Bogoliubov-de Gennes Equations Research Articles

Critical end point and universality class of neutron P23 superfluids in neutron stars

We study the thermodynamics and critical behavior of neutron $^3P_2$ superfluids in the inner cores of neutron stars. $^3P_2$ superfluids offer a rich phase diagram including uniaxial/biaxial nematic phases, the ferromagnetic phase, and the cyclic phase. Using the Bogoliubov-de Gennes (BdG) equation as superfluid Fermi liquid theory, we show that a strong (weak) magnetic field drives the first (second) order transition from the dihedral-two biaxial nematic phase to dihedral-four biaxial nematic phase in low (high) temperatures, and their phase boundaries are divided by the critical endpoint (CEP). We demonstrate that the set of critical exponents at the CEP satisfies the Rushbrooke, Griffiths, and Widom equalities, indicating a new universality class. At the CEP, the $^3P_2$ superfluid exhibits critical behavior with nontrivial critical exponents, indicating a new universality class. Furthermore, we find that the Ginzburg-Landau (GL) equation up to the 8th-order expansion satisfies three equalities and properly captures the physics of the CEP. This implies that the GL theory can provide a tractable way for understanding critical phenomena which may be realized in the dense core of realistic magnetars.

Andreev reflection in nodal-line Weyl semimetal

Based on the Bogoliubov-de Gennes equation, the quantum scattering problem through a nodal-line Weyl semimetal based normal metal/superconductor heterojunction has been theoretically studied. Since the crystallographic anisotropy in the material, two different orientations between the crystalline axis and the superconducting interface have been revealed. Considering a heterojunction with the interface paralleling to the basal plane, it is found that Andreev reflection with due to Klein-like scattering gives rise to a perfect scattering. Deviation from the critical value, Andreev reflection falls down and normal reflection goes up. While the interface is perpendicular to the basal plane, the pure intra-mode retro-Andreev reflection (RAR) and inter-mode specular Andreev reflection (SAR) are manifested at the normal incident. Moreover, the reflection coefficient exhibits the reentrant behavior with the Fermi energy. Fundamentally, such features are a consequence of the torus-like iso-energy surfaces of the nodal-line Weyl semimetals, which is in sharp contrast to the case of conventional materials, graphene, and Weyl-point semimetals. Those novel scattering processes also result in a distinctive tunneling conductance, such as the sub-gap nonmonotonic features, the interface directional dependent zero bias conductances and the reentrant behavior, which can be served as a smoking gun to distinguish the mode-resolved Andreev reflections in experiments.

Berry phase in superconducting multiterminal quantum dots

We report on the study of the non-trivial Berry phase in superconducting multiterminal quantum dots biased at commensurate voltages. Starting with the time-periodic Bogoliubov-de Gennes equations, we obtain a tight binding model in the Floquet space, and we solve these equations in the semiclassical limit. We observe that the parameter space defined by the contact transparencies and quartet phase splits into two components with a non-trivial Berry phase. We use the Bohr-Sommerfeld quantization to calculate the Berry phase. We find that if the quantum dot level sits at zero energy, then the Berry phase takes the values $\varphi_B=0$ or $\varphi_B=\pi$. We demonstrate that this non-trivial Berry phase can be observed by tunneling spectroscopy in the Floquet spectra. Consequently, the Floquet-Wannier-Stark ladder spectra of superconducting multiterminal quantum dots are shifted by half-a-period if $\varphi_B=\pi$. Our numerical calculations based on Keldysh Green's functions show that this Berry phase spectral shift can be observed from the quantum dot tunneling density of states.

Josephson current through a ferromagnetic bilayer: Beyond the quasiclassical approximation

Based on the Bogoliubov-de Gennes equations, we provide an exact numerical solution for the critical current of Josephson junctions with a composite ferromagnetic bilayer. We demonstrate that for the antiparallel orientation of the magnetic moments of the bilayer, the presence of a potential barrier at the bilayer interface results in large oscillations of the critical current as a function of ferromagnet thickness and/or exchange field. Because of this, and remarkably, in the range of small exchange field and thicknesses, the magnetism leads to the increase of the critical current. This effect is well pronounced at low temperature but disappears near $T_c$. If the potential barrier is replaced by a spin-active barrier at the bilayer interface the conventional 0-$\pi$ transition, similar to the case of an uniform ferromagnetic Josephson junction, is observed. Strikingly, for a parallel orientation of the magnetic moments of the bilayer, the presence of the spin-active barrier restores the anomalous behavior---potential barrier in the antiparallel case. These behaviors result from the resonant tunneling of Cooper pairs across the composite barrier---an effect related to the spin-dependent Fermi vector in the presence of the ferromagnets' exchange field.

