Non-Hermitian Lindhard function and Friedel oscillations

Friedel oscillations are ubiquitous around localized defects in Hermitian quantum mechanics. Here we study their fate around a non-Hermitian, imaginary impurity and evaluate the corresponding non-Hermitian Lindhard function in one dimension, where quantum effects are traditionally enhanced due to spatial confinement. Most importantly, we find that the static limit of the non-Hermitian Lindhard function has no divergence at twice the Fermi wave number and vanishes identically for all other wave numbers at zero temperature. Consequently, no Friedel oscillations are induced by a non-Hermitian, imaginary impurity to lowest order in the impurity potential at zero temperature. Our findings are corroborated numerically on a tight-binding ring by switching on a weak real or imaginary potential. We identify conventional Friedel oscillations or a heavily suppressed density response, respectively.

Figure 1

The real (dashed lines) and imaginary (solid lines) parts of the frequency-dependent Lindhard function are plotted for several momenta at $T=0$. Compared to the Hermitian Lindhard function, the roles of the real and imaginary parts are reversed.

Figure 2

Flat surface plot of the spatial and temporal evolution of the local density for $N=102$ and $V=0.1i\gamma$ (non-Hermitian, left) and $V=-0.1\gamma$ (Hermitian, right) for Eq. (10). Inside the light cone, the density can readjust in response to the external potential, potentially causing Friedel oscillations. The distinct scales for the density variations indicate the suppression of Friedel oscillations in the non-Hermitian setting, compared to the Hermitian case.

Figure 3

Friedel oscillation of the half-filled tight-binding model with PBCs and $N=1002$; the Hermitian or imaginary impurity potential is switched on at $t=0$ and $R=501$ with strength $V=0.1i\gamma$ (red, non-Hermitian) or $-0.1\gamma$ (blue, Hermitian), respectively. The Friedel oscillations are negligible for an imaginary potential, in agreement with the prediction of the non-Hermitian Lindhard function from Eq. (5) at $T=0$.