  Geometry of the wave function is a central pillar of modern solid state
physics. In this work, we unveil the wave-function geometry of two-dimensional
semimetals with band crossing points (BCPs). We show that the Berry phase of
BCPs are governed by the quantum metric describing the infinitesimal distance
between quantum states. For generic linear BCPs, we show that the corresponding
Berry phase is determined either by an angular integral of the quantum metric,
or equivalently, by the maximum quantum distance of Bloch states. This
naturally explains the origin of the $\pi$-Berry phase of a linear BCP. In the
case of quadratic BCPs, the Berry phase can take an arbitrary value between 0
and $2\pi$. We find simple relations between the Berry phase, maximum quantum
distance, and the quantum metric in two cases: (i) when one of the two crossing
bands is flat; (ii) when the system has rotation and/or time-reversal
symmetries. To demonstrate the implication of the continuum model analysis in
lattice systems, we study tight-binding Hamiltonians describing quadratic BCPs.
We show that, when the Berry curvature is absent, a quadratic BCP with an
arbitrary Berry phase always accompanies another quadratic BCP so that the
total Berry phase of the periodic system becomes zero. This work demonstrates
that the quantum metric plays a critical role in understanding the geometric
properties of topological semimetals.
