We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multi-valley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group $Z_8$. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap 8 flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U(1)\times Z_2$ symmetry-protected topological (SPT) phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between 3D topological phases protected by spatial symmetries and 2D topological phases protected by internal symmetries.
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…
We have performed angle-resolved photoemission spectroscopy (ARPES) on the (111) surface of the topological crystalline insulator SnTe. Distinct from a pair of Dirac-cone surface states across the X_bar point of the surface Brillouin zone on the (001) surface, we revealed two types of Dirac-cone surface states each centered at the G_bar and M_bar points, which originate from the bulk-band inversion at the L points. We also found that the energy location of the Dirac point and the Dirac velocity are different from each other. This ARPES experiment demonstrates the surface states on different cr…
The gapless surface Dirac cone of time reversal invariant topological insulators is protected by time reversal symmetry due to the Kramers' theorem. Spin degree of freedom is usually required since Kramers' theorem only guarantees double degeneracy for spinful fermions, but not for spinless fermions. In this paper, we present an antiferromagnetic spinless model, which breaks time reversal symmetry. Similar to time reversal invariant topological insulators, this model possesses a topologically non-trivial phase with a single surface Dirac cone, which is protected by the combination of time reve…
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…
Motivated by recent progress in crystalline symmetry protected topological (SPT) phases of interacting bosons, we study topological crystalline insulators/superconductors (TCIs) of strongly interacting fermions. We construct a class of intrinsically interacting fermionic TCIs, and show that they are beyond both free-fermion TCIs and bosonic crystalline SPT phases. We also show how these phases can be characterized by symmetry protected gapless fermion modes on the corners/hinges of an open system.
Bismuth telluride is a low energy bulk band-gap topological system with conducting surface states. Besides its very good thermoelectric properties, it also makes a very good candidate for broadband photodetectors. Here, we report temperature-dependent photo-Seebeck effect in a bulk single crystalline bismuth telluride. On light illumination, an electrically biased sample shows distinguishable contributions in the measured current due to both the Seebeck effect and the normal photo-generated carriers within a narrow layer of the sample. Detailed experiments are performed to elucidate the distin…
We study the effects of e-e interaction in a 3D Crystalline Topological Insulator by adding on-site repulsion to the single-particle Hamiltonian and solving the many-body problem within Cluster Perturbation Theory. The goal is to clarify how many body effects modify the topological phase that stems from the crystal symmetries. Tuning the strength of the on-site interaction we show that band inversion disappears in the bulk and surface states loose their metallic character.
We show that a tetragonal lattice of weakly interacting cavities with uniaxial electromagnetic response is the photonic counterpart of topological crystalline insulators, a new topological phase of atomic band insulators. Namely, the frequency band structure stemming from the interaction of resonant modes of the individual cavities exhibits an omnidirectional band gap within which gapless surface states emerge for finite slabs of the lattice. Due to the equivalence of a topological crystalline insulator with its photonic-crystal analog, the frequency band structure of the latter can be charact…
We study the topological crystalline insulator phase protected by the glide symmetry, which is characterized by the Z2 topological number. In the present paper, we derive a formula for the Z2 topological invariant protected by glide symmetry in a nonprimitive lattice, from that in a primitive lattice. We establish a formula for the glide-Z2 invariant for the space group No. 9 with glide symmetry in the base-centered lattice, by folding the Brillouin zone into that of the primitive lattice where the formula for the glide-Z2 invariant is known. The formula is written in terms of integrals of the…
We study the effect of the long-range Coulomb interaction in $j=3/2$ Dirac electrons in cubic crystals with the $O_h$ symmetry, which serves as an effective model for antiperovskite topological crystalline insulators. The renormalization group analysis reveals three fixed points that are Lorentz invariant, rotationally invariant, and $O_h$ invariant. Among them, the Lorentz- and $O_h$-invariant fixed points are stable in the low-energy limit while the rotationally invariant fixed point is unstable. The existence of a stable $O_h$-invariant fixed point of Dirac fermions with finite velocity ani…
