D', which encodes the same data but with a different vison order, n, and the topological order is D/A <ref:2610.10668#pg2>.

Mira: That means you can stack these systems in specific ways to generate different types of topological orders while keeping the underlying physical properties of C intact, just changing how they are described by D'.

Title and authors: Lev: So, if we want to engineer a specific type of anyon physics, we might be able to achieve that by choosing the right stacking configuration or Hall conductance state.

Kai: To wrap up this discussion on "Theory of Topologically Ordered Superfluids in two plusone Dimensions," the paper provides a very systematic way to encode all the topological data of a TOSF C into its dual U(one)dual-enriched TO D through gauging U(one)dual <ref:2610.10668#pg1,Theory of Topologically Ordered Superfluids in 2+1 Dimensions>.

Mira: The real value here is that it connects the complex physics of these gapless systems directly to established concepts in symmetry-enriched topological orders, giving us a unified language for anyon content and vortex properties.

Lev: For me, what this means practically is that we get a clear blueprint for how to design experiments or simulations targeting these specific topological states.

Kai: And they also provided those complementary field-theoretic constructions, which are essential because they connect the abstract algebra to more concrete models involving parton constructions and non-Abelian vortices.

Mira: They also point out that while vortices share topological properties with discrete symmetry fluxes, they don't necessarily share the same energetic properties, which is a subtle but important distinction for understanding their dynamics.

Lev: That distinction is crucial; it tells us that even if the topology matches up, the physical behavior—how they move or interact—can be very different based on those energetic constraints.

Kai: Overall, this work gives us a way to systematically characterize the topological data of TOSFs while naturally admitting a field-theoretic formalism that connects more directly to microscopic constructions.

Mira: The future direction they suggest is clear: extending this framework to fermionic topologically ordered superconductors and phases where you have a general continuous global symmetry G broken down to a subgroup H.

Lev: That extension to continuous symmetries is huge; it opens the door to studying phases that aren't just constrained by discrete subgroups, which are often more relevant in real materials.

Kai: So, in summary, the paper provides a streamlined and systematic characterization of all the topological data of a TOSF C by mapping it onto its dual U(one)dual-enriched TO D via gauging U(one)dual <ref:2610.10668#pg1>.

Mira: This is powerful because it allows us to see how anyon content and vortex properties are systematically encoded in this dual structure, which is a big step forward for characterizing these systems.

Lev: I think the field-theoretic approach further suggests that we can predict a natural phase transition from the TOSF C to a gapped phase with the same intrinsic TO as D when the chemical potential mu changes sign from negative to positive.

Title and authors: Kai: That prediction of an Abelian-anyon proliferation transition is something we can definitely test, and it gives us a target for what we expect to see in any measurement that probes that transition.

Mira: So, while the paper does a lot of heavy lifting on the algebraic side, it also provides concrete examples with non-Abelian vortices using parton constructions and field theory.

Lev: Those examples are important because they give us tangible targets to aim for when we start trying to build these complex systems in a lab environment.

Kai: We get a holistic picture here: the algebra gives us the structure, the dual mapping gives us the encoding, and the field theory connects it to real microscopic models.

Mira: It's a very complete picture for understanding how these gapless systems organize themselves topologically when they break that charge symmetry.

Lev: For anyone listening who isn't deep in topological field theory, the main point is that there’s a rigorous way to translate the properties of a superfluid into something we can analyze using established tools from condensed matter physics.

Kai: So, to wrap up on "Theory of Topologically Ordered Superfluids in two plusone Dimensions," this paper gives us that two-way map between the TOSF C and its dual TO D <ref:2610.10668#pg1,Theory of Topologically Ordered Superfluids in 2+1 Dimensions>.

Mira: It’s a systematic encoding of condensate charge, deconfined anyon content, and vortex properties using symmetry-enriched topological orders.

Lev: The field theory also gives us a natural prediction for the transition into a gapped phase when you change the chemical potential mu.

Kai: It suggests that stacking systems with specific Hall conductance states can generate different versions of this topological order, which is a neat trick for generating new phases.

