Quantum non-Markovian Hatano-Nelson model

Abstract

While considering non-Hermitian Hamiltonians arising in the presence of dissipation, in most cases, the dissipation is taken to be frequency independent. However, this idealization may not always be applicable in experimental settings, where dissipation can be frequency-dependent. Such frequency-dependent dissipation leads to non-Markovian behavior. In this work, we demonstrate how a non-Markovian generalization of the Hatano-Nelson model, a paradigmatic non-Hermitian system with nonreciprocal hopping, arises microscopically in a quasi-one-dimensional dissipative lattice. This is achieved using non-equilibrium Green's functions without requiring any approximation like weak system-bath coupling or a time-scale separation, which would have been necessary for a Markovian treatment. The resulting effective system exhibits nonreciprocal hopping, as well as uniform dissipation, both of which are frequency-dependent. This holds for both bosonic and fermionic settings. We find solely non-Markovian nonreciprocal features like unidirectional frequency blocking in bosonic setting, and a non-equilibrium dissipative quantum phase transition in fermionic setting, that cannot be captured in a Markovian theory, nor have any analog in reciprocal systems. Our results lay the groundwork for describing and engineering non-Markovian nonreciprocal quantum lattices.

Figures (5)

• Figure 1: Schematic illustration of our set-up for realizing the non-Markovian Hatano-Nelson model. Here $g$ is the nearest neighbour hopping strength in the 1D chain. Each pair of neighboring sites is connected to a dissipative auxiliary site with strength $g_b$, forming a triangular unit. One of the hoppings between the auxiliary site and the chain is complex, with a phase $\phi$. This can arise from a flux $\theta$ in the triangular unit, with $\phi=3\theta$ (see Appendix \ref{['Sec: Hamiltonian for non-Markov']}).
• Figure 2: Panels (a) and (b) show heatmaps of spectral functions $A(k,\omega)$ and $A^M(k,\omega)$ of non-Markovian and Markovian Hatano Nelson models, respectively. The horizontal dotted line in (a) corresponds to $\omega=\Delta_b$. The stars in (a) and (b) denote the point $(k_*, \omega_*)$. Panel (c) shows plots of $f_{\pm}(\omega)$ (see Eq. \ref{['G_scaling']}), and Markovian counterparts, $f_{\pm}^M(\omega)$. Panel (d) shows transmission amplitude $\tau_+(\omega)$ ($\tau_-(\omega)$) from site $1$ to $N$ ($N$ to $1$), along with their Markovian counterparts, $\tau_{\pm}^M(\omega)$. The light gray line shows the transmission for the reciprocal tight-binding model (i.e, $g_b=0$) for comparison. The vertical dashed line corresponds to $\omega=\omega_*$. Panel (e) shows the scaling of $I_{\pm}(\mu_d)$ with chain length $N$, where $\omega_*^{\pm}=\omega_*\pm 0.1 g$. $I_{\pm}^M(\mu_d)$ denotes the Markovian case, evaluated at $\mu_d=0.1g$. The black dashed lines are fits to the form $C N^{-0.5}$. We clearly see behavior consistent with Eq. \ref{['NDQPT']}. Panel (f) shows plots of $\sqrt{N}I_{+}(\mu_d)$ with $\mu_d$. The vertical dashed line shows $\mu_d=\omega_*$. This clearly highlights the NDQPT. The Markovian case is also shown by solid symbols for comparison, which scales as $N^{-0.5}$$\forall\mu_d$, despite a decay in value for $\mu_d < \Delta_c$. Parameters: $\phi=2\pi/3$, $k_*=\pi/3$, $\omega_*=\Delta_c-g$. For all plots except panel (d), $g_b=0.3g$, $\Sigma(z)=\kappa/2$, $\kappa=0.25g$, $\Delta_b=-0.5g$. For panel (d), $g_b=\kappa=0.1g$, $\Delta_b=\omega_*$. In panels (d), (e), (f) $\gamma=0.5g$. In (e), (f), inverse temperature is set to $\beta g=100$. For Markovian plots, $\Gamma(z)=g_b^2/g$.
• Figure 3: Schematic illustration of the nonreciprocal lattice system. A one-dimensional (1D) lattice chain is considered, where each site coherently interacts with its nearest neighbor with coupling strength $g$. Each pair of neighboring sites simultaneously interacts with a shared local reservoir via a dissipative mode $b_j$ at a rate $\kappa$. The coupling between the sites and the dissipative mode is described by $g_b$. Using a gauge transformation, the transformed Hamiltonian with a relative phase $\phi=3\theta$ is described on the right side of the schematic.
• Figure 4: We connect our set-up to two more baths, one at site $1$ and another at site $N$ to drive transport, in both bosonic and fermionic settings.
• Figure 5: The figure shows NESS current for Markovian case obtained from Lyapunov equation and from NEGF. Parameters: $\phi=2\pi/3, \omega_*=\Delta_c-g, \Gamma=(0.3)^2/g, \gamma=0.5g, \mu_d = \omega_*$ and inverse temperature set to $\beta g =10$.