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The shift current responses of two-dimensional systems in the gapless limit are investigated. As the energy gap becomes smaller, the interband transition probability becomes larger at the band edges. We found the divergence of the shift current response in a manner proportional to the inverse of the gap size when the system becomes a two-dimensional Weyl semimetal in the gapless limit. This behavior is different from that for electric polarization, which has a jump across the gapless case. It means that the quantitative relationship between electric polarizations and shift currents is broken in this gapless limit., Comment: 6 pages, 4 figures

In some insulators, corner charges are fractionally quantized, due to the topological invariant called a filling anomaly. The previous theories of fractional corner charges have been mostly limited to two-dimensional systems. In three dimensions, only limited cases have been studied. In this study, we derive formulas for the filling anomaly and the corner charge in various crystals with all the tetrahedral and cubic space groups. We discuss that the quantized corner charge requires the crystal shapes to be vertex-transitive polyhedra. We show that the formula of the filling anomaly is universally given by the difference between electronic and ionic charges at the Wyckoff position 1a. The fractional corner charges appear by equally distributing the filling anomaly to all the corners of the crystal. We also derive the k-space formulas for the fractional corner charge. In some cases, the corner charge is not determined solely from the irreps at high-symmetry k-points. In such cases, we introduce a new Z2 topological invariant to determine the corner charge., Comment: 14 pages, 5 figures

Antiferromagnetic (AFM) topological insulators (TIs), which host magnetically gapped Dirac-cone surface states and exhibit many exotic physical phenomena, have attracted great attention. Here, we find that the coupled surface states can be intertwined to give birth to a set of $2n$ unique new Dirac cones, dubbed intertwined Dirac cones, through the anisotropic coupling enforced by crystalline $n$-fold ($n=2, 3, 4, 6$) rotation symmetry $C_{nz}$ in the presence of a $PT$-symmetry breaking potential, for example, an electric field. Interestingly, we also find that the warping effect further drives the intertwined Dirac-cone state into a quantum anomalous Hall phase with a high Chern number ($C=n$). Then, based on first-principles calculations, we have explicitly demonstrated six intertwined Dirac cones and a Chern insulating phase with a high Chern number ($C=3$) in MnBi$_2$Te$_4$$/$(Bi$_2$Te$_3$)$_{\mathrm{m}}/$MnBi$_2$Te$_4$ heterostructures, as well as the $C=2$ and $C=4$ phases in HgS and $\alpha$-Ag$_2$Te films, respectively. This work discovers the intertwined Dirac-cone state in AFM TI thin films, which reveals a mechanism for designing the quantum anomalous Hall state with a high Chern number and also paves a way for studying highly tunable high-Chen-number flat bands of twistronics., Comment: 9 pages, 4 figures+supplemental materials

Chirality is an indispensable concept that pervades fundamental science and nature, manifesting itself in diverse forms such as chiral quasiparticles and chiral structures. Of particular interest are Weyl phonons carrying specific Chern numbers and chiral phonons doing circular motions in crystals. Up to now, Weyl and chiral phonons have been studied independently and the interpretations of chirality seem to be different in these two concepts, impeding our understanding. Here, we demonstrate that Weyl and chiral phonons are entangled in chiral crystals. Employing a typical chiral crystal of elementary tellurium (Te) as a case study, we expound on the intrinsic relationship between Chern number of Weyl phonons and pseudo-angular momentum (PAM) of chiral phonons. In light of the mutual coupling, we propose Raman scattering as a new technique to demonstrate the existence of Weyl phonons in Te, by detecting the chirality-induced energy splitting between the two constituent chiral phonon branches for Weyl phonons. By using the same experimental approach, we also observe the obstructed phonon surface states for the first time., Comment: 7 pages, 5 figures (to appear in NanoLetters)

Chiral phonons with atomic rotations converted into electron spins result in a change of spin magnetizations in crystals. In this paper, we investigate a new conversion of chiral phonons into magnons both in ferromagnets and antiferromagnets by spin models with exchange and Dzyaloshinskii-Moriya interactions. The atomic rotations in chiral phonons are treated as slow phonons, which modulate spin-spin interaction and induce time-dependent magnon excitations due to geometric effect within adiabatic approximation. We demonstrate that a non-trivial change of the number of magnons requires breaking of the spin-rotation symmetry around the spin quantization axis. As a result, the clockwise and counterclockwise chiral phonons induce a change of the magnon number with opposite signs, which corresponds to an increasing or decreasing spin magnetizaton due to the chiral nature of the atomic rotations. In particular, in antiferromagnets, the modulation of magnons due to chiral phonons generates a non-zero net magnetization by the proposed effect, which is expect to be observed in experiments., Comment: 13 pages, 5 figures

In topological phase transitions involving a change in topological invariants such as the Chern number and the $\mathbb{Z}_2$ topological invariant, the gap closes, and the electric polarization becomes undefined at the transition. In this paper, we show that the jump of polarization across such topological phase transitions in two dimensions is described in terms of positions and monopole charges of Weyl points in the intermediate Weyl semimetal phase. We find that the jump of polarization is described by the Weyl dipole at $\mathbb{Z}_2$ topological phase transitions and at phase transitions without any change in the value of the Chern number. Meanwhile, when the Chern number changes at the phase transition, the jump is expressed in terms of the relative positions of Weyl points measured from a reference point in the reciprocal space., Comment: 12pages, 10 figures

