Bulk Wave Functions Research Articles

Universal Entanglement Spectrum in One-Dimensional Gapless Symmetry Protected Topological States.

Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.

Extracting the Quantum Hall Conductance from a Single Bulk Wave Function.

We propose a new formula that extracts the quantum Hall conductance from a single (2+1)D gapped wave function. The formula applies to general many-body systems that conserve particle number, and is based on the concept of modular flow, i.e., unitary dynamics generated from the entanglement structure of the wave function. The formula is shown to satisfy all formal properties of the Hall conductance: it is odd under time reversal and reflection, even under charge conjugation, universal and topologically rigid in the thermodynamic limit. Further evidence for relating the formula to the Hall conductance is obtained from conformal field theory arguments. Finally, we numerically check the formula by applying it to a noninteracting Chern band where excellent agreement is obtained.

Chern Numbers Associated with the Periodic Toda Lattice

The periodic Toda lattice is solved by exploiting the spectral properties of the Lax operator, in which the boundary states play an important role. We show that these boundary states have a topological origin similar to that of the edge states in topological insulators, and consequently, that the bulk wave functions of the Lax operator yield nontrivial Chern numbers. This implies that the periodic Toda lattice belongs to the same topological class as the Thouless pump. We demonstrate that the cnoidal wave of the Toda lattice exhibits a Chern number of $-1$ per period.

A method for fast calculating the electronic states in 2D quantum structures based on AIIIBV nitrides

The paper presents a method for fast calculating the electronic states in two-dimensional quantum structures based on AIIIBV nitrides. The method is based on the representation of electronic states in the form of a linear combination of bulk wave functions of materials, from which quantum structures are made. The parameters and criteria for the selection of bulk wave functions that provides fast convergence of the numerical procedures for calculating the eigenvalues of the quantum Hamiltonian have been considered. The results of the calculations have been given both for one polar InGaN/GaN quantum well and for a system of several quantum wells. Being based on the full band structure of AIIIBV nitrides with a wurtzite-type crystal lattice, the proposed approach takes into account the states far from the center of the Brillouin zone, while preserving the computational efficiency of traditional methods of envelope function in approximating the effective mass.

Bound states at disclinations: an additive rule of real and reciprocal space topology

Focusing on the two-dimensional (2D) Su-Schrieffer-Heeger (SSH) model, we propose an additive rule between the real-space topological invariant s of disclinations (related to the Burgers vector B) and the reciprocal-space topological invariant p of bulk wave functions (the vectored Zak phase). The disclination-induced bound states in the 2D SSH model appear only if (s + p/2π) is nonzero modulo the lattice constant. These disclination-bound states are robust against perturbations respecting C4 point group symmetry and other perturbations within an amplitude determined by p. Besides the disclination-bound states, the proposed additive rule also suggests that a half-bound state extends over only half of a sample and a hybrid-bound state, which always have a nonvanishing component of s + p/2π.

On holography in general background and the boundary effective action from AdS to dS

We study quantum fields on an arbitrary, rigid background with boundary. We derive the action for a scalar in the holographic basis that separates the boundary and bulk degrees of freedom. A relation between Dirichlet and Neumann propagators valid for any background is obtained from this holographic action. As a simple application, we derive an exact formula for the flux of bulk modes emitted from the boundary in a warped background. We also derive a formula for the Casimir pressure on a (d − 1)-brane depending only on the boundary-to-bulk propagators, and apply it in AdS. Turning on couplings and using the holographic basis, we evaluate the one-loop boundary effective action in AdS by means of the heat kernel expansion. We extract anomalous dimensions of single and double trace CFT operators generated by loops of heavy scalars and nonabelian vectors, up to third order in the large squared mass expansion. From the boundary heat kernel coefficients we identify CFT operator mixing and corrections to OPE data, in addition to the radiative generation of local operators. We integrate out nonabelian vector fluctuations in AdS4,5,6 and obtain the associated holographic Yang-Mills β functions. Turning to the expanding patch of dS, following recent proposals, we provide a boundary effective action generating the perturbative cosmological correlators using analytical continuation from dS to EAdS. We obtain the “cosmological” heat kernel coefficients in the scalar case and work out the divergent part of the dS4 effective action which renormalizes the cosmological correlators. We find that bulk masses and wavefunction can logarithmically run as a result of the dS4 curvature, and that operators on the late time boundary are radiatively generated. More developments are needed to extract all one-loop information from the cosmological effective action.

Chiral Central Charge from a Single Bulk Wave Function.

A (2+1)-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wave function from being real valued in any local product basis.

