Time-reversal symmetry breaking (TRSB) with a Bardeen-Cooper-Schrieffer (BCS) -like superconductivity occurs in a small, but growing number of noncentrosymmetric (NCS) materials, although the mechanism is poorly understood. We present heat capacity, magnetization, resistivity, and muon spin resonance/relaxation ($μ$SR) measurements on polycrystalline samples of NCS La$_{7}$Pd$_{3}$. Transverse-field $μ$SR and heat capacity data show La$_{7}$Pd$_{3}$ is a type-II superconductor with a BCS-like gap structure, while zero-field $μ$SR results provide evidence of TRSB. We discuss the implications of these results for both the La$_{7}X_{3}$ (where $X=$ Ni, Pd, Rh, Ir) group of superconductors and other CS and NCS superconductors for which TRSB has been observed.
We study superconductivity in multilayer copper oxides, in the frame of a realistic microscopic formulation. Solving the full temperature dependent BCS gap equations, we obtain a maximum in the transition temperature Tc for M=3 or 4 CuO2 layers in the unit cell for appropriate values of the interlayer tunneling (negative pair tunneling), and via the consideration of the doping imbalance between the inner and outer layers. This is the ubiquitous experimental result for Ca intercalated copper oxides, as opposed to other intercalating elements. Further, using a restricted set of parameters, we ob…
We establish universal behavior in temperature dependencies of some observables in $(s+id)$-wave BCS superconductivity in the presence of a weak $s$ wave. There also could appear a second second-order phase transition. As temperature is lowered past the usual critical temperature $T_c$, a less ordered superconducting phase is created in $d$ wave, which changes to a more ordered phase in $(s+id)$ wave at $T_{c1}$ ($< T_c$). The presence of two phase transitions manifest in two jumps in specific heat at $T_c$ and $T_{c1}$. The temperature dependencies of susceptibility, penetration depth, and th…
We develop a concise self-consistent perturbation expansion for superconductivity where all the pair processes are naturally incorporated without drawing "anomalous" Feynman diagrams. This simplification results from introducing an interaction vertex that is symmetric in the particle-hole indices besides the ordinary space-spin coordinates. The formalism automatically satisfies conservation laws, includes the Luttinger-Ward theory as the normal-state limit, and reproduces the Bardeen-Cooper-Schrieffer theory as the lowest-order approximation. It enables us to study the thermodynamic, single-pa…
We analyze conditions of applicability of grand-canonical mean-field Bardeen-Cooper-Schrieffer theory to the evaluation of an interaction energy in the ground state of small-sized superconductors. We argue that this theory fails to describe correctly an interaction energy, when an average distance between energy levels near the Fermi energy due to the size quantization becomes of the order of the single-pair binding energy. In conventional superconductors, this quantity is much smaller than the superconducting gap.
We use in-plane tunneling spectroscopy to study the temperature dependence of the local superconducting gap $Δ(T)$ in electron-doped copper oxides with various $T_c$'s and Ce-doping concentrations. We show that the temperature dependence of $Δ(T)$ follows the expectation of the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, where $Δ(0)/k_{B}T_{c}\approx1.72\pm 0.15$ and $Δ(0)$ is the average superconducting gap across the Fermi surface, for all the doping levels investigated. These results suggest that the electron-doped superconducting copper oxides are weak coupling BCS superco…
Superfluidity and superconductivity are genuine many-body manifestations of quantum coherence. For finite-size systems the associated pairing gap fluctuates as a function of size or shape. We provide a parameter free theoretical description of pairing fluctuations in mesoscopic systems characterized by order/chaos dynamics. The theory accurately describes experimental observations of nuclear superfluidity (regular system), predicts universal fluctuations of superconductivity in small chaotic metallic grains, and provides a global analysis in ultracold Fermi gases.
