Zero-energy States Research Articles

The edge states of one-dimensional spin-$\frac{1}{2}$-like noninteracting periodic systems under the most compactly supported Wannier-type functions

Abstract A hallmark of topological insulators is topologically protected edge states, which are determined by topological invariants defined under Bloch functions. Of course, not all edge states are stable, and they are independent of topological invariants. And in a recent study, it was found that even stable protected gapless edge states cannot be explained by topological invariants in two-dimensional (2D) systems. Motivated by this idea, we carried out a thorough study in one-dimensional (1D) spin-$\frac{1}{2}$-like noninteracting periodic systems and found that all edge states can be explained by the most compactly supported Wannier-type functions (MCWTFs). All the answers are contained in the flat-band Hamiltonian defined by MCWTFs. Edge states can be obtained by solving special equations using the parameters of MCWTFs. Symmetries are the special solutions of these equations, and we have proved the relationship between zero-energy edge states and symmetry analytically in partial cases, which was only a well-known result in the past. By solving the equations, we also find that not all zero-energy edge states are determined by symmetry, which is beyond the current framework. Through numerical verification, we show that these zero-energy edge states, like those protected by symmetry, are not affected by the energy spectrum of the system, so they are only related to the Bloch functions. We also derive special properties of zero-energy edge states protected by chiral symmetry from the perspective of MCWTFs, which are consistent with the previous literature. Finally, we do a test in the 2D case and find the connection between the Chern number and the Wannier functions. Our results show that compared with topological invariants, the origin of the edge state can be understood more comprehensively from the perspective of Wannier functions.

Identifying trivial and Majorana zero-energy modes using the Majorana polarization

In this work we consider superconductor-semiconductor hybrids containing both trivial and Majorana zero modes and explore their signatures in the Majorana polarization. In particular, we consider trivial zero-energy states due to confinement and disorder, which seem to be very likely experimental scenarios. We show that the Majorana polarization is able to characterize the topological phase transition as well as the emergence of Majorana zero modes even when trivial zero-energy states proliferate. Notably, the Majorana polarization inherits direct information about spatial correlations which are then the key for distinguishing Majorana and trivial zero modes. We demonstrate the utility of the Majorana polarization in normal-superconductor junctions and superconductor-normal-superconductor Josephson junctions. Our results support the interpretation of the Majorana polarization as a real-space topological indicator. Published by the American Physical Society 2024

Generating QES potentials supporting zero energy normalizable states for an extended class of truncated Calogero Sutherland model

Motivated by recent interest in the search for generating potentials for which the underlying Schrödinger equation is solvable, we report in the recent work several situations when a zero-energy state becomes bound depending on certain restrictions on the coupling constants that define the potential. In this regard, we present evidence of the existence of regular zero-energy normalizable solutions for a system of quasi-exactly solvable (QES) potentials that correspond to the rationally extended many-body truncated Calogero–Sutherland (TCS) model. Our procedure is based upon the use of the standard potential group approach with an underlying so(2,1) structure that utilizes a point canonical transformation with three distinct types of potentials emerging having the same eigenvalues while their common properties are subjected to the evaluation of the relevant wave functions. These cases are treated individually by suitably restricting the coupling parameters.

Inhomogeneous stripe phase and tunable Majorana zero modes in Rashba s-wave superconducting systems in the presence of in-plane Zeeman field

Based on the extended Bogoliubov–de Gennes theory, the inhomogeneous stripe order and Majorana bound states are demonstrated in s-wave superconducting systems with Rashba spin–orbit interaction when the in-plane Zeeman field is applied. For a fully open square sample, topological phase transitions can be driven by the Zeeman field, and the stripe phase with spatially oscillating order parameter shows up across a critical field strength. The topological channels arise in this phase, which can be utilized to create and manipulate Majorana zero modes. Interestingly, (quasi-)one-dimensional channels with diminished pairing amplitude can be formed in narrow arms of a square loop, accompanied by the reconstruction of energy spectra of the condensate and the realization of robust Majorana zero-energy states at the ends of channels. The associated evolution of topological phases and the location of Majorana zero modes are highly sensitive to the field direction. Moreover, the effects of the rectangular aspect-ratio and the off-centered hole as well as the surface defects on Majorana end modes are explored in Rashba asymmetric loops. In comparison, the field-dependent evolution processes of low-energy levels behave more complicated because the emergence of confined topological channels can be effectively tuned by the length and width of loop arms as well as the size and position of an introduced small indentation at the outer edge. Rich patterns of Majorana corner-like states are generated for such asymmetric systems. Our theoretical predictions may shed new light on the tunability of Majorana zero modes and provide useful guidance for future experiments and applications.

