We study three particles on the two-dimensional square lattice with contact interactions of infinite strength, in every sector of total quasimomentum, for three statistics: identical bosons, two identical fermions with a distinguishable third particle, and two identical bosons with a distinguishable third particle. The atom-dimer sector associated with the infinitely degenerate eigenvalue of the pair-contact operator is analysed on the whole Brillouin zone. A Feshbach-Schur reduction gives explicit effective Hamiltonians built from the Dirichlet Laplacian and a rank-four exchange operator, and a spectral transfer theorem with explicit error bounds identifies the atom-dimer threshold of the three-body operator with that of the effective one; the eigenvalue counting function below the threshold is squeezed between those of the effective operator, so that the number of atom-dimer states bound uniformly in the coupling equals the number of bound states of the effective Hamiltonian. The threshold problem reduces to a four-by-four Hermitian matrix whose zero-energy limit is built from the two-point resistances of the square lattice and hence to a quartic polynomial. This yields the complete atom-dimer zones in closed form; the critical hopping ratio as an explicit function on the whole Brillouin zone with two-sided bounds and closed forms at all symmetry points; at most two atom-dimer states per sector with an explicit second threshold; exact onset laws of infinite order at the corner point and linear-over-logarithmic in the pure p-wave channel; and two exact dualities relating a change of statistics to a shift of quasimomentum by half a reciprocal-lattice vector. The reduction of the three-body eigenvalue problem to the lattice Skornyakov-Ter-Martirosyan equation is proved with an explicit isomorphism of eigenspaces.
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.
We investigate the formation of a Bose polaron when a single impurity in a Bose-Einstein condensate is quenched from a noninteracting to an attractively interacting state in the vicinity of a Feshbach resonance. We use a beyond-Fr\"ohlich Hamiltonian to describe both sides of the resonance and a coherent-state variational ansatz to compute the time evolution of Boson density profiles in position space. We find that on the repulsive side of the Feshbach resonance, the Bose polaron performs long-lived oscillations, which is surprising given that the two-body problem has only one bound state coupled to a continuum. They arise due to interference between multiply occupied bound states and therefore can be only found with many-body approaches, such as the coherent-state ansatz. This is a distinguishing feature of the Bose polaron compared to the Fermi polaron, where the bound state can be occupied only once. We derive an implicit equation for the frequency of these oscillations and show that it can be approximated by the energy of the two-body bound state. Finally, we consider an impurity introduced at nonzero velocity and find that, on the repulsive side, it is periodically slowed down or even arrested before speeding up again.