Topological BF Theory Constructs Dihedral Quantum Double Phases Via Spontaneous Symmetry Breaking

Nonabelian topological orders represent a potentially transformative approach to computation, but creating these exotic states of matter has proven conceptually challenging and often relies on individual, case-by-case constructions. Zhi-Qiang Gao, Chunxiao Liu, and Joel E. Moore, all from the University of California, Berkeley and Lawrence Berkeley National Laboratory, now present a systematic method for building these states, focusing on nonabelian dihedral double phases. The team constructs a topological BF theory and demonstrates its ability to accurately reproduce the complete data for anyons, the fundamental particles carrying quantum information in these systems, and to incorporate various twists in the theory. This work offers a new pathway to achieving nonabelian topological order, potentially paving the way for practical applications in emerging synthetic gauge field platforms and offering a promising alternative to existing approaches.

Defect Condensation Creates New Topological Phases

Nonabelian topological orders host exotic anyons, quasiparticles central to quantum computation, and are believed to emerge from fractional quantum Hall states and certain spin liquids. This work constructs a framework for systematically building twisted dihedral quantum double phases through spontaneous symmetry breaking, leveraging the mathematical structure of topological BF theory. The researchers demonstrate that these phases, characterised by non-trivial exchange statistics of their anyonic excitations, arise from condensing specific patterns of defects in a parent topological order, offering a pathway to engineer materials with tailored topological properties. The team establishes a connection between symmetry breaking patterns and the resulting anyon fusion rules, providing a powerful tool for classifying and understanding these complex quantum states of matter. This construction provides a robust and general method for generating a diverse range of twisted quantum double phases, potentially paving the way for realising fault-tolerant quantum computation.

Lattice Model for Non-Abelian Anyon Realization

The research team developed a systematic approach to constructing non-abelian topological order based on a continuous gauge field, beginning with a topological BF theory, a mathematical framework used to describe interactions in physics. They identified Wilson loops and twist operators within this theory as representations of anyons, quasiparticles exhibiting exotic exchange statistics crucial for topological quantum computation, and demonstrated that this BF theory accurately reproduces known anyon data and incorporates various Dijkgraaf-Witten twists, which modify the topological properties of the system. Building on this correspondence, the scientists constructed a microscopic model featuring a lattice gauge field coupled to both Ising and rotor matter, and showed that a process called Higgsing yields the desired dihedral double topological state. To understand the stability of this topological order, the team performed a perturbative renormalization group analysis, a technique used to examine how a system evolves at different energy scales.

This analysis revealed a direct connection to either a Coulomb or chiral topological phase at a stable multicritical point, and indicated the emergence of a new symmetry at this point. The researchers meticulously calculated the ground state degeneracy for both odd and even values of a parameter ‘n’ on Riemann surfaces, and confirmed that these results match those predicted by the dihedral quantum double. Further investigation involved a detailed analysis of topological spins and fusion rules, which govern the behavior of anyons when they are exchanged or combined. The scientists derived explicit transformations, known as S and T matrices, that describe how the system’s quantum states change under these operations, and demonstrated that these matrices align with those calculated from finite group data, confirming the correspondence between the S[O(2) × O(2)] BF theory and the dihedral quantum double.

Dihedral Double Order Achieved via Higgsing

Scientists have achieved a systematic construction of nonabelian dihedral double topological order, utilizing a continuous gauge field and a novel approach to realizing exotic anyons crucial for advanced computing. The research demonstrates a pathway to create these complex states of matter through a microscopic model involving a lattice gauge field coupled to both Ising and rotor matter. This model, when subjected to a process called Higgsing, yields the desired dihedral double phase, a state characterized by unique properties and potential for information processing. Experiments reveal that manipulating the interactions within this model induces phase transitions between different states of matter, including transitions from an O(2) Coulomb phase to a U(1) Coulomb phase and from a U(1) Coulomb phase to a D(Zn) phase.

These transitions are governed by the behavior of the Ising and vector fields within the model, and are characterized as 3D Ising and 3D XY transitions, respectively. Data shows that a stable multicritical point exists where four distinct phases meet, exhibiting emergent O(3) symmetry. The team’s analysis indicates that the system can transition directly between the Dn phase and the U(1) phase at this multicritical point, a result similar to observations in deconfined quantum critical points and symmetry-protected topological transitions. Measurements confirm that increasing the coupling strength induces a two-step confinement of the gauge fields, leading to the emergence of four phases with O(2), U(1), Z2, and trivial gauge structures. This breakthrough delivers a new route to realizing nonabelian topological order with promising prospects for implementation in synthetic gauge field platforms.

Dihedral Order From Continuous Gauge Fields

This work presents a systematic route to constructing nonabelian topological order, specifically the dihedral double, using continuous gauge fields. Researchers formulated a topological BF theory and demonstrated its ability to accurately reproduce the complete anyon data characteristic of these exotic phases of matter. Crucially, they established a microscopic model, a lattice gauge field coupled with specific matter fields, that realizes this topological order through a process called Higgsing. This construction provides a clear connection between continuous gauge structures and the emergence of discrete dihedral quantum double phases.

The team further investigated the properties of this constructed phase, revealing a potential direct transition to a Coulomb phase governed by established physical principles. This suggests the possibility of observing deconfined quantum criticality, a state of matter with unique properties. While the research successfully demonstrates a pathway to realizing these topological phases, the authors acknowledge that further investigation is needed to fully understand the behaviour at the multicritical point. Future work could extend this approach to create even more complex nonabelian topological orders by adapting the gauge fields used in the model, and the researchers suggest exploring experimental realization of their proposed theory and lattice model using existing platforms for creating synthetic gauge fields.