Two-body problem in a multiband lattice and the role of quantum geometry

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.

Figure 1

(a) One-body dispersion ${\varepsilon }_{s\mathbf{k}}$ is shown for the Mielke checkerboard lattice when the lower band is flat. The bands touch at the four corners of the first Brillouin zone. (b) Two-body dispersion $E_{\mathbf{q}}$ is shown for $U=5\left|t\right|$ as a function of $q_{x}a$ when $q_{y}a=0$. The conditions $F_{RR}^{\mathbf{q}}=0$ and $F_{TT}^{\mathbf{q}}=0$ are in perfect agreement with the upper and lower branches, respectively. The quadratic expansion $E_{\mathbf{q}}=E_b+q^2/\left(2m_b\right)$ is an excellent fit for the lower branch in the small-$q$ limit.

Figure 2

(a) Lowest-energy $E_b=E_{\mathbf{q}=\mathbf{0}}$ of the bound state is shown for the upper and lower branches as a function of $U$. (b) Inverse of the effective mass $m_b$ of the bound state is shown for the lower branch as a function of $U$ together with its intraband and interband contributions, where $1/m_b=1/m_b^{\mathrm{intra}}+1/m_b^{\mathrm{inter}}$. (c) Same as in (b) but with a larger region. Here, $m_b=5\pi /[U{a}^{2}ln\left(64\right|t|/U\left)\right]$ and $m_b=U/\left(8a^2{t}^2\right)$ fit very well in the small- and large-$U$ limits, respectively.