  Time reversal and inversion symmetric materials fail to yield linear and
nonlinear responses since they possess net zero Berry curvature. However,
higher-order Hall response can be generated in these systems upon constraining
the crystalline symmetries. Motivated by the recently discovered third-order
Hall (TOH) response mediated by Berry connection polarizability, namely, the
variation the Berry connection with respect to an applied electric field, here
we investigate the existence of such Hall effect in the surface states of
hexagonal warped topological insulator (e.g., Bi$_2$Te$_3$) under the
application of electric field only. Using the semiclassical Boltzmann
formalism, we investigate the effect of tilt and hexagonal warping on the Berry
connection polarizability tensor and consequently, the TOH effect provided the
Dirac cone remains gapless. We find that the magnitude of the response
increases significantly with increasing the tilt strength and warping and
therefore, they can provide the tunability of this effect. In addition, we also
explore the effect of chemical doping on TOH response in this system.
Interestingly, we show based on the symmetry analysis, that the TOH can be the
leading-order response in this system which can directly be verified in
experiments.
