Floquet topological phases coupled to environments and the induced photocurrent

We consider the fate of a helical edge state of a spin Hall insulator and its topological transition in presence of a circularly polarized light when coupled to various forms of environments. A Lindblad-type equation is developed to determine the fermion occupation of the Floquet bands. We find by using analytical and numerical methods that nonsecular terms, corresponding to two-photon transitions, lead to a mixing of the band occupations, hence the light-induced photocurrent is in general not perfectly quantized in the presence of finite coupling to the environment, although deviations are small in the adiabatic limit. Sharp crossovers are identified at driving frequencies near the Rabi frequency $\mathrm{\Omega }$ (which is the strength of light-matter coupling) and at $\frac{1}{2}\mathrm{\Omega }$ with the former resembling a phase transition.

Figure 1

The cartoon of the system, consisting of a QSH edge state with spin filtered conducting channels, interacting with circularly polarized electromagnetic field and coupled to an environment, is visualized.

Figure 2

Comparison of the edge current [in units of $e\mathrm{\Omega }/2\pi ]$ when the states are occupied based on their average energy [18], and when they are coupled to a zero temperature bath. The curves correspond to the secular approximation, which describes the infinitesimal system-bath coupling. The $s=0$ curve is understood as the limiting behavior as $s\to 0$.

Figure 3

The nonvanishing element of the density matrix in the stationary state at zero temperature ($\omega =1.1\mathrm{\Omega }$, $\alpha \to 0$ limit). The three curves correspond to different bath spectral functions $J\left(\nu \right)$. The excitations in the steady state are with stronger weight in the sub-Ohmic ($s=0$) case compared to Ohmic ($s=1$) or super-Ohmic environments ($s=2$).

Figure 4

Average stationary value of ${\rho }_z$ for various $\omega /\mathrm{\Omega }$ ratios in the XXX case ($s=1$, $\alpha =0.05$). (a) $\omega >\mathrm{\Omega }$, correspondingly ${\rho }_z$ deviates from the secular solution at critical momenta $p_{\pm }^{*}$ (${\mathrm{\Omega }}^{\prime }-\omega \approx 0$) and $p_{\pm }^{**}$ (${\mathrm{\Omega }}^{\prime }-2\omega \approx 0$). This is attributed to two-photon transitions, where the Floquet bands touch each other (bottom). (b) At lower frequency ($\mathrm{\Omega }>\omega >\frac{1}{2}\mathrm{\Omega }$) only one dangerous nonsecular term survives, giving rise to peaks at $p_{\pm }^{**}$. (c) When $\omega \lesssim \frac{1}{2}\mathrm{\Omega }$, the secular approximation still gets corrections because the dangerous frequency ${\mathrm{\Omega }}^{\prime }-2\omega$ is small at $p\approx \omega$. The actual values of $\omega /\mathrm{\Omega }$ are 1.1 in (a), 0.75 in (b), and 0.48 in (c).

Figure 5

Stationary current induced by a circular driving on a QSH edge coupled to an Ohmic bath (the vertical units are in $j_c(\omega =\mathrm{\Omega })=\frac{e\mathrm{\Omega }}{2\pi }$, $\alpha =0.1$). (a) Numerical solution. (b) Analytical results for the symmetric case with $b_x=b_y=b_z$ and ${\mathrm{\Gamma }}_x={\mathrm{\Gamma }}_y={\mathrm{\Gamma }}_z$. The secular approximation corresponds to an infinitesimal system-bath coupling, where the quantization is exact until the driving frequency reaches the Rabi frequency. The two-photon processes at finite coupling constants violate the quantization, which become effective at $\omega \sim \frac{1}{2}\mathrm{\Omega }$. This is well captured in the DFA. (c) The photon-resonances also give rise to an ac current with frequency $n\omega$, $n\in \mathbb{N}$.

Figure 6

Frequency dependence and crossovers in the transverse magnetization (in units of $\frac{\mathrm{\Omega }}{2\pi }$). In contrast to the current, the transverse magnetization oscillates with frequency $\omega$ in the $\alpha \to 0$ limit, with amplitude $M_c$ for $\omega <\mathrm{\Omega }$. (a) Deviation from the limiting value $M_c$ at finite system-bath coupling, numerical solution ($\alpha =0.1$). (b) DFA with the two dangerous frequencies and with ${\mathrm{\Omega }}^{\prime }$ is capable to reconstruct all the features in the numerical solution. (c) Other Fourier components of the magnetization.

Figure 7

Comparison of the Markovian and non-Markovian time evolution in an Ohmic environment for short (a) and long (b) times.

Figure 8

Stationary state dynamics of the density matrix on the Bloch sphere in the interaction and Schrödinger pictures. The curves show the path of $\mathbf{\rho }$ close to a critical point, where the nonsecular terms are non negligible. In the secular approximation (red dashed curve and arrow), $\mathbf{\rho }$ reaches a constant value in the interaction picture, which corresponds to a circular motion in the laboratory frame (i.e., in the Schrödinger picture). In the DFA, $\mathbf{\rho }$ draws a circle around the secular solution in the interaction picture. This circle also has a fine structure, if we go beyond the DFA (blue solid curve and arrow).