Degenerate Zero Research Articles

Ideal unconventional charge-2 Dirac fermions in an ultralightweight chiral crystal

When topology meets chirality, it is of great interest to search for possible topological properties in chiral crystals, thereby achieving the effect of making $1+1$ greater than 2. Here, we accomplish a complete list of topological charge-2 (C-2) Dirac fermions based on 65 S\"ohncke space groups. Such emergent fermions occurring in chiral crystals enjoy a higher topological charge number and exhibit more unique characteristics compared to C-0 cases, including multiple Fermi arcs extending to most of the Brillouin zone. For material realization, we propose that an ultralightweight chiral crystal ${\mathrm{N}}_{4}\mathrm{Cl}$ is a high-quality C-2 Dirac semimetal, where the seductive nodal points reside at the Fermi level and exhibit a large linear energy range to overcome the interruption of irrelevant bands. More interestingly, the doubly degenerate Landau levels are described by a Kramers-like degeneracy in such a C-2 Dirac fermion, featuring a doubly degenerate zero mode. Furthermore, the topological phase transitions from Dirac to Weyl states of ${\mathrm{N}}_{4}\mathrm{Cl}$ can be manipulated by applying shear strains. These fascinating findings will certainly evoke continued exploration of unconventional Dirac semimetals, in turn leading to extensive application prospects and uncommon insights.

Length scale formation in the Landau levels of quasicrystals

Exotic tiling patterns of quasicrystals have motivated extensive studies of quantum phenomena such as critical states and phasons. Nevertheless, the systematic understanding of the Landau levels of quasicrystals in the presence of the magnetic field has not been established yet. One of the main obstacles is the complication of the quasiperiodic tilings without periodic length scales, thus it has been thought that the system cannot possess any universal features of the Landau levels. In this paper, contrary to these assertions, we develop a generic theory of the Landau levels for quasicrystals. Focusing on the two dimensional quasicrystals with rotational symmetries, we highlight that quasiperiodic tilings induce anomalous Landau levels where electrons are localized near the rotational symmetry centers. Interestingly, the localization length of these Landau levels has a universal dependence on n for quasicrystals with n-fold rotational symmetry. Furthermore, macroscopically degenerate zero energy Landau levels are present due to the chiral symmetry of the rhombic tilings. In this case, each Landau level forms an independent island where electrons are trapped at given fields, but with field control, the interference between the islands gives rise to an abrupt change in the local density of states. Our work provide a general scheme to understand the electron localization behavior of the Landau levels in quasicrystals.

Edge channels of broken-symmetry quantum Hall states in graphene visualized by atomic force microscopy

The quantum Hall (QH) effect, a topologically non-trivial quantum phase, expanded the concept of topological order in physics bringing into focus the intimate relation between the “bulk” topology and the edge states. The QH effect in graphene is distinguished by its four-fold degenerate zero energy Landau level (zLL), where the symmetry is broken by electron interactions on top of lattice-scale potentials. However, the broken-symmetry edge states have eluded spatial measurements. In this article, we spatially map the quantum Hall broken-symmetry edge states comprising the graphene zLL at integer filling factors of {{nu }}={{0}},pm {{1}} across the quantum Hall edge boundary using high-resolution atomic force microscopy (AFM) and show a gapped ground state proceeding from the bulk through to the QH edge boundary. Measurements of the chemical potential resolve the energies of the four-fold degenerate zLL as a function of magnetic field and show the interplay of the moiré superlattice potential of the graphene/boron nitride system and spin/valley symmetry-breaking effects in large magnetic fields.

Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides

We describe and classify the thresholds of the continuous spectrum and the resulting resonances for general formally self-adjoint elliptic systems of second-order differential equations with Dirichlet or Neumann boundary conditions in domains with cylindrical and periodic outlets to infinity (in waveguides). These resonances arise because there are “almost standing” waves, that is, non-trivial solutions of the homogeneous problem which do not transmit energy. We consider quantum, acoustic, and elastic waveguides as examples. Our main focus is on degenerate thresholds which are characterized by the presence of standing waves with polynomial growth at infinity and produce effects lacking for ordinary thresholds. In particular, we describe the effect of lifting an eigenvalue from the degenerate zero threshold of the spectrum. This effect occurs for elastic waveguides of a vector nature and is absent from the scalar problems for cylindrical acoustic and quantum waveguides. Using the technique of self-adjoint extensions of differential operators in weighted spaces, we interpret the almost standing waves as eigenvectors of certain operators and the threshold as the corresponding eigenvalue. Here the threshold eigenvalues and the corresponding vector-valued functions not decaying at infinity can be obtained by approaching the threshold (the virtual level) either from below or from above. Hence their properties differ essentially from the customary ones. We state some open problems.

Engineering the topological state transfer and topological beam splitter in an even-sized Su-Schrieffer-Heeger chain

The usual Su-Schrieffer-Heeger model with an even number of lattice sites possesses two degenerate zero energy modes. The degeneracy of the zero energy modes leads to the mixing between the topological left and right edge states, which makes it difficult to implement the state transfer via topological edge channel. Here, enlightened by the Rice-Male topological pumping, we find that the staggered periodic next-nearest neighbor hoppings can also separate the initial mixed edge states, which ensures the state transfer between topological left and right edge states. Significantly, we construct an unique topological state transfer channel by introducing the staggered periodic on-site potentials and the periodic next-nearest neighbor hoppings added only on the odd sites simultaneously, and find that the state initially prepared at the last site can be transfered to the first two sites with the same probability distribution. This special topological state transfer channel is expected to realize a topological beam splitter, whose function is to make the initial photon at one position appear at two different positions with the same probability. Further, we demonstrate the feasibility of implementing the topological beam splitter based on the circuit quantum electrodynamic lattice. Our scheme opens up a new way for the realization of topological quantum information processing and provides a new path towards the engineering of new type of quantum optical device.

Coalescing Majorana edge modes in non-Hermitian {mathscr{P}}{mathscr{T}}-symmetric Kitaev chain

A single unit cell contains all the information about the bulk system, including the topological feature. The topological invariant can be extracted from a finite system, which consists of several unit cells under certain environment, such as a non-Hermitian external field. We present an exact solvable non-Hermitian finite-size Kitaev chain with {mathscr{P}}{mathscr{T}}-symmetric chemical potentials at the symmetric point. The straightforward calculation shows that there are two kinds of Majorana edge modes in this model divided by {mathscr{P}}{mathscr{T}} symmetry-broken and unbroken. The one appeared in the {mathscr{P}}{mathscr{T}} symmetry-unbroken region can be seen as the finite-size projection of the conventional degenerate zero modes in a Hermitian infinite system with the open boundary condition. It indicates a possible variant of the bulk-edge correspondence: The number of Majorana edge modes in a finite non-Hermitian system can be the topological invariant to identify the topological phase of the corresponding bulk Hermitian system.

Entanglement teleportation between distinct decohered qubits mediated via single and double Majorana wire(s)

We investigate the decoherence effects from a spin environment on two detached qubits coupled by a single Majorana wire (SMW) and double Majorana wires (DMWs) respectively when teleporting entanglement between the distinct qubits. The role of the composite system parameters on entanglement and entanglement teleportation are examined. Our results shows that strong coupling between the spin environment and the dots modifies the coupling between the qubits and the Majorana fermions (MFs) thus lifting the degenerate zero energy state for the case of the SMW. The SMW is seen to undergo a topological quantum phase transition as seen from the stable asymptotic increase and decrease in its dynamics as the coupling between the qubits and the MFs are varied. We observed that the DMWs are robust against quantum fluctuations of the spin environment with a characteristic cyclic beating phenomenon. This behavior shows that information is mediated via a channel endowed with the anti-ferromagnetic phase and the topological phase that harbors the Majorana zero-energy state. We probe the extent to which the capacitive and pure coupling that defines the DMWs should be tune for entanglement generation between the two sites and thus enhancement of the teleported state. Our description for the Majorana zero modes is in the spin model for spin based quantum applications.

