  The study of quantum geometry effects in materials has been one of the most
important research directions in recent decades. The quantum geometry of a
material is characterized by the quantum geometry tensor of the Bloch states.
The imaginary part of the quantum geometry tensor gives rise to the Berry
curvature while the real part gives rise to the quantum metric. While Berry
curvature has been well studied in the past decades, the experimental
investigation on the quantum metric effects is only at its infancy stage. In
this work, we measure the nonlinear transport of bulk MnBi${_2}$Te${_4}$, which
is a topological anti-ferromagnet. We found that the second order nonlinear
responses are negligible as required by inversion symmetry, the third-order
nonlinear responses are finite. The measured third-harmonic longitudinal
($V_{xx}^{3{\omega}}$) and transverse ($V_{xy}^{3{\omega}}$) voltages with
frequency 3w, driven by an a.c. current with frequency w, show an intimate
connection with magnetic transitions of MnBi${_2}$Te${_4}$ flakes. Their
magnitudes change abruptly as MnBi${_2}$Te${_4}$ flakes go through magnetic
transitions from an AFM state to a canted AFM state and to a FM state. In
addition, the measured $V_{xx}^{3{\omega}}$ is an even function of the applied
magnetic field B while $V_{xy}^{3{\omega}}$ is odd in B. Amazingly, the field
dependence of the third-order responses as a function of the magnetic field
suggests that $V_{xx}^{3{\omega}}$ is induced by quantum metric quadrupole and
$V_{xy}^{3{\omega}}$ is induced by Berry curvature quadrupole. Therefore, the
quadrupoles of both the real and the imaginary part of the quantum geometry
tensor of bulk MnBi${_2}$Te${_4}$ are revealed through the third order
nonlinear transport measurements. This work greatly advanced our understanding
on the connections between the higher order moments of quantum geometry and
nonlinear transport.
