We introduce a new class of out-of-equilibrium noninteracting topological phases, the topological space-time crystals. These are time-dependent quantum systems which do not have discrete spatial translation symmetries, but instead are invariant under discrete space-time translations. Similar to the Floquet-Bloch systems, the space-time crystals can be described by a frequency-domain-enlarged Hamiltonian, which is used to classify topologically distinct space-time crystals. We show that these space-time crystals can be engineered from conventional crystals with an additional time-dependent drive that behaves like a traveling wave moving across the crystal. Interestingly, we are able to construct 1D and 2D examples of topological space-time crystals based on tight-binding models which involve only one orbital, in contrast to the two-orbital minimal models for any previous discovered static or Floquet topological phases with crystalline structures.
We propose an efficient scheme for simulating the topological phases of matter based on silicon-vacancy (SiV) center arrays in phononic crystals. This phononic band gap structure allows for long-range spin-spin interactions with a tunable profile. Under a particular periodic microwave driving, the band-gap mediated spin-spin interaction can be further designed with the form of the Su-Schrieffer-Heeger (SSH) Hamiltonian. In momentum space, we investigate the topological characters of the SSH model, and show that the topological nontrivial phase can be obtained through modulating the periodic dr…
In this work, we identify a new class of Z2 topological insulator protected by non-symmorphic crystalline symmetry, dubbed a "topological non-symmorphic crystalline insulator". We construct a concrete tight-binding model with the non-symmorphic space group pmg and confirm the topological nature of this model by calculating topological surface states and defining a Z2 topological invariant. Based on the projective representation theory, we extend our discussion to other non-symmorphic space groups that allows to host topological non-symmorphic crystalline insulators.
We propose a method of measuring topological invariants of a photonic crystal through phase spectroscopy. We show how the Chern numbers can be deduced from the winding numbers of the reflection coefficient phase. An explicit proof of existence of edge states in system with nonzero reflection phase winding number is given. The method is illustrated for one- and two-dimensional photonic crystals of nontrivial topology.
We present a general approach to obtain effective field theories for topological crystalline insulators whose low-energy theories are described by massive Dirac fermions. We show that these phases are characterized by the responses to spatially dependent mass parameters with interfaces. These mass interfaces implement the dimensional reduction procedure such that the state of interest is smoothly deformed into a topological crystal, which serves as a representative state of a phase in the general classification. Effective field theories are obtained by integrating out the massive Dirac fermion…
In this Book Chapter (invited) we briefly review the basic concepts defining topological insulators and focus on elaborating on the key experimental results that revealed and established their symmetry protected (SPT) topological nature. We then present key experimental results that demonstrate magnetism, Kondo insulation, mirror chirality or topological crystalline order and superconductivity in spin-orbit topological insulator settings and how these new (bulk insulating) phases of matter arise through topological quantum phase transitions from Bloch band insulators via topological or Dirac s…
Gapless surface states of time reversal invariant topological insulators are protected by the anti-unitary nature of the time reversal operation. Very recently, this idea was generalized to magnetic structures, in which time reversal symmetry is explicitly broken, but there is still an anti-unitary symmetry operation combining time reversal symmetry and crystalline symmetry. These topological phases in magnetic structures are dubbed "topological magnetic crystalline insulators". In this work, we present a general theory of topological magnetic crystalline insulators in different types of magne…
We study the bulk-boundary correspondence for topological crystalline phases, where the crystalline symmetry is an order-two (anti)symmetry, unitary or antiunitary. We obtain a formulation of the bulk-boundary correspondence in terms of a subgroup sequence of the bulk classifying groups, which uniquely determines the topological classification of the boundary states. This formulation naturally includes higher-order topological phases as well as topologically nontrivial bulk systems without topologically protected boundary states. The complete bulk and boundary classification of higher-order to…
We propose magnonic analogs of topological crystalline insulators which possess Dirac surface states protected by the combined symmetry of time-reversal and half translation. Constructing models of the topological magnon systems, we demonstrate that the energy current flows through the systems in response to an electric field, owing to the Dirac surface states with the spin-momentum locking. We also propose a realization of the magnonic analogs of topological crystalline insulators in a magnetic compound CrI$_{3}$ with a monoclinic structure.
