Bandwidth-resonant Floquet states in honeycomb optical lattices

We investigate, within Floquet theory, topological phases in the out-of-equilibrium system that consists of fermions in a circularly shaken honeycomb optical lattice. We concentrate on the intermediate regime, in which the shaking frequency is of the same order of magnitude as the band width, such that adjacent Floquet bands start to overlap, creating a hierarchy of band inversions. It is shown that two-phonon resonances provide a topological phase that can be described within the Bernevig-Hughes-Zhang model of HgTe quantum wells. This allows for an understanding of out-of-equilibrium topological phases in terms of simple band inversions, similar to equilibrium systems.


Introduction.
A very successful modern approach to condensed-matter physics characterizes topological insulators (TIs) in terms of topological invariants. The most salient effect in these systems is the occurrence of protected metallic states at the boundary, while the bulk of the material remains insulating. This results, for instance, in the quantum spin Hall effect (QSH), which is characterized by spin-polarised, counter-propagating edge modes. Although initially conceived for graphene with spin-orbit coupling [1,2], the QSH effect was further theoretically predicted by Bernevig, Hughes, and Zhang (BHZ) [3] and experimentally observed [4] in HgTe quantum wells.
Beyond the ten-fold classification of topological matter, in terms of dimensionality and certain discrete symmetries (such as time-reversal and parity) [5], the particular band structure of TIs can, to great extent, be understood in terms of band inversion. In HgTe/CdTe quantum wells, the QSH arises from an inversion of the valence and the conduction bands at the Γ point in the first Brillouin zone. This band inversion allied with the strong spin-orbit coupling leads to an avoided crossing, and consequently a topological gap. Since the spin is a good quantum number, the BHZ model describing HgTe can be seen as two copies of the Haldane (or also BHZ) model, which was initially proposed to describe the quantum anomalous Hall (QAH) effect in graphene [6].
Whereas both the abstract classification and the abovementioned paradigmatic models with band inversion have allowed for a rather comprehensive understanding of TIs at thermodynamic equilibrium, the physical picture is far from being settled in the case of out-of-equilibrium topological phases. In this case, one gets, for example, Floquet TIs (FTIs) [7][8][9][10][11][12][13][14][15][16][17][18][19][20], which are TIs under the influence of a time-periodic perturbation. Somewhat counterintuitively, these FTIs are simplest to understand in the limit of high-frequency driving, i.e. when the driving frequency constitutes the largest energy scale in the system. In this case, the system constituents cannot follow the high-frequency perturbation, such that it remains at a quasi-equilibrium with simply renormalized lattice parameters [21,22]. From a topological point of view, the system can thus be classified within the same framework as systems at thermodynamic equilibrium.
The situation turns out to be drastically different when the driving frequency is on the same order of magnitude as the system's characteristic energy scales, e.g. the band width in a lattice model, in which case standard classification schemes break down. While there are attempts to define Chern-type topological invariants adapted for this situation [10,13,15,23,24], we show, here, in the case of artificial graphene samples irradiated by circularly polarised light, that also in the limit of moderate driving frequencies the topological aspects of the system can be understood in terms of band inversion. These band inversions occur because, with decreasing frequency, the different Floquet bands start to overlap. Because the circularly polarised light induces photon resonances, avoided crossings that generally host topological edge states occur. The polarization of the light breaks time-reversal symmetry such that the resulting FTI is in the QAH class. Furthermore, we emphasize the importance of photon resonances. It has recently been shown that one-photon resonances create an additional topological gap in the spectrum at non-zero energy, whereas two-photon resonances destroy the topological nature of the zero-energy gap, by creating counter-propagating edge states [19]. Here, we derive an effective continuum model from the out-of-equilibrium lattice model, and show that, in the vicinity of the band inversion occurring for the two-photon resonances, it turns out to be the BHZ model. Additionally, we demonstrate the validity of this model by comparing our results with a numerical solution of the full problem. In this way, we use our model to provide a link between nonequilibrium and equilibrium topological states of matter. Finally, we discuss a possible experimental implementation in honeycomb superlattices of CdSe nanocrystals [25,26].