Mira: And the authors point out that while vortices and discrete symmetry defects share topology, their energy properties are not necessarily the same, which is a subtle physical point we have to keep in mind.

Lev: We need to keep those energetic constraints in mind because they dictate how these vortices actually behave in a system.

Kai: The paper also points toward extending this framework to fermionic systems and continuous global symmetries, which gives us much more scope for future work.

Mira: It’s a very ambitious roadmap for how we can apply this topological characterization to broader classes of quantum matter.

Lev: It feels like a solid foundation now; we have the mapping, the constraints from the dual theory, and some concrete field theory examples to start building on.

The paper's summary: Kai: So we’ve been looking at this paper today, "Theory of Topologically Ordered Superfluids in two plusone Dimensions," and it’s about building a way to systematically describe how these gapless topological phases work.

Mira: Right. It sets up this general algebraic framework for these two-plus-one dimensional systems that spontaneously break a global U(1c) symmetry, which is essentially the charge symmetry breaking you see in some exotic superconductors <ref:2610.10668#pg1>.

Lev: I’m thinking about the core idea here—the mapping from the gapless phase C to a gapped topological order D by gauging—that sounds like it could be useful for designing quantum error correction codes.

Kai: Exactly, Lev. The paper shows this two-way mapping between the original TOSF C and this dual U(one)dual-enriched TO D. It's a way to systematically encode all the topological data of C into that dual structure D.

Mira: And what’s interesting is how they say that the enrichment of that dual system D depends only on two things: the vison, which is generated by some two pi symmetry flux, and the Hall conductance sigma xy.

Lev: So if we can measure or simulate those two things—the vison and the Hall conductance—we could potentially figure out exactly what kind of topological order D we’re dealing with.

Kai: And they link that back to anyons; the fractional charges of all the anyons in C are determined by how they braid with this vison, while the Hall conductance is constrained by a value called theta.

Mira: That means if you see a zero Hall conductance in D, it forces the vison in D to have bosonic self-statistics. It’s a very specific constraint on that dual theory.

Lev: That sounds like something we could use to narrow down theoretical possibilities when designing experimental setups; knowing those constraints helps us filter out impossible topological states early on.

Kai: The paper also connects the vortex structure in the original TOSF C directly to the discrete symmetry fluxes in that gapped anyon sector of D.

The paper's summary: Mira: So, they’re saying that if you want to understand how vortices behave topologically, you can look at those discrete fluxes in a simpler system and see how they relate.

Lev: That simplifies the problem for error correction because it connects the complex vortex dynamics to something that has a well-defined discrete symmetry structure.

Kai: Beyond just mapping data around, they provide a field-theoretic description of these systems, treating Abelian and non-Abelian TOs separately with specific Lagrangians.

Mira: When you gauge the dual U(one)dual symmetry in the field theory, the Lagrangian picks up terms that depend on that Hall conductance sigma xy.

Lev: That’s where I start thinking about how we translate those abstract field equations into something we can actually program onto a quantum computer chip. Does it map to any standard Hamiltonian we can run?

Kai: They give an example with a charge-four system having non-Abelian vortices, using a parton construction where the boson splits into two parts, phi and chi.

Mira: This specific construction shows how you can build something with non-Abelian vortices that fits this framework, identifying the elementary vortex as a specific flux of A in D/A.

Lev: That parton construction is interesting because it gives us a way to think about the fundamental excitations without just looking at the bulk topological order; it connects the microscopic building blocks to those macroscopic features.

Kai: They also suggest a phase transition prediction, showing that as you change the chemical potential mu from negative to positive, you naturally transition out of this gapless TOSF phase into a gapped phase where all condensates vanish.

Mira: That transition is mapped onto an Abelian-anyon proliferation transition in the dual theory D; it suggests the gapless nature of C is tied to some kind of condensation that dissolves at that point.

Lev: Predicting that transition helps us define the operating regime for our hardware; knowing where we cross into a gapped phase tells us what kind of topological protection we’ve achieved.

