Floquet topological phase transitions in a kicked Haldane-Chern insulator

Tridev Mishra, Anurag Pallaprolu, Tapomoy Guha Sarkar, and Jayendra N. Bandyopadhyay

Phys. Rev. B 97, 085405 – Published 5 February 2018

Abstract

We consider a periodically $δ$ -kicked Haldane type Chern insulator with the kicking applied in the $\hat{z}$ direction. This is known to behave as an inversion symmetry breaking perturbation, since it introduces a time-dependent staggered sublattice potential. We study here the effects of such driving on the topological phase diagram of the original Haldane model of a Hall effect in the absence of a net magnetic field. The resultant Floquet band topology is again that of a Chern insulator with the driving parameters—frequency and amplitude— influencing the inversion breaking mass $M$ of the undriven Haldane model. A family of such periodically related “Semenoff masses” is observed to occur, which support a periodic repetition of Haldane like phase diagrams along the inversion breaking axis of the phase plots. Out of these it is possible to identify two in-equivalent masses in the reduced zone scheme of the Floquet quasienergies, which form the centers of two inequivalent phase diagrams. Further, variation in the driving amplitude's magnitude alone is shown to effect the topological properties by linearly shifting the phase diagram of the driven model about the position of the undriven case, a phenomenon that allows the study of Floquet topological phase transitions in the system. Finally, we also discuss some issues regarding the modifications to Haldane's condition for preventing band overlaps at the Dirac point touchings in the Brillouin zone, in the presence of kicking.

Received 25 September 2017
Revised 10 January 2018

DOI:https://doi.org/10.1103/PhysRevB.97.085405

Physics Subject Headings (PhySH)

Floquet systems Topological phases of matter

Graphene

Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Tridev Mishra^*, Anurag Pallaprolu^†, Tapomoy Guha Sarkar^‡, and Jayendra N. Bandyopadhyay^§

Department of Physics, Birla Institute of Technology and Science, Pilani 333031, India

^*tridevmishra@pilani.bits-pilani.ac.in
^†anurag.pallaprolu@gmail.com
^‡tapomoy1@gmail.com
^§jnbandyo@gmail.com

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Issue

Vol. 97, Iss. 8 — 15 February 2018

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Images

Figure 1
Schematic illustrating the honeycomb lattice with vectors to the nearest and next-nearest neighbors of an $A$ sublattice site. A flux configuration, where $ϕ_{a}$ and $ϕ_{b}$ are fluxes through the triangles that bound each of the labeled regions, is also shown which can make the next to nearest-neighbor hoppings phase dependent with zero net flux in the hexagon, i.e., $(ϕ_{a} + ϕ_{b}) = 0$ .
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Figure 2
The ground sate Floquet bands for the driven Haldane model as seen for different driving frequencies. We consider a driving period of the form $T = 1 / v t_{1}$ where $v = 0.5$ , 1, and 2. The fixed parameters for all plots are $ϕ = 0$ , $t_{1} = 3$ , $t_{2} = 1$ , and $α_{z} \approx 0$ . Plots (a1) and (a2) are for $v = 0.5$ and $M = 2$ and 4 respectively. In these we note that, for zero driving amplitude, the band structure is very much different from that of the undriven case. At this low frequency the distortions are due to overlaps with Floquet sidebands. Plots (b1) and (b2) are for $v = 1$ and $M$ values 2 and 5 respectively. Here, the choice of frequency works for the lower $M$ value without any higher band interference but, for the larger $M$ , sideband overlaps occur. This is seen from the near flat truncation of the conductance band peak and the bump at the center of the valence band in plot (b2). Plots (c1) and (c2) are for $v = 2$ and $M$ values 2 and 5 respectively. They clearly indicate that this choice of frequency preserves the features of the Haldane spectrum when the driving is taken to zero.
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Figure 3
The behavior of the Floquet spectrum at relatively large driving amplitudes for the fixed parameters $ϕ = 0$ , $t_{1} = 3$ , and $t_{2} = 1$ . Plot (a) is for $α_{z} = π / 2$ and $M = 2$ . It shows the appearance of the tendency to flatness in the band structure, especially in the conduction band. Changing the $M$ value does affect this behavior but the flatness can be found to occur at a suitable corresponding $α_{z}$ value. Plot (b) is for $α_{z} = π$ and $M = 0$ . We see that the conduction and valence bands have completely exchanged their structures from the undriven case. It is important to note that the conduction band now shows almost complete flatness with slight bumps at the Dirac point locations which touch with the minima of the valence band when one accounts for the folding of the Floquet quasienergies. Topologically this has the effect of exactly exchanging the Chern numbers for the bands from the undriven case. This is discussed in detail in Sec. 4b.
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Figure 4
Plots of the Chern number in the $M / t_{2}$ and $ϕ$ plane for different $α_{z}$ values. The plots (a), (b), (c) and (d) are for $α_{z}$ values 0 (in the limit of weak driving), $π$ , $π / 4$ , and $π / 2$ respectively. The darkest regions indicate a Chern number of $- 1$ , the lightest ones 1, and the intermediate shade is for 0 Chern numbers. The choice of undriven Haldane model hoppings, for these plots, is fixed at $t_{1} = 3.5$ and $t_{2} = 1$ . The driving period is taken to be $T = \frac{1}{2 t_{1}}$ .
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Figure 5
Various sectional views that indicate how the topological phase diagram varies with change in driving amplitude, keeping one symmetry breaking parameter fixed and varying the other. Plot (a) indicates such variation for a fixed $ϕ = π / 3$ and illustrates the linear variation of the phases for different $M / t_{2}$ as $α_{z}$ is changed. One notes the sharp turn the phases take at $α_{z} \to 0$ , a clear consequence of the dependence of the phases on the driving magnitude $| α_{z} |$ . Plots (b) and (c) illustrate phase regions for varying $ϕ$ and $α_{z}$ , with $M / t_{2}$ choices fixed at 2 and 10 respectively. The Chern number convention for the shaded regions is the same as that for Fig. 4.
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