Subsymmetry protected topology in topological insulators and superconductors

Exploration of topology protected by a certain symmetry is central in condensed matter physics. A recent idea of subsymmetry protected (SSP) topology—remains of a broken symmetry can still protect specific topological boundary states—has been developed and demonstrated in a photonic system [], but not in topological superconductors. Here, we extend this idea further by applying subsymmetry protecting perturbation (SSPP) to one-dimensional topological insulating and superconducting systems using the Su-Schrieffer-Hegger (SSH) and Kitaev models. Using the tight-binding and low-energy effective theory, we show that the SSP boundary states retain topological properties while the SSPP results in the asymmetry of boundary states. For the SSH model, an SSP zero-energy edge state localized on one edge possesses quantized polarization. In contrast, the other edge state is perturbed to have nonzero energy, and its polarization is not quantized. For topological superconductors, SSP zero-energy Majorana boundary states for spinful Kitaev models emerge on only one edge, contrary to the conventional belief that Majorana fermions emerge at opposite edges. Our findings can be used as a platform to expand our understanding of topological materials as they broaden our understanding of the symmetry in a topological system and a method to engineer Majorana fermions. Published by the American Physical Society 2024

Vortex superlattice induced second-order topological mid-gap states in first-order topological insulators and superconductors

Topological defects such as vortex and dislocations, support zero-energy localized states as a reflection of the bulk topology, in first-order topological insulators and superconductors. Furthermore, emergent first-order topological mid-gap states have been discovered driven by the magnetic vortex superlattice. However, whether the higher-order topological mid-gap states would emerge from the first-order topological insulators and superconductors with the vortex superlattice remains elusive. In this work, we propose vortex superlattice could induce second-order topological mid-gap states with staggered lattice spacings for vortices in first-order topological insulators and superconductors. These higher-order topological mid-gap states originate from the staggered tunneling between vortex-induced bound states and the emergent π flux on vortex superlattices, as an intrinsic exhibition of the interplay between vortices and bulk topology for the first-order topological states. Our work uncovers higher-topological characteristics of topological-defect superlattice in first-order topological states, and develops a controllable environment for the creation and exploration of higher-order topological states.

Chiral Pair Density Waves with Residual Fermi Arcs in RbV3Sb5

Abstract The chiral 2 × 2 charge order has been reported and confirmed in the kagome superconductor RbV3Sb5, while its interplay with superconductivity remains elusive owing to its lowest superconducting transition temperature T C of about 0.85 K in the AV3Sb5 family (A = K, Rb, Cs) that severely challenges electronic spectroscopic probes. Here, utilizing dilution-refrigerator-based scanning tunneling microscopy down to 30 mK, we observe chiral 2 × 2 pair density waves with residual Fermi arcs in RbV3Sb5. We find a superconducting gap of 150 μeV with substantial residual in-gap states. The spatial distribution of this gap exhibits chiral 2 × 2 modulations, signaling a chiral pair density wave (PDW). Our quasi-particle interference imaging of the zero-energy residual states further reveals arc-like patterns. We discuss the relation of the gap modulations with the residual Fermi arcs under the space-momentum correspondence between PDW and Bogoliubov Fermi states.

Chiral kagome superconductivity modulations with residual Fermi arcs.

