Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach

By Yuan-Ming Lu, Ashvin Vishwanath. 2012

300 citations recorded by the source corpus.

We study topological phases of interacting systems in two spatial dimensions in the absence of topological order (i.e. with a unique ground state on closed manifolds and no fractional excitations). These are the closest interacting analogs of integer Quantum Hall states, topological insulators and superconductors. We adapt the well-known Chern-Simons {K}-matrix description of Quantum Hall states to classify such `integer' topological phases. Our main result is a general formalism that incorporates symmetries into the {{K}}-matrix description. Remarkably, this simple analysis yields the same list of topological phases as a recent group cohomology classification, and in addition provides field theories and explicit edge theories for all these phases. The bosonic topological phases, which only appear in the presence of interactions and which remain well defined in the presence of disorder include (i) bosonic insulators with a Hall conductance quantized to even integers (ii) a bosonic analog of quantum spin Hall insulators and (iii) a bosonic analog of a chiral topological superconductor, whose K matrix is the Cartan matrix of Lie group E$_8$. We also discuss interacting fermion systems where symmetries are realized in a projective fashion, where we find the present formalism can handle a wider range of symmetries than a recent group super-cohomology classification. Lastly we construct microscopic models of these phases from coupled one-dimensional systems.

Subfields: Atomic and Molecular Physics, and Optics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/ad85c9dfe40645b62e4bdef74c8cd68c89c7c891

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In Atomic and Molecular Physics, and Optics

1. E. Witten
2. Ashvin Vishwanath
3. T. Senthil
4. Chong Wang
5. Juven C. Wang
6. A. Nahum
7. X. Wen
8. Romain Vasseur

In Materials Chemistry

1. T. Senthil
2. Chong Wang
3. Ashvin Vishwanath
4. A. Nahum
5. Cenke Xu
6. M. Metlitski
7. D. Parker
8. Romain Vasseur

Citations in the corpus

• Protected edge modes without symmetry (2013, background): “[19] Yuan-Ming Lu and Ashvin Vishwanath, Theory and Classification of Interacting ‘‘Integer’’ Topological Phases in Two Dimensions: A Chern-Simons Approach, Phys.”
• Quantum Hall physics : Hierarchies and conformal field theory techniques (2016, background): “…few other semiconductor systems have also allowed observation of FQH physics, including Si MOSFETS (Furneaux et al., 1986), SiGe (Lai et al., 2004; Lu et al., 2012; Monroe et al., 1992), ZnMnO (Tsukazaki et al., 2010), and particularly graphene (Bolotin et al., 2009; Dean et al., 2011; Du et al.,…”
• Deconfined quantum critical points: symmetries and dualities (2017, background): “If we reinterpret B as a “charge” probe and B0 as a “spin” probe, this corresponds to the response of the bosonic integer quantum Hall (BIQH) state [77,78].”
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect (2012, methodology): “2D SRE topological phases were described using a simpler Chern-Simons approach [13],”
• Gapless Symmetry-Protected Topological Order (2017, methodology): “This shift in paradigm from band topology analysis of non-interacting Hamiltonians [32, 33] to strongly correlated systems led to the development of non-perturbative techniques, resulting in an essentially exhaustive classification of gapped bosonic [27, 29, 30, 34] and, to some”
• Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model (2013, unknown): “[17] Yuan-Ming Lu and Ashvin Vishwanath.”