Superconducting Pairing Amplitude Research Articles

Localization and multifractal properties of the long-range Kitaev chain in the presence of an Aubry-André-Harper modulation

In the presence of quasiperiodic potentials, the celebrated Kitaev chain presents an intriguing phase diagram with ergodic, localized and and multifractal states. In this work, we generalize these results by studying the localization properties of the Aubry-Andr\'e-Harper model in the presence of long-range hopping and superconducting pairing amplitudes. These amplitudes decay with power-law exponents $\xi$ and $\alpha$ respectively. To this end, we review and compare a toolbox of global and local characterization methods in order to investigate different types of transitions between ergodic, localized and multifractal states. We report energy-dependent transitions from ergodic to multifractal states for pairing terms with $\alpha<1$ and energy-dependent transitions from ergodic to localized states with an intermediate multifractal region for $\alpha>1$. The size of the intermediate multifractal region depends not only on the value of the superconducting pairing term $\Delta$, but also on the energy band. The transitions are not described by a mobility edge, but instead we report hybridization of bands with different types of localization properties. This leads to coexisting multifractal regimes where fractal dimensions follow different distributions.

Superconductivity in spin- 3/2 systems: Symmetry classification, odd-frequency pairs, and Bogoliubov Fermi surfaces

The possible symmetries of the superconducting pair amplitude is a consequence of the fermionic nature of the Cooper pairs. For spin-$1/2$ systems this leads to the $\mathcal{SPOT}=-1$ classification of superconductivity, where $\mathcal{S}$, $\mathcal{P}$, $\mathcal{O}$, and $\mathcal{T}$ refer to the exchange operators for spin, parity, orbital, and time between the paired electrons. However, this classification no longer holds for higher spin fermions, where each electron also possesses a finite orbital angular momentum strongly coupled with the spin degree of freedom, giving instead a conserved total angular moment. For such systems, we here instead introduce the $\mathcal{JPT}=-1$ classification, where $\mathcal{J}$ is the exchange operator for the $z$-component of the total angular momentum quantum numbers. We then specifically focus on spin-$3/2$ fermion systems and several superconducting cubic half-Heusler compounds that have recently been proposed to be spin-$3/2$ superconductors. By using a generic Hamiltonian suitable for these compounds we calculate the superconducting pair amplitudes and find finite pair amplitudes for all possible symmetries obeying the $\mathcal{JPT}=-1$ classification, including all possible odd-frequency (odd-$\omega$) combinations. Moreover, one of the very interesting properties of spin-$3/2$ superconductors is the possibility of them hosting a Bogoliubov Fermi surface (BFS), where the superconducting energy gap is closed across a finite area. We show that a spin-$3/2$ superconductor with a pair potential satisfying an odd-gap time-reversal product and being non-commuting with the normal-state Hamiltonian hosts both a BFS and has finite odd-$\omega$ pair amplitudes. We then reduce the full spin-$3/2$ Hamiltonian to an effective two-band model and show that odd-$\omega$ pairing is inevitably present in superconductors with a BFS and vice versa.

Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation

We present a detailed study of the topological properties of the Kitaev chain with long-range pairing terms and in the presence of an Aubry-Andr\'e-Harper on-site potential. Specifically, we consider algebraically decaying superconducting pairing amplitudes; the exponent of this decay is found to determine a critical pairing strength, below which the chain remains topologically trivial. Above the critical pairing, topological edge modes are observed in the central gap. For sufficiently fast decay of the pairing, these modes are identified as Majorana zero-modes. However, if the pairing term decays slowly, the modes become massive Dirac modes. Interestingly, these massive modes still exhibit a true level crossing at zero energy, which points towards an initimate relation to Majorana physics. We also observe a clear lack of bulk-boundary correspondence in the long-range system, where bulk topological invariants remain constant, while dramatic changes appear in the behavior at the edge of the system. In addition to the central gap around zero energy, the Aubry-Andr\'e-Harper potential also leads to other energy gaps at non-zero energy. As for the analogous short-range model, the edge modes in these gaps can be characterized through a 2D Chern invariant. However, in contrast to the short-range model, this topological invariant does not correspond to the number of edge mode crossings anymore. This provides another example for the weakening of the bulk-boundary correspondence occurring in this model. Finally, we discuss possible realizations of the model with ultracold atoms and condensed matter systems.

