Topological insulator (TI) thin films with surface magnetism are expected to exhibit a quantized anomalous Hall effect (QAHE) when the magnetizations on the top and bottom surfaces are parallel, and a quantized topological magnetoelectric (QTME) response when the magnetizations have opposing orientations (axion insulator phase) and the films are sufficiently thick. We present a unified picture of both effects that associates deviations from exact quantization of the QTME caused by finite thickness with non-locality in the side-wall current response function. Using realistic tight-binding model calculations, we show that in $Bi_2Se_3$ TI thin films deviations from quantization in the axion insulator-phase are reduced in size when the exchange coupling of tight-binding model basis states to the local magnetization near the surface is strengthened. Stronger exchange coupling also reduces the effect of potential disorder, which is unimportant for the QAHE but detrimental for the QTME, which requires that the Fermi energy lie inside the gap at all positions.

We suggest the tried approach of impurity band engineering to produce flat bands and additional nodes in Dirac materials. We show that surface impurities give rise to nearly flat impurity bands close to the Dirac point. The hybridization of the Dirac nodal state induces the splitting of the surface Dirac nodes and the appearance of new nodes at high-symmetry points of the Brillouin zone. The results are robust and not model dependent: our tight-binding calculations are supported by a low-energy effective model of a topological insulator surface state hybridized with an impurity band. Finally, we address the effects of electron-electron interactions between localized electrons on the impurity site. We confirm that the correlation effects, while producing band hybridization and the Kondo effect, keep the hybridized band flat. Our findings open up prospects for impurity band engineering of nodal structures and flat-band correlated phases in doped Dirac materials.

Topological insulator thin films with surface magnetism are expected to exhibit a quantized anomalous Hall effect when the magnetizations on the top and bottom surfaces are parallel, and a quantized topological magnetoelectric (QTME) response when the magnetizations have opposing orientations and the films are sufficiently thick. We present a unified picture of both effects that associates finite thickness corrections to the QTME with non-locality in the side-wall current response function. Using realistic tight-binding model calculations, we show that finite-thickness corrections in Bi2Se3 topological insulator (TI) thin films are reduced in size when the exchange coupling of band states to surface magnetization is strengthened.

Non-magnetic gap at the Dirac point of topological insulators remains an open question in the field. Here, we present angle-resolved photoemission spectroscopy experiments performed on Cr-doped Bi2Se3 and showed that the Dirac point is progressively buried by the bulk bands and a low spectral weight region in the vicinity of the Dirac point appears. These two mechanisms lead to spectral weight suppression region being mistakenly identified as an energy gap in earlier studies. We further calculated the band structure and found that the original Dirac point splits into two nodes due to the impurity resonant states and the energy separation between the nodes is the low density of state region which appears to be like an energy gap in potoemission experiments. We supported our arguments by presenting photoemission experiments carried out with on- and off- resonant photon energies. Our observation resolves the widely debated questions of apparent energy gap opening at the Dirac point without long range ferromagnetic order in topological insulators.

Driven and non-equilibrium quantum states of matter have attracted growing interest in both theoretical and experimental studies in condensed matter physics. Recent progress in realizing transient collective states in driven or pumped Dirac materials (DMs) is reviewed herein. In particular, the focus is on optically pumped DMs which are a promising platform for transient excitonic instabilities. Optical pumping combined with the linear (Dirac) dispersion of the electronic spectrum offers a knob for tuning the effective interaction between the photoexcited electrons and holes, and thus provides a way of reducing the critical coupling for excitonic instability. As a result, a transient excitonic condensate could be achieved in a pumped DM while it is not feasible in equilibrium. A unifying theoretical framework is provided for describing transient collective states in 2D and 3D DMs. The experimental signatures are described and numerical estimates of the size of the dynamically induced excitonic gaps and the values of the critical temperatures for several specific systems, are summarized. In addition, general guidelines for identifying promising material candidates are discussed. Finally, comments are provided regarding recent experimental efforts in realizing transient excitonic condensate in pumped DMs, and outstanding issues and possible future directions are outlined.

