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The geometric characteristics of Bloch wave functions play a crucial role in electronic transport properties. We show that the thermoelectric performance of materials is governed by the geometric structure of Bloch wave functions within the framework of the Boltzmann equation. The essential geometric notion is the Hilbert-Schmidt quantum distance, measuring the resemblance between two quantum states. We establish a geometric characterization of the scattering rate by extending the concept of quantum distance between two states in momentum space at a distance.Employing isotropic quadratic band touching semimetals, where one can concentrate on the role of quantum geometric effects other than the Berry curvature, we find that the response functions for electrical quantum transport and, therefore, the thermoelectric power factor can be succinctly expressed in terms of the maximum quantum distance, $d_\mathrm{max}$. Specifically, when $d_\mathrm{max}$ reaches one, the power factor doubles compared to the case with trivial geometry ($d_\mathrm{max}=0$). Our finding highlights the significance of quantum geometry in improving the performance of thermoelectric devices.

We study the influence of the quantum geometry on the magnetic responses of quadratic band crossing semimetals. More explicitly, we examine the Landau levels, quantum Hall effect, and magnetic susceptibility of a general two-band Hamiltonian that has fixed isotropic quadratic band dispersion but with tunable quantum geometry, in which the interband coupling is fully characterized by the maximum quantum distance $d_\mathrm{max}$. By continuously tuning $d_\mathrm{max}$ in the range of $0\leq d_\mathrm{max}\leq 1$, we investigate how the magnetic properties of the free electron model with $d_\mathrm{max}=0$ evolve into those of the bilayer graphene with $d_\mathrm{max}=1$. We demonstrate that despite sharing the same energy dispersion $\epsilon(p) =\pm\frac{p^2}{2m}$, the charge carriers in the free electron model and bilayer graphene exhibit entirely distinct Landau levels and quantum Hall responses due to the nontrivial quantum geometry of the wave functions.

We propose a new band engineering scheme on the biphenylene network, a newly synthesized carbon allotrope. First, we investigate the mechanism for the appearance of type II Dirac fermion in a pristine biphenylene network. We show that the essential ingredients are mirror symmetries and the stabilization of the compact localized eigenstates via destructive interference. While the former is used for the band-crossing point along high symmetry lines, the latter makes the obtained Dirac dispersion highly inclined. Then, we demonstrate that many other different kinds of Dirac fermions, such as type-I Dirac, gapped type-II Dirac, and nodal line semimetals, can be developed by fluorinating the biphenylene network periodically in various ways. In this program, the key role of the fluorine atoms is manipulating the condition of the destructive interference and mirror symmetries.

In condensed matter physics, the Kagome lattice and its inherent flat bands have attracted considerable attention for their potential to host a variety of exotic physical phenomena. Despite extensive efforts to fabricate thin films of Kagome materials aimed at modulating the flat bands through electrostatic gating or strain manipulation, progress has been limited. Here, we report the observation of a novel $d$-orbital hybridized Kagome-derived flat band in Ag/Si(111) $\sqrt{3}\times\sqrt{3}$ as revealed by angle-resolved photoemission spectroscopy. Our findings indicate that silver atoms on a silicon substrate form a Kagome-like structure, where a delicate balance in the hopping parameters of the in-plane $d$-orbitals leads to destructive interference, resulting in a flat band. These results not only introduce a new platform for Kagome physics but also illuminate the potential for integrating metal-semiconductor interfaces into Kagome-related research, thereby opening a new avenue for exploring ideal two-dimensional Kagome systems., Comment: 7 pages, 4 figures

The localized nature of a flat band is understood by the existence of a compact localized eigenstate. However, the localization properties of a partially flat band, ubiquitous in surface modes of topological semimetals, have been unknown. We show that the partially flat band is characterized by a non-normalizable compact localized state(NCLS). The partially flat band develops only in a momentum range, where normalizable Bloch wave functions can be obtained by the linear combination of the NCLSs. Outside this momentum region, a ghost flat band, unseen from the band structure, is introduced for the consistent counting argument with the full set of NCLSs. Then, we demonstrate that the Wannier function corresponding to the partially flat band exhibits an algebraic($\sim 1/r^{1+\epsilon}$ in 1D and $\sim 1/r^{3/2+\epsilon}$ in 2D) decay behavior, where $\epsilon$ is a positive number. Namely, one can have the Wannier obstruction even in a topologically trivial band if it is partially flat. Finally, we develop a construction scheme of a tight-binding model of the topological semimetal by designing an NCLS.