On localizing Futaki–Morita integrals at isolated degenerate zeros

We study the localization of Futaki–Morita integrals at isolated degenerate zeros by giving a streamlined exposition in the spirit of Bott and implementing the localization procedure for a holomorphic vector field on CPn with a maximally degenerate zero. This yields an essentially unique formula for these Futaki–Morita integral invariants without using a summation over multiple points.

Quasi-one Dimensional Topological Insulator: Möbius Molecular Devices in Peierls Transition**Supported by the National Natural Science Foundation of China under Grant No. 11504241 and the Natural Science Foundation of Shenzhen University under Grant No. 201551

We show that, assisted by the Peierls transition of lattice, as a quasi-one dimensional (Q1D) tight binding system, a Möbius molecular device can behave as a simple topological insulator. With the Peierls phase transition to form a domain wall, the solitonary zero modes exist as the ground state of this electron-phonon hybrid system, which is protected by the Z2 topology of the Möbius strip. The robustness of the ground state prevents these degenerate zero modes from their energy spectrum splitting caused by any perturbation.

Majorana wave-function oscillations, fermion parity switches, and disorder in Kitaev chains

We study the decay and oscillations of Majorana fermion wavefunctions and ground state (GS) fermion parity in one-dimensional topological superconducting lattice systems. Using a Majorana transfer matrix method, we find that Majorana wavefunction properties are encoded in the associated Lyapunov exponent, which in turn is the sum of two independent components: a `superconducting component' which characterizes the gap induced decay, and the `normal component', which determines the oscillations and response to chemical potential configurations. The topological phase transition separating phases with and without Majorana end modes is seen to be a cancellation of these two components. We show that Majorana wavefunction oscillations are completely determined by an underlying non-superconducting tight-binding model and are solely responsible for GS fermion parity switches in finite-sized systems. These observations enable us to analytically chart out wavefunction oscillations, the resultant GS parity configuration as a function of parameter space in uniform wires, and special parity switch points where degenerate zero energy Majorana modes are restored in spite of finite size effects. For disordered wires, we find that band oscillations are completely washed out leading to a second localization length for the Majorana mode and the remnant oscillations are randomized as per Anderson localization physics in normal systems. Our transfer matrix method further allows us to i) reproduce known results on the scaling of mid-gap Majorana states and demonstrate the origin of its log-normal distribution, ii) identify contrasting behavior of disorder-dependent GS parity switches for the cases of even versus odd number of lattice sites, and iii) chart out the GS parity configuration and associated parity switch points as a function of disorder strength.

Bosonic anomalies, induced fractional quantum numbers, and degenerate zero modes: The anomalous edge physics of symmetry-protected topological states

The boundary of symmetry-protected topological states (SPTs) can harbor new quantum anomaly phenomena. In this work, we characterize the bosonic anomalies introduced by the 1+1D non-onsite-symmetric gapless edge modes of 2+1D bulk bosonic SPTs with a generic finite Abelian group symmetry (isomorphic to $G=\prod_i Z_{N_i}=Z_{N_1} \times Z_{N_2} \times Z_{N_3} \times ...$). We demonstrate that some classes of SPTs (termed "Type II") trap fractional quantum numbers (such as fractional $Z_N$ charges) at the 0D kink of the symmetry-breaking domain walls; while some classes of SPTs (termed "Type III") have degenerate zero energy modes (carrying the projective representation protected by the unbroken part of the symmetry), either near the 0D kink of a symmetry-breaking domain wall, or on a symmetry-preserving 1D system dimensionally reduced from a thin 2D tube with a monodromy defect 1D line embedded. More generally, the energy spectrum and conformal dimensions of gapless edge modes under an external gauge flux insertion (or twisted by a branch cut, i.e., a monodromy defect line) through the 1D ring can distinguish many SPT classes. We provide a manifest correspondence from the physical phenomena, the induced fractional quantum number and the zero energy mode degeneracy, to the mathematical concept of cocycles that appears in the group cohomology classification of SPTs, thus achieving a concrete physical materialization of the cocycles. The aforementioned edge properties are formulated in terms of a long wavelength continuum field theory involving scalar chiral bosons, as well as in terms of Matrix Product Operators and discrete quantum lattice models. Our lattice approach yields a regularization with anomalous non-onsite symmetry for the field theory description. We also formulate some bosonic anomalies in terms of the Goldstone-Wilczek formula.

