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The geometric characteristics of Bloch wave functions play a crucial role in electronic transport properties. We show that the thermoelectric performance of materials is governed by the geometric structure of Bloch wave functions within the framework of the Boltzmann equation. The essential geometric notion is the Hilbert-Schmidt quantum distance, measuring the resemblance between two quantum states. We establish a geometric characterization of the scattering rate by extending the concept of quantum distance between two states in momentum space at a distance.Employing isotropic quadratic band touching semimetals, where one can concentrate on the role of quantum geometric effects other than the Berry curvature, we find that the response functions for electrical quantum transport and, therefore, the thermoelectric power factor can be succinctly expressed in terms of the maximum quantum distance, $d_\mathrm{max}$. Specifically, when $d_\mathrm{max}$ reaches one, the power factor doubles compared to the case with trivial geometry ($d_\mathrm{max}=0$). Our finding highlights the significance of quantum geometry in improving the performance of thermoelectric devices.

We study the influence of the quantum geometry on the magnetic responses of quadratic band crossing semimetals. More explicitly, we examine the Landau levels, quantum Hall effect, and magnetic susceptibility of a general two-band Hamiltonian that has fixed isotropic quadratic band dispersion but with tunable quantum geometry, in which the interband coupling is fully characterized by the maximum quantum distance $d_\mathrm{max}$. By continuously tuning $d_\mathrm{max}$ in the range of $0\leq d_\mathrm{max}\leq 1$, we investigate how the magnetic properties of the free electron model with $d_\mathrm{max}=0$ evolve into those of the bilayer graphene with $d_\mathrm{max}=1$. We demonstrate that despite sharing the same energy dispersion $\epsilon(p) =\pm\frac{p^2}{2m}$, the charge carriers in the free electron model and bilayer graphene exhibit entirely distinct Landau levels and quantum Hall responses due to the nontrivial quantum geometry of the wave functions.

A singular flat band(SFB), a distinct class of the flat band, has been shown to exhibit various intriguing material properties characterized by a geometric quantity of the Bloch wave function called the quantum distance. We present a general construction scheme for a tight-binding model hosting an SFB, where the quantum distance profile can be controlled. We first introduce how to build a compact localized state(CLS), a characteristic eigenstate of the flat band, providing the flat band with a band-touching point, where a specific value of the maximum quantum distance is assigned. Then, we develop a scheme designing a tight-binding Hamiltonian hosting an SFB starting from the obtained CLS, satisfying the desired hopping range and symmetries by applying the construction scheme. While the scheme can be applied to any dimensions and lattice structures, we propose several simple SFB models on the square and kagome lattices. Finally, we establish a bulk-boundary correspondence between the maximum quantum distance and the boundary modes for the open boundary condition, which can be used to detect the quantum distance via the electronic structure of the boundary states., Comment: 10 pages, 6 figures

In the standard mean-field treatment of superconductors, the electron-electron interactions are assumed to be written in terms of local density operators. However, more general interactions, such as pair-hopping interactions, may exist or may be generated in a low-energy effective Hamiltonian. In this work, we study the effect of correlated hopping interactions toward the electromagnetic response of superconductors. When only the Hamiltonian after the mean-field approximation is provided, one cannot unambiguously determine its electromagnetic response whenever such interactions are allowed. This work demonstrates that such interactions induce additional terms in the current operator, leading to modifications in the Meissner weight and optical conductivities that deviate from conventional expectations. These results underscore the need for caution when incorporating gauge fields into the BdG Hamiltonian., Comment: 6+7 pages, 3 figures; v2: discussions clarified, a new figure added

The bulk-boundary correspondence is an integral feature of topological analysis and the existence of boundary or interface modes offers direct insight into the topological structure of the Bloch wave function. While only the topology of the wave function has been considered relevant to boundary modes, we demonstrate that another geometric quantity, the so-called quantum distance, can also host a bulk-interface correspondence. We consider a generic class of two-dimensional flat band systems, where the flat band has a parabolic band-crossing with another dispersive band. While such flat bands are known to be topologically trivial, we show that the nonzero maximum quantum distance between the eigenstates of the flat band around the touching point guarantees the existence of boundary modes at the interfaces between two domains with different chemical potentials or different maximum quantum distance. Moreover, the maximum quantum distance can predict even the explicit form of the dispersion relation and decay length of the interface modes., Comment: 11pages, 5 figures