A singular flat band(SFB), a distinct class of the flat band, has been shown to exhibit various intriguing material properties characterized by a geometric quantity of the Bloch wave function called the quantum distance. We present a general construction scheme for a tight-binding model hosting an SFB, where the quantum distance profile can be controlled. We first introduce how to build a compact localized state(CLS), a characteristic eigenstate of the flat band, providing the flat band with a band-touching point, where a specific value of the maximum quantum distance is assigned. Then, we develop a scheme designing a tight-binding Hamiltonian hosting an SFB starting from the obtained CLS, satisfying the desired hopping range and symmetries by applying the construction scheme. While the scheme can be applied to any dimensions and lattice structures, we propose several simple SFB models on the square and kagome lattices. Finally, we establish a bulk-boundary correspondence between the maximum quantum distance and the boundary modes for the open boundary condition, which can be used to detect the quantum distance via the electronic structure of the boundary states., Comment: 10 pages, 6 figures

It is known that materials with broken time-reversal symmetry can have Hall response. Here we show that in addition to the conventional currents, a new type of nonlinear Hall current can occur in the time-reversal breaking materials within the second-order response to in-plane electric and vertical magnetic fields. Such a Hall response is generated by the oscillation of the electromagnetic field and has a quantum origin arising from a novel dipole associated with the Berry curvature and band velocity. We demonstrate that the massive Dirac model of LaAlO3/LaNiO3/LaAlO3 quantum well can be used to detect this Hall effect. Our work widens the theory of the Hall effect in the time-reversal breaking materials by proposing a new kind of nonlinear electromagnetic response., Comment: 6 pages, 4 figures

We investigate the origin of the ubiquitous existence of flat bands in the network superstructures of atomic chains, where one-dimensional(1D) atomic chains array periodically. While there can be many ways to connect those chains, we consider two representative ways of linking them, the dot-type and triangle-type links. Then, we construct a variety of superstructures, such as the square, rectangular, and honeycomb network superstructures with dot-type links and the honeycomb superstructure with triangle-type links. These links provide the wavefunctions with an opportunity to have destructive interference, which stabilizes the compact localized state(CLS). The CLS is a localized eigenstate whose amplitudes are finite only inside a finite region and guarantees the existence of a flat band. In the network superstructures, there exist multiple flat bands proportional to the number of atoms of each chain, and the corresponding eigenenergies can be found from the stability condition of the compact localized state. Finally, we demonstrate that the finite bandwidth of the nearly flat bands of the network superstructures arising from the next-nearest-neighbor hopping processes can be suppressed by increasing the length of the chains consisting of the superstructures., Comment: 8pages, 4figures

In recent years, the spin Hall effect has received great attention because of its potential application in spintronics and quantum information processing and storage. However, this effect is usually studied under the external homogeneous electric field. Understanding how the inhomogeneous electric field affects the spin Hall effect is still lacking. Here, we investigate a two-dimensional two-band time-reversal symmetric system and give an expression for the intrinsic spin Hall conductivity in the presence of the inhomogeneous electric field, which is shown to be expressed through gauge-invariant geometric quantities. On the other hand, when people get physical intuition on transport phenomena from the wave packet, one issue appears. It is shown that the conductivity obtained from the conventional wave packet approach cannot be fully consistent with the one predicted by the Kubo-Greenwood formula. Here, we attempt to solve this problem.

Noncontractible loop states (NLSs) are recently realized topological entity in flatband lattices, arising typically from band touching at a point where a flat band intersects one or more dispersive bands. There exists also band touching across a plane, where one flat band overlaps another all over the Brillouin zone without crossing a dispersive band. Such isolated plane-touching flat bands remain largely unexplored. For example, what are the topological features associated with such flatband degeneracy? Here, we demonstrate for the first time to our knowledge nontrivial NLSs and robust boundary modes in a system with such degeneracy. Based on a tailored photonic lattice constructed from the well-known fractal Sierpinski gasket, we theoretically analyze the wavefunction singularities and the conditions for the existence of the NLSs. We show that the NLSs can exist in both singular and nonsingular flat bands, as a direct reflection of the real-space topology. Experimentally, we observe directly such flatband NLSs in a laser-written Corbino-shaped fractal-like lattice. This work not only leads to a deep understanding of the mechanism behind the nontrivial flatband states, but also opens up new avenues to explore fundamental phenomena arising from the interplay of flatband degeneracy, fractal structures and band topology., Comment: 20 pages, 5 figures