Understanding crystal shapes is a fundamental subject in surface science. It is now well studied how chemical bondings determine crystal shapes via dependence of surface energies on surface orientations. Meanwhile, discoveries of topological materials have led us to a new paradigm in surface science, and one can expect that topological surface states may affect surface energies and crystal facets in an unconventional way. Here we show that the surface energy of glide-symmetric topological crystalline insulators (TCI) depends on the surface orientation in a singular way via the parity of the Mi…
Topological crystalline insulators define a new class of topological insulator phases with gapless surface states protected by crystalline symmetries. In this work, we present a general theory to classify topological crystalline insulator phases based on the representation theory of space groups. Our approach is to directly identify possible nontrivial surface states in a semi-infinite system with a specific surface, of which the symmetry property can be described by 17 two-dimensional space groups. We reproduce the existing results of topological crystalline insulators, such as mirror Chern i…
We present a scheme to explicitly construct and classify general topological states jointly protected by an onsite symmetry group and a spatial symmetry group. We show that all these symmetry protected topological states can be adiabatically deformed (allowing for stacking of trivial states) into a special class of states we call topological crystals. A topological crystal in, for example, three dimensions is a real-space assembly of finite-sized pieces of topological states in one and two dimensions protected by the local symmetry group alone, arranged in a configuration invariant under the s…
It is shown theoretically that a one-dimensional crystal with time reversal symmetry is characterized by a Z_{2} topological invariant that predicts the existence or otherwise of edge states. This is confirmed experimentally through the construction and simulation of a photonic crystal analogue in the microwave regime.
We study electronic states and topological invariants of (001)-films of topological crystalline insulator (TCI) Pb_{x}Sn_{1-x}Te. Gapless surface Dirac cones on bulk TCIs become gapped in thin films due to finite-size effect, which is hybridization between those on the top and bottom surfaces. We clarify that the TCI film has the strong finite-size effect as compared to three-dimensional topological insulators such as Bi2Se3. Moreover, the energy gap oscillates with the thickness of film. The oscillation stems from topological phase transitions in two dimensions. The obtained data of the topol…
Antiferromagnetic skyrmion crystals are magnetic phases predicted to exist in antiferromagnets with Dzyaloshinskii-Moriya interactions. Their spatially periodic noncollinear magnetic texture gives rise to topological bulk magnon bands characterized by nonzero Chern numbers. We find topologically-protected chiral magnonic edge states over a wide range of magnetic fields and Dzyaloshinskii-Moriya interaction values. Moreover, and of particular importance for experimental realizations, edge states appear at the lowest possible energies, namely, within the first bulk magnon gap. Thus, antiferromag…
Since topological insulators were theoretically predicted and experimentally observed in semiconductors with strong spin-orbit coupling, more and more attention has been drawn to topological materials which host exotic surface states. These surface excitations are stable against perturbations since they are protected by global or spatial/lattice symmetries. Succeeded in achieving various topological insulators, a tempting challenge now is to search for metallic materials with novel topological properties. Here we predict that orthorhombic perovskite iridates realize a new class of metals dubbe…
Topological phases, including the conventional first-order and higher-order topological insulators and semimetals, have emerged as a thriving topic in the fields of condensed-matter physics and material science. Usually, a topological insulator is characterized by a fixed order topological invariant and exhibits associated bulk-boundary correspondence. Here, we realize a new type of topological insulator in a bilayer phononic crystal, which hosts simultaneously the first-order and second-order topologies, referred here as the hybrid-order topological insulator. The one-dimensional gapless heli…
We present a theory of topological edge states in one-dimensional resonant photonic crystals with a compound unit cell. Contrary to the traditional electronic topological states the states under consideration are radiative, i.e., they decay in time due to the light escape through the structure boundaries. We demonstrate that the states survive despite their radiative decay and can be detected both in time- and frequency-dependent light reflection.
