Pairing Gap Research Articles

Electronic structure and relaxation dynamics in a superconducting topological material.

Topological superconductors host new states of quantum matter which show a pairing gap in the bulk and gapless surface states providing a platform to realize Majorana fermions. Recently, alkaline-earth metal Sr intercalated Bi2Se3 has been reported to show superconductivity with a Tc ~ 3 K and a large shielding fraction. Here we report systematic normal state electronic structure studies of Sr0.06Bi2Se3 (Tc ~ 2.5 K) by performing photoemission spectroscopy. Using angle-resolved photoemission spectroscopy (ARPES), we observe a quantum well confined two-dimensional (2D) state coexisting with a topological surface state in Sr0.06Bi2Se3. Furthermore, our time-resolved ARPES reveals the relaxation dynamics showing different decay mechanism between the excited topological surface states and the two-dimensional states. Our experimental observation is understood by considering the intra-band scattering for topological surface states and an additional electron phonon scattering for the 2D states, which is responsible for the superconductivity. Our first-principles calculations agree with the more effective scattering and a shorter lifetime of the 2D states. Our results will be helpful in understanding low temperature superconducting states of these topological materials.

Cooling compact stars and phase transitions in dense QCD

We report new simulations of cooling of compact stars containing quark cores and updated fits to the Cas A fast cooling data. Our model is built on the assumption that the transient behaviour of the star in Cas A is due to a phase transition within the dense QCD matter in the core of the star. Specifically, the fast cooling is attributed to an enhancement in the neutrino emission triggered by a transition from a fully gapped, two-flavor, red-green color-superconducting quark condensate to a superconducting crystalline or an alternative gapless, color-superconducting phase. The blue-colored condensate is modeled as a Bardeen-Cooper-Schrieffer (BCS)-type color superconductor with spin-one pairing order parameter. We study the sensitivity of the fits to the phase transition temperature, the pairing gap of blue quarks and the timescale characterizing the phase transition (the latter modelled in terms of a width parameter). Relative variations in these parameter around their best-fit values larger than 10-3 spoil the fit to the data. We confirm the previous finding that the cooling curves show significant variations as a function of compact star mass, which allows one to account for dispersion in the data on the surface temperatures of thermally emitting neutron stars.

Conventional and Unconventional Pairing and Condensates in Dilute Nuclear Matter

This contribution will survey recent progress toward an understanding of diverse pairing phenomena in dilute nuclear matter at small and moderate isospin asymmetry, with results of potential relevance to supernova envelopes and proto-neutron stars. Application of ab initio many-body techniques has revealed a rich array of temperature-density phase diagrams, indexed by isospin asymmetry, which feature both conventional and unconventional superfluid phases. At low density there exist a homogeneous translationally invariant BCS phase, a homogeneous LOFF phase violating translational invariance, and an inhomogeneous translationally invariant phase-separated BCS phase. The transition from the BCS to the BEC phases is characterized in terms of the evolution, from weak to strong coupling, of the pairing gap, condensate wave function, and quasiparticle occupation numbers and spectra. Additionally, a schematic formal analysis of pairing in neutron matter at low to moderate densities is presented that establishes conditions for the emergence of both conventional and unconventional pairing solutions and encompasses the possibility of dineutron formation.

Dirac-Hartree-Bogoliubov calculation for spherical and deformed hot nuclei: Temperature dependence of the pairing energy and gaps, nuclear deformation, nuclear radii, excitation energy, and entropy