Floquet theory. To describe out-of-equilibrium systems with a periodic time dependence, Floquet theory is used [27,28]. The time-dependent Schrödinger equation has quasiperiodic solutions where φ is a periodic function in time. Since the function ψ arXiv:1503.02635v1 [cond-mat.str-el] 9 Mar 2015 solves the time-dependent Schrödinger equation, The converse of this statement also holds, which leads us to define the Floquet Hamiltonian If H acts on the Hilbert space H, and H T is the Hilbert space of T -periodic functions, then H F acts on H ⊗ H T . The space H T is spanned by the functions |n := exp(inωt), and has inner product 1 dt/T . With respect to the states |n , we can write H F as a block matrix with elements Here, the H m−n are the Fourier modes of the original Hamiltonian H.
Model. We apply the above formalism to electrons in a honeycomb lattice irradiated by circularly polarised light. Taking into account only the nearest neighbor (NN) hopping, the bare tight-binding Hamiltonian reads [29] Here, J is the NN hopping strength, and our convention for the NN hopping vectors δ l is where a is the NN distance. In order to include coupling to circularly polarised light traveling in the z-direction, we use the vector potential where E is the magnitude of the electric field in the light wave. To obtain Eq. 5, one sets z = 0 and disregards the z-component of the vector potential. This vector potential can be incorporated into H 0 via the Peierls substitution, k →k := k + eA/ , where e is the electron charge. This yields Using Eq. (4), the matrices H n can be calculated from Eq. (6), and one obtains In this expression, J n is the Bessel function of the n th kind, and Arg gives the angle of a vector with the x-axis. Using Eq. (7), we obtain H F as an infinite block matrix If the spectrum around an energy n ω is desired, it can be computed by truncating the right-hand side of Eq. (8) around H 0 + n ω. The specific way in which one should truncate depends on the situation. For the system under consideration, H F can exhibit topological edge states, where the precise behaviour strongly depends on the radiation frequency ω. In the large-frequency limit, ω J, the Floquet bands are well separated, but one finds a hierarchy of band crossings upon decrease of ω, in the limit of ω ≈ J. These band-crossings create additional gaps in the spectrum that can be topological in nature. In the following, we focus on the band crossing in the vicinity of k = 0 and = 0, appearing in the interesting regime, ω ≈ 2.9J. In Fig. 1, the spectrum of H F is plotted for ω = 3J. The bottom of the valence band from H 0 + ω (on top) and the top of the conduction band from H 0 − ω (below) are visible with a gap between them. As ω is lowered, the valence band (on top) descends and the conduction band (on bottom) ascends; a band inversion takes place at ω ≈ 2.9J, creating an avoided crossing [ Fig. 2(a)]. At the avoided crossing, a single pair of edge states crosses the gap, as is highlighted in Fig. 2(b), where we show a zoom on a narrow energy window. It should be noted that the edge states at k = 0 are counter-propagating with respect to the ones at the Dirac points (not shown at ε = 0 to avoid confusion, but depicted at ε = ω which is equivalent to ε = 0 due to the periodicity of the Floquet spectrum), removing any topological protection of the edge states [19]. Below, we outline a derivation of a low-k, 2 × 2 effective Hamiltonian describing this band crossing. This Hamiltonian will be used to obtain the precise point of the band crossing, together with an expression for the size of the opened gap. The dispersion of the edge states will also be obtained from this model, follow-ing the procedure in Ref. [30]. Most saliently, the effective Hamiltonian gives exactly the BHZ model and thus allows for a simple understanding of Floquet side-band coupling in terms of topological transitions mediated by band inversion.