A system with a topological insulator slab sandwiched by two one-dimensional photonic crystals as a defect is investigated. A giant topological magnetoelectric effect is proposed which is characterized by two resonant peaks with half-unit transmissions and rich polarization behaviors in the photonic band gap. We also predict a giant optical Hall effect in our system, in which the transverse shift of the transmitted light could be thousand times larger than the wavelength. These effects are purely from the $Θ$-domain walls at the topological insulator interfaces and could be observed easily. Th…
We consider an interface between two strong time-reversal invariant topological insulators having surface states with opposite spin chirality, or equivalently, opposite mirror Chern number. We show that such an interface supports gapless modes that are protected by mirror symmetry. The interface states are investigated with a continuum model for the Bi2Se3 class of topological insulators that takes into account terms up to third order in the crystal momentum, which ensures that the model has the correct symmetry. The model parameters are obtained from ab initio calculations. Finally, we consid…
Madsen et al. [arXiv:1307.2577] claim that one-dimensional insulating crystals and one-dimensional insulating quasicrystals are topologically equivalent and, thus, trivial. In this comment, we clarify that in topological classification of one-dimensional systems, quasiperiodicity plays a role similar to that of a symmetry. Hence, periodic systems are out of the quasiperiodic topological space, and the equivalence between quasicrystals and crystals becomes immaterial.
We show that in conventional one-dimensional insulators excess charges created close to the boundaries of the system can be expressed in terms of the Berry phases associated with the electronic Bloch wave functions. Using this correspondence, we uncover a link between excess charges and the topological invariants of the recently classified one-dimensional topological phases protected by spatial symmetries. Excess charges can be thus used as a probe of crystalline topologies.
Time crystals form when arbitrary physical states of a periodically driven system spontaneously break discrete time-translation symmetry. We introduce one-dimensional time-crystalline topological superconductors, for which time-translation symmetry breaking and topological physics intertwine---yielding anomalous Floquet Majorana modes that are not possible in free-fermion systems. Such a phase exhibits a bulk magnetization that returns to its original form after two drive periods, together with Majorana end modes that recover their initial form only after four drive periods. We propose experim…
We investigate the spin-orbit coupling effect in a two-dimensional Wigner crystal. We show that sufficiently strong spin-orbit coupling and an appropriate sign of g-factor could transform the Wigner crystal to a topological phonon system. We demonstrate the existence of chiral phonon edge modes in finite size samples, as well as the robustness of the modes in the topological phase. We explore the possibility of realizing the topological phonon system in two-dimensional Wigner crystals confined in semiconductor quantum wells/heterostructure. We find that the spin-orbit coupling is too weak for …
We identify a one-to-one correspondence between the charge localized around a dislocation characterized by a generic Burgers vector and the Berry phase associated with the electronic Bloch waves of two-dimensional crystalline insulators. Using this correspondence, we reveal a link between dislocation charges and the topological invariants of inversion and rotation symmetry-protected insulating phases both in the absence and in the presence of time-reversal symmetry. Our findings demonstrate that dislocation charges can be used as generic probes of crystalline topologies.
The traditional systems researched in condensed matter physics always have spatial translation symmetry. However for space-time crystal systems, the spatial translation symmetry is no longer preserved and the lattice potential have space-time translation symmetry instead. We show that a rational space-time crystal system is equal to a traditional floquet system. Then, we find a way to solve the floquet equation analytically and construct an effective Hamiltonian of the rational space-time crystal system. By this effective Hamiltonian, we obtain the topological classification of gapped and gap-…
We propose a different route to time-reversal invariant Weyl semimetals employing multilayer heterostructures comprising ordinary "trivial" insulators and nontrivial insulators with \textit{pairs} of protected Dirac cones on the surface. We consider both the case of weak topological insualtors, where surface Dirac cones are pinned to time-reversal invariant momenta, and of topological crystalline insulators with unpinned surface Dirac cones. For both realizations we explain phenomenologically how the proposed construction leads to the emergence of a Weyl semimetal phase. We further formulate e…
When two-dimensional electrons are subjected to a very strong magnetic field, they are believed to form a triangular Wigner crystal. We demonstrate that, in the entire crystal phase, this crystal is very well represented by a composite-fermion-crystal wave function, revealing that it is not a simple Hartree-Fock crystal of electrons but an inherently quantum mechanical crystal characterized by a non-perturbative binding of quantized vortices to electrons, which establishes a long range quantum coherence in it. It is suggested that this has qualitative consequences for experiment.
