Oscillations Of Order Parameter Research Articles

Inhomogeneous stripe phase and tunable Majorana zero modes in Rashba s-wave superconducting systems in the presence of in-plane Zeeman field

Based on the extended Bogoliubov–de Gennes theory, the inhomogeneous stripe order and Majorana bound states are demonstrated in s-wave superconducting systems with Rashba spin–orbit interaction when the in-plane Zeeman field is applied. For a fully open square sample, topological phase transitions can be driven by the Zeeman field, and the stripe phase with spatially oscillating order parameter shows up across a critical field strength. The topological channels arise in this phase, which can be utilized to create and manipulate Majorana zero modes. Interestingly, (quasi-)one-dimensional channels with diminished pairing amplitude can be formed in narrow arms of a square loop, accompanied by the reconstruction of energy spectra of the condensate and the realization of robust Majorana zero-energy states at the ends of channels. The associated evolution of topological phases and the location of Majorana zero modes are highly sensitive to the field direction. Moreover, the effects of the rectangular aspect-ratio and the off-centered hole as well as the surface defects on Majorana end modes are explored in Rashba asymmetric loops. In comparison, the field-dependent evolution processes of low-energy levels behave more complicated because the emergence of confined topological channels can be effectively tuned by the length and width of loop arms as well as the size and position of an introduced small indentation at the outer edge. Rich patterns of Majorana corner-like states are generated for such asymmetric systems. Our theoretical predictions may shed new light on the tunability of Majorana zero modes and provide useful guidance for future experiments and applications.

Light-induced switching between singlet and triplet superconducting states

While the search for topological triplet-pairing superconductivity has remained a challenge, recent developments in optically stabilizing metastable superconducting states suggest a new route to realizing this elusive phase. Here, we devise a testable theory of competing superconducting orders that permits ultrafast switching to an opposite-parity superconducting phase in centrosymmetric crystals with strong spin-orbit coupling. Using both microscopic and phenomenological models, we show that dynamical inversion symmetry breaking with a tailored light pulse can induce odd-parity (spin triplet) order parameter oscillations in a conventional even-parity (spin singlet) superconductor, which when driven strongly can send the system to a competing minimum in its free energy landscape. Our results provide new guiding principles for engineering unconventional electronic phases using light, suggesting a fundamentally non-equilibrium route toward realizing topological superconductivity.

Superconductor/Ferromagnet Heterostructures: A Platform for Superconducting Spintronics and Quantum Computation

AbstractThe interplay between superconductivity and ferromagnetism in the superconductor/ferromagnet (SC/FM) heterostructures generates many interesting physical phenomena, including spin‐triplet superconductivity, superconducting order parameter oscillation, and topological superconductivity. The unique physical properties make the SC/FM heterostructure as promising platforms for future superconducting spintronics and quantum computation applications. In this article, important research progress of SC/FM heterostructures from superconducting spintronics to quantum computation is reviewed, and it is organized as follows. First, the progress of spin current carriers in SC/FM heterostructures including Bogoliubov quasiparticles, superconducting vortex, and spin‐triplet Cooper pairs which might be used for long‐range spin transport is discussed. Then, the π Josephson junctions and their application for constructing π qubits are described. Finally, experimental signatures of Majorana states in the SC/FM heterostructures and the theoretically proposed manipulation are briefly reviewed, which could be useful to realize fault‐tolerant topological quantum computing.

Inverse proximity effect in a topological-insulator–superconductor hybrid system

The recent experiments on ultrathin superconductors (SCs), in proximity with some substrates, have unveiled the possible enhancement and suppression of superconducting order parameters; yet their mechanisms are still ambiguous. Here we examine the inverse proximity effect between an ultrathin SC and the surface state of a topological insulator. We show the effect of the renormalized electron-phonon ($e$-ph) coupling constant $\ensuremath{\lambda}$ on superconducting order parameters is determined by two important mechanisms: the modified density of states (DOS) $N({\ensuremath{\epsilon}}_{\mathrm{F}})$ from the occupied bands near the Fermi surfaces, and the renormalized interaction ${U}_{\mathrm{eff}}$ from the unoccupied bands. We demonstrate the enhanced order parameter near the bottom of the double-well bands with appropriate interface coupling due to the enhanced DOS; while in the strong coupling regime we observe the suppressed order parameter even with finite occupation in the Fermi surface, which is the result of the conjunct influence of the suppressed DOS and the renormalized ${U}_{\mathrm{eff}}$. This picture is applied to multiple SC layers, showing that the oscillation of order parameter can be well ascribed to the oscillation of an $e$-ph coupling constant, instead of solely the DOS. Our results not only provide a theoretical basis for tuning of the order parameters in the SC by chemical potential and interface coupling, but also provide some new insight into the order parameters in a large number of SC-substrate hybrid systems via the inverse proximity effect.

