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,
L0, 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.
In the vicinity of continuous quantum phase transitions (QPTs), quantum systems become scale-invariant and can be grouped into universality classes characterized by sets of critical exponents. We have found that despite scale-invariance and universality, the experimental data still contain information related to the microscopic processes and scales governing QPTs. We have found that for many systems, the scaled data near QPTs can be approximated by the generic exponential dependence introduced in the scaling theory of localization; this dependence includes as a parameter a microscopic seeding scale of the renormalization group,
L0. We have also conjectured that for interacting systems, the temperature cuts the renormalization group flow at the length travelled by a system-specific elementary excitation over the life-time set by the Planckian time,
τP=
ℏ/kBT. We have adapted this approach for QPTs in several systems and showed that
L0 extracted from experiment is comparable to physically-expected minimal length scales, namely (i) the mean free path for metal-insulator transition in doped semiconductor Si:B, (ii) the distance between spins in Heisenberg and Ising chains, (iii) the period of an optical lattice for cold atom boson gases, and (iv) the period of a moiré superlattice for the Mott QPT in dichalcogenide bilayers. The metal-insulator transition in Si:P has been explained using a non-interacting version of the model. In two companion papers, we show that in superconducting systems,
L0 is comparable to superconducting coherence length, and in quantum Hall systems, to the magnetic length. The developed new method of data analysis identifies microscopic processes leading to QPTs and quantitatively explains and unifies a large body of experimental data.
We investigate the role of quantum fluctuations in topological quantum phase transitions of quantum spin-Hall insulators and quantum anomalous-Hall insulators. Employing the variational cluster approximation to obtain the single-particle Green's function of the interacting many-body system, we characterize different phases by direct calculation of the recently proposed topological order parameter for interacting systems. We pinpoint the influence of quantum fluctuations on the quantum spin Hall to Mott insulator transition in several models. Furthermore, we propose a general mechanism by which a topological quantum phase transition can be driven by the divergence of the self-energy induced by interactions.
The results of a microscopic theory, based on the topological concept of a θ vacuum, which show that the Coulomb potential, unlike any finite-ranged interaction potential, renders the long-standing problem of the plateau transitions in the quantum Hall regime like a non-Fermi liquid are reported. These results, which are important for quantum-phase transitions in general and composite fermion ideas in particular, provide a novel understanding of the critical exponent values that have recently been (re-)taken from a series of state-of-the-art quantum Hall samples.
We develop a method to characterize topological phase transitions for strongly correlated Hamiltonians defined on two-dimensional lattices based on the many-body Berry curvature. Our goal is to identify a class of quantum critical points between topologically nontrivial phases with fractionally quantized Hall (FQH) conductivity and topologically trivial gapped phases through the discontinuities of the many-body Berry curvature in the so-called flux Brillouin zone (fBZ), the latter being defined by imposing all possible twisted boundary conditions. For this purpose, we study the finite-size signatures of several quantum phase transitions between fractional Chern insulators and charge-ordered phases for two-dimensional lattices by evaluating the many-body Berry curvature numerically using exact diagonalization. We observe degeneracy points (nodes) of many-body energy levels at high-symmetry points in the fBZ, accompanied by diverging Berry curvature. We find a correspondence between the number and order of these nodal points, and the change of the topological invariants of the many-body ground states across the transition, in close analogy with Weyl nodes in noninteracting band structures. This motivates us to apply a scaling procedure, originally developed for noninteracting systems, for the Berry curvature at the nodal points. This procedure offers a useful tool for the classification of topological phase transitions in interacting systems harboring FQH like topological order.
Bosonic integer quantum Hall (IQH) phases are a class of symmetry-protected topological (SPT) phases that, similar to the fermionic IQH states, support a quantized Hall conductance. They, however, require interactions for their realization. Here, we study quantum Hall plateau transitions between a trivial insulator and a bosonic IQH phase, in a clean system. Generically, we find an intervening superfluid phase. The presence of additional symmetries, however, can potentially lead to a direct transition between these phases. We employ a fermionic parton description that captures both the insulating phases as well as the transition between them. The critical theory is massless QED-3 with
Nf=2 fermion flavors. The fermions have a surprisingly simple interpretation: they are vortices of the superfluid. Therefore, the universal conductivity at the transition, assuming it is continuous, equals the universal resistivity of Dirac fermions in QED-3. We also briefly comment on sigma model descriptions of the transition and the surface states of related three-dimensional SPT phases.