Bogoliubov-de Gennes Approach Research Articles

Superconducting and normal properties of the low-dimensional phases of the metallic hydrogen

In the first part of the paper we consider properties of a two-dimensional low density (n = 1) electron system with strong onsite Hubbard attraction U >W (W is the bandwidth) in the presence of a strong random potential V uniformly distributed in the range from -V to +V. In the framework of the Bogoliubov-de Gennes approach we observed an appearance of inhomogeneous state of spatially separated Fermi-Bose mixture of Cooper pairs and unpaired electrons with the formation of bosonic droplets of different size in the unpaired fermionic matrix. Software package to solve the system of Eliashberg equations that describes the transition of a metal to a superconducting state is presented. We use a version of the Eliashberg system of equations written on the imaginary axis. The system is presented in two versions: without taking into account the corrections for the chemical potential of electrons and with taking them into account. The calculations were performed for the I41/amd phase of metallic hydrogen at a pressure of 500 GPa.

Influence of d-wave superconductor orientation on Josephson current and phase difference in junctions with inhomogeneous ferromagnet

Abstract Utilizing a microscopic Bogoliubov-de Gennes approach we study Josephson effect in arbitrarily oriented d-wave superconductors connected through two ferromagnets with noncollinear magnetizations, by solving scattering problem. The critical Josephson current is calculated for various parameters of the junction. Mutual influence of d-wave superconducting electrodes orientation and angle α between magnetization in ferromagnetic bilayer of thickness d is analyzed. For various orientation of superconducting electrodes, we calculate (d,α) phase diagram, and discuss the possibility to achieve except coexistence of two stable states 0 and π, also coexistence of three stable states 0, π and π∕2, by varying the angle between magnetizations. We observe also areas of coexistence of stable and metastable states.

Half-quantum vortices on c-axis domain walls in chiral p-wave superconductors

Chiral superconductors are two-fold degenerate and domains of opposite chirality can form, separated by domain walls. There are indications of such domain formation in the quasi two-dimensional putative chiral p-wave superconductor Sr2RuO4, yet no experiment has explicitly resolved individual domains in this material. In this work, c-axis domain walls lying parallel to the layers in chiral p-wave superconductors are explored from a theoretical point of view. First, using both a phenomenological Ginzburg–Landau and a quasiclassical Bogoliubov-de Gennes approach, a consistent qualitative description of the domain wall structure is obtained. While these domains are decoupled in the isotropic limit, there is a finite coupling in anisotropic systems and the domain wall can be treated as an effective Josephson junction. In the second part, the formation and structure of half-quantum vortices on such c-axis domain walls are discussed.

Nonequilibrium quantum phase transition in a spinor quantum gas in a lattice coupled to a membrane

Recently, a novel kind of hybrid atom-optomechanical system, consisting of atoms in a lattice coupled to a membrane, has been experimentally realized [Vochezer {\it et al.,} Phys. Rev. Lett. \textbf{120}, 073602 (2018)], which promises a viable contender in the competitive field of simulating non-equilibrium many-body physics. Here we are motivated to investigate a spinor Bose gas coupled to a vibrational mode of a nano-membrane, focusing on analyzing the role of the spinor degrees of freedom therein. Through an adiabatic elimination of the degrees of freedom of the quantum oscillator, we derive an effective Hamiltonian which reveals a competition between the force localizing the atoms and the membrane displacement. We analyze the dynamical stability of the steady state using Bogoliubov-de Gennes approach and derive the stationary phase diagram in the parameter space. We investigate the non-equilibrium quantum phase transition from a localized symmetric state of the atom cloud to a shifted symmetry-broken state, where we present a detailed analysis of the effects of the spin degree of freedom. Our work presents a simple way to study the effects of the spinor degree of freedom on the non-equilibrium nonlinear phenomena that is complementary to ongoing experiments on the hybrid atom-optomechanical system.

Expanding ring-shaped Bose-Einstein condensates as analogs of cosmological models: Analytical characterization of the inflationary dynamics

We analytically study the expansion of a Bose-Einstein condensate in a ring-shaped trap with an increasing central radius. The evolution of the ground state is described using a scaling transform. Additionally, the dynamics of excited azimuthal modes over the varying ground state is analyzed through a generalization of the Bogoliubov-de Gennes approach. Our results explain some of the features observed in recent experiments focused on testing the applicability of the system as a parallel of cosmological inflationary models. The radial dynamics, which corresponds to the inflaton field of the cosmological counterpart, is analytically characterized: The expansion is found to induce the oscillatory displacement of the condensate as well as the coupled variation of the radial and vertical widths. Our findings account also for the observed redshift and emergence of the Hubble friction in the evolution of initially-prepared azimuthal modes. Our description, which traces the role of the different components of the setup in the expansion, enhances the controllability, and, therefore, the potential of the system as a ground for emulating the inflationary dynamics of cosmological models.

