We investigate the quantum criticality of topological phase transitions in a periodically-driven two-dimensional Chern insulator model hosting topological phases as well as phases with anomalous edge modes. Using a curvature renormalization group approach based on the stroboscopic Berry curvature, all phase boundaries, including those delineating phases with anomalous edge modes, can be easily obtained obviating the need for micromotion-based topological invariants. Analyzing the criticality of the different transitions in this model, we show that all belong to one of two different Floquet-engineered universality classes: a Dirac class characterized by linear gap closures and a rare nodal-loop semimetal class with quadratic gap closures. Curiously, we find that all the nodal loop phase transitions in the system are described by a unique effective theory around the high symmetry points modulo an overall scale factor. We discuss physical implications and possible measurements of the critical behavior.
A class of Aubry-Andr\'e-Harper models of spin-orbit coupled electrons exhibits a topological phase diagram where two regions belonging to the same phase are split up by a multicritical point. The critical lines which meet at this point each defines a topological quantum phase transition with a second-order nonanalyticity of the ground-state energy, accompanied by a linear closing of the spectral gap with respect to the control parameter; except at the multicritical point which supports fourth-order transitions with parabolic gap-closing. Here both types of criticality are characterized through a scaling analysis of the curvature function defined from the topological invariant of the model. We extract the critical exponents of the diverging curvature function at the non--high symmetry points in the Brillouin zone where the gap closes, and also apply a renormalization group approach to the flattening curvature function at high symmetry points. We also derive a basis-independent correlation function between Wannier states to characterize the transition. Intriguingly, we find that the critical exponents and scaling law defined with respect to the spectral gap remain the same regardless of the order of the transition.
The Kitaev model on the honeycomb lattice is a paradigmatic system known to host a wealth of nontrivial topological phases and Majorana edge modes. In the static case, the Majorana edge modes are nondispersive. When the system is periodically driven in time, such edge modes can disperse and become chiral. We obtain the full phase diagram of the driven model as a function of the coupling and the driving period. We characterize the quantum criticality of the different topological phase transitions in both the static and driven model via the notions of Majorana-Wannier state correlation functions and momentum-dependent fidelity susceptibilities. We show that the system hosts crossdimensional universality classes: although the static Kitaev model is defined on a two-dimensional (2D) honeycomb lattice, its criticality falls into the universality class of one-dimensional (1D) linear Dirac models. For the periodically driven Kitaev model, in addition to the universality class of prototype 2D linear Dirac models, an additional 1D nodal loop type of criticality exists owing to emergent time-reversal and mirror symmetries, indicating the possibility of engineering multiple universality classes by periodic driving. The manipulation of time-reversal symmetry allows the periodic driving to control the chirality of the Majorana edge states.
Recently topological states of matter have witnessed a new physical phenomenon where both edge modes and gapless bulk coexist at topological quantum criticality. The presence and absence of edge modes on a critical line can lead to an unusual class of topological phase transition between the topological and nontopological critical phases. We explore the existence of this new class of topological phase transitions in a generic model representing the topological insulators and superconductors and we show that such transition occurs at a multicritical point, i.e., at the intersection of two critical lines. To characterize these transitions we reconstruct the theoretical frameworks, which include bound state solution of the Dirac equation, winding number, correlation factors, and scaling theory of the curvature function to work for the criticality. Critical exponents and scaling laws are discussed to distinguish between the multicritical points, which separate the critical phases. Entanglement entropy and its scaling in the real space provide further insights into the unique transition at criticality revealing the interplay between fixed point and critical point at the multicriticalities.
We analyze emergent quantum multicriticality for strongly interacting, massless Dirac fermions in two spatial dimensions (
d=2) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break
O(S1) and
O(S2) symmetries in the ordered phase. Performing a renormalization-group analysis by using the $\ensuremath{\epsilon}=(3\ensuremath{-}d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an enlarged
O(S1+S2) chiral symmetry. Such a fixed point acts as an exotic quantum multicritical point (MCP), governing the continuous semimetal-insulator as well as insulator-insulator (for example, antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either
O(S1) or
O(S2) chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power-law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher-dimensional Dirac fermions is also outlined.
We explore the physics of a Chern insulator subjected to a two-step Floquet drive. We analytically obtain the phase diagram and show that the system can exhibit different topological phases characterized by the presence and chirality of edge modes in the two bulk gaps of the Floquet quasienergy spectrum, around 0 and $\ensuremath{\pi}$. We find that the phase of the system depends on the mean but not on the amplitude of the drive. The bulk topological invariants characterizing the phases can be extracted by mapping the unitary evolution within a time period to an energetically trivial but topologically nontrivial time evolution. An extensive numerical study of the bulk topological invariants in the presence of quenched disorder reveals transitions induced by strong disorder (i) from the different topological to trivial insulator phases and (ii) from a trivial to a topological Anderson insulator phase at intermediate disorder strengths. Careful analysis of level statistics of the quasienergy spectrum indicates a ``levitation-annihilation'' mechanism near these transitions.