We investigate the photon-mediated heat flow between two Josephson-coupled Bardeen-Cooper-Schrieffer (BCS) superconductors. We demonstrate that in standard low temperature experiments involving temperature-biased superconducting quantum interference devices (SQUIDs), this radiative contribution is negligible if compared to the direct galvanic one, but it largely exceeds the heat exchanged between electrons and the lattice phonons. The corresponding thermal conductance is found to be several orders of magnitude smaller, for real experiments setup parameters, than the universal quantum of therma…
Despite the multi-band spectrum of the widely-known Hofstadter butterfly, it turns out that the pairing correlations of the time-reversal-symmetric Hofstadter-Hubbard model are well-described by a single order parameter that is uniform in real space. By exploiting a BCS mean-field theory for the nearly-flat butterfly-bands regime of low magnetic-flux limit, here we reveal a number of unusual superfluid properties both in the ground state and at finite temperatures. Our thorough analysis includes but is not limited to the order parameter, condensate and superfluid fractions, and the critical BC…
Shortly after the Gor'kov microscopic derivation of the Ginzburg-Landau (GL) model via a small order parameter expansion in Bardeen-Cooper-Schrieffer theory of superconductivity, the derivation was carried to next-to-leading order in that parameter and its spatial derivatives. The aim was to obtain a generalized GL free energy that approximates the microscopic model better. Since 1960s, multiple works have claimed or implicitly assumed that this extended GL model corresponds to the free energy and has solutions in the form of local minima describing superconductivity, such as vortex solutions.…
Bardeen-Cooper-Schrieffer (BCS) theory describes a superconducting transition as a single critical point where the gap function or, equivalently, the order parameter vanishes uniformly in the entire system. We demonstrate that in superconductors described by standard BCS models, the superconducting gap survives near the sample boundaries at higher temperatures than superconductivity in the bulk. Therefore, conventional superconductors have multiple critical points associated with separate phase transitions at the boundary and in the bulk. We show this by revising the Caroli-De Gennes-Matricon …
In disordered superconductors, the local pairing field fluctuates in space, leading to the smearing of the BCS peak in the density of states and appearance of the subgap tail states. We analyze the universal mesoscopic contributions to these effects and show that they are enhanced by the Coulomb repulsion. In the vicinity of the quantum critical point, where superconductivity is suppressed by the "fermionic mechanism", strong smearing of the peak due to mesoscopic fluctuations is predicted.
Scanning tunneling spectroscopies in the subKelvin temperature range were performed on superconducting Silicon epilayers doped with Boron in the atomic percent range. The resulting local differential conductance behaved as expected for a homogeneous superconductor, with an energy gap dispersion below +/- 10%. The spectral shape, the amplitude and temperature dependence of the superconductivity gap follow the BCS model, bringing further support to the hypothesis of a hole pairing mechanism mediated by phonons in the weak coupling limit.
We demonstrate the rise-and-fall of multiple pseudogaps in the Bardeen-Cooper-Schrieffer-Bose-Einstein-condensation (BCS-BEC) crossover in two-band fermionic systems having different pairing strengths in the deep band and in the shallow band. The striking features of this phenomenon are an unusual many-body screening of pseudogap state and the importance of pair-exchange couplings, which induces multiple pseudogap formation in the two bands. The multi-band configuration suppresses pairing fluctuations and the pseudogap opening in the strongly-interacting shallow band at small pair-exchange cou…
Experimental discovery of near-room-temperature (NRT) superconductivity in highly-compressed H$_3$S, LaH$_1$$_0$ and YH$_6$ restores fundamental interest to electron-phonon pairing mechanism in superconductors. One of prerequisites of phonon-mediated NRT superconductivity in highly-compressed hydrides is strong electron-phonon interaction, which can be quantified by dimensionless ratios of Bardeen-Cooper-Schrieffer (BCS) theory vs $k{_B}T{_c}$/(${\hslash}$$ω{_l}{_n}$), where Tc is the critical temperature and $ω{_l}{_n}$ is the logarithmic phonon frequency (Mitrovic et al. 1984 Phys. Rev. B 29…
We formulate a new Bardeen-Cooper-Schrieffer (BCS)-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. With its broad applicability, this theory can be a useful tool for investigating the pairing phase transition in finite systems with the particle-number conservation. This theory provides effects of the symmetry-restoring fluctuation (SRF) for the pairing phenomena in finite fermionic systems, distinctively from those of additional quantum fluctuations. It is shown by numerical calculations that the phase trans…
