Introduction

There has been growing excitement in interplay between many-body interactions and the non-Hermitian skin effect (NHSE)1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20. This stems from the non-locality and sensitivity of the NHSE2,11,21,22,23,24,25,26,27,28, which profoundly impacts particle localization3,4,5,6 and localization dynamics7,8,10,29,30,31, which leads to unexpected consequences in the presence of quantum many-body interactions. These include the emergence of a many-fermion Fermi skin1,6, exotic spin liquids from environmental couplings3, spectral symmetry breaking9, unconventional quantum dynamics in non-Hermitian baths8,10,17 and the non-Hermitian Mott skin effect16.

In this work, we uncovered a mechanism whereby bosons can be made to condense very strongly through the triple interplay of non-Hermitian pumping, density interactions and non-trivial band topology. Due to the emergent non-locality of both the NHSE2,11,21,22,23 and the density interactions, the observed phenomenon contrasts starkly with the conventional behavior of interacting bosons, such as photons, which repel each other when they interact i.e. photon blockade32,33,34,35. In particular, we show that strong clustering, and not just boundary state accumulation, arises from an emergent caging mechanism that requires both non-Hermitian pumping and density interactions. Different from conventional onsite trapping, here the interaction is applied at x0 which might not be 1 while the bosons eventually accumulate at the distant boundary x = 1; this spatial decoupling means confinement can only arise when the non-Hermitian drift persistently pumps particles towards the left.

Results

Model

To demonstrate how non-Hermitian pumping, topology and bosonic interactions can interplay to cause unexpectedly strong bosonic clustering, we consider a minimal interacting 1D bosonic lattice model that contains topological edge modes at its boundary sites:

$$H={\sum }_{x=1}^{L}{t}_{L}{b}_{2x-1}^{{\dagger} }{b}_{2x}+{t}_{R}{b}_{2x}^{{\dagger} }{b}_{2x-1}+{\sum }_{x=1}^{L-1}{t}_{0}\left({b}_{2x}^{{\dagger} }{b}_{2x+1}+{b}_{2x+1}^{{\dagger} }{b}_{2x}\right)+\frac{U}{2}{n}_{{x}_{0}}^{2},$$
(1)

which is a Su-Schrieffer-Heeger (SSH) lattice36 with asymmetric intercell (tL and tR) hoppings and unit intracell hoppings, equipped with a density interaction \(\frac{1}{2}U{n}_{{x}_{0}}^{2}\) at site x0, U≥0. Here bx (\({b}_{x}^{{\dagger} }\)) is the bosonic annihilation (creation) operator at sites x = 1, 2, …, 2L, with \({n}_{{x}_{0}}={b}_{{x}_{0}}^{{\dagger} }{b}_{{x}_{0}}\) the boson number operator. Non-Hermitian pumping11,21,22,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69 occurs towards the left if r = tL/tR  >  1, accumulating the bosons towards the left boundary. However, when multiple indistinguishable bosons exist, this accumulation would also be affected by the repulsive (U > 0) interaction at site x0, with strength depending on bosonic occupancy. In this work, we have chosen the interaction \(\frac{1}{2}U{n}_{{x}_{0}}^{2}\) that mimics the simplest possible nonlinearity in the mean-field limit, different from the usual Bose-Hubbard interaction32,33,34,70,71 by a local density shift72,73,74,75,76.

While one might naively expect a repulsive (U > 0) density interaction to primarily suppress the NHSE, the observed behavior can be dramatically different even with just two bosons. Two distinct measures of particle accumulation can be defined: the spatial density accumulation and the many-body correlation. An initial 2-boson state \(\left\vert \phi (t=0)\right\rangle\) evolves according to the Schrodinger equation \(\left\vert \dot{\phi }(t)\right\rangle =-iH\left\vert \phi (t)\right\rangle\), and can be expressed as

$$\left\vert \phi (t)\right\rangle =\frac{1}{\sqrt{2}}{\sum }_{x=1}^{2L}{\sum }_{{x}^{{\prime} }=1}^{2L}{v}_{x{x}^{{\prime} }}(t)\left\vert (x,{x}^{{\prime} })\right\rangle ,$$
(2)

where \({v}_{x{x}^{{\prime} }}(t)={v}_{{x}^{{\prime} }x}(t)\) is the amplitude of the basis state \(\left\vert (x,{x}^{{\prime} })\right\rangle ={b}_{x}^{{\dagger} }{b}_{{x}^{{\prime} }}^{{\dagger} }\left\vert 0\right\rangle\) containing indistinguishable bosons at sites \(x,{x}^{{\prime} }\).