Pair-density-wave in the strong coupling limit of the Holstein-Hubbard model

A pair-density-wave (PDW) is a superconducting state with an oscillating order parameter. A microscopic mechanism that can give rise to it has been long sought but has not yet been established by any controlled calculation. Here we report a density-matrix renormalization-group (DMRG) study of an effective t-J-V model, which is equivalent to the Holstein-Hubbard model in a strong-coupling limit, on long two-, four-, and six-leg triangular cylinders. While a state with long-range PDW order is precluded in one dimension, we find strong quasi-long-range PDW order with a divergent PDW susceptibility as well as the spontaneous breaking of time-reversal and inversion symmetries. Despite the strong interactions, the underlying Fermi surfaces and electron pockets around the K and {K}^{prime} points in the Brillouin zone can be identified. We conclude that the state is valley-polarized and that the PDW arises from intra-pocket pairing with an incommensurate center of mass momentum. In the two-leg case, the exponential decay of spin correlations and the measured central charge c ≈ 1 are consistent with an unusual realization of a Luther-Emery liquid.

Interplay between nematicity and Bardasis-Schrieffer modes in the short-time dynamics of unconventional superconductors

Motivated by the recent experiments suggesting the importance of nematicity in the phase diagrams of ironbased and cuprate high-Tc superconductors, we study the influence of nematicity on the collective modes inside the superconducting state in a non-equilibrium. In particular, we consider the signatures of collective modes in short-time dynamics of a system with competing nematic and s- and d-wave superconducting orders. In the rotationally symmetric state, we show that the Bardasis-Schrieffer mode, corresponding to the subdominant pairing, hybridizes with the nematic collective mode and merges into a single in-gap mode, with the mixing vanishing only close to the phase boundaries. For the d-wave ground state, we find that nematic interaction suppresses the damping of the collective oscillations in the short-time dynamics. Additionally, we find that even inside the nematic s+d-wave superconducting state, a Bardasis-Schrieffer-like mode leads to order parameter oscillations that strongly depend on the competition between the two pairing symmetries. We discuss the connection of our results to the recent pump-probe experiments on high-Tc superconductors.

Hartree-Fock-Bogoliubov theory of trapped one-dimensional imbalanced Fermi systems

Ground state Hartree-Fock-Bogoliubov (HFB) theory is applied to imbalanced spin-1/2 one-dimensional Fermi systems that are spatially confined by either a harmonic or a hard-wall trapping potential. It has been hoped that such systems, which can be realized using ultracold atomic gases, would exhibit the long-sought-after Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase. The HFB formalism generalizes the standard Bogoliubov quasi-particle transformation, by allowing for Cooper pairing to exist between all possible single-particle states, and accounts for the effects of the inhomogeneous trapping potential as well as the mean-field Hartree potential. This provides an unbiased framework to describe inhomgenous densities and pairing correlations in the FFLO state of a confined 1D gas. In a harmonic trap, numerical minimization of the HFB ground state energy yields a spatially oscillating order parameter reminiscent of the FFLO state. However, we find that this state has almost no imprint in the local fermion densities (consistent with experiments that found no evidence of the FFLO phase). In contrast, for a hard-wall geometry, we find a strong signature of the spatial oscillations of the FFLO pairing amplitude reflected in the local in situ densities. In the hard wall case, the excess spins are strongly localized near regions where there is a node in the pairing amplitude, creating an unmistakeable crystalline modulation of the density.