Retro-normal reflection and specular Andreev reflection in a transition metal dichalcogenides superconducting heterojunction

Based on the Bogoliubov-de Gennes equation, the spin-resolved scattering problem through a transition metal dichalcogenides normal metal/superconductor has been investigated. It is shown that the tunneling conductances have a close relationship with the nontrivial Rashba metallic states. Without the Rashba spin–orbit interaction (RSOI), the tunneling conductances exhibit clear characteristic features for detecting the normal Rashba metal state (NRMS) and the anomalous Rashba metal state (ARMS). While in the presence of RSOI, the oscillation, hump structure, and the reentrant behavior of the tunneling conductances can be served as a smoking gun to distinguish the Rashba ring metal state (RRMS) from those nontrivial metallic states. Fundamentally, in sharp contrast to the NRMS and the ARMS where only the normal reflection and the retro-Andreev reflection can occur, the additional scattering processes of the retro-normal reflection (RNR) and the specular Andreev reflection (SAR) can take place in the present junction with the RRMS. Thus the obtained results offer a feasible way to determine the RNR, the SAR, and the nontrivial Rashba metallic states based on the transition metal dichalcogenides superconducting heterojunction.

Non-Abelian Majorana fermions in topological s-wave Fermi superfluids

By solving the Bogoliubov–de Gennes equations analytically, we derive the fermionic zero-modes satisfying the Majorana property that exist in vortices of a two-dimensional s-wave Fermi superfluid with spin–orbit coupling and Zeeman spin-splitting. The Majorana zero-mode becomes normalisable and exponentially localised to the vicinity of the vortex core when the superfluid is topologically non-trivial. We calculate the energy splitting due to Majorana hybridisation and identify that the s-wave Majorana vortices obey non-Abelian statistics.

Effect of surface defects on order modulations in mesoscopic p-wave superconducting loops

The effect of surface defects on the modulations of order parameters in mesoscopic p-wave superconducting noncircular loops with threaded magnetic flux is investigated by numerically solving the Bogoliubov-de Gennes equations self-consistently. For the case of a square loop, the sharp-jump evolution of the total energy and the sawtooth pattern of the current in flux take place when a square indentation defect is introduced at the center of the outer or inner boundary, accompanied by phase transitions between the states with diagonal-symmetric and asymmetric spatial profiles of chiral orders. Particularly, peculiar spatial patterns of the orders can occur with moving the indentation defect farther from the edge center or positioning the bulge defect at the inner or outer edge of the sample. Furthermore, the cases of two rectangular loops with different outer sizes were investigated where we consider the effect of a square indentation defect in the middle of the short or the long side of the sample. More complicated phenomena on the order modes can be found due to the aspect-ratio effect as well as the defect effect.

Numerical methods for Bogoliubov-de Gennes excitations of Bose-Einstein condensates

In this paper, we study the analytical properties and the numerical methods for the Bogoliubov-de Gennes equations (BdGEs) describing the elementary excitation of Bose-Einstein condensates around the mean field ground state, which is governed by the Gross-Pitaevskii equation (GPE). Derived analytical properties of BdGEs can serve as benchmark tests for numerical algorithms and three numerical methods are proposed to solve the BdGEs, including sine-spectral method, central finite difference method and compact finite difference method. Extensive numerical tests are provided to validate the algorithms and confirm that the sine-spectral method has spectral accuracy in spatial discretization, while the central finite difference method and the compact finite difference method are second-order and fourth-order accurate, respectively. Finally, sine-spectral method is extended to study elementary excitations under the optical lattice potential and solve the BdGEs around the first excited states of the GPE. The numerical experiments demonstrate the efficiency and accuracy of the proposed methods for solving BdGEs.