We study equilibrium crystal shapes of a topological insulator (TI), a topological crystalline insulator (TCI) protected by mirror symmetry, and a second-order topological insulator (SOTI) protected by inversion symmetry. By adding magnetic fields to the three-dimensional TI, we can realize the mirror-symmetry-protected TCI and the inversion-symmetry-protected SOTI. They each have topological boundary states in different positions: the TCI has gapless states on the surfaces that are invariant under the symmetry operation, and the SOTI has gapless states at the intersections between certain surfaces. In this paper, we discuss how these boundary states affect the surface energies and the equilibrium crystal shapes in terms of the calculations of the simple tight-binding model by using the Wulff construction. By comparing the changes in the shapes of the TI to that of the trivial insulator through the process of applying the magnetic fields, we show that the presence/absence of the topological boundary states affects the emergence of the specific facets in a different way from the trivial insulator., Comment: 13 pages, 12 figures

The non-Hermitian skin effect (NHSE) is a novel localization phenomenon in certain non-Hermitian systems with gain and/or loss. Most of previous works study the non-Hermitian skin effect in periodic systems. However, electromagnetic waves often propagate within uniform materials without periodic modulation, and it has not been clear whether the optical NHSE occurs in uniform media such as bulk materials and electromagnetic metamaterials. Here we establish the theory of the optical NHSE in non-Hermitian anisotropic media. We show that the NHSE occurs even in uniform media with appropriate anisotropy and material loss. The localization of non-Hermitian skin modes are completely determined by an effective gauge potential caused by the anisotropy of a dielectric tensor. On the basis of the theory, we propose subwavelength multilayer metamaterials as a novel platform for the optical NHSE. We also propose a new concept of stationarily-excited skin modes whose frequencies are forced to be real in non-Hermitian systems. We find that the NHSE occurs even under the condition that the frequency is forced to be real, which implies that the NHSE we propose is observable under stationary excitation. Our work presents a general theory of the NHSE in homogeneous systems, and pave the way to realize the optical NHSE in bulk materials and metamaterials., Comment: 17 pages, 7 figures

A nodal-line semimetal is a topological gapless phase containing one-dimensional degeneracies called nodal lines. The nodal lines are deformed by a continuous change of the system such as pressure and they can even change their topology, but it is not systematically understood what kind of changes of topology of nodal lines are possible. In this paper, we classify the events of topology change of nodal lines by the Morse theory and reveal that only three types of topology changes of nodal lines, i.e., creation, reconnection, and annihilation, are possible in the spinless nodal-line semimetal protected by inversion and time-reversal symmetries. They are characterized by an index having the values 0, 1, and 2 for the above three types in the Morse theory. Moreover, we extend our theory to systems with rotational symmetries and mirror symmetry and disclose the possible events of topology change of nodal lines under each symmetry., Comment: 18 pages, 18 figures

We construct a general theory of $Z_2$ topological phase transitions in two-dimensional systems with time-reversal symmetry. We investigate the possibilities of $Z_2$ topological phase transitions at band inversions at all high-symmetry points in $k$-space in all the 80 layer groups. We exclude the layer groups with inversion symmetry because the $Z_2$ topological phase transition is known to be associated with band inversions with an exchange of parities. Among the other layer groups, we find 21 layer groups with insulator-to-insulator transitions with band inversion, and this problem is finally reduced to five point groups $C_3, C_4, C_6, S_4$, and $C_{3h}$. We show how the change of the $Z_2$ topological invariant at a band inversion is entirely determined by the irreps of occupied and unoccupied bands at the high-symmetry point. For example, in the case of $C_3$, we show that the $Z_2$ topological invariants change whenever the band inversion occurs between two Kramers pairs whose $C_3$ eigenvalues are $\{e^{\pi i / 3}, e^{-\pi i / 3}\}$ and $\{-1, -1\}$. These results are not included in the theory of symmetry-based indicators or topological quantum chemistry., Comment: 15 pages, 9 figures

Since the latter half of the 20th century, the use of metal in optics has become a promising plasmonics field for controlling light at a deep subwavelength scale. Surface plasmon polaritons localized on metal surfaces are crucial in plasmonics. However, despite the long history of plasmonics, the underlying mechanism producing the surface waves is not fully understood. This study unveils the hidden symmetry protection that ensures the existence of degenerated electric zero modes. These zero modes are identified as physical origins of surface plasmon polaritons, and similar zero modes can be directly excited at a temporal boundary. In real space, the zero modes possess vector-field rotation related to surface impedance. Focusing on the surface impedance, we prove the bulk-edge correspondence, which guarantees the existence of surface plasmon polaritons even with nonuniformity. Lastly, we extract the underlying physics in the topological transition between metal and dielectric material using a minimal circuit model with duality. The transition is considered the crossover between electric and magnetic zero modes., Comment: 26 pages, 19 figures, minor corrections

The non-Bloch band theory can describe energy bands in a one-dimensional (1D) non-Hermitian system. On the other hand, whether the non-Bloch band theory can be extended to higher-dimensional non-Hermitian systems is nontrivial. In this work, we construct the non-Bloch band theory in two classes of two-dimensional non-Hermitian systems, by reducing the problem to that for a 1D non-Hermitian model. In these classes of systems, we get the generalized Brillouin zone for a complex wavevector and investigate topological properties. In the model of the non-Hermitian Chern insulator, as an example, we show the bulk-edge correspondence between the Chern number defined from the generalized Brillouin zone and the appearance of the edge states., Comment: 11pages, 7figures