We define the 2-boson density at site x = 1, 2, …, 2L by

$${\rho }_{x}(t)=\frac{\left\langle \phi (t)\right\vert {b}_{x}^{{\dagger} }{b}_{x}\left\vert \phi (t)\right\rangle }{\langle \phi (t)| \phi (t)\rangle },$$
(3)

with 0 ≤ ρx(t) ≤ 2. Associated with it is the time-averaged spatial density

$${\bar{\rho }}_{x}=\frac{1}{T}\int_{0}^{T}{\rho }_{x}({t}^{{\prime} })d{t}^{{\prime} },$$
(4)

where T is the simulation duration.

While ρx(t) and \({\bar{\rho }}_{x}\) reveal where the bosons localize on the lattice, a high value of ρx(t) or \({\bar{\rho }}_{x}\) can physically arise either due to strong single-boson localization at site x, or moderate double-boson clustering at the same site. To quantify this important distinction, we also examine the two-boson correlation probability

$${P}_{x{x}^{{\prime} }}(t)=\left\{\begin{array}{ll}| {v}_{x{x}^{{\prime} }}(t){| }^{2}\hfill \quad &\,{\mbox{if}}\,x={x}^{{\prime} },\\ | \sqrt{2}{v}_{x{x}^{{\prime} }}(t){| }^{2}\quad &\,{\mbox{if}}\,x\ne {x}^{{\prime} },\end{array}\right.$$
(5)

which represents the probability of observing a boson at sites x and \({x}^{{\prime} }\) at the same time t. To reveal underlying trends in the evolution of the two-boson correlation, we also define the time-smoothed correlation

$${\bar{P}}_{x{x}^{{\prime} }}(t)=\frac{1}{\Delta t}\int_{t-\Delta t}^{t}{P}_{x{x}^{{\prime} }}({t}^{{\prime} })d{t}^{{\prime} },$$
(6)

which removes temporal oscillations shorter than a prescribed timescale Δt.

Strong non-Hermitian Bosonic clustering

We first consider scenarios where the density interaction is at the leftmost (boundary) site x0 = 1. Figures 1a–d showcase the evolution of the dynamical density ρx(t) [Eq. (3)] and its time-average \({\bar{\rho }}_{x}\) [Eq. (4)] of three illustrative initial 2-boson state configurations.

Fig. 1: Two-boson correlation dynamics and associated spatial density distributions in a boundary-interacting system.
figure 1

The dramatic enhancement of bosonic clustering at site x = 1 is captured by P11 [Eq. (5)], which measures the probability of finding both bosons at this site. This correlation is significantly stronger (red curves in (e) when both the non-Hermitian skin effect (NHSE, r > 0) and boundary interaction (U > 0) are present, demonstrating their synergistic effect on boson trapping. This enhanced clustering is studied for different initial states (1, 1), (4, 4), and (1, 4) evolving under H [Eq. (1)]. Supporting this observation, (a−d) show the spatial density evolution ρx(t) [Eq. (3)] over time t ∈ [0, 10], with corresponding time-averaged profiles \({\bar{\rho }}_{i}\) [Eq. (4)] plotted above. The density distributions contrast four scenarios: Hermitian (r = 1) without (a) and with (b) interaction, and non-Hermitian (r = 4) without (c) and with (d) interaction. The interaction term U, despite being typically repulsive, acts as an effective trap at x0 = 1 during non-equilibrium evolution when combined with leftward NHSE, as evidenced by the bright regions in the heatmaps. Parameters: t0 = 3, and for r = 4, tL = 1.6 and tR = 0.4. While we only show up to time  <10 evolution, we have verified that this enhanced clustering behavior persists at longer times (up to t = 100). See Supplementary Note 2.

