Fractional topological insulators

Non-commuting projected position operators
3D: what would be a “pristine” extension?
r µ → f µ (r)
f 1 ( X),f 2 ( X),f 2 ( X)
X1 , X 2 , X 3
=+i 3
Nambu (quantum) 3-bracket
[ A 1 , A 2 , A 3 ]= ijk
Ai Aj Ak
=i 3 {f 1 ,f 2 ,f 3 } N ( X)+O( 5 )
{f 1 ,f 2 ,f 3 } N
(r) = µνλ ∂f1 ∂f 2 ∂f 3
(r)
∂r µ ∂r ν ∂r λ
Nambu (classical) 3-bracket
transformation preserves volume
⇒
3-brackets are unchanged
Thursday, March 22, 2012