The electric polarization as a bulk quantity is described by the modern theory of polarization in insulating systems and cannot be defined in conducting systems. Upon a gradual change of a parameter in the system, the polarization always varies smoothly as long as the gap remains open. In this paper, we focus on the two-dimensional Weyl semimetal, which hosts Weyl nodes protected by symmetries, and study the behavior of the polarization when a symmetry-breaking term $M$ is introduced and a gap opens. We show that there can be a jump between $M\to0^+$ and $M\to0^-$ limits. We find that the jump is universally described by the ``Weyl dipole" representing how the Weyl nodes with monopole charges are displaced in the reciprocal space. Our result is applicable to general two-dimensional Weyl semimetals., Comment: 8 pages, 5 figures

Excitons, which are composite boson quasi-particles composed of bound electrons and holes, have many fascinating properties and great potential in practical applications. Though experimental studies on exciton dynamics are well-developed, the ab initio simulation ones still remain vacant until two years ago. Here, we apply the density functional theory (DFT) and many-body perturbation theory (MBPT) on 2D MoSi$_2$N$_4$ to study its exciton-related physics and non-adiabatic ultrafast exciton dynamics theoretically and numerically for the first time. We calculate the photoluminescence (PL) spectra with final states as bright excitons, yet lots of them are contributed by the dark ones, and the results match the experimental ones perfectly. We also study the dark-exciton-involved processes, which were barely studied in the past but dominate in many physical processes, and obtain several main results like: (i) High scattering rates over the whole Brillouin zone (BZ) within the order of magnitude from 10$^{-2}$ fs$^{-1}$ to 10$^{-1}$ fs$^{-1}$; (ii) Thorough analysis for the dynamics of the dark excitons at $\Lambda$ valley, which have negative effective mass and the highest scattering rate among several exciton states; (iii) Simulate the time-resolved evolution of the excitons after photo-excitation with the real-time Boltzmann transport equation (rt-BTE) techniques, in which process excitons at K/K' valley play an important role; (iv) Exciton dynamics with spin-valley locking at K/K' valley are also discussed here; (v) A new approach is proposed for modulating the non-adiabatic effects for excitons, accompanied by a chiral phonon absorption/emission, by tuning the chirality of the external circularly polarized light. All the results show that the 2D material MoSi$_2$N$_4$ is an ideal platform to study the exciton-involved physics and has great application value., Comment: 8 pages, 5 figures

Weyl points, carrying a Z-type monopole charge C, have bulk-surface correspondence (BSC) associated with helical surface states (HSSs). When |C| > 1, multi-HSSs can appear in a parallel manner. However, when a pair of Weyl points carrying C = \pm 1 meet, a Dirac point carrying C = 0 can be obtained and the BSC vanishes. Nonetheless, a recent study in Ref. [arXiv:2201.03238] shows that a new BSC can survive for Dirac points when the system has time-reversal (T )-glide (G) symmetry ({\Theta}=TG), i.e., anti-parallel double/quad-HSSs associated with a new Z2-type monopole charge Q appears. In this paper, we systematically review and discuss both the parallel and anti-parallel multi-HSSs for Weyl and Dirac points, carrying two different kinds of monopole charges. Two material examples are offered to understand the whole configuration of multi-HSSs. One carries the Z-type monopole charge C, shows both local and global topology for three kinds of Weyl points, and it leads to parallel multi-HSSs. The other carries the Z2-type monopole charge Q, only showing the global topology for {\Theta}-invariant Dirac points, and it is accompanied by antiparallel multi-HSSs. These diverse topological surface states not only offer the local and global topology for the bulk degeneracy but also offer platforms to various topological phases., Comment: 9 pages, 5 figures

Properties of systems with exact n-fold screw symmetry (n=2, 3, 4, 6) have been well studied because they can be understood in terms of space groups. On the other hand, existence of materials with approximate screw symmetries, such as 7-fold and 10-fold screw symmetries, has been predicted. In this paper, we study properties of phonons in crystals with approximate screw symmetries, which will lead to unique and new physical phenomena. In a crystal with an approximate screw symmetry, we propose a method to extract information of pseudoangular momentum of phonons, which is a quantum number characterizing the properties of phonon modes under screw symmetry, based on the fact that the information of the quantum numbers defined under exact screw symmetry partially remains in the eigenvectors of approximate screw symmetric systems. As a preparation, we study a one-dimensional crystal with partially broken translation symmetry to have an enlarged unit cell, and we show how to extract information of a quantum number corresponding to the pseudoangular momentum, by studying a relative phase between neighboring atoms. We also extend this method to systems with an approximate screw symmetry, and discuss properties of the pseudoangular momentum. We apply this method to results of our first-principle calculations on candidate materials with an approximate translational symmetry or with an approximate screw symmetry, and show how this approximate symmetry is reflected in the phonon wavefunctions., Comment: 16 pages, 11 figures, to be published in Phys. Rev. B

Recently, a series of two-dimensional (2D) nonmagnetic layered materials XSi2Y4 (X=transition metals; Y=pnictogens) having similar crystal structures with transition-metal dichalcogenides (TMDs) were proposed for their potential application value. Like TMDs, we propose that chiral phonon involved valley-selective optical circular dichroism can be also obtained in XSi2Y4, and it can be further entangled with multiple Hall effects. However, it is difficult to compare such effect between XSi2Y4 and TMDs due to the non-unified conventions for pseudo-angular momentum (PAM) in the previous studies. Here we use MoSi2As4 and MoS2 as examples to establish unified convention for both phonon PAM and electronic PAM, together with showing their similarities and differences in crystal structure, band structures and valley-selective optical circular dichroism. In particular, we find in MoSi2As4, the chiral phonon emission/absorption, the spin, the valley and the transverse Hall current, can be modulated by changing either the energy or the handedness of the incident light, which show better performance than TMDs., Comment: 8 pages, 4 figures

