Summary

Chapter III: Periodically driven Topological systems

3.7 Summary

a time-ordered integral, U(k,t) = Te . As a first step in constructing the bulk topological invariant, we define an associated, “deformed” time-periodic evolution operator for the system on a torus:

U(k,t)=U(k,t)exp

iH^eff(k)t

, (3.44)

with H^eff(k) = _Tⁱ logU(k,T). Note that, by construction, U(k,T) = 1. The explicit dependence on in the above definitions comes from the necessary choice of a branch cut for log; we use a definition such that−iloge^iχ = χ if χ ∈ [0, T) and−iloge^iχ = χ−2π if χ ∈[T,2π).

With these definitions at hand, we can define the “winding number” [94], W =

Z T 0

dt

Z d²k 8π²Tr

U^†∂tU f

U^†∂k_xU,U^†∂k_yUg

. (3.45)

In Eq. (3.45), we have used the shorthand U ≡ U(k,t), and W is an integer, which can in principle depend on the quasienergy. Note that in order forW to be well defined, the quasienergy has to remain in a spectral gap ofU(k,T).

The winding number,W counts the number of Floquet edge modes that exists on a given edge at the quasienergy when considered in the cylindrical geometry.

Therefore, in Fig. 3.6, we consider a system with W±_T^π,0 = 1. The bulk Chern number between two quasienergies,landuisC_ul =W_u −W_l.

Disorder in periodically driven Hamiltonians

46

We are primarily interested in the interplay of disorder and topological behavior. In two-dimensional TIs, it has been shown [85] that ballistic edge modes are robust to disorder as long as there is a bulk mobility gap. Does this notion generalize to Floquet topological Insulators (see Chapter3)? First we consider the semiconductor quantum well models [69] for the FTI. In this 2D model, the periodic drive creates a momentum-space ring of resonances between the valence and conduction bands.

This effectively induces a band-inversion that leads to a topological phase. When disorder is present, and momentum conservation is no longer applicable, however, it is not at all clear that the band inversion argument survives. In Chapter5, we first demonstrate the robustness of the Floquet topological insulator in quantum wells to disorder and the survival of the edge modes in the quasienergy gap. Additionally, we find that the level-spacing statistics of the quasi-energies also indicate that the transition to a trivial phase is in the same universality of the quantum-Hall plateau transition.

Disorder can also induce new phases in Floquet systems. Quenched disorder may induce a Floquet topological Anderson insulator (FTAI) phase. This is the driven analog of the Topological Anderson Insulator (TAI) phase that we examined in Section2.4. The FTAI is realized in an off-resonantly driven honeycomb lattice with broken inversion symmetry [108]. The driven clean system is in a trivial phase, but the introduction of sufficiently large disorder creates a band inversion, and a topological phase. Photonic lattices made of helical waveguides in a honeycomb lattice have realized Floquet topological phases for transmission of light [92], and we expect that the same sytems will provide a realization of the honeycomb FTAI.

A similar FTAI phase also exists in the quantum-well models, which opens the way to investigating the FTAI phase in electronic systems. In contrast to the honeycomb lattice model, where the disorder simply modifies the static gaps in the vicinity of the Dirac points, in quantum wells, the disorder renormalizes the radiation potential directly, and modifies it such that the drive produces a topological gap. We explore the FTAI phase in detail in these two models in Chapters6and7.

We also show that the unique topological characteristics of periodically driven systems can lead to qualitatively new phenomena when spatial disorder is introduced.

First, it is possible for robust chiral edge states to exist in a two-dimensional driven

system whereallbulk states are Anderson localized; we refer to such a system as an anomalous Floquet-Anderson insulator (AFAI) [108]. This situation cannot occur in the absence of driving, where the existence of chiral edge states necessarily implies that there must be delocalized bulk states at some energies [36]. Crucially, in an AFAI this relation is circumvented by the periodicity of quasienergy: the edge states persist through all quasi-energies, completely wrapping around the quasienergy Brillouin zone. Moreover, the combination of these novel chiral edge states and a fully localized bulk gives rise to an intriguing non-equilbrium topological transport phenomenon: quantizednon-adiabaticcharge pumping. In Chapter 8, we discuss an explicit model that describes the AFAI phase.

HAMILTONIANS

In this chapter, we will outline the analytical and numerical methods used to an- alyze the driven-disordered systems. Let us write down the full time-dependent Hamiltonian for the system as

H(t) = Hr(t)+V_dis, (4.1) where Hr(t) is the driven part of the Hamiltonian which is periodic in time with a periodT. The disorder potential,V_disis time-independent and as defined in Chapter 1 (Eq. (1.1)). We choose the disorder potential to be diagonal in real-space. In second-quantized notation we have

V_dis =X

i

Ui1c_i^†c_i, (4.2)

Uiis a uniformly distributed number, [−U0/2,U0/2], with a variance,σ_dis² =U₀²/12.

We note that the disorder potential is chosen to be proportional to identity in the pseudospin basis.

We generalize the Born approximation (see Section 1.1) to study the disordered Floquet Green function. In Section 4.1, we calculate the perturbative correction due to weak disorder on the quasienergy density of states. Next we discuss the dif- ferent numerical methods to characterize properties of the disordered periodically driven Hamiltonians. First, in Section4.2, we outline an approximate method for the real time-evolution of wave-packets. This allows to obtain the single-particle transport properties in the presence of disorder. Second, in Section 4.3, we obtain the quasienergy eigenstates of the Floquet Hamiltonian. We obtain the Floquet Hamiltonian from the stroboscopic time-evolution operator integrated over a sin- gle time-period, T. The eigenstates and eigenvalues of the Floquet Hamiltonian provide signatures of localization and the topology of the bulk. The localization- delocalization transition as a function of disorder is examined by computing the quasienergy level-spacing statistics to identify the transitions in these systems. The topological nature of these disordered quasienergy bulk bands are investigated by computing the Bott index [73] for these bands. The Bott index is equivalent to the Chern invariant in the presence of disorder.