强耦合下三维晶格系统的有效原子-二聚体能带：精确临界质量比与统计-准动量对偶性
Effective atom-dimer bands of three-body lattice systems at strong coupling: exact critical mass ratios and a statistics-quasimomentum duality
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中文总结 AI 辅助
本研究分析强耦合方格三体系统的原子-二聚体能带，给出临界质量比闭式解及统计-准动量对偶性，揭示边缘耦合与维度嬗变机制。
中文摘要 AI 辅助
我们研究了方格上具有无限强度接触相互作用的三个粒子，涵盖所有总准动量，并考虑三种统计：三个全同玻色子、两个全同费米子加第三个粒子、以及两个全同玻色子加第三个粒子。先前的工作仅处理了三聚体和零准动量情形。这里，原子-二聚体扇区在整个布里渊区上被分析。通过Feshbach-Schur约化到对接触算子的无穷简并本征值1，我们得到了具有证明的一阶误差的显式有效哈密顿量。阈值问题归结为一个由方格的两点电阻构建的4x4厄米矩阵判据，进而归结为一个四次多项式。这给出了完整的原子-二聚体区域和布里渊区上的临界质量比，其界限在1和π/(π-2)之间，并在所有对称点处给出闭式形式。我们证明了：闭式形式的区域；零准动量处临界质量比的精确值π/(π-2)，重现了数值结果；每个准动量和质量比最多有两个原子-二聚体态；起始定律，区域边界上的本质奇点以及纯p波中的线性-对数起始；两个精确对偶性，表明统计的改变等价于准动量平移π；以及第二费米子阈值的全局范围。起始定律允许重整化群解读：逆格点格林函数是边缘s波耦合，束缚能通过维度嬗变产生，晶格尺度为16，区域边界是边缘（BCS/Kondo）类型而非Berezinskii-Kosterlitz-Thouless类型。所有常数均通过晶格求积、计算机代数、精确对角化和无限晶格求解器独立验证。
英文摘要
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.
发表机构
- Samara State Technical University（萨马拉国立技术大学）
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