We consider the two-body problem in a periodic potential, and study the bound-state dispersion of a spin-$\uparrow$ fermion that is interacting with a spin-$\downarrow$ fermion through a short-range attractive interaction. Based on a variational approach, we obtain the exact solution of the dispersion in the form of a set of self-consistency equations, and apply it to tight-binding Hamiltonians with on-site interactions. We pay special attention to the bipartite lattices with a two-point basis that exhibit time-reversal symmetry, and show that the lowest-energy bound states disperse quadratically with momentum, whose effective-mass tensor is partially controlled by the quantum metric tensor of the underlying Bloch states. In particular, we apply our theory to the Mielke checkerboard lattice, and study the special role played by the interband processes in producing a finite effective mass for the bound states in a nonisolated flat band.

中文翻译：

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多带晶格中的两体问题和量子几何的作用

我们考虑周期势中的两体问题，并研究自旋原子的束缚态色散$\uparrow$ 与自旋相互作用的费米子$\downarrow$通过短距离吸引相互作用产生费米子。基于变分方法，我们以一组自洽方程组的形式获得色散的精确解，并将其应用于具有现场相互作用的紧密结合的哈密顿量。我们特别关注具有时间反转对称性的两点基础的二分晶格，并表明最低能级束缚态随动量呈二次方分散，其有效质量张量部分地由量子的量子度量张量控制。基本的Bloch状态。特别是，我们将我们的理论应用于Mielke棋盘格，并研究了带间过程在为非隔离平坦带中的束缚态产生有限有效质量时所起的特殊作用。