Abstract
Chiral topological insulator (AIII-class) with Landau levels is constructed based on the Nambu 3-algebraic geometry. We clarify the geometric origin of the chiral symmetry of the AIII-class topological insulator in the context of non-commutative geometry of 4D quantum Hall effect. The many-body groundstate wavefunction is explicitly derived as a (l,l,l−1) Laughlin–Halperin type wavefunction with unique K-matrix structure. Fundamental excitation is identified with anyonic string-like object with fractional charge 1/(2(l−1)2+1). The Hall effect of the chiral topological insulators turns out be a color version of Hall effect, which exhibits a dual property of the Hall and spin-Hall effects.
Highlights
In the past decade, the topological insulars (TIs) with time-reversal symmetry have attracted great attentions
Since quantum Nambu geometry is closely related to the geometry of M-theory [9,10,11,12], the appearance of quantum Nambu geometry in TIs is quite intriguing, the two groups reached a contradictory conclusion about the Nambu 3-bracket description for TIs; The authors of Ref.[3] insist that 3algebra consistently describes physics of the chiral TI, while the authors of Ref.[5] advocated the 3-algebra is not appropriate because of “pathological” properties of the 3-algebra
We explored one-particle and manybody physics of the chiral TI with Landau levels based on the Nambu 3-algebraic geometry
Summary
The topological insulars (TIs) with time-reversal symmetry have attracted great attentions. The chiral IT is known as AIII-class TI which respects the chiral symmetry and lives in arbitrary odd dimension [1]. A-class and AIII-class TIs share many similar properties: Both A-class and AIII-class are classified by Z topological invariant and regularly appear in even and odd dimensions, and either of them does not respect time-reversal or particle-hole symmetry. The third question arises: (iii) Why does only AIII-class have the chiral symmetry? There are not many works about QHE in odd dimension except for the pioneering work of Nair and Randjbar-Daemi [14] where they found the Landau level spectrum depends on a “mysterious” extra parameter whose couterpart does not exist in the even dimensional case. Notice that both of the energy eigenvalue (5) and the degeneracy (6) depend on s, and exhibit the chiral symmetry with respect to s →.
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