Background: Unbound single-particle states become important in determining the properties of a hot nucleus as its temperature increases. We present relativistic mean field (RMF) for hot nuclei considering not only the self-consistent temperature and density dependence of the self-consistent relativistic mean fields but also the vapor phase that takes into account the unbound nucleon states.Purpose: The temperature dependence of the pairing gaps, nuclear deformation, radii, binding energies, entropy, and caloric curves of spherical and deformed nuclei are obtained in self-consistent RMF calculations up to the limit of existence of the nucleus.Method: We perform Dirac-Hartree-Bogoliubov (DHB) calculations for hot nuclei using a zero-range approximation to the relativistic pairing interaction to calculate proton-proton and neutron-neutron pairing energies and gaps. A vapor subtraction procedure is used to account for unbound states and to remove long range Coulomb repulsion between the hot nucleus and the gas as well as the contribution of the external nucleon gas.Results: We show that $p\ensuremath{-}p$ and $n\ensuremath{-}n$ pairing gaps in the $^{1}\mathrm{S}_{0}$ channel vanish for low critical temperatures in the range ${T}_{{c}^{p}}\ensuremath{\approx}0.6--1.1\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$ for spherical nuclei such as $^{90}\mathrm{Zr}$, $^{124}\mathrm{Sn}$, and $^{140}\mathrm{Ce}$ and for both deformed nuclei $^{150}\mathrm{Sm}$ and $^{168}\mathrm{Er}$. We found that superconducting phase transition occurs at ${T}_{{c}^{p}}=1.03{\mathrm{\ensuremath{\Delta}}}_{pp}(0)$ for $^{90}\mathrm{Zr}$, ${T}_{{c}^{p}}=1.16{\mathrm{\ensuremath{\Delta}}}_{pp}(0)$ for $^{140}\mathrm{Ce}$, ${T}_{{c}^{p}}=0.92{\mathrm{\ensuremath{\Delta}}}_{pp}(0)$ for $^{150}\mathrm{Sm}$, and ${T}_{{c}^{p}}=0.97{\mathrm{\ensuremath{\Delta}}}_{pp}(0)$ for $^{168}\mathrm{Er}$. The superfluidity phase transition occurs at ${T}_{{c}^{p}}=0.72{\mathrm{\ensuremath{\Delta}}}_{nn}(0)$ for $^{124}\mathrm{Sn}$, ${T}_{{c}^{p}}=1.22{\mathrm{\ensuremath{\Delta}}}_{nn}(0)$ for $^{150}\mathrm{Sm}$, and ${T}_{{c}^{p}}=1.13{\mathrm{\ensuremath{\Delta}}}_{nn}(0)$ for $^{168}\mathrm{Er}$. Thus, the nuclear superfluidity phase---at least for this channel---can only survive at very low nuclear temperatures and this phase transition (when the neutron gap vanishes) always occurs before the superconducting one, where the proton gap is zero. For deformed nuclei the nuclear deformation disappear at temperatures of about ${T}_{{c}^{s}}=2.0--4.0\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$, well above the critical temperatures for pairing, ${T}_{{c}^{p}}$. If we associate the melting of hot nuclei into the surrounding vapor with the liquid-gas phase transition our results indicate that it occurs at temperatures around $T=8.0--10.0\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$, somewhat higher than observed in many experimental results.Conclusions: The change of the pairing fields with the temperature is important and must be taken into account in order to define the superfluidity and superconducting phase transitions. We obtain a Hamiltonian form of the pairing field calibrated by an overall constant ${c}_{\mathrm{pair}}$ to compensate for deficiencies of the interaction parameters and of the numerical calculation. When the pairing is not zero, the states close to the Fermi energy make the principal contribution to the anomalous density that appears in the pairing field. By including temperature through the use of the Matsubara formalism, the normal and anomalous densities are multiplied by a Fermi occupation factor. This leads to a reduction in the anomalous density and in the pairing as the temperature increases. When the temperature increases $(T\ensuremath{\ge}4\phantom{\rule{0.28em}{0ex}}\mathrm{MeV})$, the effects of the vapor phase that take into account the unbound nucleon states become important, allowing the study of nuclear properties of finite nuclei from zero to high temperatures.

Thermodynamic Properties in Triangular-Lattice Superconductors

The study of superconductivity arising from doping a Mott insulator has become a central issue in the area of superconductivity. Within the framework of the kinetic-energy-driven superconducting mechanism, we discuss the thermodynamic properties in triangular-lattice superconductors. It is shown that a sharp peak in the specific-heat appears at the superconducting transition temperature Tc, and then the specific-heat varies exponentially as a function of temperature for the temperatures T<Tc due to the absence of the d-wave gap nodes at the charge-carrier Fermi surface. In particular, the upper critical field follows qualitatively the charge-carrier pair gap type temperature dependence, and has the same dome-shaped doping dependence as Tc.

