D-Algebra Structure of Topological Insulators

Abstract

In the quantum Hall effect, the density operators at different wave-vectors generally do not commute and give rise to the Girvin MacDonald Plazmann (GMP) algebra with important consequences such as ground-state center of mass degeneracy at fractional filling fraction, and W_{1 + \infty} symmetry of the filled Landau levels. We show that the natural generalization of the GMP algebra to higher dimensional topological insulators involves the concept of a D-algebra formed by using the fully anti-symmetric tensor in D-dimensions. For insulators in even dimensional space, the D-algebra is isotropic and closes for the case of constant non-Abelian F(k) ^ F(k) ... ^ F(k) connection (D-Berry curvature), and its structure factors are proportional to the D/2-Chern number. In odd dimensions, the algebra is not isotropic, contains the weak topological insulator index (layers of the topological insulator in one less dimension) and does not contain the Chern-Simons θform (F ^ A - 2/3 A ^ A ^ A in 3 dimensions). The Chern-Simons form appears in a certain combination of the parallel transport and simple translation operator which is not an algebra. The possible relation to D-dimensional volume preserving diffeomorphisms and parallel transport of extended objects is also discussed.

5 pages

References in corpus (24)

• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Fractional quantum Hall states at zero magnetic field
• Fractional quantum Hall effect in the absence of Landau levels
• Fractional Chern Insulator
• A Four Dimensional Generalization of the Quantum Hall Effect
• Fractional Quantum Hall Effect of Hard-Core Bosons in Topological Flat Bands
• Nearly flat band with Chern number C=2 on the dice lattice
• Generic Wavefunction Description of Fractional Quantum Anomalous Hall States and Fractional Topological Insulators
• An Invitation to Higher Gauge Theory
• Zoology of Fractional Chern Insulators
• Fractional topological liquids with time-reversal symmetry and their lattice realization
• Emergent Many-Body Translational Symmetries of Abelian and Non-Abelian Fractionally Filled Topological Insulators
• Fractional Chern Insulators and the W-Infinity Algebra
• Topological BF field theory description of topological insulators
• The Eight Dimensional Quantum Hall Effect and the Octonions
• Classical and Quantum Nambu Mechanics
• Categorified Symplectic Geometry and the Classical String
• Noncommutative geometry for three-dimensional topological insulators
• Time-reversal symmetric hierarchy of fractional incompressible liquids
• From Fractional Chern Insulators to a Fractional Quantum Spin Hall Effect
• Non-Linear hydrodynamics and Fractionally Quantized Solitons at Fractional Quantum Hall Edge
• Connectivity of edge and surface states in topological insulators

Cited by in corpus (19)

• Fractional Quantum Hall Physics in Topological Flat Bands
• Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators
• Origin of Model Fractional Chern Insulators in All Topological Ideal Flatbands: Explicit Color-entangled Wavefunction and Exact Density Algebra
• Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator
• The single-mode approximation for fractional Chern insulators and the fractional quantum Hall effect on the torus
• Perturbative Approach to Flat Chern Bands in the Hofstadter Model
• Electromagnetic and thermal responses of Z topological insulators and superconductors in odd spatial dimensions
• Chiral Topological Insulator on Nambu 3-Algebraic Geometry
• Relativistic Landau Models and Generation of Fuzzy Spheres
• Higher (Odd) Dimensional Quantum Hall Effect and Extended Dimensional Hierarchy
• Topological BF theory of the quantum hydrodynamics of incompressible polar fluids
• Exact correlators in the Gaussian Hermitian matrix model
• $W_{1+\infty }$ constraints for the hermitian one-matrix model
• On the signatures of non-topological patches on the surface of topological insulators
• Weyl Semimetal and Nonassociative Nambu Geometry
• Extended Dynamical Symmetries of Landau Levels in Higher Dimensions
• Higher bracket structure of density operators in Weyl fermion systems and topological insulators
• Non-Commutative Geometry in Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator
• Deformation of Matrix Geometry via Landau Level Evolution