Braiding statistics approach to symmetry-protected topological phases

By Michael Levin, Z. Gu. 2012

407 citations recorded by the source corpus.

We construct a 2D quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a "symmetry-protected topological phase." We describe a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a Z_2 gauge field and then show that the \pi-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/1eeb44c9a03182f5a6c2ee3812e92d7e9d92d167

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. Ashvin Vishwanath
2. Ryan Thorngren
3. E. Witten
4. T. Senthil
5. A. Kapustin
6. Chong Wang
7. R. Verresen
8. D. Else

In Condensed Matter Physics

1. Ryan Thorngren
2. E. Witten
3. Ashvin Vishwanath
4. A. Kapustin
5. T. Senthil
6. R. Verresen
7. Juven C. Wang
8. D. Else

Citations in the corpus

• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect (2012, background): “Levin and Gu [12] explicitly studied a specific example in 2D protected”
• Gapless Topological Phases and Symmetry-Enriched Quantum Criticality (2019, background): “[56] Michael Levin and Zheng-Cheng Gu, “Braiding statistics approach to symmetry-protected topological phases,” Phys.”
• Higher symmetry and gapped phases of gauge theories (2013, methodology): “Recently, the Dijkgraaf-Witten TQFT was used in a novel way, as a tool to classify symmetry-protected (SPT) phases without long-range entanglement [7, 8, 9].”
• Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward (2022, unknown): “(1) to the special case of a 2D product state deterministically prepares the toric code, whereas choosing a Z2 SPT phase [63] (which is itself preparable using FDLU, since the preparing circuit does not need to respect symmetry.”
• Protected edge modes without symmetry (2013, background): “[25] Michael Levin and Zheng-Cheng Gu, Braiding Statistics”
• Many-Body Quantum Magic (2020, background): “[68] Michael Levin and Zheng-Cheng Gu, “Braiding statistics approach to symmetry-protected topological phases,” Phys.”
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions (2016, background): “However, some of topological orders that have a gauge theory description of a gauge group G may be obtained by dynamically gauging the global symmetry G of SPTs [56,57].”
• Topological Phases Protected by Point Group Symmetry (2016, methodology): “There is compelling evidence that only one nontrivial such state exists [13], so we believe the Z2 × Z classification is most likely complete, unless there are additional states with intrinsic topological order not obtained from the E8 root state.”
• Pre-thermal phases of matter protected by time-translation symmetry (2016, background): “[11] Michael Levin and Zheng-Cheng Gu, “Braiding statistics approach to symmetry-protected topological phases,” Phys.”
• Superconductivity from valley fluctuations and approximate SO(4) symmetry in a weak coupling theory of twisted bilayer graphene (2018, background): “[96] Michael Levin and Zheng-Cheng Gu, “Braiding statistics approach to symmetry-protected topological phases,” Phys.”