We apply the functional path-integral approach to analyze how the presence of a spin-orbit coupling (SOC) affects the basic properties of a BCS-type paired state in a two-component Bose gas. In addition to a mean-field theory that is based on the saddle-point approximation for the inter-component pairing, we derive a Ginzburg-Landau theory by including the Gaussian fluctuations on top, and use them to reveal the crucial roles played by the momentum-space structure of an arbitrary SOC field in the stability of the paired state at finite temperatures. For this purpose, we calculate the critical …
In nonperturbative regimes, the superfluid instability in the two-dimensional Hubbard model can be described by an emergent BCS theory with small effective pairing constants. We compute the effective couplings using a controlled bold-line diagrammatic Monte Carlo approach, which stochastically sums all skeleton Feynman diagrams dressed in the one- and two-particle channels to high expansion orders, and map out the resulting superfluid ground-state phase diagram in a range of next-nearest-neighbor hopping $0 \leq t^{\prime} \leq 0.3t$, interaction strength $0 \leq U \leq 3t$, and lattice fillin…
We show how multi-level BCS Hamiltonians of finite systems in the strong pairing interaction regime can be accurately approximated using multi-dimensional shifted harmonic oscillator Hamiltonians. In the Shifted Harmonic Approximation (SHA), discrete quantum state variables are approximated as continuous ones and algebraic Hamiltonians are replaced by differential operators. Using the SHA, the results of the BCS theory, such as the gap equations, can be easily derived without the BCS approximation. In addition, the SHA preserves the symmetries associated with the BCS Hamiltonians. Lastly, for …
Convergence of Ginzburg-Landau expansions: superconductivity in the Bardeen-Cooper-Schrieffer theory and chiral symmetry breaking in the Nambu-Jona-Lasinio model
From the sixties a deep and surprising connection has followed the development of superconductivity and quantum field theory. The Anderson-Higgs mechanism and the similarities between the Dirac and Bogoliubov-de Gennes equations are the most intriguing examples. In this last analogy, the massive Dirac particle is identified with a quasiparticle excitation and the fermion mass energy with the superconducting gap energy. Here we follow further this parallelism and show that it predicts an outstanding phenomenon: the superconducting Schwinger effect (SSE). As in the quantum electrodynamics Schwin…
Starting from H. Fröhlich's second-quantized Hamiltonian for a $d$-dimensional electron gas in interaction with lattice phonons describing the quantum vibrations of a metal, we present a rigorous mathematical derivation of the superconducting state, following the principles laid out originally in 1957 by J. Bardeen, L. Cooper and J. Schrieffer. As in the series of papers written on the subject in the 90es, of which the present paper is a continuation, the representation of ions as a uniform charge background allows for a $(1+d)$-dimensional fermionic quantum-field theoretic reformulation of th…
Comparative study of quartet superfluid state: Quartet Bardeen-Cooper-Schrieffer theory and generalized Nambu-Gor'kov formalism
RIKEN Nishina Center for Accelerator-Based Science（理化学研究所西村加速器科学中心）；Department of Physics, Graduate School of Science, The University of Tokyo（东京大学理学研究科物理学系）；Quark Nuclear Science Institute, The University of Tokyo（东京大学夸克核科学研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所交叉理论数学科学中心）
We show that a topological quantum phase transition, generating flat bands and altering Fermi surface topology, is a primary reason for the exotic behavior of the overdoped high-temperature superconductors represented by $\rm La_{2-x}Sr_xCuO_4$, whose superconductivity features differ from what is described by the classical Bardeen-Cooper-Schrieffer theory [J.I. Bozović, X. He, J. Wu, and A. T. Bollinger, Nature 536, 309 (2016)]. We demonstrate that 1) at temperature $T=0$, the superfluid density $n_s$ turns out to be considerably smaller than the total electron density; 2) the critical temper…
The nonadiabatic Heisenberg model presents a nonadiabatic mechanism generating Cooper pairs in narrow, roughly half-filled "superconducting bands" of special symmetry. Here we show that this mechanism may be understood as the outcome of a special spin structure in the reciprocal space, hereinafter referred to as k-space magnetism. The presented picture permits a vivid depiction of this new mechanism highlighting the height similarity as well as the essential difference between the new nonadiabatic and the familiar Bardeen-Cooper-Schrieffer mechanism.
