The bosonic skin effect: boundary condensation in asymmetric transport

Abstract

We study the incoherent transport of bosonic particles through a one dimensional lattice with different left and right hopping rates, as modelled by the asymmetric simple inclusion process (ASIP). Specifically, we show that as the current passing through this system increases, a transition occurs, which is signified by the appearance of a characteristic zigzag pattern in the stationary density profile near the boundary. In this highly unusual transport phase, the local particle distribution alternates on every site between a thermal distribution and a Bose-condensed state with broken U(1)-symmetry. Furthermore, we show that the onset of this phase is closely related to the so-called non-Hermitian skin effect and coincides with an exceptional point in the spectrum of density fluctuations. Therefore, this effect establishes a direct connection between quantum transport, non-equilibrium condensation phenomena and non-Hermitian topology, which can be probed in cold-atom experiments or in systems with long-lived photonic, polaritonic and plasmonic excitations.

Figures (7)

• Figure 1: Asymmetric bosonic transport. (a) Sketch of the ASIP setup studied in this work. Bosons injected from a thermal particle reservoir with mean occupation number $\bar{n}_r$ on the right can incoherently hop along the lattice with asymmetric rates $\Gamma_l$ and $\Gamma_r$, before being emitted into a second reservoir with occupation number $\bar{n}_l$ on the left. A directional hopping can be imposed, for example, by applying a potential gradient with an energy offset $U$ between neighboring sites. (b) Under stationary conditions, this hopping asymmetry combined with the bosonic particle statistics results in the bosonic skin effect, i.e., the formation of a finite boundary region with a staggered density profile. The two insets show sketches of the Wigner distribution for individual lattice sites, indicating that within this boundary region, the odd sites are in a condensed state with broken $U(1)$ symmetry, while all other lattice sites exhibit a thermal distribution. See text for more details.
• Figure 2: Plots of the steady-state occupation numbers $n_p$ for a lattice of $L=15$ sites and different degrees of asymmetry: $\Gamma_A/\Gamma_l=0$ (a), $\Gamma_A/\Gamma_l=0.05$ (b), $\Gamma_A/\Gamma_l=0.17$ (c), and $\Gamma_A/\Gamma_l=1$ (d). For all plots $\bar{n}_r=10$ and two different values of $\bar{n}_l=0$ (blue lines) and $\bar{n}_l=20$ (yellow lines) have been considered. The insets show the current $J$ versus the lattice size $L$, in log-log scale and for three different values of $\bar{n}_l=0, 5, 9$. For $\Gamma_A=0$ we recover a linear population gradient and the Fourier law for the current, as expected for diffusive transport. For any $\Gamma_A>0$ and large $L$, the current becomes independent of both $L$ and $\bar{n}_l$, indicating ballistic transport. In this regime, we observe the formation of a finite boundary region of size $\xi$, as indicated by the shaded area. As the asymmetry increases, the width $\xi$ shrinks and vanishes for $\Gamma_A/\Gamma_l \simeq 0.17$. Beyond this point, a finite boundary region, but with an oscillating density profile, reappears. For all plots, we have set $\kappa_r=\kappa_l=\Gamma_l$.
• Figure 3: Dependence of the skin length $\xi$ as defined in Eq. \ref{['eq:Skindepth']} on the hopping asymmetry $\Gamma_A$ and on the thermal population of the right reservoir, $\bar{n}_r$. When $\Gamma_A$ is exactly zero (thick dark line at the bottom of the diagram), we recover the usual diffusive behavior. The dashed line corresponds to $\Gamma_A=\Gamma_A^c$, at which point $\xi=0$. Below (above) this line, the steady-state population exhibits a smooth (zigzag) profile near the left boundary. The inset shows $\xi$ along the horizontal green line at $\Gamma_A=0.5\Gamma_l$. For all points in this plot a value of $\kappa_r=\Gamma_l$ has been assumed, and the results are independent of both $\bar{n}_l$ and $\kappa_l$.
• Figure 4: (a) Plot of the second-order correlation function $g^{(2)}(0)$ for a lattice of $L=10$ sites, as obtained from a TWA simulation with 5000 trajectories. The phase-space distributions below each point indicate the distributions of the amplitudes $\alpha_p$, in the complex plane, at the final time of the simulation. (b) Distributions of the values of $|\alpha_p|^2$ and (c) plots of the coherence function $g^{(1)}(\tau)$\ref{['eq:cohefunc']} for odd (left) and even (right) sites near the boundary (all the even sites have extremely similar behavior, here we show only $p=2$ for simplicity). For all plots we have set $\kappa=\Gamma_l$, $\Gamma_r=0$, $\bar{n}_r=30$ and $\bar{n}_l=0$. For the plots in (c), we have used a reference time of $t=10\Gamma_l^{-1}$, which is sufficient to reach the steady state.
• Figure 5: (a) Evolution of the Wigner distribution of site $p=1$, when the system is initially prepared in a symmetry-broken state with $\lvert\langle\alpha_p\rangle\rvert=\sqrt{3}$ and a random phase. The three plots show the resulting phase-space distributions obtained in a TWA simulation for times $\Gamma_l t=0, 2,40$ and for $\bar{n}_r=80$. On a short timescale, the initial displacement is amplified, while phase diffusion is observed over much longer times. (b) Logarithmic plot of the ensemble-averaged amplitude of the first site for the same initial conditions, but assuming different thermal occupation numbers of the right reservoir. After a short amplification, we observe an exponential decay of the average amplitude due to phase diffusion. The dashed lines represent the analytic prediction for this decay, as given in Eq. \ref{['eq:PhaseDiffusion']}. For all plots, $\Gamma_l=\kappa_r$, $\Gamma_r=0$, $L=10$ and $\bar{n}_l=0$.