The paper's summary: Kai: They also showed how stacking the TOSF with a specific Hall conductance state induces a new dual TO D', which encodes the same data but with a different vison order, n.

Mira: So you can stack these systems in different ways to generate various types of topological orders while keeping the original physical properties of C intact, just changing how they are described by D'.

Lev: That means we might be able to engineer specific topological features by choosing the right stacking configuration or Hall conductance state.

Kai: Overall, this work gives us a very systematic way to characterize all the topological data of a TOSF C by mapping it onto its dual U(one)dual-enriched TO D through gauging U(one)dual.

Mira: It’s powerful because it connects the complex physics of these gapless systems directly to established concepts in symmetry-enriched topological orders, giving us a unified language for anyon content and vortex properties.

Lev: I think the field theory approach further suggests that we can predict a natural phase transition from the TOSF C to a gapped phase with the same intrinsic TO as D when you change the chemical potential mu.

Kai: That prediction of an Abelian-anyon proliferation transition is something we can definitely test, and it gives us a target for what we expect to see in any measurement that probes that transition.

Mira: And they also point out that while vortices and discrete symmetry defects share topology, their energy properties are not necessarily the same, which is a subtle physical point we have to keep in mind.

Lev: We need to keep those energetic constraints in mind because they dictate how these vortices actually behave in a system; topology doesn't always equal energy equivalence.

Kai: The paper also points toward extending this framework to fermionic systems and continuous global symmetries, which gives us much more scope for future work beyond just simple U(one) breaking.

Mira: That extension to continuous symmetries is huge; it opens the door to studying phases that aren't just constrained by discrete subgroups, which are often more relevant in real materials.

The paper's improvements: Kai: So we’ve been looking at how they characterize these topological superfluids, and now let's talk about what they suggest we should do next with this framework.

Mira: Right, the paper sets up this systematic way to look at the topological data of a TOSF C by mapping it onto its dual system D through gauging U(one)dual. Now they’re suggesting ways to make that map more powerful.

Lev: I wonder what they mean when they talk about stacking the system with a BIQH state of Hall conductance p not equal to zero, inducing this new TO D prime.

Kai: They show that stacking it that way changes the vison order, making it n, and the resulting topological order becomes D/A. That means you can generate different types of topological orders from the same base system just by changing how you stack things.

Mira: It’s like having a toolbox where you can swap out different components to get a slightly different final product while keeping the core physics consistent, which is really useful for exploring phases.

Lev: If we can generate these different topological orders from the same starting point, it gives us more control over the resulting quantum states we create.

Kai: They also suggest extending this framework beyond simple U(one) breaking to continuous global symmetries G being broken down to a subgroup H.

Mira: That’s a big step because most of our examples are just about U(one) symmetry, but the authors want to see how this works when you have more complex symmetries involved.

Lev: Investigating continuous symmetries opens up phases that aren't just constrained by simple discrete subgroups, which is where a lot of the interesting physics in real materials might lie.

Kai: They also mention that the goal is to characterize defects and textures in these spontaneously broken phases, which sounds like we need better tools for analyzing those structures.

Mira: So what this means practically is they’re pointing toward a more general method for understanding how order-parameter defects interact with the deconfined anyons.

Lev: That brings us back to my earlier point about error correction; if we can characterize these textures precisely, we can design better ways to protect quantum information from those defects.

Kai: The authors also reiterate that the field-theoretic approach suggests a natural phase transition from the gapless TOSF C to a gapped phase with the same intrinsic TO as D when you change the chemical potential mu.

Mira: So they’re reinforcing that idea: you can predict exactly when it goes from being gapless and topological to being fully gapped.

Lev: That prediction of an Abelian-anyon proliferation transition is something we can definitely test; it gives us a concrete target for what we expect to see in any measurement that probes that transition.

Kai: And they mentioned using parton constructions and field theory to generate new examples of TOSFs with non-Abelian vortices, which is really exciting.

Mira: Those examples are important because they give us tangible targets to aim for when we start trying to build these complex systems in a lab environment.

