Topological charge pumping in quasiperiodic systems characterized by Bott index

Abstract

We study topological charge pumping in one-dimensional quasiperiodic systems. Since these systems lack periodicity, we cannot use the conventional approach based on the topological Chern number defined in the momentum space. Here, we develop a general formalism based on a real space picture using the so-called Bott index. We extend the Bott index that was previously used to characterize quantum Hall effects in quasiperiodic systems, and apply it to topological charge pumping in quasiperiodic systems. The Bott index allows us to systematically compute topological indices of charge pumping, regardless of the detail of quasiperiodic models. We apply this formalism to the Fibonacci-Rice-Mele model which we made from Fibonacci lattice, a well-known quasiperiodic system, and Rice-Mele model. We find that these quasiperiodic systems show topological charge pumping with a multi-level behavior due to the fractal nature of the Fibonacci lattice. Such multilevel pumping behaviors can be understood by a real space renormalization group analysis.

An effective curved space-time geometric theory of generic twist angle graphene with application to a rotating bilayer configuration

“…To capture the quasiperiodic or incommensurate nature of RBG, we consider Bott index to relate the charge pumping during rotation driving [76][77][78][79][80][81][82].…”

“…(12a) is the counterpart for matrix (V −1 ) † (biorthogonal basis [83,84]). I Bott (T ) can be considered as the charge pumping per period for a given finite size L x × L y rectangle region [71,76]. The factor U x (t +dt )U −x (t ) contains the effect of the AA phase since it relates the gauge connection defining on the bundle whose base manifold isa 1d periodic time parameter in S 1 .…”

“…Numerical result shows that the quantized charge pumping will generally depend on the size and the driving frequency. To show the relation between Bott index and pumping charge consider the expectation of a wave packet center which can be expressed as [76].…”

Topological invariants in two-dimensional quasicrystals

“…Quasiperiodic systems can also have energy gaps [5][6][7][8] , while the lack of the Bloch bands makes the definition of topological numbers a nontrivial problem. Several theoretical works have been devoted to topological characterization of various one-dimensional (1D) [9][10][11][12][13][14][15][16][17][18][19][20] and two-dimensional (2D) quasiperiodic systems [21][22][23][24][25][26][27][28] . In particular, one-dimensional quasicrystals are characterized by the adiabatic charge pumping, where the number of the transfered charge under a relative slide of a single periodic structure to the other is given by the first Chern number, in an analogous manner to the quantum Hall effect.…”