Plasmons in single- and double-component helical liquids: Application to two-dimensional topological insulators

The plasmon excitations in proposed single- and double-component helical liquid (HL) models are investigated within the random-phase approximation by calculating the density-density, spin-density, and spin-spin waves. The effect due to broken time-reversal symmetry on intraband-plasmon dispersion relation in the single-component HL system is analyzed and compared to those of well-known cases, such as conventional quasi-one-dimensional electron gases and armchair graphene nanoribbons. In the long-wavelength limit, the dispersion of the collective excitations provided by the density-density response of the single-component HL matches that of spin-density response of the two-component HL.

Figure 1

Full band structure $\varepsilon \left(k\right)$ of a 2DTI with $\Delta =2B$, $A=B=1$, and $N=30$. The blue curves (1) correspond to the bulk modes, while the two green curves (2) represent the topological surface states. Each state is doubly degenerate with respect to the two eigenvalues of the ${\sigma }_x$ matrix.

Figure 2

Based on Eq. (5), the real (solid curves) and imaginary (dashed curves) parts of ${\rho }_1$ [green (1)] and ${\rho }_2$ [blue (2)] at $k=0$ as a function of $\Delta /B$ for $A=B=1$.

Figure 3

The $q$ dependence, in units of the reciprocal interatomic spacing $a$, is presented for the scaled Coulomb matrix element $V\left(q\right)/(4e^2/{\epsilon }_{s}a)$ given in Eq. (10). The calculations were done between the upper $(3/40)K_0\left(q\right)$ [curve (3)] and lower bound $-(3/40)\ln \left(q\right)$ [curve (1)] for the ratio $0\le \Delta /B\le 4$ considered in this paper, where $A=B=1$. The approximate expression in Eq. (11) at $\Delta /B=2$ is verified by curve (2) (within the region between the upper and lower bounds) in the figure, which is used to evaluate the plasmon frequency in Eq. (13).

Figure 4

Scaled plasmon excitation energy ${\omega }_p\left(q\right)/{\omega }_0$ of 2DTI [blue solid curve (2)] given by Eq. (12) with $A=B=1$. The lower blue solid curve corresponds to $\Delta =2B$, while the upper blue solid curve (1) is for $\Delta =B$ or $\Delta =3B$. The result of Eq. (13) is given by the dashed blue curve (3) in the long-wavelength limit. The conventional 1DEG plasmon dispersion is indicated by a red dashed curve (4). The particle-hole continuum collapses onto the thin black straight line (5).

Figure 5

Spectral function $\text{Im}\left[{\epsilon }^{-1}(q,\omega )\right]$ of single-component HL in 2DTI as a function of $\omega /{\omega }_0$ with and chosen $q$. Here, the curves with different values for $q$ are offset vertically for clarity.

Figure 6

Plasmon weight ${\beta }_q$ for TI-based [blue solid curve (1)] and conventional [dashed red curve (2)] 1DEGs, where ${\omega }_0$ is chosen as the same for both cases.

Figure 7

Spin-spin wave [Eq.  (29)] and the plasmons [Eq.  (24)] in TI. Thick blue curve (2) corresponds to spin wave with large critical cutoff $1/\left(2A\right)V_q\ln \left(2A{k}_c\right)\gg 1$. Thick green curve (1) is the spin wave with the cutoff in the linear regime of the dispersion $2A{k}_c=1$. The dotted blue curve (3) is the plasmon mode, while the thin black line (4) is the particle-hole excitation.