Optically Controlled Topological Phases in the Deformed $α-T_{3}$ Lattice

TL;DR

This work demonstrates Floquet-engineered topological phases in the deformed $α-T_{3}$ lattice under circularly polarized off-resonant light. The authors derive an effective Hamiltonian that includes a light-induced Haldane-like term and analyze a uniaxial deformation parameter $γ_{1}$ that moves Dirac points toward the $M$ point and drives a gap closure at $γ_{1}=2γ$, signaling a transition to a trivial phase. They compute Berry curvature, Chern numbers, Wannier charge centers, and anomalous Hall conductivity, revealing $α$-dependent Chern numbers ($C_{2}=1$ or $2$, $C_{0}=-1$ or $-2$) and a tunable phase diagram. The results establish a controllable opto-mechanical route to engineer topological phases in the $α-T_{3}$ lattice with potential impact for Floquet-enabled quantum materials.

Abstract

Haldane's tight-binding model, which describes a Chern insulator in a two-dimensional hexagonal lattice, exhibits quantum Hall conductivity without an external magnetic field. Here, we explore an $α-T_{3}$ lattice subjected to circularly polarized off-resonance light. This lattice, composed of two sublattices (A and B) and a central site (C) per unit cell, undergoes deformation by varying the hopping parameter $γ_{1}$ while keeping $γ_{2}$= $γ_{3}$= $γ$. Analytical expressions for quasi-energies in the first Brillouin zone reveal significant effects of symmetry breaking. Circularly polarized light lifts the degeneracy of Dirac points, shifting the cones from M. This deformation evolves with $γ_{1} $, breaking symmetry at $γ_{1}=2γ$, as observed in Berry curvature diagrams. In the standard case ($γ_{1}=γ$), particle-hole and inversion symmetries are preserved for $α=0$ and $% α=1$. The system transitions from a semi-metal to a Chern insulator, with band-specific Chern numbers: $C_{2}=1$, $C_{1}=0$, and $C_{0}=-1$ for $% α<1/\sqrt{2},$ shifting to $C_{2}=2$, $C_{1}=0$, and $C_{0}=-2$ when $% α\geqslant 1/\sqrt{2}.$For $γ_{1}>2γ$, the system enters a trivial insulating phase. These transitions, confirmed via Wannier charge centers, are accompanied by a diminishing Hall conductivity. Our findings highlight tunable topological phases in $α-T_{3}$ lattices, driven by light and structural deformation, with promising implications for quantum materials.

Figures (9)

• Figure 1: Schematic of a deformed $\alpha -T_{3}$ lattice exposed to circularly polarized off-resonance light, where the deformation affects only the position vector $\delta _{1}$, changing the hopping energy from $\gamma _{1}= \gamma$ to 3$\gamma$, while the hopping energies $\gamma _{1}= \gamma$ and $\gamma _{2}= \gamma$ associated with the position vectors $\delta _{2}$ and $\delta _{3}$ remain unchanged.
• Figure 2: The band structure of the irradiated and deformed $\alpha -T_{3}$ lattice is illustrated as a function of $\gamma _{1}= \beta \gamma$ along the $k_{x}$ axis in the following cases: $\bullet$ (a) $\gamma _{1}=1 \gamma$, (b)$\gamma _{1}=1.5 \gamma$, (c) $\gamma _{1}=2 \gamma$, (d) $\gamma _{1}=3 \gamma$, (e) $\gamma _{1}=4 \gamma$ with $\alpha =0$, $\bullet$ (f) $\gamma _{1}=1 \gamma$ , (g) $\gamma _{1}=1.5 \gamma$, (h)$\gamma _{1}=2 \gamma$, (i) $\gamma _{1}=3 \gamma$, (j) $\gamma _{1}=4 \gamma$ with $\alpha =\dfrac{1}{ \sqrt{2}}$, $\bullet$ (k) $\gamma _{1}=1 \gamma$, (l) $\gamma _{1}=1.5 \gamma$, (m)$\gamma _{1}=2 \gamma$, (n) $\gamma _{1}=3 \gamma$ ,(o) $\gamma _{1}=4 \gamma$ with $\alpha =1$. Sub-figure (P) represents the first Brillouin zone of the hexagonal lattice used to calculate and structure the band structure along the path ($\Gamma \rightarrow K\rightarrow M\rightarrow K^{\prime }\rightarrow \Gamma$ ). We have normalized the Hamiltonian by $\gamma$ to account for this constant in the figures. We used $\Delta/\gamma = 0.27$, with $\hbar \omega = 3\gamma$, and a field intensity amplitude $\varsigma = 0.7$. Taking $\gamma = 2.7$ eV, the frequency range considered is $\omega\in[2.7 PHz, 8.1 PHz]$, which we are considering here within a theoretical framework. This frequency range is physically accessible. Indeed, recent studies have explored controlling electric current by irradiating solid materials at petahertz frequencies REaddsp1REaddsp3REaddsp4. Furthermore, applications in the fields of ultrafast optical waveforms, digital logic, communications, and quantum computing REaddsp2REaddsp3REaddsp4 make this regime relevant to modern physics.
• Figure 3: The figure illustrates the band structure in relation to variations in the amplitude $\Delta$. The deformation parameter is controlled by the hopping energy $\gamma_{1}$. For sub-figs (a) to (i), $\gamma_{1}$ is set to 1.4$\gamma$; for sub-figs (j) to (r), it is set to 2$\gamma$; and for subfigs (s) to (z0), it is set to 3$\gamma$. These correspond to the cases $\alpha=0$ for the top plateau, $\alpha=1/\sqrt{2}$ for the middle plateau, and $\alpha=1$ for the bottom plateau. The value of the amplitude $\Delta$ is specified in each sub-fig.
• Figure 4: The Berry curvature distribution in the $k_{x}-k_{y}$ plane, corresponding to the conduction ($\nu =0$), flat ($\nu =1$) and valence ($\nu =2$) bands, is calculated for different values of parameter $\alpha$: $\alpha =0$ (the graphene case), $\alpha =0.48$, $\alpha =1/ \sqrt{2}$ (critical value corresponding to the phase transition where the band becomes dispersive at the Dirac point, as illustrated in Fig. \ref{['fig1']}-(d)), and $\alpha =1$ (dice lattice limit). Calculations are performed by setting $\Delta =0.18 \gamma$ in the standard case where $\gamma _{1}=1 \gamma$, without any deformation.
• Figure 5: The Berry curvature distribution in the presence of deformation by modifying $\gamma _{1}=2 \gamma$, while maintaining the other parameters in Fig. \ref{['fig2']}.