In this study, we theoretically show that in a helical crystal, a current is induced by chiral phonons representing the microscopic local rotation of atoms. By treating the rotational motion as a perturbation, we calculate the time-dependent current by using the adiabatic Berry phase method. The time average of the current along the helical axis becomes finite in the metallic phase but it vanishes in the insulating phase. On the other hand, the current in the hexagonal plane changes with time, but its time average vanishes due to the threefold rotation space-time symmetry. We show that the time evolutions of the current follow the space-time symmetries of the helical systems. Moreover, we explain the reason for the vanishing of the time average of the current in the insulating phase from the aspect of the Chern number in the parameter space., Comment: 12 pages, 8 figures

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 Miller index. This singular surface energy of the TCI affects equilibrium crystal shapes, resulting in emergence of unique crystal facets of the TCI. This singular dependence of the topological surface states is unique to the TCI protected by the glide symmetry in contrast to a TCI protected by a mirror symmetry. In addition, we show that such singular surface states of the TCI protected by the glide symmetries can be realized in KHgSb with first-principles calculations. Our results provide a basis for designs and manipulations of crystal facets by utilizing symmetry and topology., Comment: 6 pages, 3 figures, and Supplemental Material

In some Dirac systems with time-reversal (T) and glide (G) symmetries, multihelicoid surface states (MHSSs) appear, as discussed in various systems such as electronic and photonic ones. However, the topological nature and the conditions for the appearance of the MHSSs have not been understood. Here we show that MHSSs result from bulk-surface correspondence for the $Z_2$ monopole charge Q, which cannot be defined as a local quantity associated with the Dirac point, unlike the Z monopole charge characterizing Weyl points. The previously known formula of Q turns out to be non-gauge-invariant and thus cannot characterize the MHSSs. This shortcoming of the definition of Q is amended by redefining Q as a global topological invariant in k-space. Surprisingly, the newly defined Q, characterizing GT invariant gapless systems, is equal to the G-protected $Z_2$ topological invariant v, which is nontrivial only in T-breaking gapped systems. This global definition of Q automatically guarantees the appearance of MHSSs even when the Dirac point splits into Weyl points or a nodal ring by lowering the symmetry, as long as the GT symmetry is preserved. Q can be simplified to symmetry-based indicators when two vertical Gs are preserved, and filling-enforced topological crystalline insulators are diagnosed in several cases when a T-breaking perturbation is induced. Material candidate Li2B4O7 together with a list of space groups preserving MHSSs are also proposed., Comment: 10 pages, 5 figures, 1 table

In some non-Hermitian systems, the eigenstates in the bulk are localized at the boundaries of the systems. This is called the non-Hermitian skin effect, and it has been studied mostly in discrete systems. In the present work, we study the non-Hermitian skin effect in a continuous periodic model. In a one-dimensional system, we show that the localization lengths are equal for all the eigenstates. Moreover, the localization length and the eigenspectra in a large system are independent of the types of open boundary conditions. These properties are also found in a non-Hermitian photonic crystal. Such remarkable behaviors in a continuous periodic model can be explained in terms of the non-Bloch band theory. By constructing the generalized Brillouin zone for a complex Bloch wave number, we derive the localization length and the eigenspectra under an open boundary condition. Furthermore we show that the generalized Brillouin zone also has various physical properties, such as bulk-edge correspondence., Comment: 14 pages, 6 figures

We give a bulk-hinge correspondence for higher-order topological phases protected by rotoinversion $C_{4}\mathcal{I}$ symmetry in magnetic systems. Our approach allows us to show the emergence of the chiral hinge modes only from the information of the $C_{4}\mathcal{I}$ eigenvalues at the high-symmetry points in the Brillouin zone. In addition, based on the bulk-hinge correspondence, we propose a class of higher-order Weyl semimetals (HOWSMs) being Weyl semimetals with hinge modes and Fermi-arc surface states. The HOWSM is characterized by topological invariants for three-dimensional higher-order topological insulators, and the topological invariants are determined by the $C_{4}\mathcal{I}$ symmetry eigenvalues at the high-symmetry points. This HOWSM has chiral hinge modes as a direct consequence of the three-dimensional higher-order topology in the bulk., Comment: 16pages, 7 figures

Degenerate points/lines in the bulk band structures of crystals have become a staple of the growing number of topological materials. The bulk-boundary correspondence provides a relation between bulk topology and surface states. While line degeneracies of bulk excitations have been extensively characterized, line degeneracies of surface states are not well understood. We show that SnIP, a quasi-one-dimensional van der Waals material with a double helix crystal structure, exhibits topological nodal rings/lines in both the bulk phonon modes and their corresponding surface states. Using a combination of first-principles calculations, symmetry-based indicator theories and Zak phase analysis, we find that two neighbouring bulk nodal rings form doubly degenerate lines in their drumhead-like surface states, which are protected by the combination of time-reversal and glide mirror symmetries $\mathcal{T}\bar{M}_y$. Our results indicate that surface degeneracies can be generically protected by symmetries such as $\mathcal{T}\bar{M}_y$, and phonons provide an ideal platform to explore such degeneracies., Comment: 13 pages, 5 figures