Spin transfer torque and exchange coupling in Josephson junctions with ferromagnetic superconductor reservoirs

In this paper, the spin transfer torque (STT) and the exchange coupling of the Josephson junctions containing the interesting cases of diffusive/ballistic-triplet/singlet ferromagnetic superconductor (FS) materials are investigated. First, the diffusive FS1/Fc/FS2 structures with Fc being a junction consisting of ferromagnetic and normal metal parts as well as insulating barriers are investigated. Secondly, the ballistic Josephson junction containing the triplet chiral p/wave FS reservoirs is studied. Using the Nazarov quantum circuit theory for the diffusive structures, it is found that the antiparallel/parallel or vice versa parallel/antiparallel transition of the favorable exchange coupling takes place due to the appearance of the only out-of-plane STT. Furthermore, the analyze of the phase difference interval in which an interlayer length-induced antiparallel/parallel transition can be occurred, is performed. Afterward, the mentioned ballistic structure is dealt with solving the 16 16 Bogoliubov–de-Gennes equation. It is found that although the exchange fields of the FS are laid in the z and y direction, the STT interestingly exists in all three directions of x, y and z. This exciting finding suggests that the favorable equilibrium configuration concerning the least exchange coupling occurs in the relative exchange field direction different from 0 or .

Stability analysis of ground states in a one-dimensional trapped spin-1 Bose gas

In this work we study the stability properties of the ground states of a spin-1 Bose gas in presence of a trapping potential in one spatial dimension. To set the stage we first map out the phase diagram for the trapped system by making use of a, so-called, continuous-time Nesterov method. We present an extension of the method, which has been previously applied to one-component systems, to our multi-component system. We show that it is a powerful and robust tool for finding the ground states of a physical system without the need of an accurate initial guess. We subsequently solve numerically the Bogoliubov de-Gennes equations in order to analyze the stability of the ground states of the trapped spin-1 system. We find that the trapping potential retains the overall structure of the stability diagram, while affecting the spectral details of each of the possible ground state waveforms. It is also found that the peak density of the trapped system is the characteristic quantity describing dynamical instabilities in the system. Therefore replacing the homogeneous density with the peak density of the trapped system leads to good agreement of the homogeneous Bogoliubov predictions with the numerically observed maximal growth rates of dynamically unstable modes. The stability conclusions in the one-dimensional trapped system are independent of the spin coupling strength and the normalized trap strength over several orders of magnitude of their variation.

Gigantic T c enhancement in dirty nano-sized superconductor

Superconducting critical temperature Tc for a dirty nanosized square s-wave superconducting plate becomes five times higher than that of a pure bulk superconductor. Tc is obtained by solving the Bogoliubov-de Gennes equations with the finite element method. As the dirty superconductor, we consider a superconductor with a spatially random potential for superconducting electrons. We show that the enhancement of Tc comes from cooperation of size effects and impurity effects, both of which increase the local density of the Cooper pairs.

Impurity effects on nano-structured dirty superconductors: violation of Anderson’s Theorem

It is shown that in nano-structured s-wave superconductor, transition temperature Tc depends on random impurity potentials. This means violation of the Anderson’s theorem, which states that non-magnetic impurity does not affect the Tc of s-wave superconductor. We determine the impurity effects on Tc for nano-structured superconductor, using the finite element method to solve the Bogoliubov-de Gennes equations under spatially a random impurity potential. We show that some impurity potentials increase Tc but other impurity potentials decrease Tc. We find that the superconductor with localized order parameter shows increased Tc, which is contrary to expectation. Our results show that a dirty superconductor does not always mean weak superconductivity.

Topological crystalline superconductivity in Dirac semimetal phase of iron-based superconductors

In iron-based superconductors, band inversion of $d$- and $p$-orbitals yields Dirac semimetallic states. We theoretically investigate their topological properties in normal and superconducting phases, based on the tight-binding model involving full symmetry of the materials. We demonstrate that a Cooper pair between electrons with $d$- and $p$-orbitals relevant to the band structure yields odd-parity superconductivity. Moreover, we present the typical surface states by solving the Bogoliubov-de Gennes equation and characterize them by topological invariants defined with crystal symmetry. It is found that there appear various types of Majorana fermions such as surface flat band, Majorana quartet and M\"{o}bius twisted surface state. Our theoretical results show that iron-based superconductors are promising platforms to realize rich topological crystalline phases.