The bulk-boundary correspondence is an integral feature of topological analysis and the existence of boundary or interface modes offers direct insight into the topological structure of the Bloch wave function. While only the topology of the wave function has been considered relevant to boundary modes, we demonstrate that another geometric quantity, the so-called quantum distance, can also host a bulk-interface correspondence. We consider a generic class of two-dimensional flat band systems, where the flat band has a parabolic band-crossing with another dispersive band. While such flat bands are known to be topologically trivial, we show that the nonzero maximum quantum distance between the eigenstates of the flat band around the touching point guarantees the existence of boundary modes at the interfaces between two domains with different chemical potentials or different maximum quantum distance. Moreover, the maximum quantum distance can predict even the explicit form of the dispersion relation and decay length of the interface modes., Comment: 11pages, 5 figures

We realize fractal-like photonic lattices using cw-laser-writing technique, thereby observe distinct compact localized states (CLSs) associated with different flatbands in the same lattice setting. Such triangle-shaped lattices, akin to the first generation Sierpinski lattices, possess a band structure where singular non-degenerate and nonsingular degenerate flatbands coexist. By proper phase modulation of an input excitation beam, we demonstrate experimentally not only the simplest CLSs but also their superimposition into other complex mode structures. Furthermore, we show by numerical simulation a dynamical oscillation of the flatband states due to beating of the CLSs that have different eigenenergies. These results may provide inspiration for exploring fundamental phenomena arising from fractal structure, flatband singularity, and real-space topology.

In this work, we develop a systematic method of constructing flat-band models with and without band crossings. Our construction scheme utilizes the symmetry and spatial shape of a compact localized state (CLS) and also the singularity of the flat-band wave function obtained by a Fourier transform of the CLS (FT-CLS). In order to construct a flat-band model systematically using these ingredients, we first choose a CLS with a specific symmetry representation in a given lattice. Then, the singularity of FT-CLS indicates whether the resulting flat band exhibits a band crossing point or not. A tight-binding Hamiltonian with the flat band corresponding to the FT-CLS is obtained by introducing a set of basis molecular orbitals, which are orthogonal to the FT-CLS. Our construction scheme can be systematically applied to any lattice so that it provides a powerful theoretical framework to study exotic properties of both gapped and gapless flat bands arising from their wave function singularities., Comment: 18 pages, 9 figures, published version

Flat bands have band crossing points with other dispersive bands in many systems including the canonical flat band models in the Lieb and kagome lattices. Here we show that some of such band degeneracy points are unavoidable because of the symmetry representation (SR) of the flat band under unitary symmetry. We refer to such a band degeneracy point of flat bands as a SR-enforced band crossing. SR-enforced band crossing is distinct from the conventional band degeneracy protected by symmetry eigenvalues or topological charges in that its protection requires both specific symmetry representation and band flatness of the flat band, simultaneously. Even $n$-fold rotation $C_n$ ($n=2,3,4,6$) symmetry, which cannot protect band degeneracy without additional symmetries due to its abelian nature, can protect SR-enforced band crossings in flat band systems. In two-dimensional flat band systems with $C_n$ symmetry, when the degeneracy of a SR-enforced band crossing is lifted by a $C_n$ symmetry-preserving perturbation, we obtain a nearly flat Chern band. Our theory not only explains the origin of the band crossing points of FBs existing in various models, but also gives a strict no-go theorem for isolated FBs in a given lattice arising from the SR., Comment: 8+14 pages, 4+6 figures, published version