Non-Hermitian skin effect in a domain-wall system

The non-Hermitian skin effect is one of the most striking features in non-Hermitian physics. It reveals a novel phenomenon in a non-Hermitian system that the bulk wave function and energy spectrum are sensitively dependent on the boundary conditions. The concept of generalized Brillouin zones has been proposed to characterize bulk wave functions in such systems . Based on generalized Brillouin zones, non-Bloch topological invariants can reconstruct the non-Hermitian bulk-edge correspondence. Previous discussion of the non-Hermitian skin effect mainly focused on open boundary conditions, and the calculation of generalized Brillouin zones needs to be reconsidered under domain-wall boundary conditions. The paper introduces the related researches of the non-Hermitian skin effect in domain-wall systems, including the general form of the generalized Brillouin zone equation in a one-dimensional single-band model, non-Bloch topological invariants in non-Hermitian SSH (Su-Schieffer-Heeger) model, and the experimental realization of the non-Hermitian skin effect in one-dimensional quantum walk system.

Fragile topological insulators protected by rotation symmetry without spin-orbit coupling

We present a series of models of three-dimensional rotation-symmetric fragile topological insulators in class AI (time-reversal symmetric and spin-orbit-free systems), which have gapless surface states protected by time-reversal ($T$) and $n$-fold rotation ($C_n$) symmetries ($n=2,4,6$). Our models are generalizations of Fu's model of a spinless topological crystalline insulator, in which orbital degrees of freedom play the role of pseudo-spins. We consider minimal surface Hamiltonian with $C_n$ symmetry in class AI and discuss possible symmetry-protected gapless surface states, i.e., a quadratic band touching and multiple Dirac cones with linear dispersion. We characterize topological structure of bulk wave functions in terms of two kinds of topological invariants obtained from Wilson loops: $\mathbb{Z}_2$ invariants protected by $C_n$ ($n=4,6$) and time-reversal symmetries, and $C_2T$-symmetry-protected $\mathbb{Z}$ invariants (the Euler class) when the number of occupied bands is two. Accordingly, our models realize two kinds of fragile topological insulators. One is a fragile $\mathbb{Z}$ topological insulator whose only nontrivial topological index is the Euler class that specifies the number of surface Dirac cones. The other is a fragile $\mathbb{Z}_2$ topological insulator having gapless surface states with either a quadratic band touching or four (six) Dirac cones, which are protected by time-reversal and $C_4$ ($C_6$) symmetries. Finally, we discuss the instability of gapless surface states against the addition of $s$-orbital bands and demonstrate that surface states are gapped out through hybridization with surface-localized $s$-orbital bands.

Observation of a chiral wave function in the twofold-degenerate quadruple Weyl system BaPtGe

Topological states in quantum materials are defined by non-trivial topological invariants, such as the Chern number, which are properties of their bulk wave functions. A remarkable consequence of topological wave functions is the emergence of edge modes, a phenomenon known as bulk-edge correspondence, that gives rise to quantized or chiral physical properties. While edge modes are widely presented as signatures of non-trivial topology, how bulk wave functions can manifest explicitly topological properties remains unresolved. Here, using high-resolution inelastic x-ray spectroscopy (IXS) combined with first principles calculations, we report experimental signatures of chiral wave functions in the bulk phonon spectrum of BaPtGe, which we show to host a previously undiscovered twofold degenerate quadruple Weyl node. The chirality of the degenerate phononic wave function yields a non-trivial phonon dynamical structure factor, S(Q,$\omega$), along high-symmetry directions, that is in excellent agreement with numerical and model calculations. Our results establish IXS as a powerful tool to uncover topological wave functions, providing a key missing ingredient in the study of topological quantum matter.

Highly responsive broadband photodetection in topological insulator - Carbon nanotubes based heterostructure

Topological insulators represent a new electronic phase which comes from the topological character of the bulk wave functions in some special materials. These exotic phases can be attributed to the strong spin-orbit interaction and can be explained with the band theory of solids. Recent studies show that topological insulators (TIs) such as Bi2Te3 or Bi2Se3 have promising candidature for optoelectronic industry and some of these interesting properties have been already demonstrated such as strong light absorption, photocurrent sensitivity to the polarization of light, layer thickness and size dependent tuning in the band gap. Theoretically, it has been proposed that TIs based broad spectral photodetector has potential in analogy with that of a graphene based photodetector. Combining TIs with carbon-based nanomaterials provides novel and promising technique for photodetection, and can show broad spectral properties. Along this line, we have grown photodetector based on hybrid of TIs with carbon nanotubes. Efficient light absorption and electron hole pair generation has been observed under the illumination of visible (532 nm) and NIR (1064 nm) light. High photoresponsivity and photoconductive gain of 79.4 AW-1 and 185, respectively, along with very good detector response time were observed in our device under visible laser (532 nm) illumination.