Aspects of Floquet bands and topological phase transitions in a continuously driven superlattice

Recently the creation of novel topological states of matter by a periodic driving field has attracted great attention. To motivate further experimental and theoretical studies, we investigate interesting aspects of Floquet bands and topological phase transitions in a continuously driven Harper model. In such a continuously driven system with an odd number of Floquet bands, the bands are found to have nonzero Chern numbers in general and topological phase transitions take place as we tune various system parameters, such as the amplitude or the period of the driving field. The nontrivial Floquet band topology results in a quantized transport of Wannier states in the lattice space. For certain parameter choices, very flat yet topologically nontrivial Floquet bands may also emerge, a feature that is potentially useful for the simulation of physics of strongly correlated systems. Some cases with an even number of Floquet bands may also have intriguing Dirac cones in the spectrum. Under open boundary conditions, anomalous counter-propagating chiral edge modes and degenerate zero modes are also found as the system parameters are tuned. These results should be of experimental interest because a continuously driven system is easier to realize than a periodically kicked system.

Thermodynamic properties of the electron gas in multilayer graphene in the presence of a perpendicular magnetic field

The thermodynamic properties of the electron gas in multilayer graphene depend strongly on the number of layers and the type of stacking. Here we analyse how those properties change when we vary the number of layers for rhombohedral stacked multilayer graphene and compare our results with those from a conventional two dimensional electron gas. We show that the highly degenerate zero energy Landau level which is partly filled with electrons and partly with holes has a strong influence on the value of the different thermodynamic quantities.

POPULATION BOUNDARIES FOR COMPACT WHITE DWARF BINARIES INLISA's AMPLITUDE-FREQUENCY DOMAIN

In an earlier investigation, we proposed population boundaries for both inspiralling and mass-transferring double white dwarf (DWD) systems in the distance independent "absolute" amplitude-frequency domain of the proposed space-based gravitational-wave (GW) detector, {\it LISA}. The degenerate zero temperature mass-radius (M-R) relationship of individual white dwarf stars that we assumed, in combination with the constraints imposed by Roche geometries, permits us to identify five key population boundaries for DWD systems in various phases of evolution. Here we use the non-zero entropy donor M-R relations of \cite{DB2003} to modify these boundaries for both DWD and neutron star-white dwarf (NSWD) binary systems. We find that the mass-transferring systems occupy a larger fraction of space in ``absolute'' amplitude-frequency domain compared to the simpler T=0 donor model. We also discuss how these boundaries are modified with the new evolutionary phases found by \cite{Deloyeetal2007}. In the initial contact phase, we find that the contact boundaries, which are the result of end of inspiral evolution, would have some width, as opposed to an abrupt cut-off described in our earlier T=0 model. This will cause an overlap between a DWDs $&$ NSWDs evolutionary trajectories, making them indistinguishable with only LISA observations within this region. In the cooling phase of the donor, which follows after the adiabatic donor evolution, the radius contracts, mass-transfer rate drops and slows down the orbital period evolution. Depending upon the entropy of the donor, these systems may then lie inside the fully degenerate T=0 boundaries, but LISA may be unable to detect these systems as they might be below the sensitivity limit or within the unresolved DWD background noise.

