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Motivated by recent scanning tunneling microscopy experiments on Fe atomic line defect in iron-based high temperature superconductors, we explore the origin of the zero energy bound states near the endpoints of the line defect by employing the two-orbit four-band tight binding model. With increasing the strength of the Rashba spin-orbit coupling along the line defect, the zero energy resonance peaks move simultaneously forward to negative energy for $s_{+-}$ pairing symmetry, but split for $s_{++}$ pairing symmetry. The superconducting order parameter correction due to As(Te, Se) atoms missing does not shift the zero energy resonance peaks. Such the zero energy bound states are induced by the weak magnetic order rather than the strong Rashba spin-orbit coupling on Fe atomic line defect., Comment: 5 pages,5 figures. arXiv admin note: text overlap with arXiv:1705.06570

In this reply, I point out that the comment by Jacques H.H. Perk on my paper "Exact Solution for Three-Dimensional Ising Model" [Symmetry 2021, 13, 1837] is wrong because the author seems not to understand the Onsager operators along both {\bf X} and {\bf Y} directions and the relation between them., Comment: 2 pages, Added Reply to Reply to Symmetry 2023, 15, 375. Updated title

Three-dimensional Ising model in zero external field is exactly solved by operator algebras, similar to the Onsager's approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order-disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented., Comment: 10 pages, 2 figures

Quantum Rabi model has been exactly solved by employing the parameter-dependent unitary transformation method in both the occupation number representation and the Bargmann space. The analytical expressions for the complete energy spectrum consisting of two double-fold degenerate sub-energy spectra are presented in the whole range of all the physical parameters. Each energy level is determined by a parameter in the unitary transformation, which obeys a highly nonlinear equation. The corresponding eigenfunction is a convergent infinite series in terms of the physical parameters. Due to the level crossings between the neighboring eigenstates at certain physical parameter values, such the degeneracies could lead to novel physical phenomena in the two-level system with the light-matter interaction., Comment: 7 pages, 6 figures, NJP(accepted)

The local electronic states around a single As (Te, Se) vacancy are investigated in order to shed light on the role of ligands in a series of iron-based superconductors. Such a vacancy can produce a local hopping correction ranging from $-0.22$ eV to 0.12 eV and always induce two in-gap resonance peaks in the local density of states (LDOS) at the fixed symmetrical bias voltages, which are rather robust and irrelevant to the phase of superconducting order parameter. The LDOS images near the defect predominantly possess $0^o$ and $45^o$ stripes. These energy-dependent charge modulations created by quasiparticle interference are originated in the nesting effect between the inner (outer) hole Fermi surface around $\Gamma$ point and the inner (outer) electron Fermi surface around $M$ point., Comment: 4 pages, 5 figures

We have investigated the complex two-dimensional electron system with unequal Rashba and Dresselhaus spin-orbit interactions in the presence of a perpendicular magnetic field. The spin polarizations are obtained in a wide range of magnetic fields. It is shown that such a system is hard to be magnetized. We also find that the resonant charge and spin Hall conductances occurs simultaneously at a certain magnetic field, at which two (nearly) degenerate Landau levels are filled partly. The resonant Hall effects are universal in this type of semiconductor materials, and could have potential application for semiconductor spintronics., Comment: 4 pages, 2 figures. arXiv admin note: text overlap with arXiv:0804.1138

A pronounced local in-gap zero-energy bound state (ZBS) has been observed by recent scanning tunneling microscopy (STM) experiments on the interstitial Fe impurity (IFI) and its nearest-neighboring (nn) sites in $\mathrm{FeTe_{0.5}Se_{0.5}}$ superconducting (SC) compound. By introducing a new impurity mechanism, the so-called tunneling impurity, and based on the Bogoliubove-de Gennes (BDG) equations we investigated the low-lying energy states of the IFI and the underlying Fe-plane. We found the peak of ZBS does not shift or split in a magnetic field as long as the tunneling parameter between IFI and the Fe-plane is sufficiently small and the Fe-plane is deep in the SC state. Our results are in good agreement with the experiments. We also predicted that modulation of spin density wave (SDW), or charge density wave (CDW) will suppress the intensity of the ZBS., Comment: 5 pages, 4 figures