According to the Onsager's semiclassical quantization rule, the Landau levels of a band are bounded by its upper and lower band edges at zero magnetic field. However, there are two notable systems where the Landau level spectra violate this expectation, including topological bands and flat bands with singular band crossings, whose wave functions possess some singularities. Here, we introduce a distinct class of flat band systems where anomalous Landau level spreading (LLS) appears outside the zero-field energy bounds, although the relevant wave function is nonsingular. The anomalous LLS of isolated flat bands are governed by the cross-gap Berry connection that measures the wave-function geometry of multi bands. We also find that symmetry puts strong constraints on the LLS of flat bands. Our work demonstrates that an isolated flat band is an ideal system for studying the fundamental role of wave-function geometry in describing magnetic responses of solids., Comment: 10+8 pages, 5+3 figures

Geometry of the wave function is a central pillar of modern solid state physics. In this work, we unveil the wave-function geometry of two-dimensional semimetals with band crossing points (BCPs). We show that the Berry phase of BCPs are governed by the quantum metric describing the infinitesimal distance between quantum states. For generic linear BCPs, we show that the corresponding Berry phase is determined either by an angular integral of the quantum metric, or equivalently, by the maximum quantum distance of Bloch states. This naturally explains the origin of the $\pi$-Berry phase of a linear BCP. In the case of quadratic BCPs, the Berry phase can take an arbitrary value between 0 and $2\pi$. We find simple relations between the Berry phase, maximum quantum distance, and the quantum metric in two cases: (i) when one of the two crossing bands is flat; (ii) when the system has rotation and/or time-reversal symmetries. To demonstrate the implication of the continuum model analysis in lattice systems, we study tight-binding Hamiltonians describing quadratic BCPs. We show that, when the Berry curvature is absent, a quadratic BCP with an arbitrary Berry phase always accompanies another quadratic BCP so that the total Berry phase of the periodic system becomes zero. This work demonstrates that the quantum metric plays a critical role in understanding the geometric properties of topological semimetals., Comment: 7+9 pages, 4+1 figures, published version

We review recent progresses in the study of flat band systems, especially focusing on the fundamental physics related to the singularity of the flat band's Bloch wave functions. We first explain that the flat bands can be classified into two classes: singular and nonsingular flat bands, based on the presence or absence of the singularity in the flat band's Bloch wave functions. The singularity is generated by the band crossing of the flat band with another dispersive band. In the singular flat band, one can find special kind of eigenmodes, called the non-contractible loop states and the robust boundary modes, which exhibit nontrivial real space topology. Then, we review the experimental realization of these topological eigenmodes of the flat band in the photonic lattices. While the singularity of the flat band is topologically trivial, we show that the maximum quantum distance around the singularity is a bulk invariant representing the strength of the singularity which protects the robust boundary modes. Finally, we discuss how the maximum quantum distance or the strength of the singularity manifests itself in the anomalous Landau level spreading of the singular flat band when it has a quadratic band-crossing with another band., Comment: 43 pages, 7 figures

Corbino-geometry has well-known applications in physics, as in the design of graphene heterostructures for detecting fractional quantum Hall states or superconducting waveguides for illustrating circuit quantum electrodynamics. Here, we propose and demonstrate a photonic Kagome lattice in the Corbino-geometry that leads to direct observation of non-contractible loop states protected by real-space topology. Such states represent the "missing" flat-band eigenmodes, manifested as one-dimensional loops winding around a torus, or lines infinitely extending to the entire flat-band lattice. In finite (truncated) Kagome lattices, however, line states cannot preserve as they are no longer the eigenmodes, in sharp contrast to the case of Lieb lattices. Using a continuous-wave laser writing technique, we experimentally establish finite Kagome lattices with desired cutting edges, as well as in the Corbino-geometry to eliminate edge effects. We thereby observe, for the first time to our knowledge, the robust boundary modes exhibiting self-healing properties, and the localized modes along toroidal direction as a direct manifestation of the non-contractible loop states.

We show that flat bands can be categorized into two distinct classes, that is, singular and nonsingular flat bands, by exploiting the singular behavior of their Bloch wave functions in momentum space. In the case of a singular flat band, its Bloch wave function possesses immovable discontinuities generated by the band-crossing with other bands, and thus the vector bundle associated with the flat band cannot be defined. This singularity precludes the compact localized states from forming a complete set spanning the flat band. Once the degeneracy at the band crossing point is lifted, the singular flat band becomes dispersive and can acquire a finite Chern number in general, suggesting a new route for obtaining a nearly flat Chern band. On the other hand, the Bloch wave function of a nonsingular flat band has no singularity, and thus forms a vector bundle. A nonsingular flat band can be completely isolated from other bands while preserving the perfect flatness. All one-dimensional flat bands belong to the nonsingular class. We show that a singular flat band displays a novel bulk-boundary correspondence such that the presence of the robust boundary mode is guaranteed by the singularity of the Bloch wave function. Moreover, we develop a general scheme to construct a flat band model Hamiltonian in which one can freely design its singular or nonsingular nature. Finally, we propose a general formula for the compact localized state spanning the flat band, which can be easily implemented in numerics and offer a basis set useful in analyzing correlation effects in flat bands., Comment: 23 pages, 13 figures