Diffractive photoproduction of J/ψ and ϒ using holographic QCD: Gravitational form factors and GPD of gluons in the proton

We present a holographic analysis of diffractive photoproducton of charmonium $J/\psi$ and upsilonium $\Upsilon$ on a proton, considered as a bulk Dirac fermion, for all ranges of $\sqrt{s}$, i.e., from near threshold to very high energy. Using the bulk wave functions of the proton and vector mesons, within holographic QCD, and employing Witten diagrams in the bulk, we compute the diffractive photoproduction amplitude of $J/\psi$ and $\Upsilon$. The holographic amplitude shows elements of the strictures of vector meson dominance (VMD). It is dominated by the exchange of a massive graviton or $2^{++}$ glueball resonances near threshold, and its higher spin-j counterparts that reggeize at higher energies. Both the differential and total cross sections are controlled by the gravitational form factor $A(t)$, and compare well to the recent results reported by the GlueX collaboration near threshold and the world data at large $\sqrt{s}$. The holographic gravitational form factors, including the D-term, which is due to the exchange of massive spin-0 glueballs, are in good agreement with lattice simulations. We use it to extract the holographic pressure and shear forces inside the proton. Finally, using a pertinent integral representation of the holographic gravitational form factor $A(t)$ near threshold, and its Pomeron counterpart way above threshold, we extract the generalized parton distribution (GPD) of gluons inside the proton at different resolutions.

Topological boundaries and bulk wavefunctions in the Su–Schreiffer–Heeger model

Working in the context of the Su–Schreiffer–Heeger model, the effect of topological boundaries on the structure and properties of bulk position-space wavefunctions is studied for a particle undergoing a quantum walk in a one-dimensional lattice. In particular, we consider what happens when the wavefunction reaches a boundary at which the Hamiltonian changes suddenly from one topological phase to another and construct an exact solution for the wavefunction on both sides of the boundary. The reflection and transmission coefficients at the boundary are calculated as a function of the system’s hopping parameters, and it is shown that for some parameter ranges the transmission coefficient can be made very small. Therefore, it is possible to arrange a high degree of bulk wavefunction localization within each topological region, a fact that has information processing applications. This ‘topologically-assisted’ suppression of transitions, although not of direct topological origin itself, exists because of the presence of an abrupt change in the properties of the Hamiltonian at the topological boundary. We give a quantitative examination of the reflection and transmission coefficients of incident waves at the boundary between regions of different winding number.

Lorentzian condition in holographic cosmology

We derive a sufficient set of conditions on the Euclidean boundary theory in dS/CFT for it to predict classical, Lorentzian bulk evolution at large spatial volumes. Our derivation makes use of a canonical transformation to express the bulk wave function at large volume in terms of the sources of the dual partition function. This enables a sharper formulation of dS/CFT. The conditions under which the boundary theory predicts classical bulk evolution are stronger than the criteria usually employed in quantum cosmology. We illustrate this in a homogeneous isotropic minisuperspace model of gravity coupled to a scalar field in which we identify the ensemble of classical histories explicitly.

Photo-electrons unveil topological transitions in graphene-like systems.

The topological structure of the wavefunctions of particles in periodic potentials is characterized by the Berry curvature Ωkn whose integral on the Brillouin zone is a topological invariant known as the Chern number. The bulk-boundary correspondence states that these numbers define the number of edge or surface topologically protected states. It is then of primary interest to find experimental techniques able to measure the Berry curvature. However, up to now, there are no spectroscopic experiments that proved to be capable to obtain information on Ωkn to distinguish different topological structures of the bulk wavefunctions of semiconducting materials. Based on experimental results of the dipolar matrix elements for graphene, here we show that ARPES experiments with the appropriate x-ray energies and polarization can unambiguously detect changes of the Chern numbers in dynamically driven graphene and graphene-like materials opening new routes towards the experimental study of topological properties of condensed matter systems.