Superconductivity involving finite-momentum pairing1 can lead to spatial-gap and pair-density modulations, as well as Bogoliubov Fermi states within the superconducting gap. However, the experimental realization of their intertwined relations has been challenging. Here we detect chiral kagome superconductivity modulations with residual Fermi arcs in KV3Sb5 and CsV3Sb5 using normal and Josephson scanning tunnelling microscopy down to 30 millikelvin with a resolved electronic energy difference at the microelectronvolt level. We observe a U-shaped superconducting gap with flat residual in-gap states. This gap shows chiral 2a×2a spatial modulations with magnetic-field-tunable chirality, which align with the chiral 2a×2a pair-density modulations observed through Josephson tunnelling. These findings demonstrate a chiral pair density wave (PDW) that breaks time-reversal symmetry. Quasiparticle interference imaging of the in-gap zero-energy states reveals segmented arcs, with high-temperature data linking them to parts of the reconstructed vanadium d-orbital states within the charge order. The detected residual Fermi arcs can be explained by the partial suppression of these d-orbital states through an interorbital 2a×2a PDW and thus serve as candidate Bogoliubov Fermi states. In addition, we differentiate the observed PDW order from impurity-induced gap modulations. Our observations not only uncover a chiral PDW order with orbital selectivity but also show the fundamental space-momentum correspondence inherent in finite-momentum-paired superconductivity.

Quantized polarization and Majorana fermions beyond tenfold classification

Exploration of topology is central in condensed matter physics and applications to fault-tolerant quantum information. The bulk-boundary correspondence and tenfold classification determine the topological state compared to a vacuum. Contrary to this belief, we demonstrate that topological zero-energy domain-wall states can emerge for all forbidden 1D classes of the tenfold classification table. The guiding principle is that the difference in the topological quantities of two trivial domains can be quantized, and hence, a topologically protected state can emerge at the domain wall. Such nontrivial domain-wall states are demonstrated using generalized Su-Schrieffeer-Heeger and generalized Kitaev models, which manifest quantized polarization and Majorana fermions, respectively. The quantized Berry phase difference between the domains protects the non-trivial nature of the domain-wall states, extending the bulk-boundary correspondence, also confirmed by the tight-binding and Jackiw-Rebbi methods. Furthermore, we show that the seemingly trivial electronic and superconducting models can be transformed into their topological counterparts in the framework of the topological Fermi-liquid theory. Finally, we propose potential systems where our results may be realized, spanning from electronic and superconducting to optical systems.

Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic cubic Dirac semimetal and the peculiar case of star-shaped classical orbits

We study an anisotropic cubic Dirac semimetal subjected to a constant magnetic field. In the case of an isotropic dispersion in the x−y plane, with parameters vx=vy, it is possible to find exact Landau levels, indexed by the quantum number n, using the typical ladder operator approach. Interestingly, we find that the lowest energy level (the zero-energy state in the case of kz=0) has a degeneracy that is 3 times that of other states. This degeneracy manifests in the Hall conductivity as a step at a zero chemical potential 3/2 the size of other steps. Moreover, as n→∞, we find energies En∝n3/2, which means the nth step as a function of the chemical potential roughly occurs at a value μ∝n3/2. We propose that these exciting features could be used to experimentally identify cubic Dirac semimetals. Subsequently, we analyze the anisotropic case vy=λvx, with λ≠1. First, we consider a perturbative treatment around λ≈1 and find that energies En∝n3/2 still hold as n→∞. To gain further insight into the Landau level structure for a maximum anisotropy, we turn to a semiclassical treatment that reveals interesting star-shaped orbits in phase space that close at infinity. This property is a manifestation of weakly localized states. Despite being infinite in length, these orbits enclose a finite phase space volume and permit finding a simple semiclassical formula for the energy, which has the same form as above. Our findings suggest that both isotropic and anisotropic cubic Dirac semimetals should leave similar experimental imprints. Published by the American Physical Society 2024

Higher-order topological states in T-graphene and their realization in photonic crystals

Higher-order topological states extend the power of nontrivial topological states beyond the bulk-edge correspondence. Here we study the higher-order topological states (corner states) in an open-boundary two-dimensional T-graphene lattice. Unlike the common zero-energy corner states, our findings reveal non-zero energy corner states in such lattice systems, and the energy could be controlled by modifying the hopping parameters. Moreover, the corner states could be transferred away from the lattice corners by designing the position-specific vacancy defects. The strong robustness of the corner states is also demonstrated against the uniaxial strain and vacancy defects, respectively. A plasmonic crystal is constructed to testify to the theory, in which the corner states are realized in optical modes and their higher-order topological properties are verified. Our results open the avenue of corner-states engineering, which holds significant physical implications of higher-order topological states for the design of photonic and electronic devices with specialized functionalities.

