Pairing Amplitude Research Articles

Finite bulk Josephson currents and chirality blockade removal from interorbital pairing in magnetic Weyl semimetals

Magnetic Weyl semimetals (WSMs) have been presumed to be immune to proximity-induced spin-singlet superconducting pairing due to a chirality blockade of the regular Andreev reflection. In this work we study all possible superconducting pairing induced in a WSM sandwiched between two conventional superconductors in a Josephson junction setup. We confirm that, while conventional intra-orbital $s$-wave pairing is present on the surface of the WSM, it cannot propagate into the bulk due to the chirality blockade. However, inter-orbital $s$-wave pairing, with both even-frequency spin-singlet and odd-frequency mixed-spin triplet symmetry, propagates into the bulk of the WSM, as do several $p$-wave symmetries. To demonstrate the importance of these finite inter-orbital and $p$-wave pair amplitudes in an experimental setup, we calculate the Josephson current and find a finite and even increasing current when the chirality blockade effect for the conventional intra-orbital pairing is enhanced.

Effects of magnon-magnon interactions in a noncollinear magnet induced by combination of a symmetric and an antisymmetric exchange interaction

We study magnon-magnon interactions and their effects in a spiral magnet induced by combination of an antiferromagnetic Heisenberg interaction and a Dzyaloshinsky-Moriya interaction. We show that the main effect of magnon-magnon interactions on low-energy magnons is to renormalize the coefficient of energy dispersion. This could explain why some experiments are consistent with the noninteracting theory. We also show that although the magnon-magnon interactions induce the pair amplitude for low-energy magnons, its effect on the excitation energy is negligible. This suggests that for magnons the finite pair amplitude does not necessarily accompany the pair condensation.

Odd-frequency pairing and proximity effect in Kitaev chain systems including a topological critical point

In this paper, we investigate the relation between odd-frequency pairing and proximity effect in non-uniform Kitaev chain systems with a particular interest in the topological critical point. First, we correlate the odd-frequency pairing and Majorana fermion in a semi-infinite Kitaev chain, where we find that the spatial dependence of the odd-frequency pair amplitude coincides with that of the local density of states at low frequencies. Second, we demonstrate that, contrary to the standard view, the odd-frequency pair amplitude spreads into the bulk of a semi-infinite Kitaev chain at the topological critical point. Finally, we show that odd-frequency Cooper pairs cause the proximity effect in a normal metal/diffusive normal metal/ Kitaev chain junction even at the topological critical point. Our results hold relevance to the investigation of odd-frequency pairing and topological superconductivity in more complicated systems that involve Rashba nanowire with magnetic fields.

Effective exchange interactions for bad metals and implications for iron-based superconductors

The experimentally observed bad metal behavior in parent iron pnictides and chalcogenides suggests that these systems contain strong electronic correlations and are on the verge of a metal-to-insulator transition. The magnetic excitations in this bad-metal regime mainly derive from the incoherent part of the electronic spectrum away from the Fermi energy. We present a microscopic study of the exchange interactions in such a regime within a slave rotor approach. We find that the exchange interaction is maximized near the Mott transition. Generalizations to the multi-orbital case are discussed, as are the implications for the strength of superconducting pairing amplitude in the iron-based superconductors.

Complete evolution equation for the joint amplitude in photon-pair generation through spontaneous four-wave mixing

As four-wave-mixing-based photon-pair sources mature, accurate modelling of the photon-pair properties becomes important. Unlike spontaneous parametric down-conversion, four-wave mixing is accompanied by a number of parasitic effects such as nonlinear phase modulation. Currently, most modelling of photon-pair states are analytic in nature, which limits the number and type of effects that can be taken into account. In this work, we derive a complete, dual-pump evolution equation for the joint amplitude of photon pairs, wherein any desired effects can be included. We describe how to efficiently obtain numerical solution to this equation using a split-step approach. Lastly, we cover a few analytical solutions and compare two schemes for pure-photon generation under three different parasitic effects. We show how one scheme is highly sensitive to parasitic effects, while the other is very robust.