Josephson vortex dynamics and Fulde-Ferrell-Larkin-Ovchinnikov superconductivity in the layered organic superconductor β″−(BEDT−TTF)2SF5CH2CF2SO3

Interlayer resistance is reported in a layered organic superconductor ${\ensuremath{\beta}}^{\ensuremath{''}}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}{\mathrm{SF}}_{5}{\mathrm{CH}}_{2}{\mathrm{CF}}_{2}{\mathrm{SO}}_{3}$ with ${T}_{\mathrm{c}}\ensuremath{\approx}5$ K. At 0.7 K in parallel magnetic fields, the interlayer resistance has nonzero values in a wide field region from 2 T to the upper critical field $(\ensuremath{\sim}13\phantom{\rule{0.16em}{0ex}}\mathrm{T})$, which can be ascribed to the Josephson vortex (JV) dynamics. Significant hysteresis between 2 T and 8 T and a steplike structure between 4 T and 8 T are found, both of which become less evident with increasing temperature. In the second field-derivative curves, successive small dips are observed between 10 T and 12 T, which are ascribed to the commensurability (CM) effect between the JV lattice and the wavelength ${\ensuremath{\lambda}}_{\mathrm{FFLO}}$ of the order parameter oscillation in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. The CM effect is observed only in nearly parallel fields, showing that the FFLO phase is stabilized only in a limited field angle region. The temperature-field phase diagram and ${\ensuremath{\lambda}}_{\mathrm{FFLO}}$ values are obtained.

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.

Excitation of Higgs Mode in Superfluid Fermi Gas in BCS–BEC Crossover

In quantum many-body systems with spontaneous breaking of continuous symmetries, Higgs modes emerge as collective amplitude oscillations of order parameters. Recently, Higgs mode has been observed in the ultracold Fermi gas. In the present paper, we use the time-dependent Bogoliubov-de Gennes equations to investigate Higgs amplitude oscillations of the superfluid order parameter in a Fermi gas induced by a rapid change of the ${\it s}$-wave scattering length. In particular, we investigate the Higgs mode with different values of the initial scattering length. We find that the energy of the Higgs mode coincides with the threshold energy of the pair-breaking excitation, and exponent of the power-low decay of the Higgs mode $\gamma$ continuously changes between $\gamma=-1/2$ and $\gamma=-3/2$ through the Bardeen-Cooper-Schrieffer-Bose-Einstein condensation (BCS-BEC) crossover. Moreover, we propose the optimal ramp speed of the scattering length for observing the clearest Higgs oscillations.

Finite momentum Cooper pairing in three-dimensional topological insulator Josephson junctions

Unconventional superconductivity arising from the interplay between strong spin–orbit coupling and magnetism is an intensive area of research. One form of unconventional superconductivity arises when Cooper pairs subjected to a magnetic exchange coupling acquire a finite momentum. Here, we report on a signature of finite momentum Cooper pairing in the three-dimensional topological insulator Bi2Se3. We apply in-plane and out-of-plane magnetic fields to proximity-coupled Bi2Se3 and find that the in-plane field creates a spatially oscillating superconducting order parameter in the junction as evidenced by the emergence of an anomalous Fraunhofer pattern. We describe how the anomalous Fraunhofer patterns evolve for different device parameters, and we use this to understand the microscopic origin of the oscillating order parameter. The agreement between the experimental data and simulations shows that the finite momentum pairing originates from the coexistence of the Zeeman effect and Aharonov–Bohm flux.

Fulde-Ferrell-Larkin-Ovchinnikov superconductivity in the layered organic superconductor β”−(BEDT−TTF)4[(H3O)Ga(C2O4)3]C6H5NO2