Conventional superconductivity in quasicrystals

Motivated by a recent experimental observation of superconductivity in the Al-Zn-Mg quasicrystal, we study the low-temperature behavior of electrons moving in the quasiperiodic potential of the Ammann-Beenker tiling in the presence of a local attraction. We employ the Bogoliubov-de Gennes approach for approximants of different sizes and determine the local pairing amplitude $\Delta_{i}$ as well its spatial average, $\Delta_{0}$, the superconducting order parameter. Due to the lack of periodicity of the octagonal tiling, the resulting superconducting state is inhomogeneous, but we find no evidence of the superconductivity islands, as observed in disordered systems, with $\Delta_{i}\rightarrow0$ at $T_{c}$ for all sites. In the weak-coupling regime, we find that the superconducting order parameter depends appreciably on the approximant size only if the Fermi energy sits at a pseudogap in the noninteracting density of states, with $\Delta_{0}$ decreasing as the system size increases. These results are in line with the experimental observations for the Al-Zn-Mg quasicrystal, and they suggest that, despite their electronic structure, quasicrystals are prone to display conventional BCS-like superconductivity.

Nonuniform superconductivity and Josephson effect in a conical ferromagnet

Using the Gorkov equations, we provide an exact solution for a one-dimensional model of superconductivity in the presence of a conical helicoidal exchange field. Due to the special type of symmetry of the system, the superconducting transition always occurs into a nonuniform superconducting phase (in contrast with the Fulde-Ferrell-Larkin-Ovchinnikov state, which appears only at low temperatures). We directly demonstrate that the uniform superconducting state in our model carries a current and thus does not correspond to the ground state. We study in the framework of the Bogoliubov-de Gennes approach the properties of the Josephson junction with a conical ferromagnet as a weak link. In our numerical calculations, we do not use any approximations (such as, e.g., a quasiclassical approach), and we show a realization of an anomalous $\phi_{0}$ junction (with a spontaneous phase difference $\phi_{0}$ in the ground state). The spontaneous phase difference $\phi_{0}$ strongly increases at high values of the exchange field near the borderline with a half-metal, and it exists also in the half-metal regime.

Phase separation and proximity effects in itinerant ferromagnet/superconductor heterostructures

Heterostructures made of itinerant ferromagnets and superconductors are studied. In contrast to most previous models, ferromagnetism is not enforced as an external Zeeman field but induced in a correlated single-band model (CSBM) that displays itinerant ferromagnetism as a mean-field ground state. This allows us to investigate the influence of an adjacent superconducting layer on the properties of the ferromagnet in a self-consistent Bogoliubov-de Gennes approach. The CSBM displays a variety features not present in the Zeeman exchange model that influence the behavior of order parameters close to the interface, as e.g. phase separation and the competition between magnetism and superconducting orders.

Topological features of the thermal Hall conductivity in a chiral p-wave superconductor under strain

Motivated by experimental studies on the effect of tensile and compressive strain or uniaxial pressure in the superconducting phase of the transition metal oxide Sr2RuO4, the pairing symmetry assuming the chiral p-wave superconducting state and the transport property are investigated theoretically by means of the self-consistent Bogoliubov-de Gennes approach. The uniaxial strain leads to the Fermi surface deformation. The topological properties are sensitive to the Fermi surface geometry induced by the strain and switch around the Lifshitz transition in the γ band. The temperature dependence of the thermal Hall conductivity is connected with the amplitude of the strain. While the topological number is reflected in the temperature linear contribution in the low-temperature limit of the thermal Hall conductivity, the deviation from this behavior depends on the pairing channel. Our result may enable us to provide information on the gap function of the chiral p-wave phase.

Quantum engineering of Majorana quasiparticles in one-dimensional optical lattices

We propose a feasible way of engineering Majorana-type quasiparticles in ultracold fermionic gases on a one-dimensional (1D) optical lattice. For this purpose, imbalanced ultracold atoms interacting by the spin–orbit coupling should be hybridized with a three-dimensional Bose–Einstein condensate molecular cloud. We show that the Majorana-type excitations can be created or annihilated upon constraining the profile of a trapping potential and/or an internal scattering barier. This process is modeled within the Bogoliubov–de Gennes approach. Our study is relevant also to nanoscopic 1D superconductors, where both potentials can be imposed by electrostatic means.