We study magneto-optical (MO) properties of a semi-Dirac nanoribbon in presence of a perpendicular magnetic field using Kernel Polynomial Method (KPM) based on the Keldysh formalism in the experimental (terahertz frequency) regime. For comparison, we have also included results for the Dirac systems as well, so that the interplay of the band structure deformation and MO conductivities can be studied. We have found that the MO conductivity shows features in the semi-Dirac system that are quite distinct from the Dirac case near the ultra-violet and the visible regimes. The real parts of the longitudinal conductivities, namely Re($\sigma_{xx}$) and Re($\sigma_{yy}$) (which are different in the semi-Dirac case, as opposed to a Dirac one) present a series of resonance peaks as a function of the incident photon energy. We have also found that the absorption peaks corresponding to the $y$-direction are larger (roughly one order of magnitude) than those corresponding to the $x$-direction for the semi-Dirac case. In the case of the Hall conductivity, that is, Re($\sigma_{xy}$), there are extra peaks in the spectra compared to the Dirac case which originate from the distinct optical transitions of the carriers from one Landau level to another. We have also explored how the carrier concentration influences the MO conductivities. In the semi-Dirac case, there is the emergence of additional peaks yet again in the absorption spectrum underscoring the presence of an asymmetric dispersion compared to the Dirac case. Further, we have explored the interplay between the polarization of the incident beam and the features of the absorption spectra which can be probed in experiments. Finally, we evaluate the MO activity of the medium by computing the Faraday rotation angle, $\theta_{F}$., Comment: 15 pages, 23 figures

Chirality is a manifestation of the asymmetry inherent in nature. It has been defined as the symmetry breaking of the parity of static objects, and the definition was extended to dynamic motion such that true and false chiralities were distinguished. Recently, rotating, yet not propagating, atomic motions were predicted and observed in two-dimensional materials, and they were referred to as "chiral phonons" . A natural development would be the discovery of truly chiral phonons that propagate while rotating in three-dimensional materials. Here, we used circularly polarised Raman scattering and first-principles calculations to identify truly chiral phonons in chiral bulk crystals. This approach enabled us to determine the chirality of a crystal in a non-contact and non-destructive manner. In addition, we demonstrated that the law of the conservation of pseudo-angular momentum holds between circularly polarised photons and chiral phonons. These findings are expected to help develop ways for transferring the pseudo-angular momentum from photons to electron spins via the propagating chiral phonons in opto-phononic-spintronic devices.

Higher-order topological insulators host gapless states on hinges or corners of three-dimensional crystals. Recent studies suggested that even topologically trivial insulators may exhibit fractionally quantized charges localized at hinges or corners. Although most of the previous studies focused on two-dimensional systems, in this work, we take the initial step toward the systematic understanding of hinge and corner charges in three-dimensional insulators. We consider five crystal shapes of vertex-transitive polyhedra with the cubic symmetry such as a cube, an octahedron and a cuboctahedron. We derive real-space formulas for the hinge and corner charges in terms of the electric charges associated with bulk Wyckoff positions. We find that both the hinge and corner charges can be predicted from the bulk perspective only modulo certain fractions depending on the crystal shape, because the relaxation near boundaries of the crystal may affect the fractional parts. In particular, we show that a fractionally quantized charge $1/24$ mod $1/12$ in the unit of elementary charge can appear in a crystal with a shape of a truncated cube or a truncated octahedron. We also investigate momentum-space formulas for the hinge and corner charges. It turns out that the irreducible representations of filled bands at high-symmetry momenta are not sufficient to determine the corner charge. We introduce an additional Wilson-loop invariant to resolve this issue., Comment: 18 pages, 5 figures

For topological materials with coexistence of Weyl nodes and nodal rings, the surface-state configuration and connection are unique yet have never been studied and discussed before. In this paper, we predict a ferromagnetic (FM) material, Cs2MoCl6, with coexistence of Weyl and nodering fermions in its spinful FM electronic band structure, which is unusual since FM materials are very rare in nature and node-ring band crossings will usually open a gap when spin-orbit coupling (SOC) is taken into consideration. We find that the surface states of Cs2MoCl6 show different properties along different directions, i.e, the surface states are in the drumhead shape showing the node-ring property on the (001) surface and in the helicoid shape showing the Weyl property on the (010) surface. Interestingly, both the drumhead surface states and the helicoid surface states will cross the projected points of the Weyl and nodal ring along different directions. In particular, helicoid surface states on the (010) surface will meet the nodal ring tangentially, with their shapes change abruptly as a function of the energy. We implement both first-principle calculation and an analytical model to understand the unique surface-state connection for systems with the coexistence of Weyl nodes and nodal rings (or nodal lines). This result is universal and irrespective of the presence/absence of and time-reversal symmetry (T)., Comment: 6 pages, 4 figures

Gapped systems with glide symmetry can be characterized by a Z_2 topological invariant. We study the magnetic photonic crystal with a gap between the second and third lowest bands, which is characterized by the nontrivial glide-Z_2 topological invariant that can be determined by symmetry-based indicators. We show that under the space group No. 230 (Ia-3d), the topological invariant is equal to half of the number of photonic bands below the gap, and therefore, the band gap between the second and third lowest bands is always topologically nontrivial, and to realize the topological phase, we need to open a gap for the Dirac point at the P point by breaking time-reversal symmetry. With staggered magnetization, the photonic bands are gapped, and the photonic crystal becomes topological, whereas with uniform magnetization, a gap does not open, which can be attributed to the minimal band connectivity exceeding two in this case. By introducing the notion of Wyckoff positions, we show how the topological characteristics are determined from the structure of the photonic crystals., Comment: 16 pages, 8 figures. Supplemental Document is in the "ancillary files". To appear in Optics Express