Resistance and magnetic torque measurements are reported in a layered organic superconductor, $\ensuremath{\beta}''\ensuremath{-}{(\mathrm{BEDT}\ensuremath{-}\mathrm{TTF})}_{4}[({\mathrm{H}}_{3}\mathrm{O})\mathrm{Ga}{({\mathrm{C}}_{2}{\mathrm{O}}_{4})}_{3}]{\mathrm{C}}_{6}{\mathrm{H}}_{5}{\mathrm{NO}}_{2}$ with ${T}_{c}=4.8$ K, where BEDT-TTF stands for bis(ethylenedithio)tetrathiafulvalene. Because of the large anion between the BEDT-TTF conducting layers, the superconductivity of this salt is highly anisotropic. In magnetic fields parallel to the conducting layers for $T=0.4$ K, the magnetic torque shows a large diamagnetic signal associated with hysteresis up to $\ensuremath{\sim}21$ T, suggesting the upper critical field ${H}_{c2}\ensuremath{\gtrsim}21$ T at 0.4 K. The large reduction of the diamagnetic signal is observed above 16 T, which shows a Fulde and Ferrell and Larkin and Ovchinnikov (FFLO) phase transition. For $T=0.5$ K, the interlayer resistance has nonzero value in a wide field region up to ${H}_{c2}$, arising from the Josephson vortex dynamics. Successive dips in the second derivative curves of the resistance are observed between 16 T and ${H}_{c2}$, which are ascribed to the commensurability effect between the Josephson vortex lattice and the order parameter oscillation of the FFLO phase. The commensurability effect is observed only in nearly parallel fields, showing that the FFLO phase is stable in a very limited field angle region. The temperature-field phase diagram is determined.

Dynamical quantum phase transitions in systems with continuous symmetry breaking

Interacting many-body systems that are driven far away from equilibrium can exhibit phase transitions between dynamically emerging quantum phases, which manifest as singularities in the Loschmidt echo. Whether and under which conditions such dynamical transitions occur in higher-dimensional systems with spontaneously broken continuous symmetries is largely elusive thus far. Here, we study the dynamics of the Loschmidt echo in the three dimensional O(N) model following a quantum quench from a symmetry breaking initial state. The O(N) model exhibits a dynamical transition in the asymptotic steady state, separating two phases with a finite and vanishing order parameter, that is associated with the broken symmetry. We analytically calculate the rate function of the Loschmidt echo and find that it exhibits periodic kink singularities when this dynamical steady-state transition is crossed. The singularities arise exactly at the zero-crossings of the oscillating order parameter. As a consequence, the appearance of the kink singularities in the transient dynamics is directly linked to a dynamical transition in the order parameter. Furthermore, we argue, that our results for dynamical quantum phase transitions in the O(N) model are general and apply to generic systems with continuous symmetry breaking.

Higgs Mode in a Trapped Superfluid Fermi Gas

In quantum many-body systems with spontaneous breaking of a continuous symmetry, Higgs modes emerge as collective amplitude oscillations of order parameters. Recently, Higgs modes have been observed in superconductors and in Bose gases in optical lattices. However, it has yet to be observed in Fermi gases. In the present paper, we use the time-dependent Bogoliubov–de Gennes equations to investigate Higgs amplitude oscillations of the superfluid order parameter in a trapped Fermi gas induced by a sudden changes of the \({ s}\)-wave scattering length. In particular, we investigate the Higgs mode with different values of the initial scattering length and discuss how the frequency and damping of the Higgs mode changes around the unitarity regime.

GPU-based acceleration of free energy calculations in solid state physics

Obtaining a thermodynamically accurate phase diagram through numerical calculations is a computationally expensive problem that is crucially important to understanding the complex phenomena of solid state physics, such as superconductivity. In this work we show how this type of analysis can be significantly accelerated through the use of modern GPUs. We illustrate this with a concrete example of free energy calculation in multi-band iron-based superconductors, known to exhibit a superconducting state with oscillating order parameter (OP). Our approach can also be used for classical BCS-type superconductors. With a customized algorithm and compiler tuning we are able to achieve a 19×speedup compared to the CPU (119×compared to a single CPU core), reducing calculation time from minutes to mere seconds, enabling the analysis of larger systems and the elimination of finite size effects. Program summaryProgram title: Free_EnergyCatalogue identifier: AEVX_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEVX_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU Lesser General Public License, version 3No. of lines in distributed program, including test data, etc.: 786No. of bytes in distributed program, including test data, etc.: 6304Distribution format: tar.gzProgramming language: Fortran, CUDA C.Computer: Any with a CUDA-compliant GPU.Operating system: No limits (tested on Linux).RAM: Typically tens of megabytes.Classification: 7, 6.5.Nature of problem: GPU-accelerated free energy calculations in multi-band iron-based superconductor models.Solution method: Parallel parameter space search for a global minimum of free energy.Unusual features: The same core algorithm is implemented in Fortran with OpenMP and OpenACC compiler annotations, as well as in CUDA C. The original Fortran implementation targets the CPU architecture, while the CUDA C version is hand-optimized for modern GPUs.Running time: Problem-dependent, up to several seconds for a single value of momentum and a linear lattice size on the order of 103