Yu-Shiba-Rusinov states of impurities in a triangular lattice ofNbSe2with spin-orbit coupling

We study topography of the spin-polarized bound states of magnetic impurities embedded in triangular lattice of a superconducting host. Such states have been observed experimentally in 2H-NbSe$_{2}$ crystal [G. C. M'enard et al., Nat.Phys. 11, 1013 (2015)] and revealed the oscillating particle-hole asymmetry extending to tens of nanometers. Using the Bogoliubov-de Gennes approach we explore the Yu-Shiba-Rusinov states in presence of spin-orbit interaction. We also study the bound states of double impurities for several relative positions in a triangular lattice.

Majorana ϕ0-junction in a disordered spin-orbit coupling nanowire with tilted magnetic field

Majorana Josephson junctions in nanowire systems exhibit a pseudo-4π period current-phase relation in the clean limit. In this work, we study how this current-phase relation responds to a tilted magnetic field in a disordered Majorana Josephson junction within the Bogoliubov-de Gennes approach. We show that the tilted magnetic field induces a ϕ0 phase shift to the current-phase relation. Most importantly, we find that this ϕ0-junction behavior is robust even in the presence of disorders.

Thermal Hall conductivity of spin-triplet superconductor with time reversal symmetry breaking

The thermal Hall conductivity is investigated for the superconductor Sr2RuO4 assuming a time reversal symmetry breaking order parameter. The γ band of Sr2RuO4 has its Fermi level near the van Hove points, close to the Lifshitz transition. Within a Bogoliubov-de Gennes approach we calculate the thermal Hall conductivity for pairing symmetries including two exemplary cases, the usual chiral p-wave phase and a fx2−y2-wave phase of the structure . In the chiral p-wave phase the thermal Hall conductivity consists of a universal temperature-linear term and an exponential correction due to quasiparticle activation with a full excitation gap. Due to line-nodes in the gap we find a correction which is quadratic in temperature for fx2−y2-wave state. This difference in the corrections allows to analyze the gap structure of the superconducting phase.

Reduced-Shifted Conjugate-Gradient Method for a Green’s Function: Efficient Numerical Approach in a Nano-Structured Superconductor

We propose the reduced-shifted Conjugate-Gradient (RSCG) method, which is numerically efficient to calculate a matrix element of a Green's function defined as a resolvent of a Hamiltonian operator, by solving linear equations with a desired accuracy. This method does not calculate solution vectors of linear equations but does directly calculate a matrix element of the resolvent. The matrix elements with different frequencies are simultaneously obtained. Thus, it is easy to calculate the exception value expressed as a Matsubara summation of these elements. To illustrate a power of our method, we choose a nano-structured superconducting system with a mean-field Bogoliubov-de Gennes (BdG) approach. This method allows us to treat with the system with the fabrication potential, where one can not effectively use the kernel-polynomial-based method. We consider the d-wave nano-island superconductor by simultaneously solving the linear equations with a large number (~ 50000) of Matsubara frequencies.

Interplay between Rashba spin-orbit coupling and adiabatic rotation in a two-dimensional Fermi gas

We explore the trap profiles of a two-dimensional atomic Fermi gas in the presence of a Rashba spin-orbit coupling and under an adiabatic rotation. We first consider a non-interacting gas and show that the competition between the effects of Rashba coupling on the local density of single-particle states and the Coriolis effects caused by rotation gives rise to a characteristic ring-shaped density profile that survives at experimentally-accessible temperatures. Furthermore, Rashba splitting of the Landau levels takes the density profiles on a ziggurat shape in the rapid-rotation limit. We then consider an interacting gas under the BCS mean-field approximation for local pairing, and study the pair-breaking mechanism that is induced by the Coriolis effects on superfluidity, where we calculate the critical rotation frequencies both for the onset of pair breaking and for the complete destruction of superfluidity in the system. In particular, by comparing the results of fully-quantum-mechanical Bogoliubov-de Gennes approach with those of semi-classical local-density approximation, we construct extensive phase diagrams for a wide-range of parameter regimes in the trap where the aforementioned competition may, e.g., favor an outer normal edge that is completely phase separated from the central superfluid core by vacuum.