Higgs-like pair amplitude dynamics in superconductor–quantum-dot hybrids

We consider a quantum dot weakly tunnel coupled to superconducting reservoirs. A finite superconducting pair amplitude can be induced on the dot via the proximity effect. We investigate the dynamics of the induced pair amplitude after a quench and under periodic driving of the system by means of a real-time diagrammatic approach. We find that the quench dynamics is dominated by an exponential decay towards equilibrium In contrast, the periodically driven system can sustain coherent oscillations of both the amplitude and the phase of the induced pair amplitude in analogy to Higgs and Nambu-Goldstone modes in driven bulk superconductors.

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.

Orbital-selective superconductivity in the nematic phase of FeSe

The interplay between electronic orders and superconductivity is central to the physics of unconventional superconductors, and is particularly pronounced in the iron-based superconductors. Motivated by recent experiments on FeSe, we study the superconducting pairing in its nematic phase in a multiorbital model with frustrated spin-exchange interactions. The electron correlations in the presence of the nematic order give rise to an enhanced orbital selectivity in the superconducting pairing amplitudes. This orbital-selective pairing produces a large gap anisotropy on the Fermi surface. Our results naturally explain the striking experimental observations, and shed new light on the unconventional superconductivity of correlated electron systems in general.

Quantum oscillations and criticality in a fermionic and bosonic dimer model for the cuprates

We study quantum oscillations for a system of fermionic and bosonic dimers and compare the results to those experimentally observed in the cuprate superconductors in their underdoped regime. Based on gauge invariance, we argue that the charge carriers obey the Onsager quantization condition and quantum oscillations take on a Lifshitz-Kosevich form. We obtain the effective mass and find good qualitative agreement with experiments if we tune the model to the point where the observed mass divergence at optimum doping is associated to a van Hove singularity at which four free-dimer Fermi pockets touch pairwise in the interior of the Brillouin zone. The same van Hove singularity leads to a maximum in the d-wave superconducting pairing amplitude when anti-ferromagnetic interactions are included. Our combined results therefore suggest that a quantum critical point separating the underdoped and overdoped regimes is marked by the location of the van Hove saddle point in the fermionic dimer dispersion.

Odd-frequency superconductivity revealed by thermopower

Superconductivity is characterized by a nonvanishing superconducting pair amplitude. It has a definite symmetry in spin, momentum and frequency (time). While the spin and momentum symmetry have been probed experimentally for different classes of superconductivity, the odd-frequency nature of certain superconducting correlations has not been demonstrated yet in a direct way. Here we propose the thermopower as an unambiguous way to assess odd-frequency superconductivity. This is possible since the thermoelectric coefficient given by Andreev-like processes is only finite in the presence of odd-frequency superconductivity. We illustrate our general findings with a simple example of a superconductor-quantum dot-ferromagnet hybrid.

Universal 2Δmax/kBTc scaling decoupled from the electronic coherence in iron-based superconductors

Here we use angle-resolved photoemission spectroscopy to study superconductivity that emerges in two extreme cases, from a Fermi liquid phase (LiFeAs) and an incoherent bad-metal phase (FeTe0.55Se0.45). We find that although the electronic coherence can strongly reshape the single-particle spectral function in the superconducting state, it is decoupled from the maximum superconducting pairing amplitude, which shows a universal scaling that is valid for all FeSCs. Our observation excludes pairing scenarios in the BCS and the BEC limit for FeSCs and calls for a universal strong coupling pairing mechanism for the FeSCs.

Quench dynamics in the Aubry–André–Harper model with p-wave superconductivity

The Anderson localization phase transition in the Aubry–André–Harper (AAH) model with p-wave superconducting (SC) pairing is numerically investigated by suddenly changing the on-site potential from zero to various finite values which fall into the extended, critical and localized phase regimes shown in this model. The time evolution of entanglement entropy (EE), mean width of wave packets and Loschmidt echo of the system exhibit distinct but consistent dynamical signatures in those three phases. Specifically, the EE grows as a power function of time with the exponent of which varies in the extended phase but keeps almost unchanged in the critical phase for different quench parameters. However, if the system is in the localized phase after a quench, the EE grows much slower and will soon get saturated. The time-dependent width of wave packets in the system shows similar behaviors as the EE. In addition, from the perspective of the dynamical phase transition, we find that the Loschmidt echo oscillates and always keeps finite when the system is quenched in the extended phase. In contrast, in the critical or localized phase, the echo will reach to zero at some time intervals or will decay almost to zero after a long-time evolution. The universal features of these quantities in the critical phase of the system with various SC pairing amplitudes are also observed.

