Anomalous dynamical response of non-Hermitian topological phases

Composite topological phases with intriguing topology like Möbius strips emerge in sublattice symmetric non-Hermitian systems due to spontaneous breaking of time-reversal symmetry at some parameter regime. While these phases have been characterized by nonadiabatic complex geometric phases of multiple participating complex bands, the physical properties of these phases remain largely unknown. We explore the dynamical response of these phases by studying Loschmidt echo from an initial state of the Hermitian Su-Schrieffer-Heeger (SSH) model, which is evolved by a non-Hermitian SSH Hamiltonian after a sudden quench in parameters. Topology-changing quenches display nonanalytical temporal behavior of return rates (logarithm of the Loschmidt echo) for the non-Hermitian SSH Hamiltonian in the trivial, Möbius, and topological phase. Moreover, the dynamical topological order parameter appears only at one side of the Brillouin zone for the Möbius phase case in contrast to both sides of the Brillouin zone for quench by the trivial and topological phase of the non-Hermitian SSH model. The last feature is a dynamical signature of different symmetry constraints on the real and imaginary parts of the complex bands in the Möbius phase.

Figure 1

Summary of main results for two quench protocols from a trivial (T) phase of the Hermitian SSH model to a Möbius (M) and a topological (TP) phase of the non-Hermitian SSH model. The Pancharatnam phase ${\varphi }^G(k,t)$ is calculated for a single particle with momentum $k$ at time $t$. Parameters of the initial and final Hamiltonian are denoted by ${\delta }_{\mathrm{i}},{\theta }_{\mathrm{i}}$ and ${\delta }_{\mathrm{f}},{\theta }_{\mathrm{f}}$, respectively. The number of DTOPs per Brillouin zone is represented by ${\nu }^{\text{BZ}}$, which shows different ${\nu }^{\text{BZ}}$ for the final Hamiltonian in the M and TP phases.

Figure 2

The complex energy spectrum $E_{\pm }\left(k\right)$ with momentum $k$ (top row) and on the parametric space of $\text{Re}\left[E_{\pm }\left(k\right)\right]$ and $\text{Im}\left[E_{\pm }\left(k\right)\right]$ (middle row) in three different phases of the non-Hermitian SSH model. The contour of the end points of the real (full line) and imaginary (dashed line) parts of ${\vec{d}}_k$ is shown on the $d_k^x,d_k^y$ plane as $k$ is swept across the Brillouin zone, $k=-\pi \to \pi$ (bottom row). The parameters are $J=1,\theta =0.4$, and $\delta =-0.9$ (trivial), $-0.1$ (Möbius), and 0.9 (topological). The dot represents a value at $k=-\pi$ and the arrow indicates $k=-\pi \to \pi$ for complex energy $E_{\pm }\left(k\right)$ and Bloch vector ${\vec{d}}_k$ in the middle and bottom rows. The exceptional point and the origin of the $d_k^x,d_k^y$ plane are shown by a black star and a dot, respectively.

Figure 3

Return rate $I\left(t\right)$ in (a), DTOP over negative momentum ${\nu }_{-}\left(t\right)$ in (b), and DTOP over positive momentum ${\nu }_{+}\left(t\right)$ in (c) as a function of time. Fisher zero lines in the complex plane of $w(n,k)$ over the BZ, $k\in (-\pi ,\pi ]$ (for $n=0,1,\cdots ,4$) in (d). Points of intersection of the Fisher zero lines and the dashed black line indicate critical momentum $k_c^{n\pm }$. Dashed brown (dashed-dotted gray) vertical lines through these points correspond to critical times $t_c^{n-}\left(t_c^{n+}\right)$. Initial and final Hamiltonian parameters are ${\delta }_{\mathrm{i}}=0.9,{\theta }_{\mathrm{i}}=0,{\delta }_{\mathrm{f}}=-0.9,{\theta }_{\mathrm{f}}=0.4$.