Dynamics play an essential role in the investigation of quantum materials and matter. In this special issue, multiple facets of dynamic quantum matter are investigated and examined. The cover image shows a photon on a Dirac cone, which illustrates the interaction between photons and electrons to produce dynamic states. (Image created by Bart Olsthoorn)

We are witnessing rapid developments in the field of quantum materials with the focus on some of the most profound concepts in condensed matter, including entangled orders, quantum coherence, and quantum topology. Quantum correlations in these materials naturally reveal themselves in the time domain, since their temporal evolution is governed by the full Hamiltonian, which contains multiple interactions. Therefore, non-equilibrium quantum dynamics emerges as a design principle to create desired quantum materials and functionalities. We see a growing focus on dynamics as a way to understand and control the fundamental physical processes that emerge due to quantum coherence of entangled quantum matter. Commensurate with the developments in theory and modeling of quantum materials, we see a rapid rise of new probes for the examination of quantum matter. These new probes include ultra-fast optics, free-electron lasers, and new neutron scattering facilities with capabilities to probe quantum matter at the increasingly short time scale while maintaining high-spatial resolution. Reflecting on recent progress and excitement in the field, we organized a workshop on Dynamic Quantum Matter Dec 10–14, 2018 at the Nordic Institute for Theoretical Physics (NORDITA, https://indico.fysik.su.se/event/6504/). As a result of this conference, we put together a special issue of the Annalen der Physik on “Dynamic Quantum Matter”. We summarize the contributions in this volume. This special issue contains a rapid research letter, four feature articles, five original papers, and a perspective on the dynamics of various forms of quantum matter and materials ranging from superconductivity to magnetism, bosonic and fermionic Dirac materials ferro-states in 2D materials. One focus in this volume is on the potential for the realization of odd-frequency pair correlations. The relation of odd-frequency, or Berezinskii, pairing to dynamic order is discussed in the article by A.V. Balatsky et al.1 This rapid research letter proposes the concept of the time-independent correlators for the even- and odd-frequency pairing states that can be defined for both bosonic and fermionic quasiparticles. It demonstrates that these time independent correlators explicitly capture the existence of two distinct classes of pairing states: one corresponding to the conventional Bardeen–Cooper–Schrieffer (BCS) state and another captures the dynamic odd-frequency superconducting state. Furthermore, it offers a direct probe of the hidden Berezinskii order and illustrates it in the cases of pairings for Majorana fermions and quasiparticles in Dirac semimetals. Triola et al.2 review the recent progress in understanding the interplay of conventional BCS and odd-frequency pairing in multi-band superconductors. After starting with a simple two-band model, these authors point towards the origin of odd-frequency pairing and convey a unified picture of the emergence of this phenomenon in various systems. This contribution also addresses various experimental methods for probing odd-frequency pairing in multi-band systems. E. Rossi and C. Triola3 provide a feature article that reviews the recent work on van der Waals (vdW) systems with strong spin-orbit coupling. Focusing on a selection of vdW heterostructures and using a general effective model to describe the low energy electronic degrees of freedom, the authors examine the interesting electronic proper-ties that can arise in vdW systems formed by graphene and a topological insulator. Through a discussion of the electronic transport properties, the authors show how these interfaces exhibit much stronger spin-dependent transport effects than isolated topological insulators. Furthermore, they discuss the case in which the vdW system includes an additional superconducting layer. Here, they show that how the odd-frequency superconducting pair correlations can be designed in these systems. A significant fraction of the papers are focused on Dirac materials, examining both fermionic and bosonic nodal states. Aside from being interesting materials on their own, Dirac materials offer a new platform to probe dynamic quantum matter in topological materials. This volume contains papers investigating properties of Dirac materials both in- and out-of-equilibrium. J. Alvarez-Jimenez et al. reports, in their original article,4 a new approach to obtain the classical analogs of the quantum metric tensor and the Berry curvature. They show how this approach is advantageous over other procedures for classical systems that correspond to