We extend the emergent hydrodynamic framework for fermionic statistics in a 3+1D quantized superfluid to include non-Abelian topological excitations.Single framed vortex loops reproduce the familiar Abelian π Berry phase corresponding to fermionic exchange. Multi-component vortex networks exhibit matrix-valuednon-Abelian holonomies associated with internal symmetry groups G. Linked and knotted vortices provide a mechanical realization of both fermionic statistics andinternal quantum numbers. We discuss how effective gauge interactions emerge from topological constraints, provide energy scale estimates, and present a roadmapconnecting vortex network structures to Standard Model-like particles and forces.
We extend the emergent hydrodynamic framework for fermionic statistics in a 3+1D quantized superfluid to include non-Abelian topological excitations.Single framed vortex loops reproduce the familiar Abelian π Berry phase corresponding to fermionic exchange. Multi-component vortex networks exhibit matrix-valuednon-Abelian holonomies associated with internal symmetry groups G. Linked and knotted vortices provide a mechanical realization of both fermionic statistics andinternal quantum numbers. We discuss how effective gauge interactions emerge from topological constraints, provide energy scale estimates, and present a roadmapconnecting vortex network structures to Standard Model-like particles and forces.
AbstractWe present an emergent hydrodynamic framework for fermionic statistics within a 3 + 1-dimensional quantized superfluid effective field theory (EFT). While classical vortices in irrotational fluids are bosonic, we demonstrate that quantized vortex loops generically acquire an effective internal framing arising from the anisotropic geometry and phase-gradient structure of their cores. These framed vortex loops inhabit a configuration space C ∼= (T^3 × SO(3))N /S^N. We define the Berry connection on the Hilbert bundle of condensate states and show that the adiabatic exchange of two such defects corresponds to a non-contractible loop in SO(3). Due to π1(SO(3)) ∼= Z2, the resulting holonomy yields a topological phase π, producing the fermionic exchange sign e^iπ = −1. This provides a concrete mechanical realization of the Finkelstein–Rubinstein constraint and suggests that half-integer spin and fermionic statistics emerge as topological properties of a superfluid vacuum.
The general formulation of quantum statistical mechanics hints at interesting generalizations of the usual Bose/Fermi framework in two spatial dimensions. Anyon statistics, which is essentially a continuous interpolation between Bose and Fermi statistics, is relevant to the Fractional Quantum Hall Effect in two-dimensional (i.e., thin layer) condensed matter systems. In addition, the possibility of non-abelian statistics, in which the multi-particle wavefunction transforms as a representation of a non-abelian group under the exchange of indistinguishable particles, has been explored. Spontaneously broken non-abelian gauge theories in (2 + 1) dimensions often have stable topological defects, called non-abelian vortices, that experience non-abelian statistics. In addition, it has been suggested that degenerate quasihole multiplets in Quantum Hall systems also transform as non-abelian representations of the braid group under particle exchange. In this thesis, I explore the braiding properties of systems of two-cycle flux vortices in a residual S₃ discrete gauge group. The individual vortices are uncharged, but multi-vortex states can have Cheshire charge. The uncharged sectors all have non-vanishing bosonic subspaces, as do the non-abelian charged trivial flux sectors. A kinetic Hamiltonian for vortices on a periodic lattice is constructed. There is a modification to the translational symmetry in the periodically identified direction for non-trivial Z₂ charged sectors. The ground state energies for various three and four vortex sectors is numerically determined. Typically, the ground state is bosonic, with a gap separating it from a non-abelian subspace.
Abstract Fractional statistics is one of the most intriguing features of topological phases in 2D. In particular, the so-called non-Abelian statistics plays a crucial role towards realizing topological quantum computation. Recently, the study of topological phases has been extended to 3D and it has been proposed that loop-like extensive objects can also carry fractional statistics. In this work, we systematically study the so-called three-loop braiding statistics for 3D interacting fermion systems. Most surprisingly, we discover new types of non-Abelian three-loop braiding statistics that can only be realized in fermionic systems (or equivalently bosonic systems with emergent fermionic particles). On the other hand, due to the correspondence between gauge theories with fermionic particles and classifying fermionic symmetry-protected topological (FSPT) phases with unitary symmetries, our study also gives rise to an alternative way to classify FSPT phases. We further compare the classification results for FSPT phases with arbitrary Abelian unitary total symmetry G f and find systematical agreement with previous studies.
Recent measurements of mutual friction in strongly interacting Fermi superfluids establish a direct link between the transverse dynamics of quantized vortices and the relaxation of localized core states. Motivated by these results, we investigate the dynamics of knotted vortex filaments within an effective hydrodynamic framework. By extending the vortex Hall angle formalism to non-planar filament geometries, we derive a phenomenologically motivated scaling relation between filament topology and internal relaxation times. We show that a hierarchy of decay times emerges naturally from a topology-dependent Reynolds-like barrier, with lifetimes controlled by writhe-dependent geometric phase lags. While no claim is made of microscopic particle identity, the resulting hierarchy closely parallels the ordering observed in several physical systems exhibiting metastable topological excitations.
In this thesis, we study the non-Abelian anyons that emerge as vortices in Ki-taev's honeycomb spin lattice model. By generalizing the solution of the model, we explicity demonstrate the non-Abelian fusion rules and the braid statistics that charaterize the anyons. This is based on showing the presence of vortices leads to zero modes in the spectrum. These can acquire finite energy due to short range vortex-vortex interactions. By studying the spectral evolution as a function of the vortex seperation, we unambigously identify the zero modes with the fusion degrees of freedom of non-Abelian anyons. To calculate the non-Abelian statistics, we show how the vortex transport can be implemented through local manipulation of the couplings. This enables us to employ the eigenstates of the model to simulate a process where a vortex winds around another. The corresponding evolution of the degenerate ground state space is given by a Berry phase, which under suitable conditions coincides with the statistics. By considering a range of finite size systems, we find a physical regime where the Berry phase gives the predicted statistics of the anoyonic vortices with high fidelity. Finally we study the full-vortex sector of the model and find that is supports a previously undiscovered topological phase. This new phase emerges from the phase with non-Abelian anyons due to their interactions. To study the transitions between the different topological phases appearing in the model, we consider Fermi surface, whose topology captures the long-range properties. Each phase is found to be characterized by a distinct number of Fermi points, with the number depending on distinct global Hamiltonian symmetries. To study how the Fermi surfaces evolve into each other at phase transitions, we consider the low-energy field theory that is described by Dirac fermions. We show that phase transition driving perturbations translate to a coupling to chiral gauge fields, that always lead to Fermi point transport. By studying this transport, we obtain analytically the extended phase space of the model and its properties.
Non-invertible symmetries in quantum many-body systems generally give rise to sequential unitary circuits that move symmetry defects. In this paper, we investigate invariants defined by sequences of such circuits, which move non-invertible defects and generate a Berry phase evaluated on quantum states with defects. We show that this Berry phase generally defines an invariant under local deformations, provided that the sequential circuits preserve the locality of those deformations. This invariant also rules out a short-range-entangled state that preserves the non-invertible symmetry, thereby signaling the 't Hooft anomaly of a non-invertible symmetry purely in terms of unitary operators acting on a state. We then apply this framework to loop excitations in three spatial dimensions and identify a new loop excitation in the (3+1)D D4 topological order, which we dub a non-Abelian fermionic loop. Using the invariant of sequential circuits, we characterize the statistics of non-Abelian fermionic loops. In addition, we find a new (3+1)D mixed topological order with a single non-Abelian fermionic loop, whose long-range entanglement is protected by an invariant of sequential circuits.