Quantum quenches in a pseudo-Hermitian Chern insulator

We propose to uncover the topology of a pseudo-Hermitian Chern insulator by quantum quench dynamics. The Bloch Hamiltonian of the pseudo-Hermitian Chern insulator is defined in the basis of the $q$-deformed Pauli matrices, which are related to the representation of the deformed algebras. We show the bulk-surface duality of the pseudo-Hermitian phases, then further build a concrete relation between the static band topology and quench dynamics, in terms of the time-averaged spin textures. The results are also generalized into a fully nonequilibrium case where the postquench evolution is governed by a Floquet pseudo-Hermitian Hamiltonian. Furthermore, we propose a possible scheme to realize the seemingly challenging model in a bilayer lattice and detect the dynamics with a double-quench protocol.

Figure 1

(a) Energy spectrum of the reduced tight-binding chain of the pseudo-Hermitian Chern insulator for $M=0.8J$, $q=3$, and lattice sites $L_x=50$ when the open boundary condition is imposed along the $z$ direction. (b) The corresponding bulk spectrum of the pseudo-Hermitian Chern insulator.

Figure 2

The Hopf link formed by the preimages of two time-evolved states in momentum-time space. Here $l_x=(1,0,0)$, $l_y=(0,1,0)$, and $l_z=(0,0,1)$.

Figure 3

(a) The long-time average of the Bloch vector $\mathit{n}$ after a sudden quench from $M=\infty$ to $J$. The dashed lines indicate the BIS determined by $d_z\left(\mathbf{k}\right)=0$. (b) The dynamical spin-texture field $\mathit{g}\left(\mathit{k}\right)$ on the BIS. The winding of $\mathit{g}\left(\mathit{k}\right)$ indicates a nontrivial Chern number $C_1=-1$. The evolution time is taken as $T=30$ (in units of $1/J$).

Figure 4

(a) Quasienergy spectrum of the Floquet pseudo-Hermitian Chern insulator with open boundary condition imposed along the $z$ direction. (b) The band structure of the postquench effective Floquet Hamiltonian ${\mathcal{H}}_F$. (c) The winding of the dynamical field $\mathbf{g}\left(\mathbf{k}\right)$ on the zero BIS and $\pi$ BIS. (d) The long-time average of the Bloch vector $\mathit{n}$ after a sudden quench from a deeply trivial phase to a nontrivial Floquet pseudo-Hermitian phase. (e, f) The relative phase angle of the dynamical field $g_x\left(\mathbf{k}\right)$ and $g_y\left(\mathbf{k}\right)$ on the zero BIS and $\pi$ BIS, which characterizes the topological invariants $w_{0,\pi }=1$. The parameters in the postquench Hamiltonian are as follows: $M=8v_z$, $q=3$, $\omega =5v_z$, $T_1=0.6T$, $v_{x,y,z}=J$.

Figure 5

Schematic illustration of the bilayer cubic lattice with engineered hopping on (a) the $x\text{−}y$ plane and (b) the $y\text{−}z$ plane. The dashed lines in (a) show the next-nearest-neighbor hoppings. Sites $\gamma$ in the upper layer are dissipative, used as auxiliary sites to induce asymmetric coupling. The phases of the interlayer hopping rates contribute a staggered synthetic flux ${\varphi }_1$ and ${\varphi }_2$.

Figure 6

(a) The spectroscopic signatures of the momentum density $n_{\alpha }(\mathbf{k},t)$ for Hamiltonian (26) after the first quench at time $t=0.5$ (in units of $1/J$). (b) The corresponding tomography results of $n_{\alpha }(\mathbf{k},t)$ extracted from the TOF images after the second quench, using 30 time sampling points. (c) Density oscillations with respect to the precession time $t_p$ at momentum point $\mathbf{k}=(\pi /2,\pi /2)$ after the second quench. (d) Example image of all components of the Bloch vector $n_i$ ($i=x,y,z$) in the $k_x\text{−}{k}_y$ plane. The arrows show the winding of the in-plane components $(n_x,n_y)$, and the background color scale shows the magnitude of the out-of-plane component $n_z$. The other parameters are as follows: $v_z=5v_{x,y}=2J$, $m=J$, $q=3$, ${\mathrm{\Delta }}_{\alpha \beta }=\pi$.