Conductance signatures of odd-frequency superconductivity in quantum spin Hall systems using a quantum point contact

Topological superconductors give rise to unconventional superconductivity, which is mainly characterized by the symmetry of the superconducting pairing amplitude. However, since the symmetry of the superconducting pairing amplitude is not directly observable, its experimental identification is rather difficult. In our work, we propose a system, composed of a quantum point contact and proximity induced s-wave superconductivity at the helical edge of a two dimensional topological insulator, for which we demonstrate the presence of odd-frequency pairing and its intimate connection to unambiguous transport signatures. Notably, our proposal requires no time-reversal symmetry breaking terms. We discover the domination of crossed Andreev reflection over electron cotunneling in a wide range of parameter space, which is a quite unusual transport regime.

Tunable hybridization of Majorana bound states at the quantum spin Hall edge

Confinement at the helical edge of a topological insulator is possible in the presence of proximity-induced magnetic (F) or superconducting (S) order. The interplay of both phenomena leads to the formation of localized Majorana bound states (MBS) or likewise (under certain resonance conditions) the formation of ordinary Andreev bound states (ABS). We investigate the properties of bound states in junctions composed of alternating regions of F or S barriers. Interestingly, the direction of magnetization in F regions and the relative superconducting phase between S regions can be exploited to hybridize MBS or ABS at will. We show that the local properties of MBS translate into a particular nonlocal superconducting pairing amplitude. Remarkably, the symmetry of the pairing amplitude contains information about the nature of the bound state that it stems from. Hence, this symmetry can in principle be used to distinguish MBS from ABS, owing to the strong connection between local density of states and nonlocal pairing in our setup.

Bulk Topological Proximity Effect.

Existing proximity effects stem from systems with a local order parameter, such as a local magnetic moment or a local superconducting pairing amplitude. Here, we demonstrate that despite lacking a local order parameter, topological phases also may give rise to a proximity effect of a distinctively inverted nature. We focus on a general construction in which a topological phase is extensively coupled to a second system, and we argue that, in many cases, the inverse topological order will be induced on the second system. To support our arguments, we rigorously establish this "bulk topological proximity effect" for all gapped free-fermion topological phases and representative integrable models of interacting topological phases. We present a terrace construction which illustrates the phenomenological consequences of this proximity effect. Finally, we discuss generalizations beyond our framework, including how intrinsic topological order may also exhibit this effect.

Chiral topological superconductor and half-integer conductance plateau from quantum anomalous Hall plateau transition

We propose to realize a two-dimensional chiral topological superconducting (TSC) state from the quantum anomalous Hall plateau transition in a magnetic topological insulator thin film through the proximity effect to a conventional $s$-wave superconductor. This state has a full pairing gap in the bulk and a single chiral Majorana mode at the edge. The optimal condition for realizing such chiral TSC is to have inequivalent superconducting pairing amplitudes on top and bottom surfaces of the doped magnetic topological insulator. We further propose several transport experiments to detect the chiral TSC. One unique signature is that the conductance will be quantized into a half-integer plateau at the coercive field in this hybrid system. In particular, with the point contact formed by a superconducting junction, the conductance oscillates between $e^2/2h$ and $e^2/h$ with the frequency determined by the voltage across the junction. We close by discussing the feasibility of these experimental proposals.

