Topological phases in the non-Hermitian Su-Schrieffer-Heeger model

We address the conditions required for a $\mathbb{Z}$ topological classification in the most general form of the non-Hermitian Su-Schrieffer-Heeger (SSH) model. Any chirally symmetric SSH model will possess a “conjugated-pseudo-Hermiticity” which we show is responsible for a quantized “complex” Berry phase. Consequently, we provide an example where the complex Berry phase of a band is used as a quantized invariant to predict the existence of gapless edge modes in a non-Hermitian model. The chirally broken, $PT$-symmetric model is studied; we suggest an explanation for why the topological invariant is a global property of the Hamiltonian. A geometrical picture is provided by examining eigenvector evolution on the Bloch sphere. We justify our analysis numerically and discuss relevant applications.

Figure 1

Top: Chiral SSH with non-Hermitian hopping parameters, no potential. Bottom: Chirally broken, $PT$-symmetric SSH with non-Hermitian staggered potential.

Figure 2

One-dimensional phase diagram in $v$ for the $PT$-symmetric case $v_1=v_2=v\in \mathbb{R},w_1=w_2=w\in \mathbb{R}$ at a fixed value of $u,w>0$. The system moves in and out of the $PT$-broken phase as a function of $v$. $\nu$ is the topological index which predicts how many pairs of gapless-real-energy edge modes exist in the system. The bulk spectrum is gapless in the entire $PT$-broken phase.

Figure 3

The lower-band vectors ${\mathbf{b}}_k$ (red) on the Bloch sphere in the range $k\in \left[-\pi ,\pi \right)$ for a chiral model when $Q_{\pm }^c=0$ (left) and $Q_{\pm }^c=\pi$ (right). The north–south axis is dashed in black. Systems with distinct winding number cannot be deformed into each other without passing through the north or south poles (black dots), which can occur only at a band crossing in the chiral model. ${\mathbf{b}}_k$ are not constrained to the equator.

Figure 4

Chiral, inversion-symmetric setup: $u=0,v_1=v_2=exp\left(i\pi /5\right)sint,w_1=w_2=cost,N=100$. Edge modes are in red; bulk modes are in blue.

Figure 5

Chiral, inversion-breaking setup: $u=0,v_1=sint,v_2=exp\left(i\pi /10\right)sint,w_1=cost,w_2=exp\left(i\pi /5\right)cost,N=100$. Edge modes are in red; bulk modes are in blue.

Figure 6

$PT$-symmetric setup: $u=0.3,v_1=v_2=sint,w_1=w_2=cost,N=100$. Edge modes are in red; bulk modes are in blue.

Figure 7

(a) Magnitude of the finite-chain spectrum with periodic boundary conditions (no edge modes) for $v=expi{\chi }_{v}sint$, $w=expi{\chi }_{w}cost$, ${\chi }_v=\pi /5,{\chi }_w=0$. The magnitude of the energy gap scales as $\sqrt{t}$ away from the topological transition. (b) Bulk dispersion at transition point $t=\pi /4$ when ${\chi }_v=-{\chi }_w=0.3\left(2\pi \right)$. The spectrum at any $k$ is either purely real or imaginary.