We study symmetry protected features in the quasiparticle interference (QPI) pattern of 2D systems with mirror symmetries and time-reversal symmetry, around a single static point impurity. We show that, in the Fourier transformed local density of states (FT-LDOS), $ρ(\bq,ω)$, while the position of high intensity peaks generically depends on the geometric features of the iso-energy contour at energy $ω$, the \emph{absence} of certain peaks is guaranteed by the opposite mirror eigenvalues of the two Bloch states that are (i) on the mirror symmetric lines in the Brillouin zone (BZ) and (ii) separ…
It has been noted a long time ago that a term of the form theta (e^2/2πh) B dot E may be added to the standard Maxwell Lagrangian without modifying the familiar laws of electricity and magnetism. theta is known to particle physicists as the 'axion' field and whether or not it has a nonzero expectation value in vacuum remains a fundamental open question of the Standard Model. A key manifestation of the axion term is the Witten effect: a unit magnetic monopole placed inside a medium with non-zero theta is predicted to bind a (generally fractional) electric charge -e(theta/2 pi+n) with n integer.…
We study a two-dimensional (2D) tight-binding model of a topological crystalline insulator (TCI) protected by rotation symmetry. The model is built by stacking two Chern insulators with opposite Chern numbers which transform under conjugate representations of the rotation group, e.g. $p_\pm$ orbitals. Despite its apparent similarity to the Kane-Mele model, it does not host stable gapless surface states. Nevertheless the model exhibits topological responses including the appearance of quantized fractional charge bound to rotational defects (disclinations) and the pumping of angular momentum in …
The electronic structures of two mixed valence insulators YbB$_6$ and YbB$_12$ are studied by using the local density approximation (LDA) supplemented with the Gutzwiller method and dynamic mean field theory (DMFT). YbB$_6$ is found to be a moderately correlated Z$_2$ topological insulator, similar to SmB$_6$ but having much larger bulk band gap. Notably, YbB$_12$ is revealed to be in a new novel quantum state, strongly correlated topological crystalline Kondo insulator, which is characterized by its non-zero mirror Chern number. The surface calculations find odd (three) and even (four) number…
We study topological crystalline insulators doped with magnetic impurities, in which ferromagnetism at the surface lowers the electronic energy by spontaneous breaking of a crystalline symmetry. The number of energetically equivalent ground states is sensitive to the crystalline symmetry of the surface, as well as the precise density of electrons at the surface. We show that for a SnTe model in the topological state, magnetic states can have twofold symmetry, sixfold symmetry, or eightfold degenerate minima. We compute spin stiffnesses within the model to demonstrate the stability of ferromagn…
Topological crystalline insulators (TCI) possess electronic states protected by crystal symmetries, rather than time-reversal symmetry. We show that the transition metal oxides with heavy transition metals are able to support nontrivial band topology resulting from mirror symmetry of the lattice. As an example, we consider pyrochlore oxides of the form A$_2$M$_2$O$_7$. As a function of spin-orbit coupling strength, we find two $Z_2$ topological insulator phases can be distinguished from each other by their mirror Chern numbers, each of which indicates a different TCI. We also derive an effecti…
In addition to novel surface states, topological insulators can also exhibit robust gapless states at crystalline defects. Step edges constitute a class of common defects on the surface of crystals. In this work we establish the topological nature of one-dimensional (1D) bound states localized at step edges of the [001] surface of a topological crystalline insulator (TCI) Pb$_{0.7}$Sn$_{0.3}$Se, both theoretically and experimentally. We show that the topological stability of the step edge states arises from an emergent particle-hole symmetry of the surface low-energy physics, and demonstrate t…
We study interaction effects on the topological crystalline insulators protected by time-reversal ($T$) and reflection symmetry ($R$) in two and three spatial dimensions. From the stability analysis of the edge states with bosonization, we find that the classification of the two-dimensional SPT phases protected by $Z_2\times[\mbox{U(1)}\rtimes T]$ symmetry is reduced from $\mathbb{Z}$ to $\mathbb{Z}_4$ by interactions, where the $Z_2$ symmetry denotes the reflection whose mirror plane is the two-dimensional plane itself. By extending the approach recently proposed by Isobe and Fu, we show that…
