Effective atom-dimer bands of three-body lattice systems at strong coupling: exact critical mass ratios and a statistics-quasimomentum duality
Abstract
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.
Related
- A Framework for Teaching the Responsible and Effective Use of Generative AI in Statistics and Data Science
- Exact Nonperturbative Equilibrium Mode Statistics in Nonlinear Wave and Lattice Systems
- Robust circular cluster-based statistics for respiration-brain coupling
- Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line
- Coupling for one-dimensional subcritical and critical CBI processes with jumps
BibTeX
@misc{dolgopolov2026effective,
title = {{Effective atom-dimer bands of three-body lattice systems at strong coupling: exact critical mass ratios and a statistics-quasimomentum duality}},
author = {Mikhail V. Dolgopolov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33576},
eprint = {2609.33576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33576},
}
RIS
TY - UNPB
TI - Effective atom-dimer bands of three-body lattice systems at strong coupling: exact critical mass ratios and a statistics-quasimomentum duality
AU - Mikhail V. Dolgopolov
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33576
UR - https://arxiv.org/abs/2609.33576
AB - 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.
ER -