Entropy-Driven Structural Phase Transition in Nb$_3$Cl$_8$ via Density Functional Theory and an Effective Model

As a prototypical flat-band cluster Mott insulator on an effective triangular lattice, Nb$_3$Cl$_8$ is a potential candidate for hosting a quantum spin liquid (QSL) state. Nevertheless, a first-order structural phase transition around 90K transforms the high-temperature paramagnetic $\alpha$ phase into the low-temperature nonmagnetic $\beta$ phase, suppressing the candidate QSL regime of the $\alpha$ phase. To clarify the microscopic origin of this transition, we combine first-principles calculations with an extended Hubbard model to construct a unified free-energy framework. This framework reveals that the transition is jointly driven by phonon and spin entropy: the $\alpha$ phase is stabilized by softer phonons and larger paramagnetic spin entropy, whereas the $\beta$ phase is favored by interlayer dimerization, which hardens the phonons and quenches the spin entropy through singlet formation. Furthermore, by evaluating the pressure-dependent generalized enthalpy, we provide a thermodynamic explanation for the suppression of the transition under c-axis uniaxial pressure, where stabilizing the $\alpha$ phase may allow the candidate QSL regime of the $\alpha$ phase to be explored at low temperatures.