Dissipative Callan-Harvey mechanism in $2+1$-dimensional Dirac system: Fate of edge states along a domain wall

The Callan-Harvey mechanism in the $2+1$-dimensional ($2+1$-D) Jackiw-Rebbi model is revisited. We analyzed Callan-Harvey anomaly inflow in the massive Chern insulator (quantum anomalous Hall system) subject to an external electric field. In addition to the conventional current flowing from the bulk to the edge due to parity anomaly, we considered the dissipation of the edge charge due to the interaction with an external bosonic bath in $2+1$ D and due to an external bath of photons in $3+1$ D. In the case of a $2+1$-D bosonic bath, we found the new stationary state, which is defined by the balance between the Callan-Harvey current and the outgoing flow caused by the dissipation processes. In the case a of $3+1$-D photon bath, we found a critical electric field below which this balance state can be achieved, but above which there is no such balance. However, if one considers a more realistic model with a momentum-dependent mass $\beta p^2$, the edge state merges into the bulk when the momentum is large, leading the system to the balance. Furthermore, we estimated the photon-mediated transition rate between a $2+1$-D bulk and $1+1$-D topological edge state of the order of $1 {\mathrm{ns}}^{-1}$ at room temperature.

Figure 1

The setup of the system. As explained in the main text of Sec. 2, the fermion mass changes sign at $x=0$, and therefore the Chern number is $1/2$ at the $x>0$ region and $-1/2$ at the $x<0$ region. In the presence of a uniform electric field in the $y$ direction (green arrows), the Hall currents (blue arrows) in the two regions flow in opposite directions. Therefore, there will be charge accumulation at the edge. The red arrows denote the edge current.

Figure 2

The energy spectra for electron systems and the edge-to-bulk transition processes. The blue and black lines describe the unoccupied and occupied states, respectively. Zigzag lines represent ingoing or outgoing photons. (a) A semi-infinite electron system with one edge or domain wall. There is only one chiral mode, whose occupied states are depicted by the straight blue line. The red arrow depicts the leading-order excitation process, which absorbs one photon. The green arrows are related to the second-order relaxation process, which includes an absorption of a low-energy photon (green zigzag line) and an emission of a high-energy photon (blue zigzag line). (b) The schematic diagram of transitions in a finite-size system with two edges. The two straight blue lines represent the occupied edge states. The red and purple arrows denote the absorption and emission processes, while the red and purple zigzag lines denote the ingoing and outgoing photons, respectively.

Figure 3

(a) The black curve (marked by label 1) shows the momentum dependence of the excitation rate ${\mathrm{\Gamma }}_1\left(k_e\right)/e^2$ of the edge state $|k_e>$. It is computed according to Eq. (24), with $v/c=0.01$. The horizontal axis is $k_e/\left(m_{0}v\right)$. The red curve (marked by label 2) is the distribution function $R\left(k_e\right)/R\left(0\right)$ in the saturation state, i.e., Eq. (26), with the electric field $E_y=2\times {10}^{-5}e{m}_{0}v$ related to the red dotted line in (b). (b) The saturation momentum $K_e^{*}$ obtained from Eq. (25) as a function of electric field $E_y$, in the step-function approximation for the edge-state occupancy.

Figure 4

(a) The black curve (marked by label 1) is the excitation rate ${\mathrm{\Gamma }}_1\left(k_e\right)$, as a function of momentum $k_e$, given by Eq. (31). ns: nanosecond. The parameters are given as follows: the bulk gap $\mathrm{\Delta }=2m_0{v}^2=0.1\mathrm{eV}$, Fermi velocity $v=0.01c$, and temperature $k_{B}T=\mathrm{\Delta }/4$. The red curve (marked by label 2) is the the occupation function $R\left(k_e\right)/R\left(0\right)$ in the saturation state, with $E_y=0.75$ V/m. It is obtained from Eq. (26), with the ${\mathrm{\Gamma }}_1\left(k_e\right)$ function given by Eq. (31). (b) The saturation momentum $K_e^{*}$ in the step-function assumption for the edge occupation vs electric filed $E_y$. One observes that there is a critical electric field, around 1.5 V/m, above which the value of $K_e^{*}$ diverges.