It is pointed out that erroneous Bardeen-Cooper-Schrieffer model equations have been used by Haranath Ghosh in his recent treatment of time-reversal symmetry-breaking superconductivity. Consequently, his numerical results are misleading, and his conclusions are not to the point.
The isotope effect provided a crucial key to the development of the BCS (Bardeen-Cooper-Schrieffer) microscopic theory of superconductivity for conventional superconductors. In superconducting cooper oxides (cuprates) showing an unconventional type of superconductivity, the oxygen isotope effect is very peculiar: the exponential coefficient strongly depends on doping level. No consensus has been reached so far on the origin of the isotope effect in the cuprates. Here we show that the oxygen isotope effect in cuprates is in agreement with the bisoliton theory of superconductivity.
Departing from the assumption that pairing is induced by an retarded external 'bosonic glue', we reformulate the Bardeen-Cooper-Schrieffer theory of superconductivity as a scaling theory. Conventional superconductors correspond with the asymptotically free case but the theory can be effortlessly generalized to handle quantum critical metals. We find that superconducting Tc's are strongly enhanced in the latter when retardation is strong while the coupling is weak. It appears that this simple scaling theory does account quite well for superconducting properties near quantum critical points.
We theoretically analyze the collective modes in unconventional superconductors focusing on Bardasis-Schrieffer (BS) mode and its contribution to the third harmonic generation currents. Starting from a model with competing superconducting pairing instabilities we add fluctuations of the fields beyond saddle point approximation and calculate their response to an applied pulsed electric field. To model phase fluctuations appropriately we take into account the effect of the long-range Coulomb interaction. While the phase mode is pushed into a plasmon frequency, as known from the literature, we sh…
We consider the interface between a Bardeen-Cooper-Schrieffer superconductor and non-superconducting band insulator. We show that under certain conditions, such interfaces can have an elevated superconducting critical temperature, without increasing the strength of the pairing interaction at the interface. We identify the regimes where the interface critical temperature exceeds the critical temperature associated with a superconductor-vacuum interface.
We propose a mechanism of superconductivity from purely repulsive interactions in the strong coupling regime, where the BCS (Bardeen-Cooper-Schrieffer) mechanism such as the spin-fluctuation approach is difficult to apply. Based on the SU(2) slave-rotor representation of the Hubbard model, we find that the single energy scale for the amplitude formation of Cooper pairs and their phase coherence is separated into two energy scales, allowing the so called pseudogap state where such Cooper pairs are coherent locally but not globally, interpreted as realization of the density-phase uncertainty pri…
We predict a strong enhancement of the critical temperature in a conventional Bardeen-Cooper-Schrieffer (BCS) superconductor in the presence of a bosonic condensate of exciton-polaritons. The effect depends strongly on the ratio of the cutoff frequencies for phonon and exciton-polariton mediated BCS superconductivity, respectively. We also discuss a possible design of hybrid semiconductor-superconductor structures suitable for the experimental observation of such an effect.
A BCS (Bardeen-Cooper-Schrieffer) superconductor, which is placed out of equilibrium, can develop quantum instabilities, which manifest themselves in oscillations of the superconductor's order parameter (pairing amplitude $Δ$). These instabilities are a manifestations of the Cooper instability. Inelastic collisions are essential in resolving those instabilities. Incorporating the quantum instabilities and collisions in a unified approach based on Richardson's exact solution of the pairing Hamiltonian, we find that a BCS superconductor may end up in a state in which the spectrum has more than o…
A model describing the Anderson impurity in the Bardeen-Cooper-Schriffer superconductor is proven to exhibit hidden integrability and is diagonalized exactly by the Bethe ansatz.
Based on the Bardeen Cooper Schrieffer (BCS) theory of superconductivity, the coherent splitting of Cooper pairs from a superconductor to two spatially separated quantum dots has been predicted to generate nonlocal pairs of entangled electrons. In order to test this hypothesis, we propose a scheme to transfer the spin state of a split Cooper pair onto the polarization state of a pair of optical photons. We show that the produced photon pairs can be used to violate a Bell inequality, unambiguously demonstrating the entanglement of the split Cooper pairs.