Topological insulator nanowires with uniform cross section, combined with a magnetic flux, can host both a perfectly transmitted mode and Majorana zero modes. Here we consider nanowires with rippled surfaces---specifically, wires with a circular cross section with a radius varying along its axis---and calculate their transport properties. At zero doping, chiral symmetry places the clean wires (no impurities) in the AIII symmetry class, which results in a $\mathbb{Z}$ topological classification. A magnetic flux threading the wire tunes between the topologically distinct insulating phases, with perfect transmission obtained at the phase transition. We derive an analytical expression for the exact flux value at the transition. Both doping and disorder breaks the chiral symmetry and the perfect transmission. At finite doping, the interplay of surface ripples and disorder with the magnetic flux modifies quantum interference such that the amplitude of Aharonov-Bohm oscillations reduces with increasing flux, in contrast to wires with uniform surfaces where it is flux-independent., Comment: 12 pages, 6 figures. v2. 2 new figures and a new appendix

The bulk-boundary correspondence, a topic of intensive research interest over the past decades, is one of the quintessential ideas in the physics of topological quantum matter. Nevertheless, it has not been proven in all generality and has in certain scenarios even been shown to fail, depending on the boundary profiles of the terminated system. Here, we introduce bulk numbers that capture the exact number of in-gap modes, without any such subtleties in one spatial dimension. Similarly, based on these 1D bulk numbers, we define a new 2D winding number, which we call the pole winding number, that specifies the number of robust metallic surface bands in the gap as well as their topological character. The underlying general methodology relies on a simple continuous extrapolation from the bulk to the boundary, while tracking the evolution of Green's function's poles in the vicinity of the bulk band edges. As a main result we find that all the obtained numbers can be applied to the known insulating phases in a unified manner regardless of the specific symmetries. Additionally, from a computational point of view, these numbers can be effectively evaluated without any gauge fixing problems. In particular, we directly apply our bulk-boundary correspondence construction to various systems, including 1D examples without a traditional bulk-boundary correspondence, and predict the existence of boundary modes on various experimentally studied graphene edges, such as open boundaries and grain boundaries. Finally, we sketch the 3D generalization of the pole winding number by in the context of topological insulators., Comment: 19 pages, 11 figures

We have performed density functional theory calculation and tight binging analysis in order to investigate the mechanism for the giant Rashba-type spin splitting (RSS) observed in Bi/Ag(111). We find that local orbital angular momentum induces momentum and spin dependent charge distribution which results in spin-dependent hopping. We show that the spin-dependent interatomic-hopping in Bi/Ag(111) works as a strong effective field and induces the giant RSS, indicating that the giant RSS is driven by hopping, not by a uniform electric field. The effective field from the hopping energy difference amounts to be ~18 V/{\AA}. This new perspective on the RSS gives us a hint for the giant RSS mechanism in general and should provide a strategy for designing new RSS materials by controlling spin-dependence of hopping energy between the neighboring atomic layers.

The existence and topological classification of lower-dimensional Fermi surfaces is often tied to the crystal symmetries of the underlying lattice systems. Artificially engineered lattices, such as heterostructures and other superlattices, provide promising avenues to realize desired crystal symmetries that protect lower-dimensional Fermi surface, such as nodal lines. In this work, we investigate a Weyl semimetal subjected to spatially periodic onsite potential, giving rise to several phases, including a nodal-line semimetal phase. In contrast to proposals that purely focus on lattice symmetries, the emergence of the nodal line in this setup does not require small spin-orbit coupling, but rather relies on its presence. We show that the stability of the nodal line is understood from reflection symmetry and a combination of a fractional lattice translation and charge-conjugation symmetry. Depending on the choice of parameters, this model exhibits drumhead surface states that are exponentially localized at the surface, or weakly localized surface states that decay into the bulk at all energies., Comment: 11 pages, 8 figures, Editors' Suggestion