Though symmetry-based indicators formulae are powerful in diagnosing topological states with a gapped band structure at/between any high-symmetry points, it fails in diagnosing topological degeneracies when the compatibility condition is violated. In such cases, we can only obtain information of whether there is a band degeneracy at some high-symmetry points or along some high-symmetry lines by the compatibility condition. Under the framework of symmetry-based indicator theories, we proposed an algorithm to diagnose the topological band crossings in the compatibility condition-violating systems to obtain the whole topological information, by using the symmetry-based indicator formulae of their subgroups. In this paper, we reinterpret the algorithm in a simpler way with two material examples preserving different topological states in spinless systems with time-reversal symmetry, discuss the limitation of the symmetry-based indicator theories, and make further discussions on the algorithm applying in spinful systems with time-reversal symmetry., Comment: 7 pages, 5 figures

Higher-order topological insulators in two dimensions have states that localize at their corners, called corner states. In this paper, we reveal characteristics of the penetration depth of their corner states by using the Benalcazar-Bernevig-Hughes model. First, we show that when we change the energy of the corner states toward the end of the edge gap by adding an on-site potential to the corner site, the penetration depth along the edge diverges toward infinity while the penetration depth into the bulk remaining finite. We analytically derive the corner-state wavefunction in a form of elliptic integrals, which reproduces this anisotropic behavior of corner states. This means that corner states have two kinds of penetration depths, and they behave differently. At last, we show that hybridizations between corner states are governed by the penetration depth through interference between the corner states. It is because the corner states almost do not interfere with edge states or bulk states., Comment: 11 pages, 6 figures

Chiral phonons are the ones with nonzero polarization and can be observed only via a selective coupling with valley electrons and circularly polarized photons. In such process, a new physical quantity, i.e., pseudo-angular momentum (PAM), is required to meet the selection rule. However, phonon PAM was thought to be quantized and can be only defined in the symmorphic systems. In this work, we generalized the definition of PAM to three-dimensional non-symmorphic systems, which show distinct different properties compared with the one in symmorphic systems, e.g., PAM can be non-quantized and q-dependent but still be an observable quantity by experiments. Such new definitions and discoveries can help us to obtain chiral phonons in a broader class of systems with a nonzero group velocity and to convey information like chirality and angular momentum in solids as expected. Materials are also offered to understand the new definition and for further experimental detection., Comment: 7 pages, 4 figures and 1 table

Non-Hermitian systems show a non-Hermitian skin effect, where the bulk states are localized at a boundary of the systems with open boundary conditions. In this paper, we study dependence of the localization length of the eigenstates on a system size in a specific non-Hermitian model with a critical non-Hermitian skin effect, where the energy spectrum undergoes discontinuous transition in the thermodynamic limit. We analytically show that the eigenstates exhibit remarkable localization, known as scale-free localization, where the localization length is proportional to a system size. Our result gives a theoretical support for the scale-free localization, which has been proposed only numerically in previous works., Comment: 8 pages, 4 figures

It is known that phonons have angular momentum, and when the time-reversal symmetry (TRS) is broken, the total phonon angular momentum in the whole system becomes nonzero. In this paper, we propose that as an angular momentum of phonons for a crystal without TRS, we need to consider the canonical angular momentum, as opposed to the kinetic angular momentum in previous works. Next, we show that the angular momentum of phonons without TRS exhibits universal behaviors near the $\Gamma$ point. We focus on in-plane oscillations in two-dimensional crystals as an example. By breaking the TRS, one of the acoustic phonon branches at the $\Gamma$ point acquires a gap. We show that the angular momentum of its acoustic phonon with a gap has a peak with the height $\pm \hbar$ regardless of the details of the system. From this, we find that this peak height changes discontinuously by changing the sign of the TRS-breaking parameter., Comment: 11pages, 3figures

Based on the similarity between the chiral nanotube and the classical solenoid, we study chiral transport along the circumferential direction in a carbon nanotube. We calculate the chiral conductivity, representing a circumferential current induced by an electric field along the nanotube axis for various chiralities of carbon nanotubes. We find that the chiral conductivity in a chiral nanotube is in general non zero, and oscillates as a function of the Fermi energy. This oscillating behavior is attributed to the subband structure of the nanotubes and the warping of the Fermi surfaces., Comment: 10 pages, 5 figures

The kinetic magnetoelectric effect is an orbital analogue of the Edelstein effect and offers an additional degree of freedom to control magnetisation via the charge current. Here we theoretically propose a gigantic kinetic magnetoelectric effect in topological insulators and interpret the results in terms of topological surface currents. We construct a theory of the kinetic magnetoelectric effect for a surface Hamiltonian of a topological insulator, and show that it well describes the results by direct numerical calculation. This kinetic magnetoelectric effect depends on the details of the surface, meaning that it cannot be defined as a bulk quantity. We propose that Chern insulators and Z_2 topological insulators can be a platform with a large kinetic magnetoelectric effect, compared to metals by 5 - 8 orders of magnitude, because the current flows only along the surface. We demonstrate the presence of said effect in a topological insulator, identifying Cu$_2$ZnSnSe$_4$ as a potential candidate., Comment: 24 pages, 8 figures

In this paper, we derive a general formula for the quantized fractional corner charge in two-dimensional C_n-symmetric higher-order topological insulators. We assume that the electronic states can be described by the Wannier functions and that the edges are charge neutral, but we do not assume vanishing bulk electric polarization. We expand the scope of the corner charge formula obtained in previous works by considering more general surface conditions, such as surfaces with higher Miller index and surfaces with surface reconstruction. Our theory is applicable even when the electronic states are largely modulated near system boundaries. It also applies to insulators with non-vanishing bulk polarization, and we find that in such cases the value of the corner charge depends on the surface termination even for the same bulk crystal with C_3 or C_4 symmetry, via a difference in the Wyckoff position of the center of the C_n-symmetric crystal., Comment: 17 pages, 10 figures, to appear in Phys. Rev. B