Capacitive-like photovoltaic epiretinal stimulation enhances and narrows the network-mediated activity of retinal ganglion cells by recruiting the lateral inhibitory network

Photovoltaic retinal prostheses theoretically offer the possibility of stand-alone high-resolution electrical stimulation of the retina. However, achieving focused epiretinal stimulation is particularly challenging because of axonal activation and electrical cell coupling. Recent evidence shows that long electric pulses permit a more focal activation of retinal ganglion cells, and non-rectangular waveforms induce higher network-mediated indirect activity. Objective. The role of the pulse shape in focusing the retinal ganglion cell activation and the underlying mechanisms are not yet fully understood. Approach. To address this question, we implemented a hybrid ex vivo and in silico approach. We recorded the evoked activity of retinas explanted from retinal degeneration ten mice upon photovoltaic and electrical stimulation with rectangular or non-rectangular capacitive-like voltage pulses. We used a biophysical model to investigate the role of the pulse shape and the pulse duration on the genesis and the extent of the network-mediated activity in retinal ganglion cells. Main results. Altogether, our results suggest that non-rectangular capacitive-like voltage pulses activate more strongly the inner excitatory and inhibitory layers of the retina, when compared to a rectangular stimulation with paired pulse amplitude and duration. This feature leads to an increase of the network-mediated activity and a decrease in the network-mediated electrical receptive field of the stimulated retinal ganglion cell. Significance. These results demonstrate that recruiting the inner retinal cells with epiretinal stimulation enables us not only to bypass axonal stimulation, but also to obtain a more focal activation due to the natural lateral inhibition. The involvement of the inhibitory feedback from amacrine cells in the genesis of the network-mediated activity represents a novel biological tool with which to confine the response of the retinal ganglion cells. These results support future waveform engineering strategies and offer new perspectives on epiretinal devices to better shape prosthetic perception.

Odd-frequency superconductivity induced by nonmagnetic impurities

A growing body of literature suggests that odd-frequency superconducting pair amplitudes can be generated in normal metal-superconductor junctions. The emergence of odd-frequency pairing in these systems is often attributed to the breaking of translation invariance. In this work, we study the pair symmetry of a one-dimensional $s$-wave superconductor in the presence of a single non-magnetic impurity and demonstrate that translation symmetry breaking is not sufficient for inducing odd-frequency pairing. We consider three kinds of impurities: a local perturbation of the chemical potential, an impurity possessing a quantum energy level, and a local perturbation of the superconducting gap. Surprisingly, we find local perturbations of the chemical potential do not induce any odd-frequency pairing, despite the fact that they break translation invariance. Moreover, although odd-frequency can be induced by both the quantum impurity and the perturbation of the gap, we find these odd-frequency amplitudes emerge from entirely different kinds of scattering processes. The quantum impurity generates odd-frequency pairs by allowing one of the quasiparticles belonging to an equal-time Cooper pair to tunnel onto the impurity state and then back to the superconductor, giving rise to odd-frequency amplitudes with a temporal broadening inversely proportional to the energy level of the impurity. In contrast to this, the perturbation of the gap leads to odd-frequency pairing by "gluing-together" normal state quasiparticles from different points in space and time, leading to odd-frequency amplitudes which are very localized in the time domain.

Unconventional full-gap superconductivity in Kondo lattice with semimetallic conduction bands

A mechanism of superconductivity is proposed for the Kondo lattice which has semi-metallic conduction bands with electron and hole Fermi surfaces. At high temperatures, the $f$ electron's localized spins/pseudospins are fluctuating between electron and hole Fermi surfaces to seek for a partner to couple with. This system tries to resolve this frustration at low temperatures and chooses to construct a quantum mechanically entangled state composed of the Kondo singlet with electron surface and that with hole surface, to break the U(1) gauge symmetry. The corresponding order parameter is given by a composite pairing amplitude as a three-body bound states of localized spin/pseudospin, electron and hole. The electromagnetic response is considered, where composite pair itself does not contribute to the Meissner effect, but the induced pair between conduction electrons, which inevitably mixes due to e.g. a band cutoff effect at high energies, carries the superconducting current under the external field. Possible applications to real heavy-electron materials are also discussed.

