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.
We demonstrate that the prototypical two-dimensional Chern insulator hosts exotic quantum multicriticality in the presence of an appropriate periodic driving: a linear Dirac-like transition coexists with a quadratic nodal looplike transition. This nodal loop gap closure is characterized by an enhanced chiral-mirror symmetry that is induced by the driving procedure. The existence of multiple universality classes can be unambiguously captured by extracting critical exponents and scaling laws with a single renormalization group approach based on the curvature function of the stroboscopic Floquet Hamiltonian. This procedure is effective regardless of whether the topological phase transitions are associated with anomalous edge modes or not. We comment on possible experimental realizations of the model and detection schemes for the curvature function.
The topological phase transitions in static and periodically driven Kitaev chains are investigated by means of a renormalization group (RG) approach. These transitions, across which the numbers of static or Floquet Majorana edge modes change, are accompanied by divergences of the relevant Berry connections. These divergences at certain high-symmetry points in momentum space form the basis of the RG approach, through which topological phase boundaries are identified as a function of system parameters. We also introduce several aspects to characterize the quantum criticality of the topological phase transitions in both static and Floquet systems: a correlation function that measures the overlap of Majorana-Wannier functions, the decay length of the Majorana edge mode, and a scaling law relating the critical exponents. These indicate a common universal critical behavior for topological phase transitions, in both static and periodically driven chains. For the latter, the RG flows additionally display intriguing features related to gap closures at non-high-symmetry points due to momentarily frozen dynamics.
Novel topological properties that arose in the periodically driven system are unique, in which there are two kinds of quasienergy gaps, the zero quasienergy gap and the
π quasienergy gap. The corresponding edge modes would traverse either the zero quasienergy gap or the
π quasienergy gap, or traverse both two quasienergy gaps. And the characterization of these two kinds of edge modes might not be the same. However, in this paper, we find that both the zero edge modes and the
π edge modes in the Floquet Chern insulators can be characterized by the same topological invariant, where the corresponding Dirac mass term is periodically kicked. Particularly, we take the periodically kicked Qi-Wu-Zhang model as an illustrative example. In this model, the topology is characterized by the Floquet Chern number
CF, and there are six different topological phases in total, denoted as
CF={−10,−2,−1π,1π,2,10}. Furthermore, we find that the Floquet operator associated with the periodically kicked Qi-Wu-Zhang model reduces to a Dirac Hamiltonian in the low-energy limit. Then, the phase diagram is uncovered by examining the topology of this Dirac Hamiltonian. Additionally, we explore the orders of topological phase transitions in the context of Floquet stationary states by analyzing the von Neumann entropy of these states. Our work provides further insights into the topological phases in periodically driven systems.
We employ quench dynamics as an effective tool to probe different universality classes of topological phase transitions. Specifically, we study a model encompassing both Dirac-like and nodal loop criticalities. Examining the Kibble-Zurek scaling of topological defect density, we discover that the scaling exponent is reduced in the presence of extended nodal loop gap closures. For a quench through a multicritical point, we also unveil a path-dependent crossover between two sets of critical exponents. Bloch state tomography finally reveals additional differences in the defect trajectories for sudden quenches. While the Dirac transition permits a static trajectory under specific initial conditions, we find that the underlying nodal loop leads to complex time-dependent trajectories in general. In the presence of a nodal loop, we generically find a mismatch between the momentum modes where topological defects are generated and where dynamical quantum phase transitions occur. We also find notable exceptions where this correspondence breaks down completely.
Topological phase transitions occur in real materials as well as quantum engineered systems, all of which differ greatly in terms of dimensionality, symmetries, interactions, and driving, and hence require a variety of techniques and concepts to describe their topological properties. For instance, topology may be accessed from single-particle Bloch wave functions, Green's functions, or many-body wave functions. We demonstrate that despite this diversity, all topological phase transitions display a universal feature: namely, a divergence of the curvature function that composes the topological invariant at the critical point. This feature can be exploited via a renormalization-group–like methodology to describe topological phase transitions. This approach serves to extend notions of correlation function, critical exponents, scaling laws and universality classes used in Landau theory to characterize topological phase transitions in a unified manner.
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.