The Zak phase $\gamma$, the generalization of the Berry phase to Bloch wave functions in solids, is often used to characterize inversion-symmetric 1D topological insulators; however, since its value can depend on the choice of real-space origin and unit cell, only the difference between the Zak phase of two regions is believed to be relevant. Here, we show that one can extract an origin-independent part of $\gamma$, the so-called inter-cellular Zak phase $\gamma^{\mathrm{inter}}$, which can be used as a bulk quantity to predict the number of surface modes as follows: a neutral finite 1D tight-binding system has $n_s = \gamma^{\mathrm{inter}}/\pi$ (mod 2) number of in-gap surface modes below the Fermi level if there exists a commensurate bulk unit cell that respects inversion symmetry. We demonstrate this by first verifying that $\pm e\gamma^{\mathrm{inter}}/2\pi$ (mod $e$) is equal to the extra charge accumulation in the surface region for a general translationally invariant 1D insulator, while the remnant part of $\gamma$, the intra-cellular Zak phase $\gamma^{\mathrm{intra}}$, corresponds to the electronic part of the dipole moment of the bulk's unit cell. Second, we show that the extra charge accumulation can be related to the number of surface modes when the unit cell is inversion symmetric. This bulk-boundary correspondence using $\gamma^{\mathrm{inter}}$ reduces to the conventional one using $\gamma$ when the real-space origin is at the inversion center. Our work thereby clarifies the usage of $\gamma$ in the bulk-boundary correspondence. We study several tight binding models to quantitatively check the relation between the extra charge accumulation and the inter-cellular Zak phase as well as the bulk-boundary correspondence using the inter-cellular Zak phase., Comment: 15 pages, 9 figures

Monolayer MX$_2$ (M = Mo, W; X = S, Se) has drawn much attention recently for its possible application possibilities for optoelectronics, spintronics, and valleytronics. Its exotic optical and electronic properties include a direct band gap, circular polarization dependent optical transitions, and valence band (VB) spin band splitting at the $K$ and $-K$ points. These properties can be described within a minimal model, called the massive Dirac fermion model for which the parameters need to be experimentally determined. We propose that the parameters can be obtained from angle resolved photoemission (ARPES) data from bulk 2H-MX$_2$, instead of monolayer MX$_2$. Through tight binding calculations, we show how the electronic structure at high symmetry points evolves as the system changes from the monolayer to the three dimensional bulk 2H-MX$_2$ . We find vanishing $k_z$ dispersion and almost no change in the direct band gap at the $K$ and $-K$ points, in sharp contrast to the strong $k_z$ dispersion at the $\Gamma$ point. These facts allow us to extract the gap and spin band splitting at the $K$ point as well as the hopping energy from bulk ARPES data. We performed ARPES experiments on single crystals of MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$ at various photon energies and also with potassium evaporation. From the data, we determined the parameters for the massive Dirac fermion model for monolayer MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$., Comment: 11 pages, 6 figures, 1 table

Motivated by recent experimental efforts on three-dimensional semimetals, we investigate the static and dynamic density response of the nodal line semimetal by computing the polarizability for both undoped and doped cases. The nodal line semimetal in the absence of doping is characterized by a ring-shape zero energy contour in momentum space, which may be considered as a collection of Dirac points. In the doped case, the Fermi surface has a torus shape and two independent processes of the momentum transfer contribute to the singular features of the polarizability even though we only have a single Fermi surface. In the static limit, there exist two independent singularities in the second derivative of the static polarizability. This results in the highly anisotropic Friedel oscillations which show the angle-dependent algebraic power law and the beat phenomena in the oscillatory electron density near a charged impurity. Furthermore, the dynamical polarizability has two singular lines along $\hbar\omega = \gamma p$ and $\hbar\omega = \gamma p \sin\eta$, where $\eta$ is the angle between the external momentum $\vec{p}$ and the plane where the nodal ring lies. From the dynamical polarizability, we obtain the plasmon modes in the doped case, which show anisotropic dispersions and angle-dependent plasma frequencies. Qualitative differences between the low and high doping regimes are discussed in light of future experiments., Comment: 11 pages, 6 figures, Minor revisions are made

