# A study of the 1D Aubry-André-Harper (AAH) models and its topological properties

## Chi-Hung Weng

### (Dynamics in Complex Systems Group, Institute of Physics, University of Freiburg)

Time: Thu May 29 10:00:00 2014

Location: Building 1, Room 123

We study the 1D Aubry-André-Harper (AAH) [1,2] model, in which the
Hamiltonian is written in a tight-binding form. Since only the nearest-neighbor hopping is
considered, this quantum walk picture is simple: the quantum particle can hop from one site to its
neighbor sites within the 1D linear chain. Two AAH models are considered: The Diagonal AAH model and the Off-Diagonal AAH model, where either the diagonal part (on-site potential) or the
off-diagonal part (off-site hopping) of the Hamiltonian is written by a discrete, site-dependent cosine function with an embedded frequency and a phase shift. If the frequency is chosen to be an
irrational constant, the model enables one to study quantum walk on a 1D quasi-crystal
lattice, since the 1D lattice is then in order but without periodicity. Surprisingly, the AAH models are related to the topological insulator. For the Diagonal AAH model, it can be mapped into a 2D lattice model, which describes Bloch electrons in a 2D lattice within the precense of a magnetic field. The spectrum of this 2D lattice model is known as the “Hofstadter butterfly” [3], for which it is known that a Chern number can be assigned to each gap of the spectrum [4]. For the Off-Diagonal AAH model, if we consider the frequency to be 1/2, then the Hamiltonian has only two different hoppings, say, *v* and *w*. This is called the Su-Schrieffer-Heeger(SSH) model [5]. It is known that, for the SSH model, cases *v* > *w* and *v* < *w* are topologically different, since they can be assigned with different winding numbers (1 and 0, respectively). I will present some numerical results for the AAH models. Topological phenomena such as the topological end state (state which localized at the end of the chain) that flows from one end of the chain to another end can be seen in the plots of the Local Density of States (LDOS), as the embedded phase shift varies [6]. I also joint two 1D Off-Diagonal AAH chains from the center to form a 2D cross (which is made by one central node with four connected linear chains). In this simple 2D model, the surface states can be controlled to be localized either at the outer sites or the central site of the cross.
In addition, a modified Non-Hermitian PT symmetric Off-Diagonal AAH model is studied,
which remains in a PT non-breaking phase and possess a real spectrum. The reason why
the real spectrum is garanteed is that, there exist a similarity transformation such that the
non-Hermitian Hamiltonian of this kind can always be transformed into a Hermitian one [7].
It was known that the topological insulator is usually PT symmetric breaking [8]. Hence it
could be interesting if the PT symmetric Off-Diagonal AAH model proposed can host non-
trivial topological phase, which is currently under investigation.