Lindbladian versus Postselected Non-Hermitian Topology

Abstract

The recent topological classification of non-Hermitian `Hamiltonians' is usually interpreted in terms of pure quantum states that decay or grow with time. However, many-body systems with loss and gain are typically better described by mixed-state open quantum dynamics, which only correspond to pure-state non-Hermitian dynamics upon a postselection of measurement outcomes. Since postselection becomes exponentially costly with particle number, we here investigate to what extent the most important example of non-Hermitian topology can survive without it: the non-Hermitian skin effect and its relationship to a bulk winding number in one spatial dimension. After defining the winding number of the Lindbladian superoperator for a quadratic fermion system, we systematically relate it to the winding number of the associated postselected non-Hermitian Hamiltonian. We prove that the two winding numbers are equal (opposite) in the absence of gain (loss), and provide a physical explanation for this relationship. When both loss and gain are present, the Lindbladian winding number typically remains quantized and non-zero, though it can change sign at a phase transition separating the loss and gain-dominated regimes. This transition, which leads to a reversal of the Lindbladian skin effect localization, is rendered invisible by postselection. We also identify a case where removing postselection induces a skin effect from otherwise topologically trivial non-Hermitian dynamics.

Figures (2)

• Figure 1: Dynamics of the Lindbladian 'Hatano-Nelson' model [Eq. \ref{['eq:simple_gain_loss_model_jump_operators']}] under (a) loss only and (b) gain only in open boundary conditions. In both cases, the postselected NH Hamiltonian drives microscopic electrons towards the right edge (red arrows). For loss (a), the non-equilibrium steady state consists of all sites empty (white circles). If we add an electron into the system (blue circle), it moves to the right. For gain (b), the steady state has all sites occupied by electrons. The elementary excitations (relaxation modes) are now holes, which effectively move towards the left edge. Mathematically, this Lindbladian skin effect reversal is reflected in a Lindbladian bulk winding number that has a sign opposite to that of the postselected NH Hamiltonian.
• Figure 2: Comparison of postselected NH Hamiltonian $\mathcal{H}^{\mathrm{post}}(k)$ with Lindbladian band structure calculated from $\mathcal{H}^{\mathrm{eff}}(k)$ for the Lindbladian 'Hatano-Nelson' model Chaduteau_Data_2025 in Eq. \ref{['eq:simple_gain_loss_model_jump_operators']}. Top panels show the point gaps and their winding number [Eq. \ref{['eq:eff_and_post_windings']}] indicated by arrows for $\mathcal{H}^{\mathrm{post}}$ and $\mathcal{H}^{\mathrm{eff}}$, side by side, for both periodic and open boundary conditions. The bottom panel shows the probability density $n(x)$, summed over all eigenstates, for $\mathcal{H}^{\mathrm{post}}$ [$n_{\mathrm{post}}(x)$] and $\mathcal{H}^{\mathrm{eff}}$ [$n_{\mathrm{eff}}(x)$], side by side. (a) When loss is greater than gain ($\gamma_l > \gamma_g$), the winding number of $\mathcal{H}^{\mathrm{eff}}$ is the same as that of $\mathcal{H}^{\mathrm{post}}$, and a Lindbladian skin effect occurs on the same edge as the postselected NH skin effect in open boundary conditions. (b) When loss equals gain ($\gamma_l = \gamma_g$), the point gap of $\mathcal{H}^{\mathrm{eff}}$ collapses, while that of $\mathcal{H}^{\mathrm{post}}$ remains open, and there is no Lindbladian skin effect. (c) When gain is greater than loss ($\gamma_l < \gamma_g$), the winding number of $\mathcal{H}^{\mathrm{eff}}$ is opposite that of $\mathcal{H}^{\mathrm{post}}$, and the Lindbladian skin effect is reversed compared to the postselected NH skin effect.