Topological crystalline insulators (TCIs) are nontrivial quantum phases of matter protected by crystalline (and other) symmetries. They are originally predicted by band theories, so an important question is their stability under interactions. In this paper, by directly studying the physical bulk properties of several band-theory-based nontrivial TCIs that are conceptually interesting and/or experimentally feasible, we show they are stable under interactions. These TCIs include (1) a weak topological insulator, (2) a TCI with a mirror symmetry and its time-reversal symmetric generalizations, (3…
In the presence of crystalline symmetries, certain topological insulators present a filling anomaly: a mismatch between the number of electrons in an energy band and the number of electrons required for charge neutrality. In this paper, we show that a filling anomaly can arise when corners are introduced in $C_n$-symmetric crystalline insulators with vanishing polarization, having as consequence the existence of corner-localized charges quantized in multiples of $\frac{e}{n}$. We characterize the existence of this charge systematically and build topological indices that relate the symmetry rep…
Two-dimensional higher-order topological insulators can display a number of exotic phenomena such as half-integer charges localized at corners or disclination defects. In this paper, we analyze these phenomena, focusing on the paradigmatic example of the quadrupole insulator with $C_4$ rotation symmetry, and present a topological field theory description of the mixed geometry-charge responses. Our theory provides a unified description of the corner and disclination charges in terms of a physical geometry (which encodes disclinations), and an effective geometry (which encodes corners). We exten…
SmB6 was recently proposed to be both a strong topological insulator and a topological crystalline insulator. For this and related cubic topological Kondo insulators, we prove the existence of four different topological phases, distinguished by the sign of mirror Chern numbers. We characterize these phases in terms of simple observables, and we provide concrete tight-binding models for each phase. Based on theoretical and experimental results for SmB6 we conclude that it realizes the phase with C^+_{k_z=0}=+2, C^+_{k_z=pi}=+1, C^+_{k_x=k_y}=-1, and we propose a corresponding minimal model.
Magnetic insulators are a key resource for next-generation spintronic and topological devices. The family of layered metal halides promises ultrathin insulating multiferroics, spin liquids, and ferromagnets, but new characterization methods are required to unlock their potential. Here, we report tunneling through the layered magnetic insulator CrI3 as a function of temperature and applied magnetic field. We electrically detect the magnetic ground state and inter-layer coupling and observe a field-induced metamagnetic transition. The metamagnetic transition results in magnetoresistances of 95%,…
We prove theoretically that certain strongly correlated Kondo insulators are topological crystalline insulators with nontrivial topology protected by crystal symmetries. In particular, we find that SmB$_6$ is such a material. In addition to a nontrivial Z$_2$ topological index protected by time reversal symmetry, SmB$_6$ also has nontrival mirror Chern numbers protected by mirror symmetries. On the $(100)$ surface of SmB$_6$, the nontrivial mirror Chern numbers do not generate additional surface states beyond those predicted by the Z$_2$ topological index. However, on the $(110)$ surface, two …
We present microscopic Hamiltonians that gap the Dirac fermions on the surface of topological crystalline insulators (TCIs) protected by reflection symmetry and create symmetry-preserving states with Abelian topological orders. For TCIs with $n = 2$ copies of surface Dirac fermions, we find a fermionic $\mathbb{Z}_4$ topological order with charge-spin self duality and with a novel symmetry action on the anyons, whereby reflections with respect to two parallel planes act differently. We also analyze TCIs with $n = 1$ and $4$ copies of Dirac fermions, which taken together covers every nontrivial…
Three-dimensional, crystalline Hopf insulators are generic members of unitary Wigner-Dyson class, which can break all global discrete symmetries and point group symmetries. In the absence of first Chern number for any two-dimensional plane of Brillouin zone, the Hopf invariant $N_H \in \mathbb{Z}$. But in the presence of Chern number $N_H \in \mathbb{Z}_{2q}$, where $q$ is the greatest common divisor of Chern numbers for $xy$, $yz$, and $xz$ planes of Brillouin zone. How does $N_H$ affect topological quantization of isotropic, magneto-electric coefficient? We answer this question with calculat…
We predict a new class of topological crystalline insulators (TCI) in the anti-perovskite material family with the chemical formula A$_3$BX. Here the nontrivial topology arises from band inversion between two $J=3/2$ quartets, which is described by a generalized Dirac equation for a "Dirac octet". Our work suggests that anti-perovskites are a promising new venue for exploring the cooperative interplay between band topology, crystal symmetry and electron correlation.