quantum systems with bosonic and fermionic degrees of freedom. This approach is illustrated by applying it to multiple systems ranging from a generalized harmonic oscillator to a spin-half particle in a magnetic field, which is the proto-typical example of a fermionic system. D. Boyko et al., in their original article,5 examines the spin dynamics of Dirac bosons in the Kagome lattice. Using a spin-spin exchange Hamiltonian, they determine the conditions needed for various spin configurations with multiple nearest-neighbor interactions to modify Dirac nodes. Through their investigation of the 120° phase of the Kagome lattice, they probe the competition of exchange interactions and inversion symmetry and demonstrate how it produces a Dirac cone at the K point in the spin-wave spectrum. They also point out that bosonic Dirac nodes at finite energy require pumping and thus can be probed out-of-equilibrium. In their feature article, A. Pertsova and A.V. Balatsky6 focus on non-equilibrium states in fermionic Dirac materials, in particular on the transient collective states in optically driven or pumped Dirac materials. This work provides a promising direction for the realization of excitonic instabilities in systems, in which excitonic condensate is not achievable under equilibrium conditions. Furthermore, they describe the experimental signatures of the transient excitonic state and discuss general guidelines for the identification of potential material candidates for this dynamic phenomenon. J. A. Elias and E. A. Henriksen provide an original article7 that examines the potential of Dirac states in graphene devices with osmium adatoms. Here, they demonstrate the electronic transport of monolayer graphene devices with a sub-monolayer coating of osmium adatoms and show that osmium adatoms shift the charge neutrality point to more positive gate voltages. This effect indicates that osmium adatoms act as electron acceptors and leave the graphene hole-doped. Through an analysis of transport data, the Os adatoms appear to provide charged impurity scattering, which seems unaffected by the strong spin-orbit coupling. In their original article, P. O. Sukhachov et al.8 examines the transport properties and electron states in nanowires of Dirac and Weyl semimetals. Dirac quantum matter exhibits a four-fold symmetry. In the Weyl semimetals, these degeneracies are lowered and forms independent Weyl nodes. These authors show that the electric charge distribution in nanowires is nonuniform and explained this nonuniformity to be due to the Fermi arc states, where a significant amount of charge is located at the surface. They also examine the magnetization cur-rent in Weyl semimetal nanowires, where it is found that it does not vanish in an equilibrium state when the external fields are absent. Also, they show that DC conductivity at the surface demonstrates noticeable peaks when the Fermi level crosses the energies of the surface states. Sayed Ali Akbar Ghorashi provides an original article9 that studies the three-dimensional Luttinger semimetal in the presence of multiple non-uniform periodic kicking, focusing on the nonuniform strain and tilted quadratic band touching to demonstrate that variety of hybrid Dirac and Weyl semimetals can be realized. Y. Araki provides a topical review10 covering the interplay between Weyl electrons and the magnetic texture in what is often called magnetic Weyl semimetals. The idea of the fictitious “axial gauge fields" is examined, and the author finds it to adequately describe the effect of magnetic textures on the Weyl electrons. Furthermore, this approach claimed to account for the properties of localized electrons around magnetic domain walls. As part of this special issue, S. Barraza-Lopez provides an invited perspective11 that aims to highlight the existence of quantum paraelectric, quantum paraelastic, and other structural phases determined by zero-point nuclear motion in two-dimensional materials. It is argued that many two-dimensional materials are hosting structural degeneracies that can stimulate additional studies on these materials and phases. While not necessarily associated with Dirac and Weyl materials, this review of dynamics in paraelectric materials helps to broaden the impact of dynamic states within the realm of quantum materials. We are grateful to all participants and contributors for making this exciting volume possible and to the editors and staff at Annalen der Physik for the steadfast work on this issue. We acknowledge the financial support of NORDITA that enabled this conference. Work of A.V.B. and P.H. was supported by the VILLUM FONDEN via the Centre of Excellence for Dirac Materials (Grant No. 11744), the European Research Council under the European Union's Seventh Framework Program Synergy HERO, and the VR. The work of J.T.H. was supported by the Institute for Materials Science at Los Alamos National Laboratory.