Figure 4

Return rate $I\left(t\right)$ in (a), DTOP over negative momentum ${\nu }_{-}\left(t\right)$ in (b), and DTOP over positive momentum ${\nu }_{+}\left(t\right)$ in (c) as a function of time. Fisher zero lines in the complex plane of $w(n,k)$ over the BZ, $k\in (-\pi ,\pi ]$ (for $n=0,1,\cdots ,4$) in (d). Parameters for topological to Möbius quench are ${\delta }_{\mathrm{i}}=0.9,{\theta }_{\mathrm{i}}=0\Rightarrow {\delta }_{\mathrm{f}}=-0.1,{\theta }_{\mathrm{f}}=0.4$. Vertical dashed-dotted gray lines show critical times ($t_c^{n+}$) corresponding to positive critical momenta ($k_c^{n+}$). $I\left(t\right)$ and DTOPs in the Hermitian limit (${\theta }_{\text{f}}=0$) are depicted by pink dotted lines.

Figure 5

Return rate $I\left(t\right)$ in (a), DTOP over negative momentum ${\nu }_{-}\left(t\right)$ in (b), and DTOP over positive momentum ${\nu }_{+}\left(t\right)$ in (c) as a function of time. Fisher zero lines in the complex plane of $w(n,k)$ over the BZ, $k\in (-\pi ,\pi ]$ (for $n=0,1,\cdots ,4$) in (d). Parameters for trivial to topological quench are ${\delta }_{\mathrm{i}}=-0.9,{\theta }_{\mathrm{i}}=0\Rightarrow {\delta }_{\mathrm{f}}=0.9,{\theta }_{\mathrm{f}}=0.4$. DTOPs in the positive and negative momentum of the BZ ($\pm$) are aligned with the critical times $t_c^{n\pm }$ (vertical lines) and the corresponding critical momenta $k_c^{n\pm }$.

Figure 6

Return rate $I\left(t\right)$ in (a), DTOP over negative momentum ${\nu }_{-}\left(t\right)$ in (b), and DTOP over positive momentum ${\nu }_{+}\left(t\right)$ in (c) as a function of time. Fisher zero lines in the complex plane of $w(n,k)$ over the BZ, $k\in (-\pi ,\pi ]$ (for $n=0,1,\cdots ,4$) in (d). Parameters for topological to Möbius quench are ${\delta }_{\mathrm{i}}=-0.9,{\theta }_{\mathrm{i}}=0\Rightarrow {\delta }_{\mathrm{f}}=-0.1,{\theta }_{\mathrm{f}}=0.4$. Vertical lines indicate critical times $t_c^{n+}$ for critical momenta $k_c^{n+}$ in the BZ ($k\in [0,\pi ]$). $I\left(t\right)$ and DTOPs in the Hermitian limit (${\theta }_{\text{f}}=0$) are depicted by pink dotted lines.

Figure 7

The real (a,c) and imaginary (b,d) energy spectrum of the NSSH model as a function of intercell hopping parameter $w$ for periodic $\eta =1$ (a,b) and special $\eta =1/L$ (c,d) boundary. The parameters are $v_l=v_r=1, w_l=w{e}^{\theta }, w_r=w, \theta =0.4, L=100$.

Figure 8

The spatial profile of right eigenvectors (${\left|\right|\psi \rangle |}^2$) corresponding to the energies (a) 0.69, (b) $0.124i$, and (c) 0.0087 for $v_l=v_r=1, w_r=w, w_l=w{e}^{\theta }, \theta =0.4,L=100, \eta =1/L$ and (a) $w=0.25$, (b) $w=0.75$, (c) $w=1.25$.

Figure 9

The boundary contribution to return rate ($I_B$) in (a,c) and the dynamics of edge occupancy ($n_{a_1,b_L}$) in (b,d) as a function of time for a quench from a topological to a trivial phase (${\delta }_{\text{i}}=-{\delta }_{\text{f}}=0.9, {\theta }_{\text{i}}=0, {\theta }_{\text{f}}=0.4$) in (a,b), and a trivial to a topological phase (${\delta }_{\text{i}}=-{\delta }_{\text{f}}=-0.9, {\theta }_{\text{i}}=0, {\theta }_{\text{f}}=0.4$) in (c,d) for a chain of $L=500$.

Figure 10

The boundary contribution to the return rate ($I_B$) in (a,c) and the dynamics of edge occupancy ($n_{a_1,b_L}$) in (b,d) as a function of time for a quench from a topological (${\delta }_{\text{i}}=0.9$) to a Möbius phase in (a,b) and a trivial (${\delta }_{\text{i}}=-0.9$) to a Möbius phase in (c,d) for a chain of $L=500$. The other parameters are ${\delta }_{\text{f}}=-0.1, {\theta }_{\text{i}}=0, {\theta }_{\text{f}}=0.4$.