We study three particles on the square lattice with contact interactions of infinite strength, for all total quasimomenta, for three statistics: three identical bosons, two identical fermions plus a third particle, and two identical bosons plus a third particle. Previous work treated only the trimer and zero quasimomentum. Here the atom-dimer sector is analyzed on the whole Brillouin zone. A Feshbach-Schur reduction onto the infinitely degenerate eigenvalue 1 of the pair-contact operator gives explicit effective Hamiltonians with proved first-order error. The threshold problem reduces to a 4x4 Hermitian matrix criterion built from the two-point resistances of the square lattice, and hence to a quartic polynomial. This yields the complete atom-dimer zone and the critical mass ratio on the Brillouin zone, with bound between 1 and pi/(pi-2), and closed forms at all symmetry points. We prove: the zones in closed form; the critical mass ratio with exact value pi/(pi-2) at zero quasimomentum, reproducing a numerical value; at most two atom-dimer states for every quasimomentum and mass ratio; the onset laws, an essential singularity on the zone boundary and a linear-over-logarithm onset in the pure p-wave; two exact dualities showing that a change of statistics is equivalent to a shift of quasimomentum by pi; and the global range of the second fermionic threshold. The onset laws admit a renormalization-group reading: the inverse lattice Green function is a marginal s-wave coupling, binding energies arise by dimensional transmutation with lattice scale 16, and the zone boundary is of marginal (BCS/Kondo) type rather than of Berezinskii-Kosterlitz-Thouless type. All constants are verified independently by lattice quadrature, computer algebra, exact diagonalisation, and infinite-lattice solvers.
We study three particles on the square lattice with contact interactions of infinite strength, for all total quasimomenta, for three statistics: three identical bosons, two identical fermions plus a third particle, and two identical bosons plus a third particle. Previous work treated only the trimer and zero quasimomentum. Here the atom-dimer sector is analyzed on the whole Brillouin zone. A Feshbach-Schur reduction onto the infinitely degenerate eigenvalue 1 of the pair-contact operator gives explicit effective Hamiltonians with proved first-order error. The threshold problem reduces to a 4x4 Hermitian matrix criterion built from the two-point resistances of the square lattice, and hence to a quartic polynomial. This yields the complete atom-dimer zone and the critical mass ratio on the Brillouin zone, with bound between 1 and pi/(pi-2), and closed forms at all symmetry points. We prove: the zones in closed form; the critical mass ratio with exact value pi/(pi-2) at zero quasimomentum, reproducing a numerical value; at most two atom-dimer states for every quasimomentum and mass ratio; the onset laws, an essential singularity on the zone boundary and a linear-over-logarithm onset in the pure p-wave; two exact dualities showing that a change of statistics is equivalent to a shift of quasimomentum by pi; and the global range of the second fermionic threshold. The onset laws admit a renormalization-group reading: the inverse lattice Green function is a marginal s-wave coupling, binding energies arise by dimensional transmutation with lattice scale 16, and the zone boundary is of marginal (BCS/Kondo) type rather than of Berezinskii-Kosterlitz-Thouless type. All constants are verified independently by lattice quadrature, computer algebra, exact diagonalisation, and infinite-lattice solvers.
We compare two three-particle lattice models on the square lattice at total quasimomentum zero: three identical bosons with attractive contact interactions, and a two-fermion plus one distinguishable particle system with attractive interspecies contact. Using the Birman--Schwinger principle combined with symmetry reduction and strong-coupling asymptotics, we distinguish shared methodology from model-dependent spectral conclusions. For bosons, the ground-state energy behaves as minus three times the coupling plus six minus three over twice the coupling plus higher-order corrections, and exactly two bound states exist below the threshold for large coupling; the second lies at a finite distance 0.03542 below the threshold, a genuine eigenvalue rather than a threshold resonance. For fermions, the critical mass ratio is pi divided by pi minus two, approximately 2.75194; above it, a doubly degenerate level appears at strong coupling. Near onset, the binding energy follows a linear-over-logarithmic law. The results are compared with numerical finite-grid Faddeev computations and with recent three-dimensional classifications of threshold states.
We study three identical bosons with contact attraction of strength mu on the square lattice at total quasimomentum K = 0 and K = pi. First, at K = 0, for all sufficiently large mu there are exactly two eigenvalues below the essential spectrum. The trimer branch follows from Kato perturbation theory at the simple isolated eigenvalue 3 of the pair-contact operator, uniformly in K; the atom-dimer branch follows from a Feshbach-Schur reduction whose limiting equation coincides with the second Birman-Schwinger (Faddeev) eigenvalue of the finite-rank principal part. The spectral gap between the two branches grows linearly in mu. Second, at K = pi the trimer branch is unchanged up to O(mu^-2), the two-particle threshold equals -mu + 4 exactly, and the gap to it is 2mu - 2 plus lower order corrections. The second bound state disappears because the exchange operator changes sign, S_pi = -S_0: the s-channel eigenvalue that exceeds 1 at K = 0 becomes negative, while the only positive atom-dimer channel at K = pi is the odd p-channel, bounded by 2 - 4/pi < 1. Consequently the operator has exactly one eigenvalue below the threshold minus 36/(mu - 10). Third, the ground state is exponentially localized in the relative coordinates. The decay exponent is positive unconditionally; under a non-cancellation hypothesis it is bounded above by log(3mu) plus lower order terms, while finite-volume computations indicate the slowest tail is the atom-dimer dissociation channel with decay rate one half log(2mu). A dictionary between the principal-part method, Kato theory, and the atom-dimer threshold matrix is given, together with the three-dimensional constants for comparison.