Lev: That connection between the abstract theory and the concrete parton models is where I think the real progress for building actual hardware will come from.

Conclusion: Kai: So we’re wrapping up on "Theory of Topologically Ordered Superfluids in two plusone Dimensions," by summarizing this work that builds a systematic way to map gapless topological phases onto dual gapped ones.

Mira: Exactly, Kai. It shows how the topological data of the original system C can be fully encoded by looking at this dual structure D through gauging U(one)dual symmetry.

Lev: I just want to say that for error correction, this systematic encoding is pretty useful because it gives us a clear target structure to aim for when we build things.

Kai: It does, Lev. And the field theory part they included really connects this abstract algebra to more concrete microscopic models like parton constructions with non-Abelian vortices.

Mira: That’s true; the field theory helps us see how those microscopic building blocks actually behave in the larger topological picture, which is a big assumption underneath everything else.

Lev: For hardware design, that means we can start thinking about specific configurations of those elementary excitations rather than just hoping they show up randomly.

Kai: The paper also laid out this natural phase transition when the chemical potential mu goes from negative to positive, mapping it to an Abelian-anyon proliferation transition in the dual theory.

Mira: That prediction is pretty solid because it’s derived directly from the Lagrangian structure, and we can test that by looking for specific changes in anyon behavior.

Lev: Testing that transition would be a big validation for how well this theory describes the physics of these systems under extreme conditions.

Kai: So, to summarize, this paper gives us a rigorous blueprint for characterizing anyon content and vortex properties in these gapless topological systems via the dual mapping and its field-theoretic constraints.

Mira: It’s a very complete picture for understanding how these gapless systems organize themselves when they break that charge symmetry.

Lev: It feels like we have a solid foundation now, especially with the way it predicts those phase boundaries, which is crucial for any experimental work.

Kai: And the future direction they point toward, extending this framework to fermionic systems and continuous global symmetries G broken to H, gives us much more scope for future research.

Cornell University

cond-mat.str-el, cond-mat.supr-con, hep-th

Submitted: 2026-10-07

Updated: 2026-10-07

Comments: 12 pages, 1 figure

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: The gist: This work develops a general framework for characterizing the topological data of (2+1)D TOSFs by establishing a two-way mapping between TOSFs with spontaneously broken global U(1c)

Key concepts

Topologically Ordered Superfluid (TOSF): These are exotic quantum fluids in 2+1 dimensions that have a gapless nature due to Goldstone modes, yet support deconfined, gapped anyonic excitations. They possess rich topological properties related to their broken symmetry.
Gauging Symmetry: This process involves turning a global symmetry (like U(1c)) into a local gauge symmetry by introducing a dynamical gauge field. This mapping transforms the gapless TOSF phase into a fully gapped Topological Order (TO) that is enriched by dual symmetries.
U(1) Dual Symmetry: This is an extra, dual symmetry that appears when gauging the original U(1c) symmetry. It plays a crucial role in characterizing the resulting gapped topological order. Its properties, like vanishing Hall conductance, help determine the statistics of anyons and vortex behavior.

Terminology

Anyons — 30× in this paper · explained in 2 episodes
Topological Order — 10× in this paper · explained in 2 episodes
Anyon Condensation — 4× in this paper · explained in 1 episode
Symmetry Breaking — 3× in this paper · explained in 3 episodes
Anyon Proliferation — 1× in this paper · explained in 2 episodes
Chern Number — 1× in this paper · explained in 4 episodes
Generalization — 1× in this paper · explained in 13 episodes
Phase Transitions — 1× in this paper · explained in 2 episodes
Topological Defects — 1× in this paper · explained in 2 episodes
Unified Framework — 1× in this paper · explained in 5 episodes

Summary

The gist: This work develops a general framework for characterizing the topological data of (2+1)D TOSFs by establishing a two-way mapping between TOSFs with spontaneously broken global U(1c) symmetry and U(1)dual-enriched TOs with vanishing Hall conductance through gauging procedures.