This letter investigates a flow-free, pseudospin-based acoustic topological insulator. Zone folding, a strategy originated from photonic crystal, is used to form double Dirac cones in phononic crystal. The lattice symmetry of the phononic crystal is broken by tuning the size of the center "atom" of the unit cell in order to open the nontrivial topological gap. Robust sound one-way propagation is demonstrated both numerically and experimentally. This study provides a flexible approach for realizing acoustic topological insulators, which are promising for applications such as noise control and w…
Topological crystalline phases in electronic structures can be generally classified using the spatial symmetry characters of the valence bands and mapping them onto appropriate symmetry indicators. These mappings have been recently applied to identify thousands of topological electronic materials. There can exist, however, topological crystalline non-trivial phases that go beyond this paradigm: they cannot be identified using spatial symmetry labels and consequently lack any classification. In this work, we achieve the first of such classifications showcasing the paradigmatic example of two-di…
Topological insulators, first observed in electronic systems, have inspired many analogues in photonic and phononic crystals in which remarkable one-way propagation edge states are supported by topologically nontrivial bandgaps. Such bandgaps can be achieved by breaking the time-reversal symmetry to lift the degeneracy associated with Dirac cones at the corners of the Brillouin zone. Here, we report on our construction of a phononic crystal exhibiting a Dirac-like cone in the Brillouin zone center. We demonstrate that simultaneously breaking the time-reversal symmetry and altering the geometri…
Topological crystalline insulators represent a novel topological phase of matter in which the surface states are protected by discrete point group-symmetries of the underlying lattice. Rock-salt lead-tin-selenide alloy is one possible realization of this phase which undergoes a topological phase transition upon changing the lead content. We used scanning tunneling microscopy (STM) and angle resolved photoemission spectroscopy (ARPES) to probe the surface states on (001) Pb$_{1-x}$Sn$_{x}$Se in the topologically non-trivial (x=0.23) and topologically trivial (x=0) phases. We observed quasiparti…
A Z2 topological insulator protected by time-reversal symmetry is realized via spin-orbit interaction driven band inversion. For example, the topological phase in the Bi-Sb system is due to an odd number of band inversions. A related spin-orbit system, the (Pb/Sn)Te class, has been known to contain an even number of inversions based on band theory. Here we experimentally investigate the possibility of a mirror symmetry protected topological crystalline insulator phase in the (Pb/Sn)Te class of materials which has been theoretically predicted to exist in its non-alloyed version. Our experimenta…
Recently, the topological physics in artificial crystals for classical waves has become an emerging research area. In this Letter, we propose a unique bilayer design of sonic crystals that are constructed by two layers of coupled hexagonal array of triangular scatterers. Assisted by the additional layer degree of freedom, a rich topological phase diagram is achieved by simply rotating scatterers in both layers. Under a unified theoretical framework, two kinds of valley-projected topological acoustic insulators are distinguished analytically, i.e., the layer-mixed and layer-polarized topologica…
While the experimental progress on three dimensional topological insulators is rapid, the development of their two dimensional counterparts has been comparatively slow, despite their technological promise. The main reason is materials challenges of the to date only realizations of two-dimensional topological insulators, in semiconductor quantum wells. Here we identify a two dimensional topological insulator in a material which does not face similar challenges and which is by now most widely available and well-charaterized: graphene. For certain commensurate interlayer twists graphene multilaye…
We present an all-dielectric photonic crystal structure that supports two-dimensionally confined helical topological edge states. The topological properties of the system are controlled by the crystal parameters. An interface between two regions of differing band topologies gives rise to topological edge states confined in a dielectric slab that propagate around sharp corners without backscattering. Three dimensional finite-difference time-domain calculations show these edges to be confined in the out-of-plane direction by total internal reflection. Such nanoscale photonic crystal architecture…
