Full counting statistics in the many-body Hatano-Nelson model

We study non-Hermitian many-body physics in the interacting Hatano-Nelson model with an open boundary condition. The violation of reciprocity, resulting from an imaginary vector potential, induces the non-Hermitian skin effect and causes exponential localization for all single-particle eigenfunctions in the noninteracting limit. Nevertheless, the density profile of the interacting system becomes only slightly tilted relative to the average filling. The Friedel oscillations exhibit a beating pattern due to the modification of the Fermi wave number. The probability distribution of particles over any finite interval is the normal distribution, whose mean scales with the imaginary vector potential and the variance is symmetric to the center of the chain. This is confirmed by several numerically exact methods even for relatively small systems. These features are expected to be generic not only for fermions, which naturally repel each other due to Pauli's exclusion principle, but for interacting bosons as well. Our findings indicate that many-body effects can significantly alter and conceal the single-particle properties and the skin effect in non-Hermitian systems.

Figure 1

Sum of absolute squares of amplitudes per site [12] of all right single-particle eigenstates of the noninteracting $H_{\text{HN}}$ with $U=0$, $N=100$, and OBC with $ha=0.1$ (upper blue) and 0.2 (lower red), exhibiting the non-Hermitian skin effect.

Figure 2

Real-space density profile and Friedel oscillations for $U/J=-0.8:0.2:0.8$ as indicated in the panels, $N=100$, and OBC with $ha=0.1$. The open circles denote the numerical data from DMRG, while the red solid line represents bosonization from Eq. (16) with $c=1/2$, $\delta =0$, and $L=N\alpha$. The black dashed line depicts the long-wavelength contribution from Eq. (15).

Figure 3

Probability distribution of the density of particles in the first half of the chain ($N/2$ sites) from ED ($N⩽28$) and DMRG ($N=80$) for $U/J=-0.8:0.2:0.8$ and for $ah=0.1$ and $N=28$ (circle), 26 (square), for $ah=0.2$ and $N=28$ (triangle), 24 (star), and 80 (pentagram), and for $ah=0.3$ and $N=80$ (dot). The black solid line denotes the standard normal distribution.