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arXiv:2511.03613 · Physical Review A

Directional quantum walks of two bosons on the Hatano-Nelson lattice

Sk Anisur, Kartik Singh, Sayan Choudhury

quant-phcond-mat.quant-gascond-mat.stat-mechSubmitted Nov 5, 2025• Announced Nov 6, 2025• 0 citations
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Abstract

We theoretically investigate the interplay of interactions and non-Hermiticity in the dynamics of two bosons on the one-dimensional Hatano-Nelson lattice with non-reciprocal tunneling. We find that the non-reciprocity in the tunneling leads to the formation of an asymmetric density cone during the time-evolution of the system; the degree of asymmetry can be tuned by tuning the non-reciprocity parameter, $δ$. Next, we analyze the dynamics of this system in the presence of a static external force and demonstrate that non-Hermiticity leads to asymmetric two-particle Bloch oscillations. Interestingly, when $F=0$ ($F \ne 0$), strong interactions leads to the formation of an inner density-cone (density-hourglass) structure; this inner structure also becomes asymmetric in the presence of non-Hermiticity. We further analyze the spatial correlations and establish that the system exhibits non-reciprocal bunching (anti-bunching) in the presence of weak (strong) interactions. Finally, we examine the growth of the Quantum Fisher Information, $F_Q$, with time, and demonstrate that $F_Q \propto t^α$ where $α\sim 3$. This feature persists for both one- and two-particle walks, thereby demonstrating that this system can be employed as a quantum-enhanced sensor for detecting weak forces.