We propose a phenomenological theory for topological crystalline insulators with time reversal and $C_4$ symmetries. First, we introduce a fictitious space and transformation of electromagnetic field operators. This transformation leaves the speed of light unchanged but changes the elementary charge to $\sqrt{2}e$. Then we formulate the theory of topological crystalline insulators in terms of transformed fields in this fictitious space as 3D BF theory containing $π$-flux excitations. It is known that a 3D BF theory with half flux quantum excitations describes low energy properties of time reve…
Topological crystalline insulators (TCIs) are insulating materials whose topological property relies on generic crystalline symmetries. Based on first-principles calculations, we study a three-dimensional (3D) crystal constructed by stacking two-dimensional TCI layers. Depending on the inter-layer interaction, the layered crystal can realize diverse 3D topological phases characterized by two mirror Chern numbers (MCNs) ($μ_1,μ_2$) defined on inequivalent mirror-invariant planes in the Brillouin zone. As an example, we demonstrate that new TCI phases can be realized in layered materials such as…
Dynamical quantum phase transitions (DQPTs) are topologically characterized in quantum quench dynamics in topological systems. In this paper, we study Loschmidt amplitudes and DQPTs in quantum quenches in mirror-symmetric topological phases. Based on the topological classification of mirror-symmetric insulators, we show that mirror symmetry creates symmetry-protected DQPTs. If mirror symmetry is present, topologically robust DQPTs can occur in quantum quenches, even in high-dimensional time-reversal invariant systems. Then, we also show that symmetry-protected DQPTs occur in quenches in two-di…
Topological crystalline insulators are new states of matter in which the topological nature of electronic structures arises from crystal symmetries. Here we predict the first material realization of topological crystalline insulator in the semiconductor SnTe, by identifying its nonzero topological index. We predict that as a manifestation of this nontrivial topology, SnTe has metallic surface states with an even number of Dirac cones on high-symmetry crystal surfaces such as {001}, {110} and {111}. These surface states form a new type of high-mobility chiral electron gas, which is robust again…
We study what kind of topological crystalline insulator phase emerges from nodal line semimetal phases in the face-centered cubic system which is not indicated by topological indices when spin-orbit coupling (SOC) is introduced. We construct an effective model which hosts two different nodal lines phases, and calculated mirror Chern numbers in it by introducing SOC. As a result, we find that the two nodal line phases with different nodal line configurations are mapped to different topological crystalline insulator phases. This result shows that turning off SOC and checking the nodal line confi…
The surface of a three-dimensional topological electron system often hosts symmetry-protected gapless surface states. With the effect of electron interactions, these surface states can be gapped out without symmetry breaking by a surface topological order, in which the anyon excitations carry anomalous symmetry fractionalization that cannot be realized in a genuine two-dimensional system. We show that for a mirror-symmetry-protected topological crystalline insulator with mirror Chern number $n=4$, its surface can be gapped out by an anomalous $\mathbb Z_2$ topological order, where all anyons c…
We introduce a novel class of interaction-enabled topological crystalline insulators in two- and three-dimensional electronic systems, which we call "topological crystalline magnet." It is protected by the product of the time-reversal symmetry $\mathcal{T}$ and a mirror symmetry or a rotation symmetry $\mathcal{R}$. A topological crystalline magnet exhibits two intriguing features: (i) it cannot be adiabatically connected to any Slater insulator and (ii) the edge state is robust against coupling electrons to the edge. These features are protected by the anomalous symmetry transformation proper…
In this work, we use Wannier functions to analyze topological phase transitions in one dimensional photonic crystals. We first review the construction of exponentially localized Wannier functions in one dimension, and show how to numerically construct them for photonic systems. We then apply these tools to study a photonic analog of the Su-Schrieffer-Heeger model. We use photonic Wannier functions to construct a quantitatively accurate approximate model for the topological phase transition, and compute the localization of topological defect states. Finally, we discuss the implications of our w…
We analyze the robustness of corner modes in topological photonic crystals, taking a $C_6$-symmetric breathing honeycomb photonic crystal as an example. First, we employ topological quantum chemistry and Wilson loop calculations to demonstrate that the topological properties of the bulk crystal stem from an obstructed atomic limit phase. We then characterize the topological corner modes emerging within the gapped edge modes employing a semi-analytical model, determining the appropriate real space topological invariants. For the first time, we provide a detailed account of the effect of long-ra…
