Non-Hermitian Multipole Skin Effects Challenge Localization

Abstract

We study the effect of quenched disorder on the non-Hermitian skin effect in systems that conserve a U(1) charge and its associated multipole moments. In particular, we generalize the Hatano-Nelson argument for a localization transition in disordered, non-reciprocal systems to the interacting case. When only U(1) charge is conserved, we show that there is a transition between a skin effect phase, in which charges cluster at a boundary, and a many-body localized phase, in which charges localize at random positions. In the dynamics of entanglement, this coincides with an area to volume law transition. For systems without boundaries, the skin effect becomes a delocalized phase with a unidirectional current. If dipoles or higher multipoles are conserved, we show that the non-Hermitian skin effect remains stable to arbitrary disorder. Counterintuitively, the system is therefore always delocalized under periodic boundary conditions, regardless of disorder strength.

Figures (7)

• Figure 1: (a) The charge skin effect (OBC) and persistent charge current (PBC) give way to Anderson localization as disorder strength is increased (b) The dipole skin effect (OBC) and persistent dipole currents (PBC) are stable to disorder.
• Figure 2: Non-interacting Hatano-Nelson model, $W=4$. (a) Phase diagram as a function of disorder strength $W$ and non-Hermiticity $g$. Dotted and dashed lines show the scaling of the transition at weak and strong disorder, respectively. (b) An extensive dipole moment in many-body OBC eigenstates indicates the charge skin effect. (c) A finite fraction of PBC energies with a nonzero imaginary part, $f_{imag}$, indicates delocalization (insets: sample single-particle spectra), (d) The imbalance $\mathcal{I}(t)$, after evolving a Néel state in PBC, does not go to zero in the localized phase at long times. Quantities with an overline are averaged over disorder and mid-spectrum eigenstates.
• Figure 3: Interacting charge-conserving Hatano-Nelson model, $V=1, W=8$ (a) Eigenstate dipole moment $P$ divided by its maximum value. For $g<g_c$ the system is in an MBL phase and $\overline{P}/P_{max}\rightarrow 0$, while this limit is finite in the skin effect phase for $g>g_c$. The dipole moment $P$ is averaged over disorder and mid-spectrum eigenstates. (b) Volume-to-area law entanglement transition in long-time steady states of the dynamics (inset: entanglement entropy fluctuations peak at the transition). The bipartite von Neumann entropy $S_{L/2}(t)$ is averaged over disorder and random initial states.
• Figure 4: Dipole-conserving Hatano-Nelson model, $W=5.5$, $t_1 = 0.3 t_0$. (a) The dipole skin effect manifests as extensive quadrupole moment in OBC eigenstates, and is present for any $g>0$ (b) The fraction of PBC energies with nonzero imaginary part tends to a finite value, indicating delocalization (inset: sample energy spectrum). Both quantities are averaged over disorder and mid-spectrum eigenstates. (c) Time evolution of imbalance starting from the Néel state in PBC with $L=24$. Imbalance remains finite at long times for $g=0$ (inset: different system sizes), while it decays for $g>0$.
• Figure S1: Skin effect order parameters for models without a similarity transformation (a) The charge-conserving Hatano-Nelson model with next-nearest neighbor hoppings, $\tilde{H}_\text{charge}(g)$, has a transition between Anderson localization and the skin effect as measured by eigenstate dipole moment ($W=4$). (b) The dipole-conserving model $\tilde{H}_\text{dip}(g)$ appears to exhibit the dipole skin effect for any $g>0$, as measured by eigenstate quadrupole moment. The data collapse reasonably well when non-reciprocity is rescaled by (c) $L^{1/4}$ ($g_c = 0.38$) for the charge-conserving case and (d) $\sqrt{L}$ ($g_c = 0$) for the dipole-conserving case.