Low-energy effective theory. To write down an effective theory, we must extract the relevant energy bands from H F . This is done by diagonalizing H 0 , and for simplicity we keep terms up to second order in k. We define the unitary transformation and consider the transformed Hamiltoniañ whereH n = U H n U by construction. From Eqs. (4) and (9), one finds the identitỹ It follows that for ω ≈ 3J and k ≈ 0, the matrix elements (H 0 + ω) 2,2 and (H 0 − ω) 1,1 are much smaller than all the other energy scales in the problem. They are the zeroth order energies of the two bands near the band crossing shown in Figs. 1, and 2. To obtain an effective Hamiltonian in the low energy limit that also includes the coupling between these two bands, we therefore include the matrix elements from H F that couple them: The effective Hamiltonian in Eq. (11) is correct up to k 2 , and up to second-order photon processes. A comparison with numerical calculations shows that this approximation is sufficient to accurately describe both the gap size and the presence of the topological states; see Fig. 2(b) for example. The bulk bands are quadratic in nature for every band-crossing, but the number of topological states in the gap varies. Due to the nature of our approximation, the topological behaviour of the other band crossings cannot be captured (for example, the one-photon resonance gap at ω/2 has two pairs of edge states, which cannot be described within the BHZ model). Using the definition ofH 2 , one obtains

and thus the Hamiltonian
where These expressions are correct up to order (eEa/ ω) 2 . The presence of the Bessel function of the second kind, J 2 in Eq. (15) shows that the opening of a gap at the band inversion is a second-order photon process. It should be noted that in some systems of interest (such as graphene), the NN hopping energy is negative, yielding minus the Hamiltonian in Eq. (6). Propagating this sign through the calculations, one finds that A in Eq. (15) obtains an additional minus sign, but the spectrum remains unaffected.
Results. From the effective Hamiltonian in Eq. 12, a variety of results can be obtained for the gap at = 0 and k = 0, for values of ω at which the band inversion occurs. Following Ref. [30], one can derive an explicit solution for the edge state in the infinite half-plane. Using perturbation theory to linear order in k, the edge states then disperse as i.e. the edge states have a velocity quadratic in the applied electric field strength E.
From H eff , an expression for the gap size ∆ can also be derived, and one obtains Filling the parameter values ω = 2.7J and eEa = J into Eq. 16 yields a gap size ∆ = 0.033J, which is in good agreement with the numerical results shown in Fig. 2(b).
Conclusion. In conclusion, we have investigated electrons on a honeycomb lattice irradiated by circularly polarised light in the intermediate regime, where the light frequency is on the order of the band width ω ≈ 3J. In this particular regime, the system in characterized by a substantial overlap between the Floquet side bands, and a series of band inversions can be created that generally host topological edge states. We have concentrated on the crossing associated with two photon resonances, at ω ≈ 2.9J, and we have shown that the relevant effective continuum model is just the BHZ model for HgTe quantum wells. This allows for an understanding of topological phases in out-of-equilibrium systems in terms of well-studied effective models, and especially in terms of band inversion, now between adjacent Floquet bands.
From an experimental point of view, one might think of graphene as a natural candidate for testing the occurrence of the topological phases and the associated edge states that we have analyzed here. However, the relevant hopping parameter J = 2.8 eV and the NN bond length a = 1.4Å in graphene would require unphysically large frequencies beyond the THz regime, and a very high field strength of E ≈ 5.3 · 10 10 V/m. A more realistic system to test our theory would be a selfassembled honeycomb lattice of CdSe nanocrystals [25,26], which hosts an s-band exhibiting a dispersion similar to that of graphene. The hopping parameter in these artificial structures depends on the diameter and the contact area of the nanocrystals. A hopping parameter J = 25 meV, that is roughly two orders of magnitude smaller than that in graphene, has been theoretically predicted for nanocrystals with a diameter of 3.4 nm [25]. By using light with E = 10 7 V/m and ω = 65 meV, a gap of 1.5 meV is obtained for these parameters, which is 6% of the hopping J, and, therefore, reachable within stateof-the-art experiments.