Classification of topological quantum matter with symmetries

Topological materials have become the focus of intense research in recent years, since they exhibit fundamentally new physical phenomena with potential applications for novel devices and quantum information technology. One of the hallmarks of topological materials is the existence of protected gapless surface states, which arise due to a nontrivial topology of the bulk wave functions. This review provides a pedagogical introduction into the field of topological quantum matter with an emphasis on classification schemes. We consider both fully gapped and gapless topological materials and their classification in terms of nonspatial symmetries, such as time-reversal, as well as spatial symmetries, such as reflection. Furthermore, we survey the classification of gapless modes localized on topological defects. The classification of these systems is discussed by use of homotopy groups, Clifford algebras, K-theory, and non-linear sigma models describing the Anderson (de-)localization at the surface or inside a defect of the material. Theoretical model systems and their topological invariants are reviewed together with recent experimental results in order to provide a unified and comprehensive perspective of the field. While the bulk of this article is concerned with the topological properties of noninteracting or mean-field Hamiltonians, we also provide a brief overview of recent results and open questions concerning the topological classifications of interacting systems.

Wave functions of symmetry-protected topological phases from conformal field theories

We propose a method for analyzing two-dimensional symmetry protected topological (SPT) wavefunctions using a correspondence with conformal field theories (CFTs) and integrable lattice models. This method generalizes the CFT approach for the fractional quantum Hall effect wherein the wavefunction amplitude is written as a many-operator correlator in the CFT. Adopting a bottom-up approach, we start from various known microscopic wavefunctions of SPTs with discrete symmetries and show how the CFT description emerges at large scale, thereby revealing a deep connection between group cocyles and critical, sometimes integrable, models. We show that the CFT describing the bulk wavefunction is often also the one describing the entanglement spectrum, but not always. Using a plasma analogy, we also prove the existence of hidden quasi-long-range order for a large class of SPTs. Finally, we show how response to symmetry fluxes is easily described in terms of the CFT.

TopologicalD+p-wave superconductivity in Rashba systems

We show two-dimensional "strong" topological superconductivity in d-wave superconductors (SCs). Although the topological invariant of the bulk wave function cannot be defined in d_{x^2-y^2}-wave and d_{xy}-wave SCs because of nodal excitations, the bulk energy spectrum of d-wave SCs on a substrate is fully gapped in a magnetic field. Then the superconducting state is specified by a nontrivial Chern number, and hence topologically nontrivial properties are robust against disorders and interactions. We discuss high-temperature topological superconductivity in cuprate SCs recently fabricated on a substrate. Furthermore, we show that the three-dimensional noncentrosymmetric d-wave SC is a Weyl SC hosting topologically protected Weyl nodes. Noncentrosymmetric heavy-fermion SCs, such as CeRhSi_3 and CeIrSi_3, are candidates for Weyl SCs.

Plasma-Wave Terahertz Detection Mediated by Topological Insulators Surface States.

Topological insulators (TIs) represent a novel quantum state of matter, characterized by edge or surface-states, showing up on the topological character of the bulk wave functions. Allowing electrons to move along their surface, but not through their inside, they emerged as an intriguing material platform for the exploration of exotic physical phenomena, somehow resembling the graphene Dirac-cone physics, as well as for exciting applications in optoelectronics, spintronics, nanoscience, low-power electronics, and quantum computing. Investigation of topological surface states (TSS) is conventionally hindered by the fact that in most of experimental conditions the TSS properties are mixed up with those of bulk-states. Here, we activate, probe, and exploit the collective electronic excitation of TSS in the Dirac cone. By engineering Bi2Te(3-x)Sex stoichiometry, and by gating the surface of nanoscale field-effect-transistors, exploiting thin flakes of Bi2Te2.2Se0.8 or Bi2Se3, we provide the first demonstration of room-temperature terahertz (THz) detection mediated by overdamped plasma-wave oscillations on the "activated" TSS of a Bi2Te2.2Se0.8 flake. The reported detection performances allow a realistic exploitation of TSS for large-area, fast imaging, promising superb impacts on THz photonics.

Bulk invariants and topological response in insulators and superconductors with nonsymmorphic symmetries

In this work we consider whether nonsymmorphic symmetries such as a glide plane can protect the existence of topological crystalline insulators and superconductors in three dimensions. In analogy to time-reversal symmetric insulators, we show that the presence of a glide gives rise to a quantized magnetoelectric polarizability, which we compute explicitly through the Chern-Simons 3-form of the bulk wave functions for a glide symmetric model. Our approach provides a measurable property for this insulator and naturally explains the connection with mirror symmetry protected insulators and the recently proposed $Z_2$ index for this phase. More generally, we prove that the magnetoelectric polarizability becomes quantized with any orientation-reversing space group symmetry. We also construct analogous examples of glide protected topological crystalline superconductors in classes D and C and discuss how bulk invariants are related to quantized surface thermal-Hall and spin-Hall responses.