Recent cold atom experiments have realized one-dimensional anyons and enabled the tuning of 1D~statistics between bosons and fermions. Here, we analyze the symmetries, integrability, and resulting degeneracies of the underlying anyon-Hubbard model of finite length. Our results reveal a switching between symmetry classes AI, BDI, and CI in dependence on system size, particle number, and boundary conditions, and show that two anyons with periodic boundaries are integrable, while two anyons with open boundary conditions are not. We include a comprehensive analysis of all model limits, especially of interacting bosons and pseudofermions and resolve spectral signatures. We additionally reveal an exactly solvable doublon state that hides in the continuum of scattering states and the exact solution of the nullspace of two noninteracting anyons. The uncovered symmetries shape the fundamental properties of the one-dimensional anyons at hand, and the predicted states are accessible in state-of-the-art experiments.
Recent cold atom experiments have realized one-dimensional anyons and enabled the tuning of 1D~statistics between bosons and fermions. Here, we analyze the symmetries, integrability, and resulting degeneracies of the underlying anyon-Hubbard model of finite length. Our results reveal a switching between symmetry classes AI, BDI, and CI in dependence on system size, particle number, and boundary conditions, and show that two anyons with periodic boundaries are integrable, while two anyons with open boundary conditions are not. We include a comprehensive analysis of all model limits, especially of interacting bosons and pseudofermions and resolve spectral signatures. We additionally reveal an exactly solvable doublon state that hides in the continuum of scattering states and the exact solution of the nullspace of two noninteracting anyons. The uncovered symmetries shape the fundamental properties of the one-dimensional anyons at hand, and the predicted states are accessible in state-of-the-art experiments.
Low-dimensional quantum systems can host anyons, particles with exchange statistics that are neither bosonic nor fermionic. However, the physics of one-dimensional (1D) anyons remains largely unexplored. Using ultracold bosonic Rb-87 atoms in an optical lattice, we realize and explore the anyon-Hubbard model, which hosts 1D anyons and features a rich phase diagram. We implement the model using techniques from Floquet engineering, realizing the statistical phase in the form of a density-dependent Peierls phase. Using our high-resolution imaging system, we explore both the dynamical and ground state behavior in the anyon-Hubbard model. In the first set of experiments, we probe 1D anyons via two-particle quantum walks, observing the anyonic Hanbury Brown-Twiss effect, as well as the formation of bound states without on-site interactions. Once interactions are introduced, we observe spatially asymmetric transport in contrast to the symmetric dynamics of bosons and fermions. In the next set of experiments, we adiabatically prepare and probe ground states in the anyon-Hubbard model. First, we observe continuous fermionization in the density profile with increasing statistical phase. Then, we probe the chirality of the bound states in the anyon-Hubbard model, drawing similarities to a continuum model of 1D anyons. This work constitutes the first experimental study of the anyon-Hubbard model, and lays the foundation for exploration in the strongly correlated many-body regime.
Fractional statistics give rise to quantum behaviors that differ fundamentally from those of bosons and fermions. While two-dimensional anyons play a major role in strongly correlated systems and topological quantum computing, the nature of their one-dimensional (1D) counterparts remains the subject of intense debate, with renewed interest fueled by recent experimental progress. Theoretically, 1D anyons are predicted to host exotic many-body phases and quantum phase transitions, yet experimental signatures have remained elusive. Using ultracold atoms in an optical lattice, we prepare two-body ground states of the 1D anyon-Hubbard model by combining Hamiltonian engineering via quasiperiodic drives and adiabatic state manipulation. We uncover the effects of statistical interactions that lead to pseudo-fermionization and to the formation of chiral bound states when particles remain close together. Our results establish a link between lattice and continuum realizations of anyon models, and mark important steps towards the precise control of 1D anyons in both equilibrium and out-of-equilibrium settings.
Quantum statistics in low-dimensional systems predicts anyonic particles with fractional exchange statistics which are neither that of bosons nor fermions. While anyons are typically found in two dimensions as excitations of topologically-ordered states of matter, anyon-like exchange statistics has also been discussed in one dimension, for instance, in the context of the anyon-Hubbard model (AHM), the physics of which has recently been observed in experiment [Kwan et al., arXiv:2306.01737; Dhar et al., arXiv:2412.21131; and Bakkali-Hassani et al., arXiv:2602.20421]. The AHM can be formulated in terms of bosons featuring density-dependent Peierls phases, described by a statistical phase angle θ, which controls asymmetric transport and the formation of dynamically bound pairs at finite momentum. Here, we show theoretically that the AHM also hosts exact two-body bound states in the continuum (BICs) for arbitrary θ=0, and genuine three- and four-body bound states. Unlike conventional bound states stabilized by attractive (or repulsive) interactions, which are energetically localized with a large effective mass, these clusters here are bound by a purely kinematic mechanism endowing them with fast chiral transport properties. We provide a simple variational approximation to the three-body bound states and explain their binding mechanism. Moreover, we show that the signatures of three-body bound states in the AHM can be directly probed experimentally from the expansion dynamics starting from three localized particles.