At an interface between a topological insulator (TI) and a conventional superconductor (SC), superconductivity has been predicted to change dramatically and exhibit novel correlations. In particular, the induced superconductivity by an s-wave SC in a TI can develop an order parameter with a p-wave component. Here we present experimental evidence for an unexpected proximity-induced novel superconducting state in a thin layer of the prototypical TI, Bi_2Se_3, proximity coupled to Nb. From depth-resolved magnetic field measurements below the superconducting transition temperature of Nb, we observe a local enhancement of the magnetic field in Bi_2Se_3 that exceeds the externally applied field, thus supporting the existence of an intrinsic paramagnetic Meissner effect arising from an odd-frequency superconducting state. Our experimental results are complemented by theoretical calculations supporting the appearance of such a component at the interface which extends into the TI. This state is topologically distinct from the conventional Bardeen-Cooper-Schrieffer state it originates from. To the best of our knowledge, these findings present a first observation of bulk odd-frequency superconductivity in a TI. We thus reaffirm the potential of the TI-SC interface as a versatile platform to produce novel superconducting states.

A first-principles investigation of the optical response of the Weyl Semimetals MoTe2 and WTe2 is presented. The approach, based on combining two formulations, allows to both separate the intraband and interband parts of the optical conductivity and to distinguish between the bulk and surface contributions to the optical response. It is found that the response is truly anisotropic, with peaks that can be associated with interband transitions involving either bulk or surface states. The role of the relaxation time, and the relation of the calculated results with available experimental measurements, are also discussed. Furthermore, the approach reported is transferable to any system, topologically trivial or non-trivial, thus addressing the long-standing need for comprehensive characterization of the optical response.

The quantum anomalous Hall effect (QAHE), characterized by dissipationless quantized edge transport, relies crucially on a nontrivial topology of the electronic bulk band structure and a robust ferromagnetic order that breaks time-reversal symmetry. Magnetically doped topological insulators (TIs) satisfy both these criteria, and are the most promising quantum materials for realizing the QAHE. Because the spin of the surface electrons aligns along the direction of the magnetic-impurity exchange field, only magnetic TIs with an out-of-plane magnetization are thought to open a gap at the Dirac point (DP) of the surface states, resulting in the QAHE. Using a continuum model supported by atomistic tight-binding and first-principles calculations of transition-metal doped Bi2Se3, we show that a surface-impurity potential generates an additional effective magnetic field which spin polarizes the surface electrons along the direction perpendicular to the surface. The predicted gap-opening mechanism results from the interplay of this additional field and the in-plane magnetization that shifts the position of the DP away from the Gamma point. This effect is similar to the one originating from the hexagonal warping correction of the band structure but is one order of magnitude stronger. Our calculations show that in a doped TI with in-plane magnetization the impurity-potential-induced gap at the DP is comparable to the one opened by an out-of-plane magnetization.

Driven and non-equilibrium quantum states of matter have attracted growing interest in both theoretical and experimental studies in condensed matter physics. We review recent progress in realizing transient collective states in driven or pumped Dirac materials (DMs). In particular, we focus on optically-pumped DMs which have been theoretically proposed as a promising platform for observation of a transient excitonic instability. Optical pumping combined with the linear (Dirac) dispersion of the electronic spectrum offers a knob for tuning the effective interaction between the photoexcited electrons and holes, and thus provides a way of reducing the critical coupling for excitonic instability. As a result, a transient excitonic condensate could be achieved in a pumped DM while it is not feasible in equilibrium. We provide a unifying theoretical framework for describing transient collective states in two- and three-dimensional DMs. We describe experimental signatures of the transient excitonic state and summarize numerical estimates of the magnitude of the effect, namely the size of the dynamically-induced excitonic gaps and the values of the critical temperatures for several specific systems. We also discuss general guidelines for identifying promising material candidates.Finally, we comment recent experimental efforts in realizing transient excitonic condensate in pumped DMs and outline outstanding issues and possible future directions.

We present the result of an ab initio search for new Dirac materials among inverse perovskites. Our investigation is focused on the less studied class of lanthanide antiperovskites containing heavy f-electron elements in the cation position. Some of the studied compounds have not yet been synthesized experimentally. Our computational approach is based on density functional theory calculations which account for spin-orbit interaction and strong correlations of the f-electron atoms. We find several promising candidates among lanthanide antiperovskites which host bulk Dirac states close to the Fermi level. Specifically, our calculations reveal massive three-dimensional Dirac states in materials of the class A(3)BO, where A=Sm, Eu, Gd, Yb, and B=Sn, Pb. In materials with finite magnetic moment, such as Eu3BO (B=Sn, Pb), the degeneracy of the Dirac nodes is lifted, leading to appearance of Weyl nodes.