Odd-frequency superconducting pairing in Kitaev-based junctions

We investigate odd-frequency superconducting correlations in normal-superconductor (NS) and short superconductor-normal-superconductor (SNS) junctions with the S region described by the Kitaev model of spinless fermions in one dimension. We demonstrate that, in both the trivial and topological phases, Andreev reflection is responsible for the coexistence of even- and odd-frequency pair amplitudes at interfaces, while normal reflections solely contribute to odd-frequency pairing. At NS interfaces we find that the odd-frequency pair amplitude exhibits large, but finite, values in the topological phase at low frequencies. This enhancement is due to the emergence of a Majorana zero mode at the interface, but notably there is no divergence and a finite odd-frequency pair amplitude also exists outside the topological phase. We also show that the local density of states and local odd-frequency pairing can be characterized solely by Andreev reflections deep in the topological phase. Moreover, in the topological phase of short SNS junctions, we find that both even- and odd-frequency amplitudes capture the emergence of topological Andreev bound states. For a superconducting phase difference $0<\phi<\pi$ the odd-frequency magnitude exhibits a linear frequency ($\sim |\omega|$) dependence at low-frequencies, while at $\phi=\pi$ it develops a resonance peak ($\sim 1/|\omega|$) due to the protected Majorana zero modes.

Numerically exact study of Weyl superconductivity

We study the interplay of interactions and topology in a pseudo-spin Weyl system, obtained from a minimally modified Hubbard model, using the numerically exact auxiliary-field quantum Monte Carlo method complemented by mean-field theory. We find that the pseudo-spin plays a key role in the pairing mechanism, and its effect is reflected in the structure of the pairing amplitude. An attractive on-site interaction leads to pairing between quasiparticles carrying opposite spin and opposite topological charge, resulting in the formation of real-spin singlet pairs that are a mixture of pseudo-spin singlet and pseudo-spin triplet. Our results provide a detailed characterization of the exotic pairing behavior in this system, and represent an important step towards a more complete understanding of superconductivity in the context of topological band structures, which will help guide searches for topological superconductivity in real materials and ultracold atoms.

Bounds on the Superconducting Transition Temperature: Applications to Twisted Bilayer Graphene and Cold Atoms

Understanding the material parameters that control the superconducting transition temperature $T_c$ is a problem of fundamental importance. In many novel superconductors, phase fluctuations determine $T_c$, rather than the collapse of the pairing amplitude. We derive rigorous upper bounds on the superfluid phase stiffness for multi-band systems, valid in any dimension. This in turn leads to an upper bound on $T_c$ in two dimensions (2D), which holds irrespective of pairing mechanism, interaction strength, or order-parameter symmetry. Our bound is particularly useful for the strongly correlated regime of low-density and narrow-band systems, where mean field theory fails. For a simple parabolic band in 2D with Fermi energy $E_F$, we find that $k_BT_c \leq E_F/8$, an exact result that has direct implications for the 2D BCS-BEC crossover in ultra-cold Fermi gases. Applying our multi-band bound to magic-angle twisted bilayer graphene (MA-TBG), we find that band structure results constrain the maximum $T_c$ to be close to the experimentally observed value. Finally, we discuss the question of deriving rigorous upper bounds on $T_c$ in 3D.