Engineering multiform quantum state transfers and topological devices in a splicing Su–Schrieffer–Heeger chain

We propose to implement the various kinds of topological quantum state transfers (QSTs) in a splicing Su–Schrieffer–Heeger (SSH) chain, where the transfer forms vary with the introduction of the on-site potential and are distinguished by different output ports. The splicing SSH chain without the on-site potential shows an interface symmetry in chain configuration and supports a zero-energy gap state to realize a symmetric state transfer from the central site to the two ends with equal probabilities. After introducing the on-site potential on one end or two ends, we further obtain two asymmetric versions of the splicing SSH chain. The existence of the on-site potential leads to the transformation of the zero-energy gap state into a nonzero gap state, and the nonzero gap state exhibits different distribution characteristics in two asymmetric versions. Accordingly, three QST channels with different forms in a splicing SSH chain are designed, each of which possesses the robustness against the disorder of the couplings and the inaccuracy of the evolution time. In addition, we demonstrate the scalability of a multiform QST protocol with high fidelity. The protocol with different transfer forms is expected to realize diverse functional quantum devices, which effectively improves the utilization of a splicing SSH chain and economizes the physical resource.

Theory of Majorana Zero Modes in Unconventional Superconductors

Abstract Majorana fermions are spin-1/2 neutral particles that are their own antiparticles; they were initially predicted by Ettore Majorana in particle physics but their observation still remains elusive. The concept of Majorana fermions has been borrowed by condensed matter physics, where, unlike particle physics, Majorana fermions emerge as zero-energy quasiparticles that can be engineered by combining electrons and holes and have therefore been called Majorana zero modes. In this review, we provide a pedagogical explanation of the basic properties of Majorana zero modes in unconventional superconductors and their consequences in experimental observables, putting a special emphasis on the initial theoretical discoveries. In particular, we first show that Majorana zero modes are self-conjugated and emerge as a special type of zero-energy surface Andreev bound states at the boundary of unconventional superconductors. We then explore Majorana zero modes in 1D spin-polarized p-wave superconductors, where we address the formation of topological superconductivity and the physical realization in superconductor–semiconductor hybrids. In this part we highlight that Majorana quasiparticles appear as zero-energy edge states, exhibiting charge neutrality, spin-polarization, and spatial nonlocality as unique properties that can already be seen from their energies and wavefunctions. Next, we discuss the analytically obtained Green’s functions of p-wave superconductors and demonstrate that the emergence of Majorana zero modes is always accompanied by the formation of odd-frequency spin-triplet pairing as a unique result of the self-conjugate nature of Majorana zero modes. We finally address the signatures of Majorana zero modes in tunneling spectroscopy, including the anomalous proximity effect, and the phase-biased Josephson effect.

Mean-field elastic moduli of a three-dimensional, cell-based vertex model

The mechanics of a foam depends on bubble shape, bubble network topology, and the material at hand, be it metallic or polymeric, for example. While the shapes of bubbles are the consequence of minimizing surface area for a given bubble volume in a space-filling packing, if one were to consider biological tissue as a foam-like material, the zoology of observed shapes of cells perhaps motivates different energetic contributions. Building on earlier two-dimensional results, here, we focus on a mean field approach to obtain the elastic moduli for an ordered, three-dimensional vertex model. We use the space-filling shape of a truncated octahedron and an energy functional containing a restoring surface area spring and a restoring volume spring. The tuning of the three-dimensional shape index exhibits a rigidity transition via a compatible–incompatible transition. Specifically, for smaller shape indices, both the target surface area and volume cannot be achieved, while beyond some critical value of the three-dimensional shape index, they can be, resulting in a zero-energy state. In addition to analytically determining the location of the transition in mean field, we find that the rigidity transition and the elastic moduli depend on the parameterization of the cell shape. This parameterization effect is more pronounced in three dimensions than in two dimensions given the zoology of shapes that a polyhedron can take on (as compared to a polygon). We also uncover nontrivial dependence of the elastic moduli on the deformation protocol in which some deformations result in affine motion of the vertices, while others result in nonaffine motion. Such dependencies on the shape parameterization and deformation protocol give rise to a nontrivial shape landscape and, therefore, nontrivial mechanical response even in the absence of topology changes.