Topological phases have been explored in various fields in physics such as spintronics, photonics, liquid helium, correlated electron system and cold-atomic system. This leads to the recent foundation of emerging materials such as topological band insulators, topological photonic crystals and topological superconductors/superfluid. In this paper, we propose a topological magnonic crystal which provides protected chiral edge modes for magnetostatic spin waves. Based on a linearized Landau-Lifshitz equation, we show that a magnonic crystal with the dipolar interaction acquires spin-wave volume-m…
Magneto-optics based current imaging technique compares the nature of topological current distribution in a single crystal and thin film of topological insulator material, Bi2Se3. The single crystal, at low temperatures, has uniform topological surface current sheets which are about 3.6 nm thick. With increasing temperature, the current partially diverts into the crystal bulk and concomitantly, the sheet break up into a patchy network of high and low current density regions. The temperature dependence of the high current density areas shows that the surface to bulk transformation in the crysta…
We demonstrate that multiple topological transitions can occur, with high-sensitivity, by continuous change of the geometry of a simple 2D dielectric-frame photonic crystal consisting of circular air-holes. By changing the radii of the holes and/or the distance between them, multiple transitions between normal and topological photonic band gaps (PBGs) can appear. The time-reversal symmetric topological PBGs resemble the quantum spin-Hall insulator of electrons and have two counter-propagating edge states. We search for optimal topological transitions, i.e., sharp transitions sensitive to the g…
Topological photonics harnesses the physics of topological insulators to control the behavior of light. Photonic modes robust against material imperfections are an example of such control. In this work, we propose a soft-matter platform based on nematic liquid crystals that supports photonic topological insulators. The orientation of liquid crystal molecules introduces an extra geometric degree of freedom which in conjunction with suitably designed structural properties, leads to the creation of topologically protected states of light. The use of soft building blocks potentially allows for rec…
Skyrmions are spatially localized magnetic swirls which carry a nonzero integer topological charge. We study crystals of skyrmions in a two-dimensional ferromagnet model with chiral interactions induced by the presence of broken inversion symmetry. We show that non-linear quartic mode-coupling terms allowed by symmetry enhances the Zeeman-field range over which the skyrmion-crystal phase remains stable. Furthermore, it leads to a significant dependence of the lattice constant of this spin crystal over this wide field range. Conduction electrons coupled to such a tunable spin crystal are shown …
We classify discrete-rotation symmetric topological crystalline superconductors (TCS) in two dimensions and provide the criteria for a zero energy Majorana bound state (MBS) to be present at composite defects made from magnetic flux, dislocations, and disclinations. In addition to the Chern number that encodes chirality, discrete rotation symmetry further divides TCS into distinct stable topological classes according to the rotation eigenspectrum of Bogoliubov-de Gennes quasi-particles. Conical crystalline defects are shown to be able to accommodate robust MBS when a certain combination of the…
In band theory of solids, degeneracies manifest themselves as point, line, and surface objects. The topological nature of band degeneracies has recently been appreciated since the recognition of gapless topological quantum phases of matter. Based on first-principles calculations, we report the topological semimetal phase in osmium oxyhalide OsOF$_5$ that exhibits zero-dimensional Weyl points, one-dimensional nodal lines, and two-dimensional nodal surfaces. The semimetal phase is ensured by the electron filling constraint while band degeneracies of various dimensionalities are protected by the …
The celebrated tenfold-way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of non-spatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly-interacting electrons' non-crystalline topological phases for all symmetry classes. Over recent years, a plethora of topological phenomena with diverse surface states has been discovered in crystalline materials. In this paper, we obtain an exhaustive classification of topologically distinct groundstates as well as topologica…
