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We investigate signatures of topological quantum phase transitions (TQPTs)between the $Z_2$ quantum spin liquids (QSLs). In two spatial dimensions, $Z_2$QSLs and their TQPTs are only well defined at zero temperature ($T=0$), and itis imperative to clarify their observable signatures under nonzerotemperatures. Here, we present the vestiges of TQPTs between $Z_2$ QSLs withMajorana fermions in terms of thermal Hall conductivity $\kappa_{xy}$ atnonzero temperatures. The $\kappa_{xy}/T$ shows characteristic temperaturedependences around TQPTs. We argue that an exponential upturn near $T=0$ andthe peak of $\kappa_{xy}/T$ around massive excitation energy are observablesmoking-gun signals of the TQPTs. Quantum critical fan-shape temperaturedependences are uncovered across TQPTs. We also perform the parton mean-fieldanalysis on a modified Kitaev model with next-nearest neighbor interactionsfinding TQPTs between the phases with different Chern numbers and theirvestiges self-consistently. We discuss the implication of our results to therecent experiments in $\alpha$-RuCl$_3$.

Vestiges of Topological Phase Transitions in Kitaev Quantum Spin Liquids