Based on a two-orbit four-band tight binding model, we investigate the low-lying electronic states around the interstitial excess Fe ions in the iron-based superconductors by using T-matrix approach. It is shown that the local density of states at the interstitial Fe impurity (IFI) possesses a strong resonance inside the gap, which seems to be insensitive to the doping and the pairing symmetry in the Fe-Fe plane, while a single or two resonances appear at the nearest neighboring (NN) Fe sites. The location and height of the resonance peaks only depend on the hopping $t$ and the pairing parameter $\Delta_I$ between the IFI and the NN Fe sites. These in-gap resonances are originated in the Andreev's bound states due to the quasiparticle tunneling through the IFI, leading to the change of the magnitude of the superconducting order parameter. When both $t$ and $\Delta_I$ are small, this robust zero-energy bound state near the IFI is consistent with recent scanning tunneling microscopy observations., Comment: 4 pages, 5 figures

By employing T-matrix approach, we investigate a nonmagnetic impurity located on the surface of three-dimensional topological insulators, where the time reversal symmetry is preserved. It is shown that the images of the local density of states (LDOS) around the single impurity have the dip-hump structures with six-fold symmetry at different bias voltages. The peaks are produced by quasiparticle interference while the local minima at the backscattering wave vectors are due to the absence of backscatterings. With increasing the bias voltage, the peaks and dips move forward to the location of the impurity. These dips at the backscattering wave vectors in the LDOS spectra can be regarded as a signature of the time reversal symmetry in the topological surface states, which could be observed by scanning tunneling microscopy., Comment: 4 pages, 4 figures

The eigenstates in the presence of a step defect (SD) along $x$ or y axis on the surface of topological insulators are exactly solved. It is shown that unlike the electronic states in conventional metals, the topological surface states across the SD can produce spin rotations. The magnitudes of the spin rotations depend on the height and direction of the SD. The oscillations of local density of states (LDOS) are characterized by a wave vector connecting two points on the hexagonal constant-energy contour at higher energies. The period of the oscillation caused by the SD along y axis is $\sqrt 3$ ($\frac{1}{\sqrt 3}$) times that induced by the SD along x axis at a larger positive (negative) bias voltage. With increasing the bias voltage, the period of the oscillation, insensitive to the strength of the SD, becomes smaller. At lower energies near the Fermi surface, the two types of wave vectors coexist in the LDOS modulations. These results are consistent qualitatively with recent observations of scanning tunneling microscopy., Comment: 4 pages, 4 figures

The impurity-induced quasiparticle interference(QPI) in the parent compounds of iron-pnictide superconductors is investigated based on a phenomenological two-orbital four-band model and T-matrix method. We find the QPI is sensitive to the value of the magnetic order which may vary from one compound to another. For small value of the magnetic order, the pattern of oscillation in the local density of states (LDOS) induced by the QPI exhibits two-dimensional characteristics, consistent with the standing wave state observed in the 1111 compound. For larger value of the magnetic order, the main feature of the spatial modulation of the LDOS is the existence of one-dimensional stripe structure which is in agreement with the nematic structure in the parent compound of the 122 system. In both cases the system shows $C_2$ symmetry and only in the larger magnetic order case, there exist in-gap bound states. The corresponding QPI in $q$-space is also presented. The patterns of modulation in the LDOS at nonzero energies are attributed to the interplay between the underlying band structure and Fermi surfaces., Comment: 9 pages,12 figures

The electronic and magnetic structures in the normal and superconducting states of iron pnictides are investigated by solving self-consistently the Bogoliubov-de Gennes equation. It is shown that strong electron correlations can induce domain walls, which separate regions with different spin density wave orders. At zero or low electron doping, $90^\circ$ domain walls are formed while anti-phase domain walls are produced at higher electron doping. On the domain walls, larger electron densities are always present. The results agree qualitatively with recent observations of scanning tunneling microscopy and superconducting quantum interference device microscopy., Comment: 5 pages,5 figures

This is a reply to the comment by Daghofer and Moreo.

We study theoretically the coexistence of the spin-density-wave (SDW) and superconductivity in electron-doped iron-pnictide superconductors based on the two orbital model and Bogoliubov-de Gennes equations. The phase diagram is mapped out and the evolution of the Fermi surface as the doping varies is presented. The local density of states (LDOS) has also been calculated from low to high doping. We show that the strength of the superconducting coherent peak at the positive energy gets enhanced and the one at the negative energy is suppressed by the SDW order in the underdoped region. Several features of our results are in good agreement with the experiments., Comment: 5 pages, 3 figures