Recently optical lattices allow us to observe phase transition without the uncertainty posed by complex materials, and the simulations of these systems are an excellent bridge between materials-based condensed matter physics and cold atoms. In this way, the computational physics related to many-body problems have increased in importance. Using the density matrix renormalization group method, we studied a Hubbard model for anyons, which is an equivalent to a variant of the Bose-Hubbard model in which the bosonic hopping depends on the local density. This is an exact mapping between anyons and bosons in one dimension. The anyons interlope between bosons and fermions. For two anyons under particle exchange, the wave function acquires a fractional phase e iθ . We conclude that this system exhibits two phases: Mott-insulator and superfluid. We present the phase diagram for some angles. The Mott lobe increases with an increase of the statistical. We observed a reentrance phase transition for all lobes. We showed that the model studied is in the same universality class as the Bose-Hubbard model with two-body interactions.
In textbook physics, all particles are categorized as either bosons or fermions. When exchanging two bosons, the corresponding wavefunction is dressed with an exchange phase φ=0, whereas for fermions the exchange phase is φ=π, reflecting the well-known symmetrization postulate of quantum mechanics. While this is true in three spatial dimensions, one- and two-dimensional systems can host particles with non-trivial exchange phase 0≤φ<2π, so-called anyons, which exhibit exchange statistics intermediate between those of bosons and fermions. Here, we consider a Hubbard-type model for one-dimensional anyons, consisting in a discrete lattice on which anyons can tunnel and interact.\\ We first address the case of two anyons on an infinite lattice and solve the Schrödinger equation analytically, deriving bound- and scattering state solutions. We demonstrate the existence of scattering resonances related to the anyonic exchange phase. We confirm this via numerical simulations of two-wavepacket scattering dynamics, where we observe Wigner time delays and lateral shifts. These two phenomena are accurately explained by studying the behavior of the scattering phase in the analytical solution. By further introducing a potential barrier, we model two-anyon interference on a beamsplitter and show that the exchange phase allows to interpolate between bosonic bunching and fermionic antibunching behaviors.\\ We further examine many-anyon systems from the perspective of quantum chaos. Comparing the eigenvalue statistics, in particular the mean level-splitting-ratio with random matrix predictions, allows us to identify the non-interacting limit of the anyon-Hubbard model as chaotic, in sharp contrast to the bosonic case, which we attribute to the effective interaction mediated by the exchange phase. We further consider the collective delocalization of eigenstates in the site basis via their fractal dimensions as an additional indicator of quantum chaos. Lastly, we discuss signatures of many-body interference in the long-time average variance Δnj of a single site's population, and the dependence of these on the exchange phase. In the chaotic regime, we find that Δnj neither significantly depends on the initial state, nor on the exchange phase.
While elementary particles obey either bosonic or fermionic exchange statistics, generalized exchange statistics that interpolate between bosons and fermions---applicable to quasiparticles---constitute an intriguing topic from both the fundamental and practical points of view. This work develops a scattering framework for two identical one-dimensional (1D) bosonic anyons and two identical 1D fermionic anyons with zero-range contact interactions. The two-body system with zero-range interactions, in both free space and under external confinement, is used to illustrate the recently proposed bosonic-anyon---fermionic-anyon mapping (R. Hidalgo-Sacoto et al., Phys. Rev. A 112, 063309 (2025)), which connects the eigenstates of bosonic anyons to those of fermionic anyons and vice versa. Performing explicit calculations for two-particle systems, the momentum distributions and the off-diagonal correlations of the single-particle density matrix for bosonic anyons and fermionic anyons are confirmed to be distinct. We also confirm the previously derived asymptotic coefficients of the momentum distribution tail at orders ${k}^{\ensuremath{-}2}$ and ${k}^{\ensuremath{-}3}$ for two harmonically confined anyons. Nonuniversal contributions at order ${k}^{\ensuremath{-}4}$ are discussed.