Collective modes for helical edge states interacting with quantum light

We investigate the light-matter interaction between the edge state of a 2D topological insulator and quantum electromagnetic field. The interaction originates from the Zeeman term between the spin of the edge electrons and the magnetic field, and also through the Peierls substitution. The continuous U(1) symmetry of the system in the absence of the vector potential reduces into discrete time reversal symmetry in the presence of the vector potential. Due to light-matter interaction, a superradiant ground state emerges with spontaneously broken time reversal symmetry, accompanied by a net photocurrent along the edge, generated by the vector potential of the quantum light. The spectral function of the photon field reveals polariton continuum excitations above a threshold energy, corresponding to a Higgs mode and another low energy collective mode due to the phase fluctuations of the ground state. This collective mode is a zero energy Goldstone mode that arises from the broken continuous U(1) symmetry in the absence of the vector potential and acquires finite a gap in the presence of the vector potential. The optical conductivity of the edge electrons is calculated using the random phase approximation by taking the fluctuation of the order parameter into account. It contains the collective modes as a Drude peak with renormalized effective mass, which moves to finite frequencies as the symmetry of the system is lowered by the inclusion of the vector potential.

Figure 1

Schematic illustration of a quantum spin Hall insulator with spin filtered edge state placed inside an optical cavity. The green ellipselike object represents the quantum spin Hall insulator that supports the edge states, while the blue and red arrows the spin-momentum locked electrons on the edge. The wave vector of the cavity mode is perpendicular to the direction of the edge electrons.

Figure 2

The contourplot for the total ground-state energy. In panel (a) $g_A=0$, the energy exhibits a mexican hat structure with $\phi \in [0,2\pi ]$ one can sweep through the ground-state manifold with no energy cost. In panel (b) $\rho g_A=0.1$ the minimum bends toward the real axis when $\phi =0$ and $\pi$, so tunneling between the two degenerate ground states will require a finite amount of energy. The other parameters used: $\rho W=100,\rho \omega =1,\rho g=1$.

Figure 3

The mean photon number density as the function of the Zeeman coupling. The parameters used: $\rho \omega =1,\rho g_A=0.1$.

Figure 4

The real root of $F\left({\mathrm{\Omega }}_0\right)=0$ as the function of increasing vector potential coupling strength $\left(g_A\right)$. This is understood as the energy requirement for phase fluctuations and as the energy of the gapped Goldstone mode. The parameters used: $\rho \omega =1,\rho g=0.4$. The inset figure is the spectral weight of the Goldstone mode as the function of the Zeeman coupling with parameters: $\rho W=100,\rho \omega =1$.

Figure 5

The contour plot of the spectral function $ln\left(\omega A\right(\mathrm{\Omega }\left)\right)$ as the function of the Zeeman coupling $\rho \tilde{g}$ with $g_A=0.1$. The energy of the Goldstone mode ${\mathrm{\Omega }}_0$ is shown as the red solid line in the $\mathrm{\Omega }<2\left|\mathrm{\Delta }\right|$ regime. The white dashed line denotes the minimum excitation energy $2\left|\mathrm{\Delta }\right|$, above which the polariton continuum is formed. The parameters used: $\rho W=100,\rho \omega =1$.

Figure 6

The frequency dependent conductivity of the edges. The solid black is the single particle result, the red contains the collective terms with $\rho g_A=0$, and the blue contains all terms with $\rho g_A=0.1$. The Goldstone peaks are also depicted; as $g_A$ is nonzero the peak moves to frequency ${\mathrm{\Omega }}_0$ and its weight increases. Further parameters: $\rho \omega =1,\rho W=100,\rho g=0.4$.

Figure 7

The effective mass as the function of the Zeeman interaction strength is plotted and it resembles a step function. The arrow indicates that the increase of $g_A$ shifts this step, thus the effective mass becomes infinite at larger Zeeman coupling strength. This means that the interaction involving the vector potential is making the collective modes more stable at larger $g$. The parameters used: $\rho \omega =1,\rho W=100.$