Critical mass ratio and phase transition in three-particle lattice systems: comparison of bosonic and fermionic cases
Annotatsiya
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.