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For the initial state (1, 1) (Top Row) where both bosons are already on x0 = 1, the density interaction indeed repels the bosons in the Hermitian limit, as evidenced in the suppressed \({\bar{\rho }}_{x}\) density at x = x0 = 1 across the Hermitian interacting vs. non-interacting cases (orange vs. blue). Somewhat expected, this suppression vanishes when the NHSE counteracts the repulsion by pumping the bosons towards x0 = 1 (green and red). However, for the initial state (4, 4) (Center Row) where both bosons are initially far from x0 = 1, the boson density oscillates and spreads out, failing to accumulate appreciably at x0 = 1 except when both the non-Hermiticity and interaction are present (red). Indeed, in this non-equilibrium scenario, the U > 0 density term behaves more like a local potential well that traps the ρx(t) at x = x0, rather than repulsion. Qualitatively similar behavior is observed in the density evolution for initial state (1, 4) (Bottom Row), when one of the bosons is initially already at x0 = 1 and is as such unaffected by the NHSE.

We observe the strong two-boson clustering at boundary site x0 = 1 when non-Hermiticity (r > 1) and the density interaction (U > 0) are both present. This is revealed in the dynamical P11(t) [Eq. (5)] plots in Fig. 1e, which shows the probability of both bosons condensing at x = 1. From the Top Row, when both bosons are prepared at x = 1 (initial state = (1,1)), they can only remain there if the interaction and non-Hermiticity are simultaneously present (red). But most surprising is what happens when at least one boson is initially away from x = 1 (Center and Bottom Rows): we observe overwhelmingly higher P11 only when non-Hermiticity and interaction simultaneously interplay (red), compared to having either on their own (yellow, green). This suggests that trapping both bosons and suppressing photon blockade32,33,34 hinges on an emergent consequence of this interplay, a fact not evident from the density evolution alone [Fig. 1a–d].

To quantitatively characterize this two-boson clustering, i.e., P11(t), we plot in Fig. 2a the time-averaged clustering probability \({\bar{P}}_{11}={\bar{P}}_{11}(T)\) with Δt = T [Eq. (6)], which is the correlation P11 smoothed over the entire simulation duration T. The heatmap of \({\bar{P}}_{11}\) in the interaction strength vs. non-Hermiticity U-r plane [Fig. 2a] reveals significantly enhanced \({\bar{P}}_{11}\) (colored) for certain (U, r) regions, compared to the r = 1 Hermitian limit where \({\bar{P}}_{11}\) essentially vanishes (black). While \({\bar{P}}_{11}\) generally increases with r, optimal clustering \({\bar{P}}_{11}\) occurs at windows of U that are highly dependent on the initial state, as will be explained below.

Fig. 2: Two-boson clustering and its spectral origin induced by a boundary density interaction at x0  =  1 [Eq. (1)].
figure 2

a Displays the time-averaged clustering probability \({\bar{P}}_{11}\) at the boundary (x = 1) as a function of the non-Hermitian hopping asymmetry r and the interaction strength U. b Shows the corresponding real-valued two-boson spectrum for r = 4 (tL = 1.6, tR = 0.4) with t0 = 3, revealing five distinct bands. In this model, increasing U produces a group of eigenstates at E ≈ U that exhibit marked clustering at the boundary (indicated in red by ψ11). This additional modulation in the spectrum-stemming from the model’s topological band structure-governs the overlap between the chosen initial state and the edge modes. For instance, the (1,1) initial state projects strongly onto a hybrid band (band 4) that combines bulk and topological characters, resulting in a suppressed \({\bar{P}}_{11}\) (dashed black circle), whereas the (1,4) initial state preferentially overlaps with the topological mode, leading to enhanced clustering (solid black line). In contrast, the bulk states in bands 3 and 5 (circled in light blue) promote clustering for bulk initial states such as (4,4).