Signatures of nonadiabatic BCS state dynamics in pump-probe conductivity

We theoretically study the pump-probe response of nonequilibrium BCS superconductors coupled to optical phonons. For ultrashort pump pulses a nonadiabatic regime emerges, which is characterized by oscillations of the superconducting order parameter as well as by the generation of coherent phonons. Using the density-matrix formalism, we compute the pump-probe response in the nonadiabatic regime of the coupled Bogoliubov quasiparticle-phonon system and determine the signatures of the order parameter and of the phonon oscillations in the pump-probe conductivity. We find that the nonadiabatic dynamics of the BCS superconductor reflects itself in oscillations of the pump-probe response as functions of delay time between pump and probe pulses. We argue that from the analysis of this oscillatory behavior both frequency and decay time of the algebraically decaying order-parameter oscillations can be inferred. Similarly, the coherent phonons are evidenced in the pump-probe conductivity by oscillations with the frequency of the phonons. Remarkably, we find that the oscillatory response in the pump-probe conductivity is resonantly enhanced when the frequency of the order-parameter oscillations is tuned to the phonon energy.

The Fulde–Ferrell–Larkin–Ovchinnikov State in Pnictides

Fe-based superconductors (FeSC) exhibit all the properties of systems that allow the formation of a superconducting phase with oscillating order parameter, called the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase. By the analysis of the Cooper pair susceptibility in two-band FeSC, such systems are shown to support the existence of a FFLO phase, regardless of the exhibited order parameter symmetry. We also show the state with nonzero Cooper pair momentum, in superconducting FeSC with ∼cos(k x )⋅cos(k y ) symmetry, to be the ground state of the system in a certain parameter range.

The Fulde–Ferrell–Larkin–Ovchinnikov State in Quantum Rings

The Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state is a super-conducting phase with oscillating order parameter in real space. It is an effect of broken translation symmetry of the system. Similarly, for the quantum rings, an angular FFLO phase comes into existence and breaks the rotation symmetry. We are pondering possible realizations of non-trivial FFLO in superconducting quantum rings. To find distribution of the order parameter in real space, we use the Bogoliubov–de Gennes equations. On the basis of numerical results, we analyse distribution of the order parameter for different quantum rings as a function of their geometrical sizes.

Resonant generation of coherent phonons in a superconductor by ultrafast optical pump pulses

We study the generation of coherent phonons in a superconductor by ultrafast optical pump pulses. The nonequilibrium dynamics of the coupled Bogoliubov quasiparticle-phonon system after excitation with the pump pulse is analyzed by means of the density-matrix formalism with the phonons treated at a full quantum kinetic level. For ultrashort excitation pulses, the superconductor exhibits a nonadiabatic behavior in which the superconducting order parameter oscillates. We find that in this nonadiabatic regime the generation of coherent phonons is resonantly enhanced when the frequency of the order-parameter oscillation is tuned to the phonon energy, a condition that can be achieved in experiments by varying the integrated pump pulse intensity.

Heating of quasiparticles driven by oscillations of the order parameter in short superconducting microbridges

We predict ``heating'' of quasiparticles driven by order parameter oscillations in the resistive state of short superconducting microbridges. The finite relaxation time of the magnitude of the order parameter $|\ensuremath{\Delta}|$ and the dependence of the spectral functions both on $|\ensuremath{\Delta}|$ and the supervelocity $Q$ are the origin of this effect. Our results are opposite to those of Aslamazov and Larkin [Zh. Eks. Teor. Fiz. 70, 1340 (1976)] and Schmid et al. [Phys. Rev. B 21 5076 (1980)] where ``cooling'' of quasiparticles was found.