Quantum criticality at finite temperature for two-dimensional $JQ_3$ models on the square and the honeycomb lattices.

We study the quantum criticality at finite temperature for three two-dimensional (2D) $JQ_3$ models using the first principle nonperturbative quantum Monte Carlo calculations (QMC). In particular, the associated universal quantities are obtained and their inverse temperature dependence are investigated. The considered models are known to have quantum phase transitions from the N\'eel order to the valence bond solid. In addition, these transitions are shown to be of second order for two of the studied models, with the remaining one being of first order. Interestingly, we find that the outcomes obtained in our investigation are consistent with the mentioned scenarios regarding the nature of the phase transitions of the three investigated models. Moreover, when the temperature dependence of the studied universal quantities is considered, a substantial difference between the two models possessing second order phase transitions and the remaining model is observed. Remarkably, by using the associated data from both the models that may have continuous transitions, good data collapses are obtained for a number of the considered universal quantities. The findings presented here not only provide numerical evidence to support the results established in the literature regarding the nature of the phase transitions of these $JQ_3$ models, but also can be employed as certain promising criterions to distinguish second order phase transitions from first order ones for the exotic criticalities of the $JQ$-type models. Finally, based on a comparison between the results calculated here and the corresponding theoretical predictions, we conclude that a more detailed analytic calculation is required in order to fully catch the numerical outcomes determined in our investigation.