Diffusion Monte Carlo study of strongly interacting two-dimensional Fermi gases

Ultracold atomic Fermi gases have been a popular topic of research, with attention being paid recently to two-dimensional (2D) gases. In this work, we perform T=0 ab initio diffusion Monte Carlo calculations for a strongly interacting two-component Fermi gas confined to two dimensions. We first go over finite-size systems and the connection to the thermodynamic limit. After that, we illustrate pertinent 2D scattering physics and properties of the wave function. We then show energy results for the strong-coupling crossover, in between the Bose-Einstein Condensation (BEC) and Bardeen-Cooper-Schrieffer (BCS) regimes. Our energy results for the BEC-BCS crossover are parametrized to produce an equation of state, which is used to determine Tan's contact. We carry out a detailed comparison with other microscopic results. Finally, we calculate the pairing gap for a range of interaction strengths in the strong coupling regime, following from variationally optimized many-body wave functions.

Statistical properties ofPu243, andPu242(n,γ)cross section calculation

The level density and gamma-ray strength function (gammaSF) of 243Pu have been measured in the quasi-continuum using the Oslo method. Excited states in 243Pu were populated using the 242Pu(d,p) reaction. The level density closely follows the constant-temperature level density formula for excitation energies above the pairing gap. The gammaSF displays a double-humped resonance at low energy as also seen in previous investigations of actinide isotopes. The structure is interpreted as the scissors resonance and has a centroid of omega_{SR}=2.42(5)MeV and a total strength of B_{SR}=10.1(15)mu_N^2, which is in excellent agreement with sum-rule estimates. The measured level density and gammaSF were used to calculate the 242Pu(n,gamma) cross section in a neutron energy range for which there were previously no measured data.

Pairing properties of realistic effective interactions

We investigate the pairing properties of an effective shell-model interaction defined within a model space outside 132 Sn and derived by means of perturbation theory from the CD-Bonn free nucleon-nucleon potential. It turns out that the neutron pairing component of the effective interaction is significantly weaker than the proton one, which accounts for the large pairing gap difference observed in the two-valence identical particle nuclei 134 Sn and 134 Te. The role of the contribution arising from one particle-one hole excitations in determining the pairing force is discussed and its microscopic structure is also analyzed in terms of the multipole decomposition.

Cassiopeia A and direct Urca cooling

We model neutron star cooling, in particular the current rapid cooldown of the neutron star Cas A, with a microscopic nuclear equation of state featuring strong direct Urca processes and using compatible nuclear pairing gaps as well as effective masses. Several scenarios are possible to explain the features of Cas A, but only large and extended proton $^1S_0$ gaps and small neutron $^3PF_2$ gaps are able to accommodate also the major part of the complete current cooling data. We conclude that the possibility of strong direct Urca processes cannot be excluded from the cooling analysis.

Direct observation of spin–orbit coupling in iron-based superconductors

Spin-orbit coupling (SOC) is a fundamental interaction in solids which can induce a broad spectrum of unusual physical properties from topologically non-trivial insulating states to unconventional pairing in superconductors. In iron-based superconductors (IBS) its role has so far been considered insignificant with the models based on spin- or orbital fluctuations pairing being the most advanced in the field. Using angle-resolved photoemission spectroscopy we directly observe a sizeable spin-orbit splitting in all main families of IBS. We demonstrate that its impact on the low-energy electronic structure and details of the Fermi surface topology is much stronger than that of possible nematic ordering. Intriguingly, the largest pairing gap is always supported exactly by SOC-induced Fermi surfaces.

Antiferromagnetic nuclear spin helix and topological superconductivity inC13nanotubes

We investigate the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction arising from the hyperfine coupling between localized nuclear spins and conduction electrons in interacting $^{13}$C carbon nanotubes. Using the Luttinger liquid formalism, we show that the RKKY interaction is sublattice dependent, consistent with the spin susceptibility calculation in noninteracting carbon nanotubes, and it leads to an antiferromagnetic nuclear spin helix in finite-size systems. The transition temperature reaches up to tens of mK, due to a strong boost by a positive feedback through the Overhauser field from ordered nuclear spins. Similar to GaAs nanowires, the formation of the helical nuclear spin order gaps out half of the conduction electrons, and is therefore observable as a reduction of conductance by a factor of 2 in a transport experiment. The nuclear spin helix leads to a density wave combining spin and charge degrees of freedom in the electron subsystem, resulting in synthetic spin-orbit interaction, which induces non-trivial topological phases. As a result, topological superconductivity with Majorana fermion bound states can be realized in the system in the presence of proximity-induced superconductivity without the need of fine tuning the chemical potential. We present the phase diagram as a function of system parameters, including the pairing gaps, the gap due to the nuclear spin helix, and the Zeeman field perpendicular to the helical plane.