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Contributions from topological bands

Intriguingly, for a density interaction at the x0 = 1 boundary, it turns out that the enhanced boundary bosonic clustering is not just due to the interaction-NHSE interplay, but also relies crucially on SSH topology. This is evident from the 2-boson band structure plot in Fig. 2b, where each eigenenergy E is colored according to the overlap ψ11 = 〈(1, 1)∣ψ〉 of its corresponding eigenstate \(\left\vert \psi \right\rangle\) with \(\left\vert (1,1)\right\rangle ={({b}_{1}^{{\dagger} })}^{2}\left\vert 0\right\rangle\). Computed for our Hamiltonian H [Eq. (1)] at strong non-Hermitian hopping asymmetry r = 4, it features 5 bands, with bands 2 and 4 from the hybridization of bulk and topological bands. This follows from the single-boson band structure36,77 with two symmetrically gapped bulk bands separated by in-gap topological zero modes (see Supplementary Note 1). The effect of the U density interaction is to induce high ψ11 overlap (red) successively from bands 3−5, as U is increased. It is noteworthy that the robust dynamical trends persist even in systems with such entirely real energy spectra, indicating deeper underlying physical principles. In contrast, dynamics in systems with complex energies (ImE > 0) are more straightforward, being dominated by states with the largest imaginary energy components.

To illustrate, with the initial state (1,1), suppressed \({\bar{P}}_{11}\) at U ≈ 3 corresponds to high ψ11 overlap with band 4 [circled in black in Figs. 2a, b]. This is because band 4 contains one bulk and one topological boson, which is not consistent with maintaining both bosons at x0 = 1. As such, the same interaction strength of U ≈ 3 leads to enhanced \({\bar{P}}_{11}\) (dashed black) for the initial state (1,4), where only one boson overlaps with the topological boundary mode. Likewise, U ≈ 1 or U ≈ 5 favors the \({\bar{P}}_{11}\) bosonic clustering for the bulk initial state (4,4), as circled in light blue, since that is when the ψ11 overlap with the bulk bands (3 and 5) is strongest. Although the basic non-Hermitian mechanisms (such as the NHSE and interactions) lead to clustering phenomena in all models, in the nontrivial SSH model the topological structure introduces additional spectral modulation, which in turn governs the overlap between the initial state and specific edge states.

Two-boson interactions as caging mechanisms

Interestingly, our density interaction U can also act as a non-local cage when it is situated away from the boundary site x = 1. In Fig. 3, we demonstrate how this interaction largely confines the evolved state to the left of x0 i.e. x ≤ x0, if the initial state has both bosons in the same region. And, more elaborately, for an initial state with both bosons lying to the right of x0, Fig. 4 shows how the interaction will first act as a barrier to the evolved state, but then eventually still trap the particle flux that leaked through it due to NHSE pumping. Overall, this caging mechanism hence further enhances the enhanced boson clustering at the boundary. We shall fix the density interaction to be at x0 = 3, and consider initial states (2, 2) and (4, 4) in Fig. 3 and 4 respectively.

Fig. 3: Strong boundary clustering \({\bar{P}}_{11}\) from interaction-induced caging, for initial state (2,2) within a x0 = 3 cage for a 5 unit cell chain.
figure 3

Snapshots of the time-smoothed (Δt = 2) two-boson correlation probability \({\bar{P}}_{11}\) [Eq. (6)] for (a1) non-interacting and (a2) interacting cases. The L-shaped cage in (a2) traps the bosons tightly. (b1) Evolution of probability of both bosons at the left boundary, with greatly enhanced \({\bar{P}}_{11}(t)\) only for the non-Hermitian interacting case (red). (b2) When only one boson is present, the boundary density \({\bar{\rho }}_{1}(t)\) remains low regardless of whether a U barrier is present, showcasing that the caging mechanism is an interaction effect (See Supplementary Video 1).