We study three-particle Schrödinger operators on the two-dimensional lattice Z 2 and show that a critical mass ratio γ c ≈ 2.75194 governs the existence of a bound trimer in the fermionic 2 + 1 configuration (two identical fermions and a third particle). For γ < γ c there is a topological prohibition (Pauli suppression) of a three-body bound state, whereas for γ > γ c a doubly degenerate eigenvalue emerges below the essential spectrum with the strong-coupling asymptotics z(γ, λ) = −λ+e 0 (γ)+O(λ −1 ). Within a unified framework based on the Birman–Schwinger principle and strong-coupling asymptotic analysis, we compare this behaviour with the bosonic case of three identical particles, where two bound states exist below the essential spectrum and the ground-state energy satisfies z 1 s (µ) = −3µ + C 2 + O(µ −1 ). The resulting second-order phase transition with respect to the mass ratio γ is relevant for the design of experiments on fermionic trimers in optical lattices and for modelling excitonic complexes and defect-bound states in two-dimensional nanomaterials, where the critical value γ c serves as a design guideline for the observability of three-body bound states. We also outline a modified three-particle lattice model with two competing interaction channels, for which the Birman–Schwinger analysis naturally leads to a Landau-type scenario of a first-order phase transition in the space of trimer bound states. In the bosonic case we prove a strong-coupling theorem describing the existence and asymptotics of trimer bound states, while in the fermionic 2+1 case we establish a spectral phase-transition theorem that identifies an explicit critical mass ratio γ c separating the trimer and non-trimer regimes.
We present a rigorous strong-coupling Birman-Schwinger analysis of the three-boson lattice Schroedinger operator on Z^2 at the exceptional quasimomentum K = pi. First, we provide an exact closed-form benchmark for the fiber Fredholm determinant, valid for every quasimomentum K, obtained via a uniform elliptic reduction. Second, we formulate and prove a general order-matching criterion determining whether a leading-order Fredholm determinant asymptotic, with relative error O(1/mu), suffices to fix the constant-order additive energy correction, or if the next-order refinement is required. Applying the criterion to the formal branch z = -2mu + d, we identify an algebraic crossing -2mu + 6 + 8/mu + O(mu^{-2}), but demonstrate that this crossing does not correspond to a true eigenvalue of the full Hamiltonian. The actual ground state obeys the rigorous variational bounds -3mu <= z_1^{pi,s}(mu) <= -3mu + 6, so that z_1^{pi,s}(mu) = -3mu + O(1), the same leading branch as at K = 0; direct finite-volume diagonalisation confirms the refined asymptotic -3mu + 6 + O(mu^{-1}) and spectral gap 2mu - 2 + O(mu^{-1}). We independently confirm that the known K = 0 constant C approximately 3.96458 requires no analogous refinement. Finally, we compare the asymptotic precision levels achieved across recent lattice few-body models and draw a structural parallel with the parity-based classification of topological band insulators at time-reversal-invariant momenta.
Bound States of 2+1 Fermionic Trimers on Lattice at Strong Couplings
Abstract In this paper, we investigate the bound states of 2+1 2 + 1 fermionic trimers on a three-dimensional lattice at strong coupling. Specifically, we analyze the discrete spectrum of the associated three-body discrete Schrödinger operator Hγ,λ(K), H γ , λ ( K ) , focusing on energies below the continuum and within its gap. Depending on the quasi-momentum K , we show that if the mass ratio γ>0 γ > 0 between the identical fermions and the third particle is below a certain threshold, the operator lacks a discrete spectrum below the essential spectrum for sufficiently large coupling λ>0. λ > 0 . Conversely, if γ γ exceeds this threshold, Hγ,λ(K) H γ , λ ( K ) admits at least one eigenvalue below the essential spectrum. Similar phenomena are observed in the neighborhood of the two-particle branch of the essential spectrum, which resides within the gap and grows sublinearly as λ→+∞. λ → + ∞ . For K=0, K = 0 , the mass ratio thresholds are explicitly calculated and it turns out that, for certain intermediate mass ratios and large couplings, bound states emerge within the gap, although ground states are absent.
We calculate root-mean-square radii for a three-body system confined to two spatial dimensions and consisting of two identical bosons (A) and one distinguishable particle (B). We use zero-range two-body interactions between each of the pairs, and focus thereby directly on universal properties. We solve the Faddeev equations in momentum space and express the mean-square radii in terms of first-order derivatives of the Fourier transforms of densities. The strengths of the interactions are adjusted for each set of masses to produce equal two-body bound-state energies between different pairs. The mass ratio, A=mB/mA, between particles B and A are varied from 0.01 to 100, providing a number of bound states decreasing from 8 to 2. Energies and mean-square radii of these states are analyzed for small A by use of the Born-Oppenheimer potential between the two heavy A particles. For large A the radii of the two bound states are consistent with a slightly asymmetric three-body structure. When A approaches thresholds for binding of the three-body excited states, the corresponding mean-square radii diverge inversely proportional to the deviation of the three-body energy from the two-body thresholds. The structures at these three-body thresholds correspond to bound AB dimers and one loosely bound A particle.