Atomic-Limit Mott Insulator in [4]Triangulene Frameworks.

Triangulene, one unique class of zigzag-edged triangular graphene molecules, has attracted tremendous research interest. In this work, as an ultimate phase of the Mott insulator, we present the realization of the atomic-limit Mott insulator in experimentally synthesized [4]triangulene frameworks ([4]-TGFs) from first-principles calculations. The frontier molecular orbitals of the nonmagnetic [4]triangulene consist of three coupled corner modes. After the isolated [4]triangulene is assembled into [4]-TGF, one special enantiomorphic flat band is created through the coupling of these corner modes, which is identified to be a second-order topological insulator with half-filled topological corner states at the Fermi level. Moreover, [4]-TGF prefers an antiferromagnetic ground state under Hubbard interactions, which further splits these metallic zero-energy states into an atomic-limit Mott insulator with spin-polarized corners. Since the fractional filling of topological corner states is a smoking-gun signature of higher-order topology, our results demonstrate a universal approach to explore the atomic-limit Mott insulators in higher-order topological materials.

Hidden skyrmion on the spin direction of the Majorana zero mode

The spin information of Majorana zero mode (MZM) has important application value in distinguishing MZM from other zero energy states. We study the spin direction of the Majorana zero mode (MZM) localized in a vortex state in a topological superconductor. We find that the topological number of the magnetic structure of the MZM is zero. However, with an increase in the vortex radius, the magnetic structure of the electron oscillates between the antiskyrmion and skyrmion (starting with the antiskyrmion), while that of the hole oscillates between the skyrmion and antiskyrmion (starting with the skyrmion). The total magnetic structure of the electron is a antiskyrmion (the topological number is −1) while that of the hole is a skyrmion (the topological number is 1). We hope these new features provide more characters for detection and regulation of MZM.

A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states

The one-to-one relation between the winding number and the number of robust zero-energy edge states, known as bulk-boundary correspondence, is a celebrated feature of 1d systems with chiral symmetry. Although this property can be explained by the K-theory, the underlying mechanism remains elusive. Here, we demonstrate that, even without resorting to advanced mathematical techniques, one can prove this correspondence and clearly illustrate the mechanism using only Cauchy’s integral and elementary algebra. Furthermore, our approach to proving bulk-boundary correspondence also provides clear insights into a kind of system that doesn’t respect chiral symmetry but can have robust left or right zero-energy edge states. In such systems, one can still assign the winding number to characterize these zero-energy edge states.

Characterization of zero-energy corner states in higher-order topological systems with chiral symmetry

Characterization of zero-energy corner states in higher-order topological systems with chiral symmetry

Power-duality in path integral formulation of quantum mechanics

Power duality in Feynman's path integral formulation of quantum mechanics is investigated. The power duality transformation consists of a change in coordinate and time variables, an exchange of energy and coupling, and a classical angular momentum replacement. Two physical systems connected by the transformation form a power-dual pair. The propagator (Feynman's kernel) expressed by Feynman's path integral cannot be form-invariant under the transformation, whereas the promotor constructed by modifying Feynman's path integral is found form-invariant insofar as the angular momentum is classical. Upon angular quantization, the power duality breaks down. To save the notion of power duality, the idea of quasi power duality is proposed, which constitutes of an ad hoc angular momentum replacement. The power-dual invariant promotor leads to the quasi-dual invariant Green function. A formula is proposed, which determines the Green function for one of a dual pair by knowing the Green function for the other. As examples, the Coulomb-Hooke dual pair and a family of two-term confinement potentials for a zero-energy state are discussed.

Self-purification and entanglement revival in lambda matter

In this study, we explore the dynamics of entanglement in an ensemble of qutrits with a lambda-type level structure interacting with single-mode bosons. Our investigation focuses on zero-energy states within the subspace of totally symmetric wave functions. Remarkably, we observe a universal two-stage dynamics of entanglement with intriguing revival behavior. The revival of entanglement is a consequence of the self-purification process, where the quantum state relaxes and converges universally to a special dark state within the system.