We show that in crystalline insulators point group symmetry alone gives rise to a topological classification based on the quantization of electric polarization. Using C3 rotational symmetry as an example, we first prove that the polarization is quantized and can only take three inequivalent values. Therefore, a Z3 topological classification exists. A concrete tight-binding model is derived to demonstrate the Z3 topological phase transition. Using first-principles calculations, we identify graphene on BN substrate as a possible candidate to realize the Z3 topological states. To complete our ana…
The interaction of spin-polarized one-dimensional (1D) topological edge modes localized along single-atomic steps of the topological crystalline insulator $Pb_{0.7}Sn_{0.3}Se(001)$ has been studied systematically by scanning tunneling spectroscopy. Our results reveal that the coupling of adjacent edge modes sets in at a step{to{step distance $d_{ss} < 25$ nm, resulting in a characteristic splitting of a single peak at the Dirac point in tunneling spectra. Whereas the energy splitting exponentially increases with decreasing $d_{ss}$ for single-atomic steps running almost parallel, we find no sp…
In the recently discovered topological crystalline insulators (TCIs), topology and crystal symmetry intertwine to create surface states with a unique set of characteristics. Among the theoretical predictions for TCIs is the possibility of imparting mass to the massless Dirac fermions by breaking crystal symmetry, as well as a Lifshitz transition with a change of Fermi surface topology. Here we report high resolution scanning tunneling microscopy studies of a TCI, Pb1-xSnxSe. We demonstrate the formation of zero mass Dirac fermions protected by crystal symmetry and the mechanism of mass generat…
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…
Crystalline topological insulators owe their topological character to the protection that certain boundary states acquire because of certain point-group symmetries. We first show that a Hermitian operator obeying supersymmetric quantum mechanisms (SUSY QM) delivers the entanglement spectrum. We then show that such an entanglement spectrum that is compatible with a certain point-group symmetry obeys a certain local spectral symmetry. The latter result is applied to the stability analysis of four fermionic non-interacting Hamiltonians, the last of which describes graphene with a Kekule distortio…
Crystalline topological phases have recently attracted a lot of experimental and theoretical attention. Key advances include the complete elementary band representation analyses of crystalline matter by symmetry indicators and the discovery of higher-order hinge and corner states. However, current classification schemes of such phases are either implicit or limited in scope. We present a new scheme for the explicit classification of crystalline topological insulators and superconductors. These phases are protected by crystallographic point group symmetries and are characterized by bulk topolog…
The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs) featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We ide…
We reconsider the problem of surface states spectrum in type One Dirac metals. We find that the surface states, despite being gapped, always form branches terminating at Dirac points. Furthermore, we consider evolution of the surface states in the case, when rotational symmetry is broken, and as a result, Dirac points are gapped. We find, that in this case, special role is played by mirror symmetry relative to the plane connecting Dirac points. When it is present, the resulting gapped state is a topological crystalline insulator, which surface spectrum can contain either one or three Dirac poi…