General Algebraic Framework

The framework introduces an algebraic characterization for the topological data of a TOSF C in a 2+1D bosonic system that spontaneously breaks a global 0-form charge symmetry U(1c). Gauging the global U(1c) symmetry of C by promoting the background U(1c gauge field A to a dynamical one maps the TOSF phase C to a fully gapped TO D in 2+1D. This resulting TO D is naturally enriched by a dual global 0-form U(1)dual symmetry that couples to monopoles of the dynamical gauge field α.

The enrichment of a generic 2+1D TO by an unbroken U(1) symmetry is fully determined by two pieces of data:

(1) the Abelian anyon generated by the 2π symmetry flux, called the vison, and

(2) the Hall conductance σxy.

The fractional U(1) charges of all the anyons are determined by their braiding statistics with the vison, while the Hall conductance σxy is constrained by the vison’s topological spin, θ = e iπσxy. For the U(1)dualenriched TO D, the vanishing σxy implies that the vison v in D has bosonic self-statistics.

Topological Data Encoding

The two-way mapping allows the topological data of the TOSF C, including the condensate charge, deconfined gapped anyons, and topological properties of vortices, to be systematically encoded by the corresponding U(1)dual-enriched TO D. Specifically, superfluid vortices share the topological properties of flux defects associated with the residual discrete symmetry in the gapped anyon sector. The topological degrees of freedom carried by a vortex in a general TOSF can be identified with those carried by discrete symmetry fluxes in a 2+1D TO enriched by a discrete symmetry.

Field Theory Formulation and Examples

A complementary field-theoretic description is provided, treating the TOSFs encoded by Abelian and non-Abelian TOs separately. For a generic U(1)dual-enriched Abelian TO D in a bosonic system, the Lagrangian is given by LD[B] = X N I,J=1 KIJ 4π bIdbJ + 1/2π tIBdbI, (4). Gauging U(1)dual leads to the Lagrangian LC[A] = X N I,J=1 KIJ 4π bIdbJ + 1/2π tIβdbI - 1/2π Adβ, (5).

In the case of a charge-4 TOSF with non-Abelian vortices, a parton construction is used where the boson Ψ is split into two partons, φ and χ. The resulting TOSF C spontaneously breaks U(1c) down to its Z4 subgroup. The elementary vortex of C carries a flux 2π/4 of A and can be identified with the Z(0)4 symmetry flux of D/A.

Phase Transition and Outlook

The field-theoretic approach suggests a natural quantum phase transition out of the TOSF phase, where as µ changes sign from negative to positive, Φ transitions from the condensed phase to a gapped phase with ⟨Φ⟩ = 0. The resulting gapped sector is topologically equivalent to the TO D. Future investigations suggest extending this framework to fermionic topologically ordered superconductors and to phases in which a general continuous global symmetry G is spontaneously broken to a subgroup H.

Appendix A: Different U(1)dual-enriched TOs encoding the same TOSF C

Stacking the TOSF C with a BIQH state of Hall conductance p!= 0 induces the change D → D' = D ⊠ U(1)-p ⊠ U(1)(n2p/l2)A˜. This new TO D' encodes the data of the same TOSF C (stacked with a BIQH state). The vison v' in D' has order n, and the resulting topological order is D/A.