The low energy band structure of the FeAs based superconductors is fitted by a tight binding model with two Fe ions per unit cell and two degenerate orbitals per Fe ion. Based on this, superconductivity with extended s-wave pairing symmetry of the form $\cos k_x +\cos k_y$ is examined. The local density of states near an impurity is also investigated by using T-matrix approach. For the nonmagnetic scattering potential, we found that there exist two major resonances inside the gap. The height of the resonance peaks depends on the strength of the impurity potential. These in-gap resonances are originated in the Andreev's bound states due to the quasiparticle scattering between the hole Fermi surfaces around $\Gamma$ point with positive order parameter and the electron Fermi surfaces around $M$ point with negative order parameter., Comment: 5 pages, 5 figures

We have studied tunneling-mediated local density of states (LDOS) of the surface layer of a bilayer cuprate, where a Zn impurity is located on the second Cu-O layer. When the tunneling strength between two Cu-O layers is larger than a critical value, the LDOS on the site just above the Zn impurity first exhibits a resonant peak near the Fermi surface. The larger the tunneling strength, the stronger the resonant peak. It is also shown that the height of the resonant peak oscillates decreasingly with the distance from the site just above the Zn impurity. The location of the resonant peak in the surface LDOS depends on doping, energy gap, and the tunneling strength, and has an opposite bias voltage to that on its nearest neighboring sites. The results could be tested by the STM experiments and be used to further understand the electronic properties of high temperature superconductors., Comment: 5 pages, 4 figures

We have studied transport properties in a two-dimensional electron gas with equal Rashba and Dresselhaus spin-orbit interactions under a perpendicular magnetic field. By employing the exact solution for this system, we found resonant charge and spin Hall conductances at a certain magnetic field, where all the nearest-neighboring Landau levels cross. Near this value of magnetic field, there exists a resonant spin polarization, which can also induce resonant charge and spin Hall effects., Comment: 4 pages, 2 figures

We study the effect of the Coulomb interaction on the local density of states (LDOS) and its Fourier component in disordered cuprates. It is shown that the Coulomb interaction suppresses strongly the maximum value of the LDOS induced by the dopant impurity at each energy and expands significantly the Friedel oscillation in real space. The existence of the Coulomb interaction with a moderate strength yields an energy-dependent checkerboard LDOS modulation around the impurity, which is very different from that produced by pure quasiparticle interference. The orientation and transformation of the checkerboard pattern with energy and the relations among the modulation vectors, dopings and the bias voltages agree qualitatively with the recent STM experiments., Comment: 4 pages, 3 figures

We study a two-dimensional electron system in the presence of both Rashba and Dresselhaus spin-orbit interactions in a perpendicular magnetic field. Defining two suitable boson operators and using the unitary transformations we are able to obtain the exact Landau levels in the range of all the parameters. When the strengths of the Rashba and Dresselhaus spin-orbit interactions are equal, a new analytical solution for the vanishing Zeeman energy is found, where the orbital and spin wave functions of electron are separated. It is also shown that in this case the Zeeman and spin-orbit splittings are independent of the Landau level index $n$. Due to the Zeeman energy, new crossing between the eigenstates $|n, k, s=1, \sigma>$ and $|n+1, k, s^\prime =-1, \sigma^\prime>$ is produced at certain magnetic field for larger Rashba spin-orbit coupling. This degeneracy leads to a resonant spin Hall conductance if it happens at the Fermi level., Comment: 4 pages, 1 figure

The local density of states (LDOS) and its Fourier component induced by a unitary impurity in a supercurrent-carrying d-wave superconductor are investigated. Both of these quantities possess a reflection symmetry about the line passing through the impurity site and along the supercurrent if it is applied along the antinodal or nodal direction. With increasing supercurrent, both the coherence and resonant peaks in the LDOS are suppressed and slightly broadened. Under a supercurrent along the antinodal direction, the coherence peaks split into double peaks. The modulation wavevectors associated with elastic scatterings of quasiparticles by the defect from one constant-energy piece of the Fermi surface to another are displayed as bright or dark spots in the Fourier space of the LDOS image, and they may be suppressed or enhanced, and shifted depending on the applied current and the bias voltage., Comment: 5 pages, 6 figures

The low-temperature conductance (G) characteristics between a normal metal and a clean superconductor (S) carrying a supercurrent $I_s$ parallel to the interface is theoretically investigated. Increasing $I_s$ causes lowering and broadening of (1) coherence peaks of s-wave S, and d-wave S at (100) contact, (2) midgap-states-induced zero-bias conductance peak for d-wave S at (110) contact, and (3) Andreev-reflection-induced enhancement of $G$ within the gap near the metallic-contact limit. Novel features found include a current-induced central peak and a three-humped structure at intermediate barrier strength, etc., Comment: 4 pages, 4 figures