Local dynamics and gravitational collapse of a self-gravitating magnetized Fermi gas

We use the Bianchi-I spacetime to study the local dynamics of a magnetized self-gravitating Fermi gas. The set of Einstein–Maxwell field equations for this gas becomes a dynamical system in a 4D phase space. We consider a qualitative study and examine numeric solutions for the degenerate zero temperature case. All dynamic quantities exhibit similar qualitative behavior in the 3D sections of the phase space, with all trajectories reaching a stable attractor whenever the initial expansion scalar H 0 is negative. If H 0 is positive the trajectories end up in a curvature singularity that can be, depending on initial conditions, isotropic or anisotropic. In particular, if the initial magnetic field intensity is sufficiently large the collapsing singularity will always be anisotropic and pointing in the same direction of the field.

Late-time oscillatory behaviour for self-gravitating scalar fields

This paper investigates the late-time behaviour of certain cosmological models where oscillations play an essential role. Rigorous results are proved on the asymptotics of homogeneous and isotropic spacetimes with a linear massive scalar field as source. Various generalizations are obtained for nonlinear massive scalar fields, k-essence models and f(R) gravity. The effect of adding ordinary matter is discussed as is the case of nonlinear scalar fields whose potential has a degenerate zero.

Nonexponential dynamics and competition between quasi-trapped states in an N-atomic micromaser

Transient processes are examined for the reduced density matrix (RDM) of the quantum mode of a micromaser pumped by clusters of N two-level atoms. The RDM dynamics consists of the fast and slow stages. The hierarchy of time scales is explained by the fact that the spectrum of the operator of RDM evolution contains groups of eigenvalues concentrated near zero and unity. A convenient basis for describing the fast stage is provided by a set of eigenvectors and adjoint vectors of Jordan cells related to the degenerate zero eigenvalue. The dynamics of the adjoint vector as a function of the number of passed clusters is nonexponential. One adjoint vector transforms jumpwise into the neighboring vector upon the passage of the next cluster. The slow stage of dynamics is controlled by the quasi-trapped states (QTSs) of the field. These states are the generalization of the trapped states of the ideal one-atom model and can be represented as linear combinations of the long-lived eigenvectors of the evolution operator with eigenvalues close to unity. An important feature of the QTSs is their stability to fluctuations of the number of atoms in clusters and to the field relaxation rate.

Holomorphic vector fields with totally degenerate zeroes

A zero set of a holomorphic vector field is totally degenerate, if the endomorphism of the conormal sheaf induced by the vector field is identically zero. By studying a class of foliations generalizing foliations of -actions, we show that if a projective manifold admits a holomorphic vector field with a smooth totally degenerate zero component, then the manifold is stably birational to that component of the zero set. When the vector field has an isolated totally degenerate zero, we prove that the manifold is rational. This is a special case of Carrell's conjecture.

On partner potentials in parasupersymmetric quantum mechanics

The constraint on the two superpotentials characterizing a parasuperhamiltonian is shown to be a Riccati equation which helps us for studying partner potentials. We point out that only three different types of parasuperspectra can appear. The simplest type admitting a unique non degenerate zero energy value can be associated with different models already discussed in other contexts.

Analysis of chemical kinetic systems over the entire parameter space. III. A wet combustion system

A simple generalization of Semenov thermal explosion theory is studied here; one exothermic reaction taking place at the same time as a second one involving water as a reactant. Evaporation and condensation are also considered. The resulting two ordinary differential equations containing three parameters are treated by recently developed methods outlined in the first paper of this series. This model is a simple representation of the combustion of wet cellulosic materials such as bagasse, cotton and sawdust. The theory predicts an amazing variety of behaviour arising in association with the degenerate singularities that occur. These include degeneracies up to and including the quartic fold, H3 2 bifurcation and degenerate double zero eigenvalue points.