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Fig. 4: Persistence of strong boundary clustering \({\bar{P}}_{11}\) from interaction-induced caging, even for initial state (4,4) outside a x0 = 3 cage.
figure 4

Snapshots of the time-smoothed (Δt = 2) two-boson correlation probability \({\bar{P}}_{11}\) (Eq. (6)) for (a1) non-interacting and (a2) interacting cases. The bosons still gradually penetrate the L-shaped cage due to the NHSE towards (1,1), and remain trapped in it after that (t = 3, 8) (See Supplementary Video 2). (b1) Eventually, \({\bar{P}}_{11}(t)\) is ultra-enhanced only for the non-Hermitian interacting case (red), with (b2) negligible enhancement of boundary localization by U in the single-boson case, similar to (b1,b2) of Fig. 3. (c1)--(c3) Time-averaged two-boson probability \({\bar{P}}_{{{\rm{cage}}}}\) within the x0 × x0 cage as a function of hopping asymmetry r. For initial states (2, 2) and (4, 4) with coincident bosons, the U = 1 cases exhibit much larger \({\bar{P}}_{{{\rm{cage}}}}\) due to interaction-induced caging. But for (2, 4), non-coincident initial bosons do not interact appreciably and the effect of interactions is negligible.

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In stark contrast to a usual trap, the repulsive term at x0 only erects an L-shaped barrier in the two-particle configuration space. It is the unidirectional non-Hermitian drift that drives bosons across this barrier and finally confine them at the boundary x = 1, well away from the interaction site. This spatially separated, drift-assisted confinement constitutes the essence of interaction-induced caging.

This caging mechanism is most intuitively represented in the 2D configuration space \((x,{x}^{{\prime} })\), where x and \({x}^{{\prime} }\) represent the positions of the two bosons (Here \((x,{x}^{{\prime} })\) and \(({x}^{{\prime} },x)\) refer to the same state due to bosonic statistics71,74,78,79,80). A key observation is that the nonlinear term \(\frac{1}{2}U{n}_{{x}_{0}}^{2}\) can be interpreted as a non-local L-shaped “cage” [Figs. 3a and 4a], with potential “walls”71,74,78,79,80 of height \(\frac{1}{2}U\) across the entire lines (x, x0) and (x0, x) where only one boson is at x0. The L-shaped energy barrier in the two-particle configuration space appears only when both interaction and NHSE are present. These walls cross at (x0, x0), where double bosonic occupancy at x0 gives an even higher potential barrier of 2U. Figure 3a compares the dynamical evolution of an initial state (2, 2) in the (a1) absence and (a2) presence of the density interaction U. In Fig. 3(a1) with U = 0, the smoothed probability cloud \({\bar{P}}_{x{x}^{{\prime} }}(t)\) [Eq. (6)] of finding the bosons at \((x,{x}^{{\prime} })\) spreads out freely and reflects against the boundaries, even though it is slightly amplified towards the (1, 1) corner due to leftwards NHSE pumping. However, in Fig. 3a2, this spreading is drastically contained within the 3 × 3 L-shaped cage formed by nonzero U at x0 = 3 [see Supplementary Video 1]. This caging, which requires both the U interaction and directed amplification towards (1, 1), indeed leads to far enhanced boundary boson clustering \({\bar{P}}_{11}(t)\), as shown in the red curve in Fig. 3(b1).

Interestingly, the caging mechanism protects the \({\bar{P}}_{11}\) clustering even when the initial state is outside the cage. In Fig. 4a with both bosons initially to the right of x0 = 3 at (4, 4), their flux \({\bar{P}}_{x{x}^{{\prime} }}(t)\) still gradually enters the 3 × 3 cage due to the NHSE pumping towards (1, 1) [see Supplementary Video 2]. Comparing the non-interacting (a1) with the interacting (a2) cases between t = 0 to 2.2, the interaction-induced potential walls only slow the diffusion into the cage slightly. However, once the bosons have entered the cage, they are subject to the same trapping mechanism as in Fig. 3, thereby also experiencing eventual enhanced clustering at x = 1 [red in Fig. 4(b1)].