Comments
9 pages, 5 figures

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We theoretically investigate the interplay of interactions and non-Hermiticity in the dynamics of two bosons on the one-dimensional Hatano-Nelson lattice with non-reciprocal tunneling. We find that the non-reciprocity in the tunneling leads to the formation of an asymmetric density cone during the time-evolution of the system; the degree of asymmetry can be tuned by tuning the non-reciprocity parameter, δδδ. Next, we analyze the dynamics of this system in the presence of a static external force and demonstrate that non-Hermiticity leads to asymmetric two-particle Bloch oscillations. Interestingly, when F=0F=0F=0 (F≠0F \ne 0F=0), strong interactions leads to the formation of an inner density-cone (density-hourglass) structure; this inner structure also becomes asymmetric in the presence of non-Hermiticity. We further analyze the spatial correlations and establish that the system exhibits non-reciprocal bunching (anti-bunching) in the presence of weak (strong) interactions. Finally, we examine the growth of the Quantum Fisher Information, FQF_QFQ​, with time, and demonstrate that FQ∝tαF_Q \propto t^αFQ​∝tα where α∼3α\sim 3α∼3. This feature persists for both one- and two-particle walks, thereby demonstrating that this system can be employed as a quantum-enhanced sensor for detecting weak forces.
Recent experiments on quantum walks (QWs) demonstrated a full control over the statistics-dependent walks of single particles and two particles in one-dimensional lattices. However, little is known about the general characterization of QWs at the many-body level. Here, we rigorously study QWs, Bloch oscillations, and the quantum Fisher information for three indistinguishable bosons and fermions in one-dimensional lattices using a time-evolving block decimation algorithm and many-body perturbation theory. We show that such strongly correlated QWs not only give rise to statistics-and-interaction-dependent ballistic transports of scattering states and of two- and three-body bound states but also allow a quantum enhanced precision measurement of the gravitational force. In contrast to the QWs of the fermions, the QWs of three bosons exhibit strongly correlated Bloch oscillations, which present a surprising time scaling t^{3} of the Fisher information below a characteristic time t_{0} and saturate to the fundamental limit of t^{2} for t>t_{0}.
Anyons, namely particles obeying fractional quantum statistics that interpolate between bosons and fermions, possess a lot of new and exotic physical properties related to the particle exchange statistics. In this work, we explore the few-body quantum dynamics and quantum correlations of indistinguishable anyons with on-site interactions in onedimensional lattices within the scheme of three-body continuous-time quantum walks. By employing a time-evolving block decimation (TEBD) algorithm, we numerically calculate the dynamical evolution process of the local density distribution of anyons among the whole lattice. Numerical simulations shown in the main text mainly focus on a threebody initial state as ψ|(t=0)＞=â†-1â†0â†1|0＞, in which three particles are located on neighbouring sites at lattice centre. This choice of initial state features that the three particles influence one another most strongly, while we have also implemented numerical simulations on other choices of three-body initial states as are discussed in appendix. It is shown that the local density distribution of anyons is dramatically altered by fractional particle statistics with new dynamical structure showing up during the time evolution. For free anyons, an inner cone emerges as the statistical parameter increases, while the outer cone remains robust all along. When the on-site interaction joins in, the structure of the inner cone is further modified with new features. Specifically, for interaction of finite strength, an exotic dynamical asymmetry in real space, is clearly demonstrated during the time evolution of the local density distribution for particles within the fractional statistics regime. However, for boson limit and pseudofermion limit, the time evolution of the local density distribution keeps symmetric as the three-body initial state. And remarkably, the dynamical asymmetry is interactiondependent manifested as the local density distribution of anyons favors opposite side of the lattice for repulsive and attractive interaction, respectively. Moreover, when the on-site interaction is further increased to hard-core limit, the dynamical asymmetry will then be largely suppressed. We also calculate the density-density correlations for anyons before they reach the lattice boundary to reveal the interesting effect of fractional statistics on quantum correlations. It is shown that the inner cone corresponds to co-walking of anyons, while the outer cone is related to individual walking and is immune to the variation of statistical parameter. Furthermore, the exotic real-space asymmetry originated from the interplay of fractional statistics and finite interaction is also shown up in the density-density correlations.
We investigate the dynamics of two bosons in a tilted one-dimensional optical lattice with both standard and density-assisted tunnelling. The relative sign of the two tunnelling processes strongly affects the Bloch dynamics, leading to distinct reinforcing and competing regimes. We examine how the tunnelling processes generate correlations and find that, in the absence of onsite interactions, the early-time entanglement growth follows the same scaling with the density-assisted tunnelling amplitude. Finite onsite interaction changes this picture by breaking the symmetry between the reinforcing and competing regimes and enhancing the correlation differently in two regimes. This shows how the interplay between the two tunnelling mechanisms can change dyanmics and entanglement in a controlled few-body setting. Particularly, at equal tunnelling amplitudes, the doublon and singlon sectors decouple completely for arbitrary onsite interaction, lattice tilting, and initial state.
We investigate the quantum quench dynamics of the interacting Hatano-Nelson model with open boundary conditions using both Abelian bosonization and numerical methods. Specifically, we follow the evolution of the particle density and current profile in real space over time by turning the imaginary vector potential on or off in the presence of weak interactions. Our results reveal spatiotemporal Friedel oscillations in the system with light cones propagating ballistically from the open ends, accompanied by local currents of equal magnitude for both switch-off and -on protocols. Remarkably, the bosonization method accurately accounts for the density and current patterns with a single overall fitting parameter. The continuity equation is satisfied by the long-wavelength part of the density and current, despite the nonunitary time evolution when the Hatano-Nelson term is switched on.
The single orbital, one-dimensional, Hatano-Nelson Hamiltonian provides deep insight into the physics of non-Hermiticity, resulting from asymmetric left/right hopping, and its connections to localization. In the absence of disorder, its single particle eigenvalues EαE_αEα​ lie on an ellipse in the complex plane whose extent in the imaginary direction is controlled by the degree of asymmetry. When randomness is introduced, two sets of real eigenvalues emerge at the extremes of the largest and smallest real part of EαE_αEα​. These real eigenvalues are associated with localized eigenvectors. For spinless fermions, increasing near-neighbor interactions first cause a transition to a charge density wave phase, and ultimately, on finite lattices, a collapse of all eigenvalues to the real axis. In this paper, we explore the presence of real eigenvalues in the interacting, two-particle sector for the spinful case (Hubbard model) in a two-chain (two-band) geometry with a Hermitian interchain hopping. Our key results are to obtain the ``phase" diagrams for the existence of a purely real spectrum, as a function of the interaction strength, degree of non-Hermiticity, and interchain hopping. We study the sensitivity to boundary conditions of the spectral properties of our two-chain model with winding number analysis and explore the relationship between PBC doublon states and OBC skin modes. To address the question of stability in such non-equilibrium systems, we solve the dynamics at low filling according to Lindbladian evolution and find that the non-Hermitian description is able to qualitatively describe such systems.
We investigate the localization behavior of two-particle Hubbard model in the presence of non-reciprocal tunneling and non-Hermitian bound states can be obtained with strong repulsive interaction. Remarkably, the interaction induced bound state localization (BSL) can compete with non-Hermitian skin effect (NHSE) and give rise to diverse density profiles. Via the quantum scattering methods in the center of mass frame, the system can be mapped to an effective two dimensional (2D) lattice with the two-particle interaction contributing to a defective line. For the bound states of the largest eigen-energy, in contrast to the Hermitian cases, where the maximal localization center is pinned around the center of lattice, NHSE can lead to a faded diagonal line localization. For the unbound scattering states, unlike the single corner localization in the 2D Hatano-Nelson model, interaction can force the total localization to split into multiple centers. To make the system topological nontrivial, we further include terms taking the form of two-photon tunneling and the non-Hermitian photon bound pairs are also observed, which demonstrates a competition among NHSE, BSL and edge localization. Finally, we propose the experimental simulations via the platforms of electrical circuits. Our works shed light on the crossover study of quantum optics and non-Hermitian many body physics.