Using the non‐equilibrium Green's function method and the Keldysh formalism, we study the effects of spin–orbit interactions and time‐reversal symmetry breaking exchange fields on non‐equilibrium quantum transport in graphene armchair nanoribbons. We identify signatures of the quantum spin Hall (QSH) and the quantum anomalous Hall (QAH) phases in non‐equilibrium edge transport by calculating the spin‐resolved real space charge density and local currents at the nanoribbon edges. We find that the QSH phase, which is realized in a system with intrinsic spin–orbit coupling, is characterized by chiral counter‐propagating local spin currents summing up to a net charge flow with opposite spin polarization at the edges. In the QAH phase, emerging in the presence of Rashba spin–orbit coupling and a ferromagnetic exchange field, two chiral edge channels with opposite spins propagate in the same direction at each edge, generating an unpolarized charge current and a quantized Hall conductance G = 2e2/h. Increasing the intrinsic spin–orbit coupling causes a transition from the QAH to the QSH phase, evinced by characteristic changes in the non‐equilibrium edge transport. In contrast, an antiferromagnetic exchange field can coexist with a QSH phase, but can never induce a QAH phase due to a symmetry that combines time‐reversal and sublattice translational symmetry.

The quantum anomalous Hall effect (QAHE) has recently been reported to emerge in magnetically doped topological insulators. Although its general phenomenology is well established, the microscopic origin is far from being properly understood and controlled. Here, we report on a detailed and systematic investigation of transition metal (TM) doped Sb2Te3. By combining density functional theory calculations with complementary experimental techniques, i.e., scanning tunneling microscopy, resonant photoemission, and x-raymagnetic circular dichroism, we provide a complete spectroscopic characterization of both electronic and magnetic properties. Our results reveal that the TM dopants not only affect the magnetic state of the host material, but also significantly alter the electronic structure by generating impurity-derived energy bands. Our findings demonstrate the existence of a delicate interplay between electronic and magnetic properties in TM doped topological insulators. In particular, we find that the fate of the topological surface states critically depends on the specific character of the TM impurity: while V-and Fe-doped Sb2Te3 display resonant impurity states in the vicinity of the Dirac point, Cr and Mn impurities leave the energy gap unaffected. The single-ion magnetic anisotropy energy and easy axis, which control the magnetic gap opening and its stability, are also found to be strongly TM impurity dependent and can vary from in plane to out of plane depending on the impurity and its distance from the surface. Overall, our results provide general guidelines for the realization of a robust QAHE in TM doped Sb2Te3 in the ferromagnetic state.

Recently it was suggested that transient excitonic instability can be realized in optically pumped two-dimensional (2D) Dirac materials (DMs), such as graphene and topological insulator surface states. Here we discuss the possibility of achieving a transient excitonic condensate in optically pumped three-dimensional (3D) DMs, such as Dirac and Weyl semimetals, described by nonequilibrium chemical potentials for photoexcited electrons and holes. Similar to the equilibrium case with long-range interactions, we find that for pumped 3D DMs with screened Coulomb potential two possible excitonic phases exist, an excitonic insulator phase and the charge density wave phase originating from intranodal and internodal interactions, respectively. In the pumped case, the critical coupling for excitonic instability vanishes; therefore the two phases coexist for arbitrarily weak coupling strengths. The excitonic gap in the charge density wave phase is always the largest one. The competition between screening effects and the increase of the density of states with optical pumping results in a rich phase diagram for the transient excitonic condensate. Based on the static theory of screening, we find that under certain conditions the value of the dimensionless coupling constant screening in 3D DMs can be weaker than in 2D DMs. Furthermore, we identify the signatures of the transient excitonic condensate that could be probed by scanning tunneling spectroscopy, photoemission, and optical conductivity measurements. Finally, we provide estimates of critical temperatures and excitonic gaps for existing and hypothetical 3D DMs.