High critical temperature nodal superconductors as building block for time-reversal invariant topological superconductivity

We study possible applications of high critical temperature nodal superconductors for the search for Majorana bound states in the DIII class. We propose a microscopic analysis of the proximity effect induced by d-wave superconductors on a semiconductor wire with strong spin-orbit coupling. We characterize the induced superconductivity on the wire employing a numerical self-consistent tight-binding Bogoliubov-de Gennes approach, and analytical considerations on the Green's function. The order parameter induced on the wire, the pair correlation function, and the renormalization of the Fermi points are analyzed in detail, as well as the topological phase diagram in the case of weak coupling. We highlight optimal Hamiltonian parameters to access the nontrivial topological phase which could display time-reversal invariant Majorana doublets at the boundaries of the wire.

Anderson localization of Cooper pairs and Majorana fermions in an ultracold atomic Fermi gas with synthetic spin-orbit coupling

We theoretically investigate two-particle and many-particle Anderson localizations of a spin-orbit coupled ultracold atomic Fermi gas trapped in a quasi-periodic potential and subjected to an out-of-plane Zeeman field. We solve exactly the two-particle problem in a finite length system by exact diagonalization and solve approximately the many-particle problem within the mean-field Bogoliubov-de Gennes approach. At a small Zeeman field, the localization properties of the system are similar to that of a Fermi gas with conventional $s$-wave interactions. As the disorder strength increases, the two-particle binding energy increases and the fermionic superfluidity of many particles disappears above a threshold. At a large Zeeman field, where the interatomic interaction behaves effectively like a $p$-wave interaction, the binding energy decreases with increasing disorder strength and the resulting topological superfluidity shows a much more robust stability against disorder than the conventional $s$-wave superfluidity. We also analyze the localization properties of the emergent Majorana fermions in the topological phase. Our results could be experimentally examined in future cold-atom experiments, where the spin-orbit coupling can be induced artificially by using two Raman lasers, and the quasi-periodic potential can be created by using bichromatic optical lattices.

Helical Spin Texture and Interference of Majorana Bound States in One-Dimensional Topological Superconductor

We investigate one-dimensional (1D) Majorana bound states (MBSs) realized in terms of the helical edge states of a 2D quantum spin-Hall insulator (QSHI) in a heterostructure with a superconducting substrate and two ferromagnetic insulators (FIs). By means of Bogoliubov-de Gennes approach we demonstrate that there is a helical spin texture in the MBS wave function with a pitch proportional to the Fermi momentum of the helical edge states of QSHI. Moreover, simultaneous detection on local density of states by scanning tunneling microscopy and spectroscopy at a position close to one FI edge and at the midpoint between two FIs can not only map out the energy spectrum $\pm E \cos(\phi/2)$, but also prove experimentally that the two quasiparticle excitations do not mix with each other as protected by the parity conservation associated with the MBSs.

Evolution of Density of States and a Spin-Resolved Checkerboard-Type Pattern Associated with the Majorana Bound State.

In terms of the Bogoliubov-de Gennes approach, we investigate the Majorana bound state (MBS) in a vortex of proximity-induced superconductivity on the surface of a topological insulator. Mapping out the local density of states (LDOS) of quasiparticle excitations as a function of energy and distance from the vortex center, it is found that the spectral distribution evolves from a V shape to a Y shape with the emergence of a MBS upon variation of the chemical potential, consistent with the STM/STS measurement in a very recent experiment [Xu et al., Phys. Rev. Lett. 114, 017001 (2015)] on a Bi(2)Te(3) thin layer on the top of NbSe(2). Moreover, we demonstrate that there is a checkerboard-type pattern in the relative LDOS between the spin-up and -down channels, where the quantum mechanical wave function of the MBS manifests itself clearly as a single quantum state. Therefore, a spin-resolved STM/STS technique is expected to be able to provide phase-sensitive evidence for a MBS in the vortex core of a topological superconductor.

Zero-energy peak and triplet correlations in nanoscale superconductor/ferromagnet/ferromagnet spin valves

Using a self-consistent Bogoliubov-de Gennes approach, we theoretically study the proximity-induced density of states (DOS) in clean SFF spin-valves with noncollinear exchange fields. Our results clearly demonstrate a direct correlation between the presence of a zero energy peak (ZEP) in the DOS spectrum and the persistence of spin-1 triplet pair correlations. By systematically varying the geometrical and material parameters governing the spin-valve, we point out to experimentally optimal system configurations where the ZEPs are most pronounced, and which can be effectively probed via scanning tunneling microscopy. We complement these findings in the ballistic regime by employing the Usadel formalism in the full proximity limit to investigate their diffusive SFF counterparts. We determine the optimal normalized ferromagnetic layer thicknesses which result in the largest ZEPs. Our results can serve as guidelines in designing samples for future experiments.