An Explicit Wavefunction of the Interacting Non-Hermitian Spin-1/2 1D System

TL;DR

This work addresses how non-Hermitian spin-orbit coupling (SOC) affects many-body physics in a 1D spin-1/2 fermion system by constructing an explicit Bethe-ansatz (BA) wavefunction. In the dilute limit, the BA solution simplifies to a Slater-determinant product times a Jastrow factor, with a resonance encoded by a complex momentum $k_r=(\chi_r+i\eta_r)/L$ and $\eta_r=\ln(1+g/\alpha)$. A clear two-body resonance appears, and for many bodies the dominant channels yield a universal up-down phase shift $e^{\eta_r}$, while channels involving diffractive reflections are suppressed. As $\eta_r$ increases, the system undergoes a first-order phase transition from a uniform configuration to phase separation, indicating the resonance is enhanced by repulsive interactions. The paper also connects to experimental realizations via spin-dependent NHSE lattices, and discusses robustness under particle loss within a Lindblad framework, offering a practical route to observe non-Hermitian many-body resonance phenomena.

Abstract

We present an explicit Bethe-ansatz wavefunction to a 1D spin-$\frac{1}{2}$ interacting fermion system, manifesting a many-body resonance resulting from the interplay between interaction and non-Hermitian spin-orbit coupling. In the dilute limit, the Bethe-ansatz wavefunction is factorized into Slater determinants and a Jastrow factor. An effective thermodynamic distribution is constructed with an effective Hamiltonian including a repulsion resulting from Pauli's exclusion principle and a distinctive zigzag potential arising from the resonance. The competition between these effects leads to a transition from a uniformly distributed configuration to a phase separation. Clustering of particles with identical spins is observed in the latter phase, demonstrating that the many-body resonance effect is enhanced by the repulsive interaction.

Figures (7)

• Figure 1: The 2-body wavefunction $\varphi(x_1, x_2)$ with particles of opposite spins by fixing the spin-up one at $x_2=0$. The parameter values are $\chi_{1}=\chi_{2}=0$, $m=1$, $\alpha=1$ and $L=50$. The peak appears at $x_1 \sim \ln (L \lambda_s^{-1}) / ( L \lambda_s^{-1} )$. In the case of $L \gg \lambda_s$, the peak is located at $x_1=x_2$ and $\varphi$ becomes discontinuous. It implies that the spin-down particle tends to lie on the right side of the spin-up particle, forming a resonant pair on the ring. The wavefunction becomes more localized as the strength of repulsive interaction $g$ increases, which indicates that the resonance is enhanced by the repulsive interaction.
• Figure 2: The construction of the Bethe-ansatz wavefunction. It is composed of many 'plane waves'. Each plane wave is a product of Slater determinants, due to the Fermi statistics. The 'plane waves' in the same column or the same row are connected by transmission or reflection respectively. Due to the diffractive reflection described by Eq. (\ref{['Diffractive reflection']}), the reflected descendants of $A_i$ are all suppressed.
• Figure 3: The Monte-Carlo simulations of the average 'spin dipole' $\langle p \rangle$ at $L=100$ with different particle numbers. As $\eta_r = \ln(1+g/\alpha)$ increases, $\langle p \rangle/L$ jumps from zero to a finite value, indicating that the transition is of 1st order. ($a$) and ($b$) shows the configuration of the two phases. If $\alpha$ is fixed, the more localized ($b$) phase emerges only when $g$ is big enough, which implies that the many-body resonance is enhanced by the repulsive interaction.
• Figure 4: The probability distribution of four-particle wavefunction with $x_2, x_3, x_4$ fixed. From (a) to (f), we decrease the value of $x_2$, with $x_3=0$, $x_4=40$, $\Delta x = x_2 - x_3$. Here we take $\chi_{1, 2}=\chi_{3, 4}=\pm 2\pi/L$, $m=1$, $g=1$, $\alpha=3$, $L=100$. The normalization of the original wavefunction and the approximated wavefunction differs by a ratio 0.00765. As a comparison, $\lambda_s/L = 0.00333$.
• Figure 5: ($a$) and ($b$) shows the Pauli repulsion energy $V(r)$ and the resonance energy $W(r)$ respectively. They are periodical functions due to the periodical boundary condition. ($c$) illustrates the configurations with the locally minimal energy in the zero temperature limit. In the first one, four particles form two up-down pairs with distance $L/2$. The size of the pair is negligible; In the second one, particles with identical spins form two adjacent clusters with size $a= L \tan^{-1} (\frac{\pi}{2} \eta_r)/\pi$.