We classify interacting topological insulators and superconductors with order-two crystal symmetries (reflection and twofold rotation), focusing on the case where interactions reduce the noninteracting classification. We find that the free-fermion $\mathbb{Z}_2$ classifications are stable against quartic contact interactions, whereas the $\mathbb{Z}$ classifications reduce to $\mathbb{Z}_N$, where $N$ depends on the symmetry class and the dimension $d$. These results are derived using a quantum nonlinear $σ$ model (QNLSM) that describes the effects of the quartic interactions on the boundary m…
Superconductors derived from topological insulators and topological crystalline insulators by doping have long been considered to be candidates as topological superconductors. Pb$_{0.5}$Sn$_{0.5}$Te is a topological crystalline insulator with mirror symmetry protected surface states on (001), (011) and (111) oriented surfaces. The superconductor (Pb$_{0.5}$Sn$_{0.5}$)$_{0.7}$In$_{0.3}$Te is induced by In doping in Pb$_{0.5}$Sn$_{0.5}$Te, and is thought to be a topological superconductor. Here we report the first scanning tunneling spectroscopy measurement of the superconducting state as well a…
A new class of materials, Topological Crystalline Insulators (TCIs) have been shown to possess exotic surface state properties that are protected by mirror symmetry. These surface features can be enhanced if the surface-area-to-volume ratio of the material increases, or the signal arising from the bulk of the material can be suppressed. We report the experimental procedures to obtain high quality crystal boules of the TCI, SnTe, from which nanowires and microcrystals can be produced by the vapour-liquid-solid (VLS) technique. Detailed characterisation measurements of the bulk crystals as well …
We report detailed investigations of the properties of a superconductor obtained by substituting In at the Sn site in the topological crystalline insulator (TCI), SnTe. Transport, magnetization and heat capacity measurements have been performed on crystals of Sn0.6In0.4Te, which is shown to be a bulk superconductor with Tc(onset) at ~4.70(5) K and Tc(zero) at ~3.50(5) K. The upper and lower critical fields are estimated to be μ0Hc2(0) = 1.42(3) T and μ0Hc1(0) = 0.90(3) mT respectively, while κ = 56.4(8) indicates this material is a strongly type II superconductor.
Topological crystalline insulators in IV-VI compounds host novel topological surface states consisting of multi-valley massless Dirac fermions at low energy. Here we show that strain generically acts as an effective gauge field on these Dirac fermions and creates pseudo-Landau orbitals without breaking time-reversal symmetry. We predict the realization of this phenomenon in IV-VI semiconductor heterostructures, due to a naturally occurring misfit dislocation array at the interface that produces a periodically varying strain field. Remarkably, the zero-energy Landau orbitals form a flat band in…
ThTaN$_3$, a rare cubic perovskite nitride semiconductor, has been studied using {\it ab initio} methods. Spin-orbit coupling (SOC) results in band inversion and a band gap of 150 meV at the zone center. In spite of the trivial $Z_2$ indices, two pairs of spin-polarized surface bands cross the gap near the zone center, indicating that this system is a topological crystalline insulator with the mirror Chern number of $|{\cal C}_m|=2$ protected by the mirror and $C_4$ rotational symmetries. Additionally, SOC doubles the Seebeck coefficient, leading to a maximum of $\sim$400 $μ$V/K at 150 K for c…
The thermal conductivity of the doped topological crystalline insulator, Sn$_{0.6}$In$_{0.4}$Te superconducting single crystal with $T_c =$ 4.1 K, was measured down to 50 mK. It is found that the residual linear term $κ_0/T$ is negligible in zero magnetic field. The $κ_0/T$ shows a slow field dependence at low magnetic field. These results suggest that the superconducting gap is nodeless, unless there exist point nodes with directions perpendicular to the heat current. Due to its high-symmetry fcc crystal structure of Sn$_{0.6}$In$_{0.4}$Te, however, such point nodes can be excluded. Therefore…