This caging mechanism is an emergent consequence of few-body density interactions, and cannot be replicated by an effective potential barrier in the single-particle context. Plotted in Figs. 3b2 and 4b2 are the evolutions of the densities \({\bar{\rho }}_{x = 1}\) of a single boson in the same 1D SSH chain of Eq. (1), but with an on-site potential U at x0 = 3. The U ≠ 0 cases (orange) do not trap the boson at x = 1 boundary any more than the U = 0 cases (blue), with the boundary boson density remaining far lower than the \({\bar{P}}_{11}(t)\) of the interacting NHSE 2-boson cases [red in Figs. 3b1 and 4b1].

The efficacy of the caging mechanism is further corroborated by plots of \({\bar{P}}_{{{\rm{cage}}}}={\sum}_{x,{x}^{{\prime} }=1}^{{x}_{0}}{\bar{P}}_{x{x}^{{\prime} }}\), the time-averaged two-boson probability within the cage \(x,{x}^{{\prime} }\le {x}_{0}\). As evident in Fig. 4c1–c3, \({\bar{P}}_{{{\rm{cage}}}}\) increases steadily with the NHSE strength r, testimony to the crucial role of non-Hermitian pumping. Comparing Fig. 4c1 and c2, we see that the density interaction (U > 0, orange) significantly enhances \({\bar{P}}_{{{\rm{cage}}}}\), particularly when both bosons are already initially inside the cage [Fig. 4c2]. Saliently, however, the interaction U has a negligible effect if one boson is initially inside the cage and the other outside such that they interact minimally, as in the initial state (2, 4) for x0 = 3 [Figs. 4c3].

Discussion

Although we have explicitly considered only two particles, the caging mechanism that leads to strong boundary clustering holds for rather general density interactions with arbitrary numbers of bosons. With N bosons, each Fock state is indexed by a lattice point \(\overrightarrow{x}={({x}_{1},\ldots ,{x}_{N})}^{T}\) living in the configuration space \({{\mathbb{R}}}^{N}\)81,82,83,84, and the density takes distinct integer values in the various hyperplanes, depending on the bosonic occupancy. Given a generic interaction that is a multinomial in the densities n1, n2, … at various sites, a monomial \(U{n}_{{x}_{1}}{n}_{{x}_{2}}\ldots {n}_{{x}_{M}}\) gives different energy offset in the corresponding hyperplanes, such as to form the barriers of a “hyper-cage”. For instance, with 4 bosons, an interaction \(U{n}_{{x}_{0}}{n}_{{x}_{0}^{{\prime} }}\) gives an energy U in the planes where two bosons at fixed at x0 and \({x}_{0}^{{\prime} }\), 2U along lines where three bosons are fixed at either x0 or \({x}_{0}^{{\prime} }\), etc. Due to the non-locality of the NHSE2,11,22,23, caging can occur as long as translation invariance is broken in the configuration space, even if the interaction-induced barriers do not completely enclose any region. As a concrete example beyond two particles, we have illustrated in the Supplementary Note 5 that the caging effect persists for three-boson systems under various initial conditions.

We have revealed how the interplay between the NHSE, topology, and boson-boson interactions can counter-intuitively lead to strong bosonic clustering. This particle clustering strongly exceeds that of NHSE-pumped free bosons, being also facilitated by the interaction-induced hybridization of topological and bulk states and an emergent non-local caging mechanism. Our additional analysis with site-wide interactions (Supplementary Note 3) confirms that isolated interactions are more effective in inducing stronger clustering, as uniform interactions tend to delocalize the bosons. Our findings generalize to higher numbers of bosons as well as generic density interactions and topologies, bringing forth a approach to trapping and controlling bosons. Experimentally, demonstrations of non-Hermitian systems have been maturely built upon mechanical arrays85,86, photonic29,60,87,88,89, electrical circuit22,53,77,90 and quantum circuit platforms6. Various platforms can potentially implement the setup for locally interacting bosons to incorporate the \({n}_{x}^{2}\) term, including optical lattices91,92,93,94,95,96, photonic systems76,97,98,99,100,101,102,103, circuit QED104,105,106,107,108,various quantum simulators70,94,105,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127 as well as photonic resonator arrays, as further elaborated in Supplementary Note 4.