Bose polaron in spherical trap potentials: Spatial structure and quantum depletion

We investigate how the presence of a localized impurity in a Bose-Einstein condensate of trapped cold atoms that interact with each other weakly and repulsively affects the profile of the condensed and excited components at zero temperature. By solving the Gross-Pitaevskii and Bogoliubov-de Gennes equations, we find that an impurity-boson contact attraction (repulsion) causes both components to change in spatial structure in such a way as to be enhanced (suppressed) around the impurity, while slightly declining (growing) in a far region from the impurity. Such behavior of the quantum depletion of the condensate can be understood by decomposing the impurity-induced change in the profile of the excited component with respect to the radial and azimuthal quantum number. A significant role of the centrifugal potential and the "hole" excitation level is thus clarified.

Reexamining Ginzburg-Landau theory for neutron P23 superfluidity in neutron stars

The Ginzburg-Landau (GL) effective theory is a useful tool to study a superconductivity or superfluidity near the critical temperature, and usually the expansion up to the fourth-order in terms of order parameters is sufficient for the description of the second-order phase transition. In this paper, we discuss the GL equation for the neutron $^{3}P_{2}$ superfluidity relevant for interior of neutron stars. We derive the GL expansion up to the eighth order in the condensates and find that this order is necessary for the system to have the unique ground state, unlike the ordinary cases. Starting from the $LS$ potential, which provides the dominant attraction between two neutrons at the high density, we derive the GL equation in the path-integral formalism, where the auxiliary field method and the Nambu-Gor'kov representation are used. We present the detailed description for the trace calculation necessary in the derivation of the GL equation. As numerical results, we show the phase diagram of the neutron $^{3}P_{2}$ superfluidity on the plane spanned by the temperature and magnetic field, and we find that the eighth-order terms lead to a first-order phase transition, whose existence was predicted in the Bogoliubov-de Gennes equation but has not been found thus far within the framework of the GL expansion up to the sixth order. The first-order phase transition will affect the interior structures inside the neutron stars.

Immiscible and miscible states in binary condensates in the ring geometry

We report detailed investigation of the existence and stability of mixed and demixed modes in binary atomic Bose–Einstein condensates with repulsive interactions in a ring-trap geometry. The stability of such states is examined through eigenvalue spectra for small perturbations, produced by the Bogoliubov–de Gennes equations, and directly verified by simulations based on the coupled Gross–Pitaevskii equations, varying inter- and intra-species scattering lengths so as to probe the entire range of miscibility–immiscibility transitions. In the limit of the one-dimensional (1D) ring, i.e. a very narrow one, stability of mixed states is studied analytically, including hidden-vorticity (HV) modes, i.e. those with opposite vorticities of the two components and zero total angular momentum. The consideration of demixed 1D states reveals, in addition to stable composite single-peak structures, double- and triple-peak ones, above a certain particle-number threshold. In the 2D annular geometry, stable demixed states exist both in radial and azimuthal configurations. We find that stable radially-demixed states can carry arbitrary vorticity and, counter-intuitively, the increase of the vorticity enhances stability of such states, while unstable ones evolve into randomly oscillating angular demixed modes. The consideration of HV states in the 2D geometry expands the stability range of radially-demixed states.

Full proximity treatment of topological superconductors in Josephson-junction architectures

Experiments on planar Josephson junction architectures have recently been shown to provide an alternative way of creating topological superconductors hosting accessible Majorana modes. These zero-energy modes can be found at the ends of a one-dimensional channel in the junction of a two-dimensional electron gas (2DEG) proximitized by two spatially separated superconductors. The channel, which is below the break between the superconductors, is not in direct contact with the superconducting leads, so that proximity coupling is expected to be weaker and less well-controlled than in the simple nanowire configuration widely discussed in the literature. This provides a strong incentive for this paper which investigates the nature of proximitization in these Josephson architectures. At a microscopic level we demonstrate how and when it can lead to topological phases. We do so by going beyond simple tunneling models through solving self-consistently the Bogoliubov-de Gennes equations of a heterostructure multicomponent system involving two spatially separated $s$-wave superconductors in contact with a normal Rashba spin-orbit-coupled 2DEG. Importantly, within our self-consistent theory we present ways of maximizing the proximity-induced superconducting gap by studying the effect of the Rashba spin-orbit coupling, chemical potential mismatch between the superconductor and 2DEG, and sample geometry on the gap. Finally, we note (as in experiment) a Fulde-Ferrell-Larkin-Ovchinnikov phase is also found to appear in the 2DEG channel, albeit under circumstances which are not ideal for topological superconducting phase.