In this work, we present an analytical theory of strongly correlated Wigner crystals (WCs) in the lowest Landau level (LLL) by constructing an approximate, but accurate effective two-body interaction for composite fermions (CFs) participating in the WCs. This requires integrating out the degrees of freedom of all surrounding CFs, which we accomplish analytically by approximating their wave functions by delta functions. This method produces energies of various strongly correlated WCs that are in excellent agreement with those obtained from the Monte Carlo simulation of the full CF crystal wave functions. We compute the compressibility of the strongly correlated WCs in the LLL and predict discontinuous changes at the phase boundaries separating different crystal phases., Comment: 9 pages, 6 figures

We investigate novel Landau level structures of semi-metals with nodal ring dispersions. When the magnetic field is applied parallel to the plane in which the ring lies, there exist almost non-dispersive Landau levels at the Fermi level (E_F = 0) as a function of the momentum along the field direction inside the ring. We show that the Landau levels at each momentum along the field direction can be described by the Hamiltonian for the graphene bilayer with fictitious inter-layer couplings under a tilted magnetic field. Near the center of the ring where the inter-layer coupling is negligible, we have Dirac Landau levels which explain the appearance of the zero modes. Although the inter-layer hopping amplitudes become finite at higher momenta, the splitting of zero modes is exponentially small and they remain almost flat due to the finite artificial in-plane component of the magnetic field. The emergence of the density of states peak at the Fermi level would be a hallmark of the ring dispersion., Comment: 5 pages, 2 figures

Motivated by recent experiments on Pr2Ir2O7, we provide a theory of quantum oscillations in the Luttinger model with quadratic band touching, modelled for the spin-orbit-coupled conduction electrons in pyrochlore iridates. The magneto- and Hall resistivities are computed for electron- and hole-doped systems, and the corresponding Shubnikov de Haas (SdH) signals are investigated. The SdH signals are characterized by aperiodic behaviors that originate from the unconventional Landau level structures of the Luttinger model near the neutrality point, such as the inter-Landau level crossing, nonuniform Landau level spacings and non-parabolic dispersions along the applied magnetic-field direction. The aperiodic SdH signals observed in the paramagnetic state of Pr2Ir2O7 are shown to be consistent with such behaviors, justifying the use of the Luttinger model and the quadratic band touching spectrum as excellent starting points for physics of pyrochlore iridates. The implications of these results are discussed in light of recent theoretical and experimental devel- opments in these systems.

We examine Kondo effect in the periodic Anderson model for which the conduction band is of multi-orbital character and subject to mirror symmetry breaking field imposed externally. Taking p-orbital-based toy model for analysis, we find the Kondo pairing symmetry of p-wave character emerges self-consistently over some regions of parameter space and filling factor even though only the on-site Kondo hybridization is assumed in the microscopic Hamiltonian. The band structure in the Kondo-hybridized phase becomes nematic, with only two-fold symmetry, due to the p-wave Kondo coupling. The reduced symmetry should be readily observable in spectroscopic or transport measurements for heavy fermion system in a multilayer environment such as successfully grown recently., Comment: 5 pages, 4 figures

Intricate interplay between the periodicity of the lattice structure and that of the cyclotron motion gives rise to a well-known self-similar fractal structure of the energy eigenvalue, known as the Hofstadter butterfly, for an electron moving in lattice under magnetic field. Evolving from the $n=0$ Landau level, the central band of the Hofstadter butterfly is especially interesting since it may hold a key to the mysteries of the fractional quantum Hall effect observed in graphene. While the entire Hofstadter butterfly can be in principle obtained by solving Harper's equations numerically, the weak-field limit, most relevant for experiment, is intractable due to the fact that the size of the Hamiltonian matrix, that needs to be diagonalized, diverges. In this paper, we develop an effective Hamiltonian method that can be used to provide an accurate analytic description of the central Hofstadter band in the weak-field regime. One of the most important discoveries obtained in this work is that massless Dirac particles always exist inside the central Hofstadter band no matter how small the magnetic flux may become. In other words, with its bandwidth broadened by the lattice effect, the $n=0$ Landau level contains massless Dirac particles within itself. In fact, by carefully analyzing the self-similar recursive pattern of the central Hofstadter band, we conclude that massless Dirac particles should occur under arbitrary magnetic field. As a corollary, the central Hofstadter band also contains a self-similar structure of recursive Landau levels associated with such massless Dirac particles. To assess the experimental feasibility of observing massless Dirac particles inside the central Hofstadter band, we compute the width of the central Hofstadter band as a function of magnetic field in the weak-field regime., Comment: 17 pages, 9 figures

