Topological invariants in dissipative extensions of the Su-Schrieffer-Heeger model

We investigate dissipative extensions of the Su-Schrieffer-Heeger model with regard to different approaches of modeling dissipation. In doing so, we use two distinct frameworks to describe the gain and loss of particles: One uses Lindblad operators within the scope of Lindblad master equations, and the other uses complex potentials as an effective description of dissipation. The reservoirs are chosen in such a way that the non-Hermitian complex potentials are $\mathcal{PT}$-symmetric. From the effective theory we extract a state which has similar properties as the nonequilibrium steady state following from Lindblad master equations with respect to lattice site occupation. We find considerable similarities in the spectra of the effective Hamiltonian and the corresponding Liouvillian. Further, we generalize the concept of the Zak phase to the dissipative scenario in terms of the Lindblad description and relate it to the topological phases of the underlying Hermitian Hamiltonian.

Figure 1

Sketch of the SSH model and its extensions for $n=8$ sites. Top row: The Hermitian model consists of double-well unit cells (sites $A,B$) with intratunneling $t_1$ joined by the intercell hopping $t_2$. Lower side: In this paper we study dissipative extensions of the SSH model where sites marked with a plus (minus) sign indicate single-particle gain (loss). The two patterns with dissipation only among the boundary sites ($U_1$, middle row) and alternating gain and loss ($U_2$, bottom row) are motivated by previous works [5, 14, 16, 17, 28].

Figure 2

Complex single-particle energy spectrum of the $\mathcal{PT}$-symmetric Hamiltonian $H_{\text{eff}}^{\left(U_2\right)}$ for different dissipative strengths with highlighted differences between trivial ($\theta =2\pi /3$, dark blue points) and nontrivial ($\theta =\pi /3$, additional light orange points) dimerization. Additional features in the nontrivial lattice configuration are caused by zero-energy edge modes.

Figure 3

Rapidity spectrum of the Liouvillian shape matrix with dissipative couplings according to $U_2$. The decay rates $\pm \mathrm{Re}\left(\beta \right)$ of the NMM shown in the left panel are independent of the lattice dimerization (see Appendix pp2). By contrast, the imaginary parts $\mathrm{Im}\left(\beta \right)$ depend only on the Hamiltonian and are presented in the right panel for different dimerizations $\theta$. We emphasize that the imaginary rapidity spectrum exactly reproduces the energies of the Hermitian SSH model.

Figure 4

Lattice occupation corresponding to the NESS (blue dots) and MBS (open orange circles) of the SSH model with alternating gain and loss ($\gamma =0.5,1.4,2.5$ from top to bottom). Left panels: Nontrivial dimerization ($\theta =\pi /3$). Right panels: Trivial dimerization ($\theta =2\pi /3$).

Figure 5

Complex Zak phase $\nu$ of the SSH model ($t=\mathrm{\Delta }=1$) with alternating gain and loss. Left panel: Real part of the complex Zak phase obtained from the effective description using a complex $\mathcal{PT}$-symmetric potential. The hatched area marks the $\mathcal{PT}$-broken parameter regime in which the real part of the complex Zak phase is not quantized (see Appendix pp1 for details). Right panel: Zak phase characterizing an NMM band of the Bloch Liouvillian obtained from a description of the system using an LME.

Figure 6

Complex single-particle eigenvalue spectrum of the $\mathcal{PT}$-symmetric Hamiltonian of the SSH model subject to the complex potential $U_1$ with $\theta =\pi /3$ (blue and green points in the left panel, and right top) and $\theta =2\pi /3$ (blue and orange points in the left panel, and right bottom). The colors of the imaginary part branches on the right correspond to the real part of the complex energies shown on the left.

Figure 7

Rapidity spectrum of the Liouvillian of the SSH model with gain and loss at the edges (Lindblad operators $L_{\mu }^{\left(U_1\right)}$) with $\theta =\pi /3$ (left top, and blue and green points in the right panel) and $\theta =2\pi /3$ (left bottom, and blue and orange points in the right panel). The colors of the real part branches correspond to the imaginary part of the rapidities shown on the right.

Figure 8

Comparison of the lattice site occupation profile of the NESS (blue circles) and the corresponding MBS (open orange circles) for $t=\mathrm{\Delta }=1$ and $\theta =\pi /3$ on the left and $\theta =2\pi /3$ on the right and $\gamma =0.25\text{and}4$ from top to bottom.

Figure 9

Complex energy spectrum of the $\mathcal{PT}$-symmetric Hamiltonian $H_{\text{eff}}^{\left(U_2\right)}$ with 64 sites subject to disorder of the form (C1) for 100 random realizations at $\gamma =0.5$ for $\theta =\pi /3\text{and}2\pi /3$ in the top and bottom row. Gray dots indicate the extreme scenario where ${\xi }_m=1$ for all $m$.

Figure 10

Rapidity spectrum of the Liouvillian describing the dissipative SSH model with 64 sites with alternating gain and loss subject to disorder of the form (C1) for 100 random realizations for $\theta =\pi /3\text{and}2\pi /3$ in the top and bottom row. Note that the shown spectrum does not depend on $\gamma$. Gray dots indicate the extreme scenario where ${\xi }_m=1$ for all $m$.