Switch effect and 0- π transition in Ising superconductor Josephson junctions

We theoretically study the Josephson current in Ising superconductor-half-metal-Ising superconductor junctions. By solving the Bogoliubov-de Gennes equations, the Josephson currents contributed by the discrete Andreev levels and the continuous spectrum are obtained. For very short junctions, because the direct tunneling of the Cooper pair dominates the Josephson current, the current-phase difference relation is independent of the magnetization direction, which is the same as the conventional superconductor-ferromagnet-superconductor junctions. On the other hand, when the length of the half-metal is similar to or greater than the superconducting coherence length, the spin-triplet Josephson effect occurs and dominates the Josephson current. In this case, the current-phase difference relations show the strong magnetoanisotropic behaviors with the period \pi. When the magnetization direction points to the $\pm$ z directions, the current is zero regardless of the phase difference. However, the current has a large value when the magnetization direction is parallel to the junction plane, which leads to a perfect switch effect of the Josephson current. Furthermore, we find that the long junctions can host both the 0 state and \pi state, and the $0$-$\pi$ transitions can be achieved with the change of the magnetization direction. The physical origins of the switch effect and $0$-$\pi$ transitions are interpreted from the perspectives of the spin-triplet Andreev reflection, the Ising pairing order parameter and the Ginzburg-Landau type of free energy. In addition, the influences of the chemical potential, the magnetization magnitude, and the strength of the Ising spin-orbit coupling on the switch effect and $0$-$\pi$ transitions are also investigated. Furthermore, the two-dimensional Josephson junctions are also investigated and we show that the spin-triplet Josephson effect can exist always.

Evanescent modes and step-like acoustic black holes

We consider a model of an acoustic black hole formed by a quasi-one dimensional Bose–Einstein condensate with a step-like horizon. This system is analyzed by solving the corresponding Bogoliubov–de Gennes equation with an appropriate matching condition at the jump. When the step is between a subsonic and supersonic flow, a sonic horizon develops and in addition to the scattering coefficients we compute the distribution of the accompanying analogue Hawking radiation. Additionally, in response to the abrupt variation in flow and non-linear Bogoliubov dispersion relation, evanescent solutions of the Bogoliubov–de Gennes equation also appear and decay out from the horizon. We bound this decay length and show that these modes produce a modulation of observables outside the event horizon by their interference with outgoing Hawking flux. We go further and find specific superpositions of ingoing eigenmodes which exhibit coherent cancellation of the Hawking flux outside the horizon but nevertheless have evanescent support outside the black hole. We conclude by speculating that when quasiparticle interactions are included, evanescent modes may yield a leakage of information across the event horizon via interactions between the real outgoing Hawking flux and the virtual evanescent modes, and that we may expect this as a generic feature of models which break Lorentz invariance at the UV (Planck) scale.

Exact surface-wave spectrum of a dilute quantum liquid

We consider a dilute gas of bosons with repulsive contact interactions, described on the mean-field level by the Gross-Pitaevski\ifmmode \check{i}\else \v{i}\fi{} equation, and bounded by an impenetrable ``hard'' wall (either rigid or flexible). We solve the Bogoliubov-de Gennes equations for excitations on top of the Bose-Einstein condensate analytically, by using matrix-valued hypergeometric functions. This leads to the exact spectrum of gapless Bogoliubov excitations localized near the boundary. The dispersion relation for the surface excitations represents for small wave numbers $k$ a ripplon mode with fractional power law dispersion for a flexible wall, and a phonon mode (linear dispersion) for a rigid wall. For both types of excitation we provide, for the first time, the exact dispersion relations of the dilute quantum liquid for all $k$ along the surface, extending to $k\ensuremath{\rightarrow}\ensuremath{\infty}$. The small wavelength excitations are shown to be bound to the surface with a maximal binding energy $\mathrm{\ensuremath{\Delta}}=\frac{1}{8}{(\sqrt{17}\ensuremath{-}3)}^{2}m{c}^{2}\ensuremath{\simeq}0.158\phantom{\rule{0.16em}{0ex}}m{c}^{2}$, which both types of excitation asymptotically approach, where $m$ is mass of bosons and $c$ bulk speed of sound. We demonstrate that this binding energy is close to the experimental value obtained for surface excitations of helium II confined in nanopores, reported in Phys. Rev. B 88, 014521 (2013).