Anyons with arbitrary exchange phases exist on 1D lattices in ultracold gases. Yet, known continuum theories in 1D do not match. We derive the continuum limit of 1D lattice anyons via interacting bosons. The theory maintains the exchange phase periodicity fully analogous to 2D anyons. This provides a mapping between experiments, lattice anyons, and continuum theories, including Kundu anyons with a natural regularization as a special case. We numerically estimate the Luttinger parameter as a function of the exchange angle to characterize long-range signatures of the theory and predict different velocities for left- and right-moving collective excitations.
Anyons with arbitrary exchange phases exist on 1D lattices in ultracold gases. Yet, known continuum theories in 1D do not match. We derive the continuum limit of 1D lattice anyons via interacting bosons. The theory maintains the exchange phase periodicity fully analogous to 2D anyons. This provides a mapping between experiments, lattice anyons, and continuum theories, including Kundu anyons with a natural regularization as a special case. We numerically estimate the Luttinger parameter as a function of the exchange angle to characterize long-range signatures of the theory and predict different velocities for left- and right-moving collective excitations.
Nonrelativistic anyons in one dimension possess generalized exchange statistics in which the exchange of two identical anyons generates a phase that is governed by the spatial ordering of the particles and the statistical parameter $\ensuremath{\alpha}$. Working in the continuum, we demonstrate the existence of two distinct types of one-dimensional (1D) anyons, namely, bosonic anyons and fermionic anyons. We identify a many-body Hamiltonian with additive two-body zero-range interactions that supports bosonic and fermionic anyon eigenstates, which are, for arbitrary interaction strength, related through a generalized bosonic-anyon--fermionic-anyon mapping, an extension of the celebrated Bose-Fermi mapping for zero-range interacting 1D systems. The momentum distributions of bosonic and fermionic anyons are distinct: while both feature ${k}^{\ensuremath{-}2}$ and ${k}^{\ensuremath{-}3}$ tails, the associated prefactors differ. Our work reveals intricate connections between the generalized exchange statistics, the universal two- and three-body Tan contacts of systems consisting of N identical particles, and the emergence of statistics-induced chiral symmetry breaking.
Low-dimensional quantum systems can host anyons, particles with exchange statistics that are neither bosonic nor fermionic. Despite indications of a wealth of exotic phenomena, the physics of anyons in one dimension (1D) remains largely unexplored. Here, we realize Abelian anyons in 1D with arbitrary exchange statistics using ultracold atoms in an optical lattice, where we engineer the statistical phase via a density-dependent Peierls phase. We explore the dynamical behavior of two anyons undergoing quantum walks, and observe 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. Our work forms the foundation for exploring the many-body behavior of 1D anyons.
Anyons obeying fractional exchange statistics arise naturally in two dimensions: Hard-core two-body constraints make the configuration space of particles not simply-connected. The braid group describes how topologically-inequivalent exchange paths can be associated to non-trivial geometric phases for abelian anyons. Braid-anyon exchange statistics can also be found in one dimension (1D), but this requires broken Galilean invariance to distinguish different ways for two anyons to exchange. However, recently it was shown that an alternative form of exchange statistics can occur in 1D because hard-core three-body constraints also make the configuration space not simply-connected. Instead of the braid group, the topology of exchange paths and their associated non-trivial geometric phases are described by the traid group. In this article we propose a first concrete model realizing this alternative form of anyonic exchange statistics. Starting from a bosonic lattice model that implements the desired geometric phases with number-dependent Peierls phases, we then define anyonic operators so that the kinetic energy term in the Hamiltonian becomes local and quadratic with respect to them. The ground-state of this traid-anyon-Hubbard model exhibits several indications of exchange statistics intermediate between bosons and fermions, as well as signs of emergent approximate Haldane exclusion statistics. The continuum limit results in a Galilean invariant Hamiltonian with eigenstates that correspond to previously constructed continuum wave functions for traid anyons. This provides not only an a-posteriori justification of our lattice model, but also shows that our construction serves as an intuitive approach to traid anyons, i.e. anyons intrinsic to 1D.
One-dimensional anyons, whose exchange statistics can be continuously tuned between bosonic and fermionic limits, are among the most significant frontier discoveries in condensed matter and ultracold atomic physics. Existing theories describe the statistical behavior of anyons at the level of mathematical structures or effective models, but none addresses the fundamental question of the physical origin of the exchange factor. This work provides a first-principles physical explanation for anyon exchange statistics within the Huaxia Spacetime Total Conservation Theory. In H-SET, particles are helical dislocation loops in the space medium. In a one-dimensional channel, two dislocation loops cannot bypass each other and must directly penetrate. The penetration operation leaves an irreversible twist in the spatial elastic field, and the degree of twisting determines the exchange factor. Starting from basic formulas of dislocation elasticity theory, we rigorously derive the quantitative relationship between the exchange factor and the effective spatial elastic modulus. Because the effective elastic modulus can be continuously tuned via Feshbach resonance, the exchange factor is naturally and continuously tunable between the bosonic and fermionic limits, in quantitative agreement with the core finding of a landmark experiment. We further provide four quantitative predictions that can be tested by independent experiments, including the quantitative mapping between the exchange factor and the scattering length, the momentum-distribution asymmetry ratio, the linear constraint on the Haldane statistical parameter, and the braid-group locking mechanism for the two-dimensional braiding phase. All derivations proceed from the established constants of H-SET, without introducing any free parameters specifically adjusted for the anyon phenomenon.
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.