Symmetry and Topology in the Non-Hermitian Kitaev chain

Abstract

We investigate the non-Hermitian Kitaev chain with non-reciprocal hopping amplitudes and asymmetric superconducting pairing. We work out the symmetry structure of the model and show that particle-hole symmetry (PHS) is preserved throughout the entire parameter regime. As a consequence of PHS, the topological phase transition point of a finite open chain coincides with that of the periodic (infinite) system. By explicitly constructing the zero-energy wave functions (Majorana modes), we show that Majorana modes necessarily occur as reciprocal localization pairs accumulating on opposite boundaries, whose combined probability density exhibits an exact cancellation of the non-Hermitian skin effect for the zero energy modes. Excited states, by contrast, generically display skin-effect localization, with particle and hole components accumulating at opposite ends of the system. At the level of bulk topology, we further construct a $\mathbb{Z}_2$ topological invariant in restricted parameter regimes that correctly distinguishes the topological and trivial phases. Finally, we present the topological phase diagram of the non-Hermitian Kitaev chain across a broad range of complex parameters and delineate the associated phase boundaries.

Figures (5)

• Figure 1: A caricature of the non-Hermitian Kitaev chain with non-reciprocal left and right hopping amplitudes, $t_{1}$ and $t_{2}$ and asymmetric superconducting pairing terms $\Delta_{1}e^{-i\varphi_1}$ and $\Delta_{2}e^{i\varphi_2}$. Each site is labeled by a positive integer $j (\in \mathbb{Z}^{+})$ and has a finite on-site (chemical) potential $\mu$.
• Figure 2: Energy bands of the finite ($L=50$) non-Hermitian Kitaev chain with open boundary conditions (OBC) as function of the chemical potential, $\mu$. Non-Hermiticity is introduced by choosing non-reciprocal hopping parameters i.e., left hopping parameter $t_1 = 1$ and right hopping parameter, $t_{2}=1.5$. The superconducting pairing terms are $\Delta_{1} = 2 = \Delta_{2}$ and $\varphi_{1}=0=\varphi_{2}$ cf., Eq. \ref{['eq:non_herm_kitaev']}. (a) Absolute value of the energy bands as we vary the chemical potential $\mu$. The zero modes (or the topological phase) appear whenever the chemical potential is fine tuned to lie between $\mu_{1}=-\frac{1}{2}(t_{1}+t_{2})$ and $\mu_{2}=\frac{1}{2}(t_{1}+t_{2})$. As a result of breaking Hermiticity, one obtains in general, a complex energy spectrum i.e. non-zero real part (b) and non-zero imaginary part (c) of the energy eigenvalues.
• Figure 3: Probability density of the zero modes (doubly degenerate ground states) and first excited state in non-Hermitian Kitaev chain of length $L=50$, with open boundary conditions (OBC) in the topological phase i.e., $|\mu| <\frac{1}{2}(t_{1}+t_{2})$. The chemical potential is $\mu=0.2$ and the superconducting pairing terms are set to $\Delta_{1} = 2 =\Delta_{2}$ with $\varphi_{1}=0=\varphi_{2}$. The zero modes ($\psi_{0}^{(1)}$ and $\psi_{0}^{(2)}$) do not show any skin-effect with non-reciprocal hoppings. However, the first excited state ($\psi_{1}$) shows skin-effect (right or left localization) depending on the hopping parameters. (a) Sum of probability densities of zero mode wave functions, $\psi_{0}^{(1)}$ and $\psi_{0}^{(2)}$ with hopping parameters $t_{1} = 1.5$ (left hopping), $t_{2}=1$ (right hopping). (b) Sum of probability densities of zero mode wave functions with reciprocal hoppings $t_{1}= 1=t_{2}$. (c) Sum of probability densities of zero mode wave functions with reversed reciprocities in hopping i.e., $t_{1}=1$ and $t_{2}=1.5$. (d) Probability density of the first excited state showing left localization (blue) with $t_{1} (=1.5)>t_{2}(=1)$, right localization (red) with $t_{1} (=1)<t_{2}(=1.5)$ in contrast to the case with reciprocal hoppings $t_{1}=1=t_{2}$ (green).
• Figure 4: Probability density of zero mode wavefunctions and bulk mode wavefunctions of the non-Hermitian Kitaev chain with $L=50$, $\mu=0.2$, $t_1 =1+i$, $t_2=0.5+i$, $\Delta_1=2+i$ and $\Delta_2=1+i$. The zero modes are immune to the NHSE whereas the particle and hole wavefunctions show an equal and opposite skin effect. (a) Sum of probability densities of the zero modes $\psi_0^{(1)}$ and $\psi_0^{(2)}$ is symmetric and localized at the edges. (b) The dashed red line denotes the probability density of the smallest positive energy (first excited state for a particle i.e., $+E>0$) state and the solid blue line is the probability density of the smallest negative energy (first excited state for a hole i.e., $-E<0$) state.
• Figure 5: Topological phase diagram for non-Hermitian Kitaev chain in the complex chemical potential ($\mu$) plane. The quantities $E_{0}^{(1)}$ and $E_{0}^{(2)}$ are the two lowest eigen-energies (absolute values) of the non-Hermitian Kitaev chain, Eq. \ref{['eq:non_herm_kitaev']}. Vanishing of the two (eigen-energies, $E_{0}^{(1)}$ and $E_{0}^{(2)}$) corresponds to the Majorana-edge modes and hence marks the topological phase. (a) Phase diagram with non-reciprocal real hopping parameters. For obtaining the phase diagram we have used the following specific parameters, $t_{1}=1,\,t_{2}=2, \,\Delta_{1}=2,\,\Delta_{2} = 2, \,\text{and} \,\varphi_{1}=0=\varphi_{2}$. (b) Phase diagram with non-reciprocal complex hopping parameters with equal imaginary parts i.e., $\text{Im}(t_{1})=\text{Im}(t_{2})$. Specific choice of parameters for obtaining the phase diagram are as follows, $t_{1}=1 + i,\,t_{2}=2 + i, \,\Delta_{1}=2,\,\Delta_{2} = 2, \,\text{and} \,\varphi_{1}=0=\varphi_{2}$.