Pairing correlations in hot nuclei

The phase transition of pairing correlations is one of the interesting phenomena of hot nuclei and is always connected with broken symmetry. In this work, the pairing correlations in hot nuclei $^{162}$Dy are investigated in terms of the thermodynamical properties by covariant density functional theory. The thermodynamical quantities are evaluated by the canonical ensemble theory and the paring correlations are treated by a shell-model-like approach, in which the particle number is conserved exactly. An S-shaped heat capacity curve as a function of temperature has been obtained, which indicates a pairing transition from the superfluid to the normal state. The properties of hot nuclei are studied in terms of defined seniority component. It is found that the one-pair-broken ($s$= 2) states play crucial roles in the appearance of the S shape of the heat capacity curve. Meanwhile, we could predict that higher seniority sates may lead to higher term S shape of heat capacity at high temperature. Moreover, compared with the calculation results of finite temperature BCS, the pairing gap varies smoothly with the temperature due to the effect of the particle-number conservation, which indicates a gradual transition from the superfluid to the normal state.

Magicity of neutron-rich nuclei within relativistic self-consistent approaches

The formation of new shell gaps in intermediate mass neutron-rich nuclei is investigated within the relativistic Hartree–Fock–Bogoliubov theory, and the role of the Lorentz pseudo-vector and tensor interactions is analyzed. Based on the Foldy–Wouthuysen transformation, we discuss in detail the role played by the different terms of the Lorentz pseudo-vector and tensor interactions in the appearing of the N=16, 32 and 34 shell gaps. The nuclei 24O, 48Si and 52,54Ca are predicted with a large shell gap and zero (24O, 52Ca) or almost zero (48Si, 54Ca) pairing gap, making them candidates for new magic numbers in exotic nuclei. We find from our analysis that the Lorentz pseudo-vector and tensor interactions induce very specific evolutions of single-particle energies, which could clearly sign their presence and reveal the need for relativistic approaches with exchange interactions.

Crossover between few and many fermions in a harmonic trap

The properties of a balanced two-component Fermi gas in a one-dimensional harmonic trap are studied by means of the coupled-cluster method. For few fermions we recover the results of exact diagonalization, yet with this method we are able to study much larger systems. We compute the energy, the chemical potential, the pairing gap, and the density profile of the trapped clouds, smoothly mapping the crossover between the few-body and many-body limits. The energy is found to converge surprisingly rapidly to the many-body result for every value of the interaction strength. Many more particles are instead needed to give rise to the nonanalytic behavior of the pairing gap, and to smoothen the pronounced even-odd oscillations of the chemical potential induced by the shell structure of the trap.

The Effect of Local and Non-Local Potentials on BCS Solutions for Some Even-Even Nuclei

We study the like-particle pairings using the BCS approximation applied for different eveneven nuclear isotopes using different realistic potentials for the NN interaction. First we solve the Schrodinger equation using a deformed axially symmetric Woods-Saxon potential. For the pairing interaction the interaction G-matrix is calculated using local potentials (Argonne V18 and Nijmegen II) and non-local potentials (Bonn-CD and Nijmegen I). A study of the effect of these different realistic potentials on the pairing strengths, average pairing gaps, Fermi energies, and occupation probabilities of protons and neutrons is introduced. We find that the pairing strength values calculated by the local potentials are greater than that calculated by the non-local potentials. We find that only one of the BCS outputs, the pairing energy gap, is slightly affected by using different types of realistic potentials.

Bulk disorder in the superconductor affects proximity-induced topological superconductivity