Information geometric analysis of long range topological superconductors

The Uhlmann connection is a mixed state generalisation of the Berry connection. The latter has a very important role in the study of topological phases at zero temperature. Closely related, the quantum fidelity is an information theoretical quantity which is a measure of distinguishability of quantum states. Moreover, it has been extensively used in the analysis of quantum phase transitions. In this work, we study topological phase transitions in 1D and 2D topological superconductors with long-range hopping and pairing amplitudes, using the fidelity and the quantity closely related to the Uhlmann connection. The drop in the fidelity and the departure of from zero signal the topological phase transitions in the models considered. The analysis of the ground state fidelity susceptibility and its associated critical exponents are also applied to the study of the aforementioned topological phase transitions.

Odd-frequency pairing in a superconductor coupled to two parallel nanowires

We study the behavior of Cooper pair amplitudes that emerge when a two-dimensional superconductor is coupled to two parallel nanowires, focusing on the conditions for realizing odd-frequency pair amplitudes in the absence of spin-orbit coupling or magnetism. In general, any finite tunneling between the superconductor and the two nanowires induces odd-frequency spin-singlet pair amplitudes in the substrate as well as a substantial odd-frequency interwire pairing, both of which vanish locally. Interestingly, in the regime of strong superconductor-nanowire tunneling, we find that the presence of two nanowires allows for the conversion of non-local odd-frequency pairing to local even-frequency pairing. By studying this higher-order symmetry conversion process, we are able to identify a notable effect of the odd-frequency pairing in the superconductor on local quantities accessible by experiments. Specifically, we find that the odd-frequency pairing plays a direct role in the emergence of certain subgap features in the local density of states, and, importantly, it is responsible for a reduction of the maximum Josephson current between the two nanowires, measurable using Josephson scanning tunneling microscopy. We discuss ways to control the sizes of these effects induced by odd-frequency superconductivity by tuning the parameters describing the nanowires.

Fate of a strongly correlated d -wave superconductor in a Zeeman field: The Fulde-Ferrel-Larkin-Ovchinnikov perspective

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase is an unconventional superconducting state found under the influence of strong Zeeman field. This phase is identified by finite center-of-mass momenta in the Cooper pairs, causing the pairing amplitude to oscillate in real space. Repulsive correlations, on the other hand, smear out spatial inhomogeneities in d-wave superconductors. We investigate the FFLO state in a strongly correlated d-wave superconductor within a consolidated framework of Hartree-Fock-Bogoliubov theory and Gutzwiller approximation. We find that the profound effects of strong correlations lie in shifting the BCS-FFLO phase boundary towards a lower Zeeman field and thereby enlarging the window of the FFLO phase. In the FFLO state, our calculation features a sharp mid-gap peak in the density of states, indicating the formation of strongly localized Andreev bound states. We also find that the signatures of the FFLO phase survive even in the presence of an additional translational symmetry breaking competing order in the ground state. This is demonstrated by considering a broken symmetry ground state with a simultaneous presence of the d-wave superconducting order and a spin-density wave order, often found in unconventional superconductors.

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.

Finite-duration interaction quench in dilute attractively interacting Fermi gases: Emergence of preformed pairs

We investigate the non-equilibrium behavior of BCS superconductors subjected to slow ramps of their internal interaction strength. We identify three dynamical regimes as a function of ramp duration. For short ramp times, these systems become non-superconducting; however, fermions with opposite momenta remain paired albeit with reduced amplitudes, and the associated pair amplitude distribution is non-thermal. In this first regime, the disappearance of superconductivity is due to the loss of phase coherence between pairs. By contrast, for intermediate ramp times, superconductivity survives but the magnitude of the order parameter is reduced and presents long-lived oscillations. Finally, for long ramp times, phase coherence is almost fully retained during the slow interaction quench, and the steady-state is characterized by a thermal-like pair amplitude distribution. Using this approach, one can therefore dynamically tune the coherence between pairs in order to control the magnitude of the superconducting order parameter and even engineer a non-equilibrium state made of pre-formed pairs.