In recent research, it has been shown that non-Hermitian systems exhibit sensitivity to boundaries, and it is caused by the non-Hermitian skin effect. In this work, we construct the non-Bloch band theory in bosonic Bogoliubov--de Gennes (BdG) systems. From our theory, we can calculate the generalized Brillouin zone and the energy spectrum in such systems with open boundary conditions in the thermodynamic limit, and we can thus discuss its non-Hermitian nature, despite Hermiticity of an original Hamiltonian. In fact, we find that the bosonic Kitaev-Majorana chain exhibits rich aspects of the non-Hermitian skin effect, such as instability against infinitesimal perturbations and reentrant behavior, in terms of the non-Bloch band theory. This result indicates that our theory is powerful tool for studying non-Hermitian nature in bosonic BdG systems., Comment: 9 pages, 3 figures

Perovskite oxides exhibit a rich variety of structural phases hosting different physical phenomena that generate multiple technological applications. We find that topological phonons, i.e., nodal rings, nodal lines, and Weyl points, are ubiquitous in oxide perovskites in terms of structures (tetragonal, orthorhombic, and rhombohedral), compounds (BaTiO3, PbTiO3, and SrTiO3), and external conditions (photoexcitation, strain, and temperature). In particular, in the tetragonal phase of these compounds, all types of topological phonons can simultaneously emerge when stabilized by photoexcitation, whereas the tetragonal phase stabilized by thermal fluctuations only hosts a more limited set of topological phonon states. Additionally, we find that the photoexcited carrier concentration can be used to tune the topological phonon states and induce topological transitions even without associated structural phase changes. Overall, we propose oxide perovskites as a versatile platform in which to study topological phonons and their manipulation with light., Comment: 26 pages, 5 figures

In this paper, we review our non-Bloch band theory in one-dimensional non-Hermitian tight-binding systems. In our theory, it is shown that in non-Hermitian systems, the Brillouin zone is determined so as to reproduce continuum energy bands in a large open chain. By using simple models, we explain the concept of the non-Bloch band theory and the method to calculate the Brillouin zone. In particular, for the non-Hermitian Su-Schrieffer-Heeger model, the bulk-edge correspondence can be established between the topological invariant defined from our theory and existence of the topological edge states., Comment: 12 pages, 6 figures

In various topological phases, nontrivial states appear at the boundaries of the system. In this paper, we investigate anomalous dielectric response caused by such states caused by the pi Zak phase. First, by using the one-dimensional Su-Schrieffer-Heeger model, we show that, when the system is insulating and the Zak phase is pi, the polarization suddenly rises to a large value close to e/2, by application of an external electric field. The pi Zak phase indicates existence of half-filled edge states, and we attribute this phenomenon to charge transfer between the edge states at the two ends of the system. We extend this idea to two- and three-dimensional insulators with the pi Zak phase over the Brillouin zone, and find similar anomalous dielectric response. We also show that diamond and silicon slabs with (111) surfaces have the pi Zak phase by ab intio calculations, and show that this anomalous response survives even surface reconstruction involving an odd number of original surface unit cells. Another material example with an anomalous dielectric response is polytetrafluoroethylene (PTFE), showing plateaus of polarization at +e by ab initio calculation, in agreement with our theory., Comment: 19 pages, 15 figures

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 Berry curvatures and Berry phases in the k-space. We also derive a formula of the glide-Z2 invariantwhen the inversion symmetry is added, and the space group becomes No. 15. This reduces the formula into the Fu-Kane-like formula, expressed in terms of the irreducible representations at high-symmetry points in $k$ space. We also construct these topological invariants by the layer-construction approach, and the results completely agree with those from the k-space approach., Comment: 21 pages, 4 figures

In crystals with time-reversal symmetry but without inversion symmetry, the phonon angular momentum can be generated by the temperature gradient, and it is called phonon thermal Edelstein effect. On the other hand, when both symmetries are broken and their product is conserved, the phonon angular momentum for any phonon modes at any wave vectors vanishes, and the phonon thermal Edelstein effect does not occur. In this paper, we propose another mechanism of generation of the phonon angular momentum. We show that in such crystals the electric field generates the phonon angular momentum, via the lattice distortion due to the electric field. This effect is in the same symmetry class with the magnetoelectric effect, and we call this effect phonon rotoelectric effect. We discuss the temperature dependence of the phonon angular momentum generated by the temperature gradient and by the electric field in the high- and the low-temperature limits., Comment: 8 pages, 3 figures

We theoretically investigate the microscopic mechanism of conversion between the electron spin and the microscopic local rotation of atoms in crystals. In phonon modes with angular momenta, the atoms microscopically rotate around their equilibrium positions in crystals. In a simple toy model with phonons, we calculate the spin expectation value by using the adiabatic series expansion. We show that the time-averaged spin magnetization is generated by the microscopic local rotation of atoms via the spin-orbit interaction. On the other hand, in the system with a simple vibration of atoms, time-averaged spin magnetization becomes zero due to the time-reversal symmetry. Moreover, the magnitude of the time-averaged spin magnetization depends on the inverse of the difference of instantaneous eigenenergy, and we show that it becomes smaller in band insulators with a larger gap., Comment: 11 pages, 7 figures, to be published in Phys. Rev. Research

