Universal nonanalytic behavior of the Hall conductance in a Chern insulator at the topologically driven nonequilibrium phase transition

We study the Hall conductance of a Chern insulator after a global quench of the Hamiltonian. The Hall conductance in the long time limit is obtained by applying the linear response theory to the diagonal ensemble. It is expressed as the integral of the Berry curvature weighted by the occupation number over the Brillouin zone. We identify a topologically driven nonequilibrium phase transition, which is indicated by the nonanalyticity of the Hall conductance as a function of the energy gap $m_f$ in the post-quench Hamiltonian ${\widehat{H}}_f$. The topological invariant for the quenched state is the winding number of the Green's function $W$, which equals the Chern number for the ground state of ${\widehat{H}}_f$. In the limit $m_f\to 0$, the derivative of the Hall conductance with respect to $m_f$ is proportional to $ln|m_f|$, with the constant of proportionality being the ratio of the change of $W$ at $m_f=0$ to the energy gap in the initial state. This nonanalytic behavior is universal in two-band Chern insulators such as the Dirac model, the Haldane model, or the Kitaev honeycomb model in the fermionic basis.

Figure 1

The Hall conductance $C_{\text{neq}}$ as a function of $(M_f,B_f)$ for different $(M_i,B_i)$. (Left panel) The initial state has a nonzero Chern number with $M_i,B_i>0$. (Right panel) The Chern number of the initial state is zero with $M_i>0$ but $B_i<0$.

Figure 2

$\partial C_{\text{neq}}/\partial M_f$ as a function of $ln|M_f|$. The different types of lines and points represent $\partial C_{\text{neq}}/\partial M_f$ for different $(M_i,B_i,B_f)$. We simultaneously plot $\partial C_{\text{neq}}/\partial M_f$ in the limit $M_f\to 0^{+}$ and $M_f\to 0^{-}$, which are undistinguishable at small $|M_f|$.

Figure 3

Schematic diagram of the Haldane model. The black and empty circles denote the “$A$” and “$B$” sites, respectively. The circle arrow at the center of the dotted lines connecting three $A$ sites or three $B$ sites shows the direction of hopping with the matrix element $t_2{e}^{i\varphi }$. The six vectors ${\vec{a}}_s$ and ${\vec{b}}_s$ with $s=1,2,3$ are marked, which are used for expressing the Hamiltonian in momentum space.

Figure 4

The reciprocal lattice of the Haldane model. $\mathrm{\Gamma }=(0,0)$ denotes the origin. The hexagon surrounding $\mathrm{\Gamma }$ point is the first Brillouin zone. The black (empty) circles denote the singularities at which the energy gap closes for $M=M_c^{-}$ $\left(M=M_c^{+}\right)$. We choose the Brillouin zone surrounded by the dashed lines to calculate the Hall conductance. $\vec{q}$ is the singularity inside this Brillouin zone as $M=M_c^{+}$, and the shadow around $\vec{q}$ represents the region contributing to the nonanalyticity of the Hall conductance.

Figure 5

The Hall conductance and its derivative in the vicinity of $M_f=M_f^c$. The former is plotted in the left panel, while the latter is plotted in the right panel. In the right panel, the solid lines denote $\frac{d{C}_{\text{neq}}}{d{M}_f}$ obtained from numerical integration, while the dashed one is a straight line with slope $\frac{1}{2|M_i-M_i^c|}$. The parameters are set to $M_i=0,{\varphi }_i={\varphi }_f=\pi /6,t_{2i}=0.5$, and $t_{2f}=1$.

Figure 6

Lattice of the Kitaev honeycomb model. Spin-1/2 degrees of freedom reside on the vertices of a honeycomb lattice. The anisotropic nearest-neighbor interaction depends on the link type $(x,y$, or $z)$.

Figure 7

The reciprocal lattice of the Kitaev honeycomb model. $\mathrm{\Gamma }=(0,0)$ denotes the origin. The black and the empty circles denote the singularities at which the energy gap closes as there is no magnetic field. The dashed lines surround the Brillouin zone that we choose to calculate the Chern number and the Hall conductance. $\overrightarrow{q_1}$ and ${\vec{q}}_2$ are two singularities inside this Brillouin zone. The shadows around them represent the regions contributing to the nonanalyticity of the Hall conductance.

Figure 8

(Left panel) The function $C_{\text{neq}}\left(M_f\right)$ for a quench starting from $M_i=1/2$ obtained by numerical integration (adaptive Simpson's) of Eq. (22) over a half unit cell and doubling the result. (Right panel) Derivative of $C_{\text{neq}}$ with respect to $M_f$ for a quench from $M_i=1/2$ and one quench from $M_i=1/4$. The dots denote the numerical results, while the lines are fits of the form $\left(-|M_i-M_i^c{|}^{-1}ln|M_f-M_f^c|+\text{const.}\right)$.