We investigate effects of ordinary nonmagnetic disorder in the bulk of a superconductor on magnetic adatom-induced Shiba states and on the proximity-induced superconductivity in a nanowire that is tunnel coupled to the bulk superconductor. Within the formalism of self-consistent Born approximation, we show that, contrary to widespread belief, the proximity-induced topological superconductivity can be adversely affected by the bulk superconducting disorder even in the absence of any disorder in the nanowire (or the superconductor-nanowire interface) when the proximity tunnel coupling is strong. In particular, bulk disorder can effectively randomize the Shiba-state energies. In the case of a proximate semiconductor nanowire, we numerically compute the dependence of the effective disorder and pairing gap induced on the wire as a function of the semiconductor-superconductor tunnel coupling. We find that the scaling exponent of the induced disorder with respect to coupling is always larger than that of the induced gap, implying that at weak coupling, the proximity-induced pairing gap dominates, whereas at strong coupling, the induced disorder dominates. These findings bring out the importance of improving the quality of the bulk superconductor itself (in addition to the quality of the nanowire and the interface) in the experimental search for solid-state Majorana fermions in proximity-coupled hybrid structures and, in particular, points out the pitfall of pursuing strong coupling between the semiconductor and the superconductor in a goal toward having a large proximity gap. In particular, our work establishes that the bulk superconductor in strongly coupled hybrid systems for Majorana studies must be in the ultraclean limit, since otherwise the bulk disorder is likely to completely suppress all induced topological superconductivity effects.

Spin-triplet superconductivity in a weak-coupling Hubbard model for the quasi-one-dimensional compoundLi0.9Mo6O17

The purple bronze Li$_{0.9}$Mo$_6$O$_{17}$ is of interest due to its quasi-one-dimensional electronic structure and the possible Luttinger liquid behavior resulting from it. For sufficiently low temperatures, it is a superconductor with a pairing symmetry that is still to be determined. To shed light on this issue, we analyze a minimal Hubbard model for this material involving four molybdenum orbitals per unit cell near quarter filling, using asymptotically exact perturbative renormalization group methods. We find that spin triplet odd-parity superconductivity is the dominant instability. Approximate nesting properties of the two quasi-one-dimensional Fermi surfaces enhance certain second-order processes, which play crucial roles in determining the structure of the pairing gap. Notably, we find that the gap has accidental nodes, i.e. it has more sign changes than required by the point-group symmetry.

Magnetic response of energy levels of superconducting nanoparticles with spin-orbit scattering

Discrete energy levels of ultrasmall metallic grains are extracted in single-electron-tunneling-spectroscopy experiments. We study the response of these energy levels to an external magnetic field in the presence of both spin-orbit scattering and pairing correlations. In particular, we investigate $g$-factors and level curvatures that parametrize, respectively, the linear and quadratic terms in the magnetic-field dependence of the many-particle energy levels of the grain. Both of these quantities exhibit level-to-level fluctuations in the presence of spin-orbit scattering. We show that the distribution of $g$-factors is not affected by the pairing interaction and that the distribution of level curvatures is sensitive to pairing correlations even in the smallest grains in which the pairing gap is smaller than the mean single-particle level spacing. We propose the level curvature in a magnetic field as a tool to probe pairing correlations in tunneling spectroscopy experiments.

Thermodynamics of pairing transition in hot nuclei

The pairing correlations in hot nuclei $^{162}\text{Dy}$ are investigated in terms of the thermodynamical properties by covariant density functional theory. The thermodynamical quantities are evaluated by the canonical ensemble theory and the paring correlations are treated by a shell-model-like approach, in which the particle number is conserved exactly. An S-shaped heat capacity curve as a function of temperature has been obtained. The properties of hot nuclei, such as entropy and level density are studied in terms of defined seniority component. It is found that the one-pair-broken states play crucial roles in the appearance of the S shape of the heat capacity curve. Moreover, due to the effect of the particle-number conservation, the pairing gap varies smoothly with the temperature, which indicates a gradual transition from the superfluid to the normal state.

Evidence for superconductivity in Li-decorated monolayer graphene

Monolayer graphene exhibits many spectacular electronic properties, with superconductivity being arguably the most notable exception. It was theoretically proposed that superconductivity might be induced by enhancing the electron-phonon coupling through the decoration of graphene with an alkali adatom superlattice [Profeta G, Calandra M, Mauri F (2012) Nat Phys 8(2):131-134]. Although experiments have shown an adatom-induced enhancement of the electron-phonon coupling, superconductivity has never been observed. Using angle-resolved photoemission spectroscopy (ARPES), we show that lithium deposited on graphene at low temperature strongly modifies the phonon density of states, leading to an enhancement of the electron-phonon coupling of up to λ ≃ 0.58. On part of the graphene-derived π*-band Fermi surface, we then observe the opening of a Δ ≃ 0.9-meV temperature-dependent pairing gap. This result suggests for the first time, to our knowledge, that Li-decorated monolayer graphene is indeed superconducting, with Tc ≃ 5.9 K.