Here we comprehensively investigate Landau levels, Hofstadter butterfly and transport properties of a semi-Dirac nanoribbon in a perpendicular magnetic field using a recently developed real-space implementation of the Kubo formula based on Kernel Polynomial Method. A Dirac ribbon is considered to compare and contrast our results for a semi-Dirac system. We find that the Landau levels being non-equidistant from each other for the semi-Dirac case (true for a Dirac as well), the flatness of the energy bands vanishes in the bulk and becomes dispersive for a semi-Dirac ribbon in contrast to a Dirac system. This feature is most discernible for intermediate values of the external field. We further compute the longitudinal ($\sigma_{xx}$ and $\sigma_{yy}$) and the transverse or Hall ($\sigma_{xy}$) conductivities where the Hall conductivity shows a familiar quantization, namely, $\sigma_{xy} \propto 2n$ (the factor `2' includes the spin degeneracy) which is highly distinct from a Dirac system, such as graphene. We also observe anisotropic behavior in magneto-transport in a semi-Dirac ribbon owing to the dispersion anomalies in two different longitudinal directions. Our studies may have important ramifications for monolayer phosphorene.

We study positions of chiral hinge states in higher-order topological insulators (HOTIs) with inversion symmetry. First, we exhaust all possible configurations of the hinge states in the HOTIs in all type-I magnetic space groups with inversion symmetry by studying dependence of the sign of the surface Dirac mass on surface orientations. In particular, in the presence of glide symmetry, for particular surface orientations, the surface Dirac mass changes sign by changing the surface terminations. By applying this result to a layered antiferromagnet (AFM), we find a difference in the hinge states between the cases with an even and odd number of layers. In the case of an even number of layers, which does not preserve inversion symmetry, positions of hinge states are not inversion symmetric. Nonetheless, these inversion-asymmetric hinge states result from the bulk topology. We show that their inversion-asymmetric configurations are uniquely determined from the symmetries and the topological invariant., Comment: 15 pages, 9 figures

In this paper, we introduce Berry curvature, topological Chern number and topological chiral edge mode, that emerge from a hybridization between magnon and electromagnetic wave in a ferromagnet insulator. By focusing on the energy conservation, we first reformulate the Landau-Lifshitz-Maxwell equation into a Hermitian eigenvalue equation. From the eigenvalue equation, we define the Berry curvature of the magnon-photon coupled waves. We show that the Berry curvature thus introduced shows a prominent peak around a hybridization point between magnon mode and photon mode, and a massive hybrid mode takes a non-zero Chern number ($\pm 1$) due to the magnon-photon coupling. In accordance with the non-zero Chern number, the topological edge modes emerge inside the hybridization gap at a domain wall between two ferromagnetic insulators with opposite magnetizations., Comment: 10 pages, 4 figures

In higher-order topological insulators, bulk and surface electronic states are gapped, while there appear gapless hinge states protected by spatial symmetry. Here we show by ab initio calculations that the La apatite electride is a higher-order topological crystalline insulator. It is a one-dimensional electride, in which the one-dimensional interstitial hollows along the $c$ axis support anionic electrons, and the electronic states in these one-dimensional channels are well approximated by the one-dimensional Su-Schrieffer-Heeger model. When the crystal is cleaved into a hexagonal prism, the 120$^\circ$ hinges support gapless hinge states, with their filling quantized to be 2/3. This quantization of the filling comes from a topological origin. We find that the quantized value of the filling depends on the fundamental blocks that constitute the crystal. The apatite consists of the triangular blocks, which is crucial for giving nontrivial fractional charge at the hinge., Comment: 13 pages, 11 figures

Conventional Weyl nodes are twofold band crossings that carry a unit monopole charge, which can exist in condensed matter systems with the protection of translation symmetry. Unconventional Weyl nodes are twofold/multifold band crossings carrying a quantized monopole charge larger than one, and their existence needs the protection of additional crystalline symmetries. Studies on unconventional Weyl nodes are already very comprehensive, such as twofold Weyl nodes with Chern number of C=2/C=3 and fourfold/sixfoldWeyl nodes with C=4. Yet in this paper, we propose a newfound twofold unconventional Weyl node with C=4, which can exist in any chiral cubic systems with integer spin. Such kind of twofold quadruple Weyl node has a cubic band dispersion along [111] direction and will evolve into a fourfold quadruple Weyl node after considering spin-orbit coupling. In this paper, we exhaust all the possible chiral cubic space groups and corresponding k-points, which can have twofold quadruple Weyl nodes. We also propose a series of LaIrSi-type materials that both have twofold quadruple Weyl nodes in electronic systems and the phonon spectra., Comment: 11 pages, 9 figures

Energy bands of non-Hermitian crystalline systems are described in terms of the generalized Brillouin zone (GBZ) having unique features which are absent in Hermitian systems. In this paper, we show that in one-dimensional non-Hermitian systems with both sublattice symmetry and time-reversal symmetry such as the non-Hermitian Su-Schrieffer-Heeger model, a topological semimetal phase with exceptional points is stabilized by the unique features of the GBZ. Namely, under a change of a system parameter, the GBZ is deformed so that the system remains gapless. It is also shown that each energy band is divided into three regions, depending on the symmetry of the eigenstates, and the regions are separated by the cusps and the exceptional points in the GBZ., Comment: 11 pages, 6 figures

Theories of symmetry-based indicators and topological quantum chemistry, while powerful in diagnosing gapped topological materials, cannot be directly applied to diagnosing band degeneracies at high-symmetry momenta due to the violation of the compatibility conditions. Here we design a recursive protocol that utilizes indicators of maximal subgroups to infer the topological nature of band degeneracies at high-symmetry lines. For demonstration, the method is used to predict the existence of iso-energy Weyl points and a node-line cage, respectively, in the phonon bands of In2Te and ZrSiO., Comment: 6 pages, 5 figures