Exact solution to the Haldane-BCS-Hubbard model along the symmetric lines: Interaction-induced topological phase transition

We propose a Haldane-BCS-Hubbard model on a honeycomb lattice, which is composed of two copies of the Haldane model of the quantum anomalous Hall effect, an equal-spin pairing term and an onsite Hubbard interaction term. For any interaction strength, this model is exactly solvable along the symmetric line where the hopping and pairing amplitudes are equal to each other. The ground state of the Haldane-BCS-Hubbard model is a topological superconducting state at weak interaction with two chiral Majorana edge states. A strong interaction drives the system across a topological quantum phase transition to a topologically trivial superconductor. A $\mathbb{Z}_{2}$ symmetry of the Hamiltonian, which is a composition of the bond-centered inversion and a gauge transformation, is spontaneously broken by the interaction, resulting a finite antiferromagnetic order in the $y$-direction.

Non-adiabatic Dynamics in d + id-Wave Fermionic Superfluids

We consider a problem of non-adiabatic dynamics of a 2D fermionic system with $d+id$-wave symmetry of paring amplitude. Under the mean-field approximation, we determine the asymptotic behavior of the pairing amplitude following a sudden change of coupling strength. We also study an extended $d+id$ pairing system for which the long-time asymptotic states of the pairing amplitude in the collisionless regime can be determined exactly. By using numerical methods, we have identified three non-equilibrium steady states described by different long-time asymptotes of the pairing amplitude for both the non-integrable and the integrable versions of $d+id$-wave models. We found that despite of its lack of integrability, long-time dynamics resulting from pairing quenches in the non-integrable $d+id$ model are essentially similar to the ones found for its exactly-integrable extended $d+id$ model. We also obtain the long-time phase diagram of the extended $d+id$ model through the Lax construction that exploits underlying integrability showing that the dynamic phases obtained by numerics are consistent with the dynamics of the exactly integrable approach. Both models describe a topological fermionic system with a topologically non-trivial BCS phase appearing at weak coupling strength. We show that the presence of oscillating order parameter region in the chiral $d+id$ pairing dynamics differs from the d-wave ($d_{x^2-y^2}$), which may be used to probe pairing symmetries of chiral superconductors.

Significant variables extraction of post-stroke EEG signal using wavelet and SOM kohonen

Stroke patients require a long recovery. One success of the treatment given is the evaluation and monitoring during recovery. One device for monitoring the development of post-stroke patients is Electroencephalogram (EEG). This research proposed a method for extracting variables of EEG signals for post-stroke patient analysis using Wavelet and Self-Organizing Map Kohonen clustering. EEG signal was extracted by Wavelet to obtain Alpha, beta, theta, gamma, and Mu waves. These waves, the amplitude and asymmetric of the symmetric channel pairs are features in Self Organizing Map Kohonen Clustering. Clustering results were compared with actual clusters of post-stroke and no-stroke subjects to extract significant variable. These results showed that the configuration of Alpha, Beta, and Mu waves, amplitude together with the difference between the variable of symmetric channel pairs are significant in the analysis of post-stroke patients. The results gave using symmetric channel pairs provided 54-74% accuracy.

Odd-frequency pairs in chiral symmetric systems: Spectral bulk-boundary correspondence and topological criticality

Odd-frequency Cooper pairs with chiral symmetry emerging at the edges of topological superconductors are a useful physical quantity for characterizing the topological properties of these materials. In this work, we show that the odd-frequency Cooper pair amplitudes can be expressed by a winding number extended to a nonzero frequency, which is called a `spectral bulk-boundary correspondence,' and can be evaluated from the spectral features of the bulk. The odd-frequency Cooper pair amplitudes are classified into two categories: the amplitudes in the first category have the singular functional form $\sim 1/z$ (where $z$ is a complex frequency) that reflects the presence of a topological surface Andreev bound state, whereas the amplitudes in the second category have the regular form $\sim z$ and are regarded as non-topological. We discuss the topological phase transition by using the coefficient in the latter category, which undergoes a power-law divergence at the topological phase transition point and is used to indicate the distance to the critical point. These concepts are established based on several concrete models, including a Rashba nanowire system that is promising for realizing Majorana fermions.