We show, by way of tight-binding and first-principles calculations, that a one-to-one correspondence between electron's crystal momentum k and non-zero orbital angular momentum (OAM) is a generic feature of surface bands. The OAM forms a chiral structure in momentum space much as its spin counterpart in Rashba model does, as a consequence of the inherent inversion symmetry breaking at the surface but not of spin-orbit interaction. Circular dichroism (CD) angle-resolved photoemission (ARPES) experiment is an efficient way to detect this new order, and we derive formulas explicitly relating the CD-ARPES signal to the existence of OAM in the band structure. The cases of degenerate p- and d-orbital bands are considered., Comment: 5 pages, 1 figure

Rashba-split surface band is characterized by a one-to-one correspondence between the electron's momentum k and its spin orientation. Here we show that a similar correspondence between momentum and orbital angular momentum (OAM) must exist on surface bands once the inversion symmetry is broken. The correspondence is valid even when there is no spin-orbit interaction. Tight-binding and first-principles calculations are presented to support our claim. As a method to detect such OAM-momentum correspondence, we propose the circular dichroism (CD) experiment using the angle-resolved photoemission (ARPES) setup. CD-ARPES experiment performed on Cu surface confirms the existence of chiral OAM. A new concept of "orbital Galvanic effect" is proposed., Comment: 5 pages, 2 figures, ARPES experiment on Cu included; author list updated, This version will not be published. A newer version, which is submitted, is available at arXiv:1112.1821

There has been tremendous recent progress in realizing topological insulator initiated by the proposal of Kane and Mele for the graphene system. They have suggested that the odd $Z_2$ index for the graphene manifests the spin filtered edge states for the graphene nanoribbons, which lead to the quantum spin Hall effect(QSHE). Here we investigate the role of the spin-orbit interaction both for the zigzag and armchair nanoribbons with special care in the edge geometry. For the pristine zigzag nanoribbons, we have shown that one of the $\sigma$ edge bands located near E=0 lifts up the energy of the spin filtered chiral edge states at the zone boundary by warping the $\pi$-edge bands, and hence the QSHE does not occur. Upon increasing the carrier density above a certain critical value, the spin filtered edge states are formed leading to the QSHE. We suggest that the hydrogen passivation on the edge can recover the original feature of the QSHE. For the armchair nanoribbon, the QSHE is shown to be stable. We have also derived the real space effective hamiltonian, which demonstrates that the on-site energy and the effective spin orbit coupling strength are strongly enhanced near the ribbon edges. We have shown that the steep rise of the confinement potential thus obtained is responsible for the warping of the $\pi$-edge bands., Comment: 20 pages, 6 figures

Electronic structures of the zigzag bilayer graphite nanoribbons(Z-BGNR) with various ribbon width $N$ are studied within the tight binding approximation. Neglecting the inter-layer hopping amplitude $\gamma_4$, which is an order of magnitude smaller than the other inter-layer hopping parameters $\gamma_1$ and $\gamma_3$, there exist two fixed Fermi points $\pm k^*$ independent of the ribbon width with the peculiar energy dispersion near $k^*$ as $\ve (k) \sim \pm (k-k^*)^N$. By investigating the edge states of the Z-BGNR, we notice that the trigonal warping of the bilayer graphene sheets are reflected on in the edge state structure. With the inclusion of $\gamma_4$, the above two Fermi points are not fixed, but drift toward the vicinity of the Dirac point with the increase of the width $N$ as shown by the finite scaling method and the peculiar dispersions change to the parabolic ones. The edge magnetism of the Z-BGNR is also examined by solving the half-filled Hubbard Hamiltonian for the ribbon using the Hartree-Fock approximation. We have shown that within the same side of the edges, the edge spins are aligned ferromagnetically for the experimentally relevant set of parameters., Comment: 22 pages, 7 figures; Corrections are added concerning the edge magnetism