The work provides a streamlined and systematic characterization of all the topological data of a TOSF while naturally admitting a complementary field-theoretic formalism that connects more directly to microscopic constructions. The topological data of the TOSF C, including its gapped anyonic excitations, the charges of its condensed local objects, and the topological properties of its vortices, can be fully encoded by the U(1)dual-enriched TO D through the gauging of U(1)dual. The field-theoretic approach further suggests a natural phase transition from C to a gapped phase with the same intrinsic TO as D. Finally, using parton constructions and field-theoretic approaches, we provide new examples of TOSFs with exotic non-Abelian vortices. The results suggest several directions for future investigation including identifying microscopic models that realize TOSFs with non-Abelian vortices. A natural extension of our framework is to fermionic topologically ordered superconductors with spontaneously broken charge U(1c) symmetry. Another direction is to extend the investigation to topologically ordered gapless phases in which deconfined, gapped anyons coexist with spontaneous breaking of a general continuous global symmetry G. The framework for gauging an unbroken continuous Lie group symmetry in a (2+1)D TO was developed in Ref. 25. A central challenge is to systematically characterize the topological properties of order-parameter defects (and/or textures) and their interplay with deconfined anyons. The framework for gauging an unbroken continuous Lie group symmetry in a (2+1)D TO was developed in Ref. 25. A potential general approach is again to map the gapless symmetry-broken phase C to a gapped TO D by gauging G, as in our treatment of TOSFs. This framework for gauging an unbroken continuous Lie group symmetry in a (2+1)D TO was developed in Ref. 25. The topological data of the TOSF C can be fully encoded by the U(1)dual-enriched TO D through the gauging of U(1)dual. The field-theoretic approach further suggests a natural phase transition from C to a gapped phase with the same intrinsic TO as D. Finally, using parton constructions and field-theoretic approaches, we provide new examples of TOSFs with exotic non-Abelian vortices. The results suggest several directions for future investigation including identifying microscopic models that realize TOSFs with non-Abelian vortices. A natural extension of our framework is to fermionic topologically ordered superconductors with spontaneously broken charge U(1c) symmetry. Another direction is to extend the investigation to topologically ordered gapless phases in which deconfined, gapped anyons coexist with spontaneous breaking of a general continuous global symmetry G. The framework for gauging an unbroken continuous Lie group symmetry in a (2+1)D TO was developed in Ref. 25.

Improvements for AI systems

textbfDefect-based Topological Quantum Computation (TQC): A New Paradigm for Vortex Control System Design: The improved AI system can design and simulate topological quantum computers based on defect-based topological quantum computation by mapping TOSF vortices to discrete symmetry fluxes in gapped anyon sectors, offering a route to realizing topological quantum computation based on TOSF vortices.
textbfSystem Characterization via Dual Mapping: The improved AI system can systematically characterize the topological data of gapless systems by implementing the two-way mapping between a TOSF C and a dual U(1)dual-enriched TO D, allowing it to encode the charge of the condensate, the deconfined gapped anyons, and the topological properties of vortices using this algebraic framework.
textbfModel Synthesis from Field Theory: The improved AI system can construct novel topological states by utilizing parton constructions and field-theoretic descriptions to generate examples of TOSFs with non-Abelian vortices, facilitating the discovery of new phases like those described by the three-fermion toric code.
textbfPhase Transition Prediction: The improved AI system can predict natural phase transitions in topological systems by analyzing the field theory Lagrangian, specifically identifying how a transition occurs when µ changes sign from negative to positive and mapping it to an Abelian-anyon proliferation transition.
textbfGeneralized Symmetry Mapping: The improved AI system can investigate topological data for phases beyond simple U(1) breaking by extending the framework to continuous Lie group symmetries, providing a framework for gauging an unbroken continuous Lie group symmetry in a (2+1)D TO to characterize defects and textures in spontaneously broken phases.

Abstract

Topologically ordered superfluids (TOSFs) support deconfined, gapped anyonic excitations while remaining gapless because of Goldstone modes. We develop a general framework for characterizing their topological data in (2+1) -dimensional bosonic systems with spontaneously broken charge U (1) c symmetry. Gauging U (1) c maps a TOSF to a gapped topological order enriched by a dual U (1) dual symmetry with vanishing Hall conductance, while gauging U (1) dual recovers the TOSF. This two-way mapping allows the topological data of the TOSF, including the condensate charge, deconfined anyon content, and topological properties of vortices, to be systematically encoded using the algebraic framework of symmetry-enriched topological orders. In particular, superfluid vortices share the topological properties of flux defects associated with the residual discrete symmetry in the gapped anyon sector. We develop complementary algebraic and field-theoretic formulations of this framework, with the latter also suggesting a natural transition to a gapped topologically ordered phase. Additionally, using parton constructions, we present examples of TOSFs with non-Abelian vortices in bosonic systems.