  In time-reversal invariant systems, all charge Hall effects predicted so far
are extrinsic effects due to the dependence on the relaxation time. We explore
intrinsic Hall signatures by studying quantum noise spectrum of the Hall
current in time-reversal invariant systems, and discover intrinsic thermal Hall
noises in both linear and nonlinear regimes. As the band geometric
characteristics, quantum geometric tensor and Berry curvature play critical
roles in various Hall effects, so are their quantum fluctuations. It is found
that the thermal Hall noise in linear order of the electric field is purely
intrinsic, and the second-order thermal Hall noise has both intrinsic and
extrinsic contributions. In particular, the intrinsic part of the second-order
thermal Hall noise is a manifestation of the quantum fluctuation of quantum
geometric tensor, which widely exists as long as Berry curvature is nonzero.
These intrinsic thermal Hall noises provide direct measurable means to band
geometric information, including Berry curvature related quantities and quantum
fluctuation of quantum geometric tensor.
