Quantum phase transitions beyond critical exponents: Microscopic scales and universal role of the Planckian time

Abstract

Over the past four decades, experimental studies of quantum phase transitions (QPTs) have been primarily focused on identifying these transitions in various materials and determining the critical exponents and universality classes associated with them. Since the critical exponents originate from the scale invariance, they are not explicitly tied to the microscopic physics of QPTs, which remains poorly understood for many systems. We have developed a model of QPTs, which addresses this deficiency. The model is based on the experimentally supported conjecture that the scaled data across a QPT depend, in a universal manner, on two system-specific length scales: the microscopic scale, ${L}_{0}$, typically known from general considerations (e.g., mean-free path, lattice constant) and the dephasing length, ${L}_{\ensuremath{\varphi}}(T)$. Using this model, we have found that experimental data in the quantum critical regime of 18 distinct material classes can be quantitatively described by assuming that ${L}_{\ensuremath{\varphi}}(T)$ is determined by the distance traveled by system-specific excitations during the Planckian time, ${\ensuremath{\tau}}_{P}=\ensuremath{\hbar}/{k}_{B}T$. In this work, we demonstrate how the model explains microscopic physics of QPTs in doped semiconductors and amorphous alloys, materials with Ising and Heisenberg spin chains, cold boson gases in optical traps, and the Mott transitions in organic semiconductors. The dynamics of excitations is determined by parameters such as the effective mass and diffusion coefficient, $D$, of the corresponding noninteracting or weakly renormalized system. For doped semiconductors and amorphous alloys, the noncritical behavior of $D$ in the quantum critical regime sharply contrasts with zero-temperature diffusion coefficient, which goes to zero at the transition. We also analyze the results of numerical studies of the Mott transition with dynamical mean-field theory, interpreting them as if they were experimental data. For these results, we find that, as expected, the microscopic scale matches the dimensionless lattice constant, ${L}_{0}\ensuremath{\sim}1$. Finally, we report that the metal-insulator transition in Si:P, Ge:Sb, and Ge:As can be explained using a noninteracting version of the model.