  The signature of quantum phase transition is generally wiped out at finite
temperature. A few quantities that have been observed to carry this signature
through a nonanalytic behavior are also limited to low temperatures only. With
an aim to identify a suitable dynamical quantity at a high temperature, we have
recently constructed a function from quantum fidelity, which has the potential
to bear a nonanalytic signature at the quantum critical point beyond low
temperature regime. In this paper, we elaborate our earlier work and
demonstrate the behavior of the corresponding rate function and the robustness
of the nonanalyticity for a number of many-body Hamiltonians in different
dimensions. We have also shown that our rate function reduces to that used in
the demonstration of the dynamical quantum phase transition (DQPT) at zero
temperature. It has been further observed that, unlike DQPT, the long time
limit of the rate function can faithfully detect the equilibrium quantum phase
transition as well.