Probing vortex Majorana fermions and topology in semiconductor/superconductor heterostructures

We investigate the local density of states, spectral function, and superconducting pair amplitudes for signatures of Majorana fermions in vortex cores in ferromagnetic and spin-orbit coupled semiconductor-superconductor heterostructures. We show that the Majorana fermion quasiparticle momentum distribution is always symmetrically distributed at a finite radius around a high symmetry point, thereby providing a necessary condition for a low-energy state to be a Majorana fermion. In real space profiles of the local density of states through the vortex core the Majorana fermion, together with other finite-energy vortex states, form a characteristic x-shape structure only present at non-trivial topology. Moreover, we find that the Mexican hat band structure property of the topologically non-trivial phase translates into multiple high-intensity band edges and also vortex core states located above the superconducting gap in the local density of states. Finally, we find no strong correlation between odd-frequency pairing and the appearance of Majorana fermions, but odd-frequency pairing exists as soon as ferromagnetism is present. In fact, we find that the only vortex superconducting pair amplitude directly related to any phase transition, is the appearance of certain spin-triplet $p$-wave pairing components in the vortex core at a pre-topological vortex core widening transition.

Nonperturbative phase diagram of interacting disordered Majorana nanowires

We develop a Gaussian variational approach in replica space to investigate the phase diagram of a one-dimensional interacting disordered topological superconducting wire in the strong-coupling regime. This method allows for a nonperturbative treatment in the disorder strength, electron-electron interactions, and the superconducting pairing amplitude. We find only two stable phases: a topological superconducting phase and a glassy, nontopological localized phase, characterized by replica symmetry breaking.

Separating pairing from quantum phase coherence dynamics above the superconducting transition by femtosecond spectroscopy

In classical superconductors an energy gap and phase coherence appear simultaneously with pairing at the transition to the superconducting state. In high-temperature superconductors, the possibility that pairing and phase coherence are distinct and independent processes has led to intense experimental search of their separate manifestations. Using femtosecond spectroscopy methods we now show that it is possible to clearly separate fluctuation dynamics of the superconducting pairing amplitude from the phase relaxation above the critical transition temperature. Empirically establishing a close correspondence between the superfluid density measured by THz spectroscopy and superconducting optical pump-probe response over a wide region of temperature, we find that in differently doped Bi2Sr2CaCu2O8+δ crystals the pairing gap amplitude monotonically extends well beyond Tc, while the phase coherence shows a pronounced power-law divergence as T → Tc, thus showing that phase coherence and gap formation are distinct processes which occur on different timescales.

Effect of a spin-active interface on proximity-induced superconductivity in topological insulators

We examine the effect of a spin-active interface on the symmetry of proximity-induced superconducting pairing amplitudes in topological insulators. We develop a model to investigate the leading-order contribution to the pairing amplitude considering three different kinds of spin-active interfaces: (i) those that induce spin-dependent scattering phases, (ii) those that flip the spin of incident electrons, and (iii) interfaces that both induce spin-dependent phases and flip the spins of incident electrons. We find that in cases (i) and (iii) odd-frequency triplet pairing is induced in the TI while for case (ii) no odd-frequency pairing is induced to leading order. We compare our results to those for normal metals and ferromagnetic materials finding that the nontrivial spin structure of the TI leads to qualitatively different behavior.

Interference effect on a Josephson junction with magnetic interfaces

We study the proximity effect in a superconductor-normal-superconductor (SNS) layer with spin-active interfaces, paying particular attention to the effect of interference on the transition between 0 and π states of a Josephson junction. We compute the supercurrent density in the ballistic limit by using the Green’s function formalism without making the quasiclassical approximation to study the interference effect due to coherent scattering of quasiparticles from the interfaces. The supercurrent is strongly influenced by various factors such as the thickness of the normal layer, the strength of the interface potential, and the orientation of the interface spin. We also discuss the symmetry of the induced superconducting pair amplitude for the 0 and the π states. The detailed features of the effect of interference on the Josephson junction can be important for applications of quantum devices.

Many-body calculation of the spatial extent of the wave function of a nonmagnetic impurity in ad-wave high-temperature superconductor using thet−Jmodel

Scanning tunneling microscopy (STM) by providing images of the effects of individual zinc impurities in cuprate superconductors with unprecedented atomic resolution offers a stringent test to models of correlated fermions for high-temperature superconductors. Using a t-J model supplemented by variational Monte Carlo many-body techniques, the spatial dependence of the hole density and of the valence bond and superconducting pairing amplitudes around the impurity are computed. Our calculation supplemented by the theory of interlayer tunneling by I. Martin [Phys. Rev. Lett. 88, 097003 (2002)] can explain most features of the STM experiments including the cross-shaped spatial structure and the impurity resonant peak.