The Bardeen-Cooper-Schrieffer (BCS) mean-field theory of the pairing interaction breaks down for nuclei and ultra-small metallic grains (nanoparticles). Finite-temperature pairing correlations in such finite-size systems can be calculated beyond the BCS theory in an auxiliary-field Monte Carlo approach. We identify thermal signatures of pairing correlations in both nuclei and nanoparticles that depend on the particle-number parity.
In this paper we study a model of s-wave marginal Fermi liquid superconductor at two dimensions. Besides the usual Bardeen-Cooper-Schrieffer(BCS) point attractive interaction, the fermions in our system also interact {\em via} a long range repulsive interaction $v(q)=g_{D}/q^2$. We find that although superconductivity is not destroyed by the long-ranged repulsion interaction (demonstrated by the existence of Meissner effect), nevertheless the BCS pairing function $Δ_{BCS}$ is strongly suppressed resulting in power-law correlation $<Δ_{BCS}(\vec{x})Δ_{BCS}(0)>\sim|\vec{x}|^{-2γ}$. This observat…
We report the observation of two gaps in the superconductor SmFeAsO$_{0.9}$F$_{0.1}$ (F-SmFeAsO) with $T_c=51.5K$ as measured by point-contact spectroscopy. Both gaps decrease with temperature and vanish at $T_c$ and the temperature dependence of the gaps are described by the theoretical prediction of the Bardeen-Cooper-Schrieffer (BCS) theory. A zero-bias conductance peak (ZBCP) was observed, indicating the presence of Andreev bound states at the surface of F-SmFeAsO. Our results strongly suggest an unconventional nodal superconductivity with multiple gaps in F-SmFeAsO.
The BCS theory injected two powerful ideas into the collective consciousness of theoretical physics: pairing and spontaneous symmetry breaking. In the 50 years since the seminal work of Bardeen, Cooper, and Schrieffer, those ideas have found important use in areas quite remote from the stem application to metallic superconductivity. This is a brief and eclectic sketch of some highlights, emphasizing relatively recent developments in QCD and in the theory of quantum statistics, and including a few thoughts about future directions. A common theme is the importance of symmetry {\it transmutation}…
Recent experiments on Nb-doped SrTiO$_3$ have shown that the superconducting energy gap to the transition temperature ratio maintains the Bardeen-Cooper-Schrieffer (BCS) value throughout its superconducting dome. Motivated by these and related studies, we show that the Cooper pairing mediated by a single soft transverse-optical phonon is the most natural mechanism for such a superconducting dome given experimental constraints, and present the microscopic theory for this pairing mechanism. Furthermore, we show that this mechanism is consistent with the $T^2$ resistivity in the normal state. Las…
A scanning tunneling microscope is used to explore the evolution of electron and Cooper pair transport across single Mn-phthalocyanine molecules adsorbed on Pb(111) from tunneling to contact ranges. Normal-metal as well as superconducting tips give rise to a gradual transition of the Bardeen-Cooper-Schrieffer energy gap in the tunneling range into a zero-energy resonance close to and at contact. Supporting transport calculations show that in the normal-metal - superconductor junctions this resonance reflects the merging of in-gap Yu-Shiba-Rusinov states as well as the onset of Andreev reflecti…
We review recent results concerning the mathematical properties of the Bardeen-Cooper-Schrieffer (BCS) functional of superconductivity, which were obtained in a series of papers partly in collaboration with R. Frank, E. Hamza, S. Naboko, and J.P. Solovej. Our discussion includes, in particular, an investigation of the critical temperature for a general class of interaction potentials, as well as a study of its dependence on external fields. We shall explain how the Ginzburg-Landau model can be derived from the BCS theory in a suitable parameter regime.
Based on an exact solution of the Bardeen-Cooper-Schrieffer type Hamiltonian on a spherical surface, we calculate the specific heat for the electron system with pair correlations on a sphere. We find that it is possible to extract from the specific heat a temperature above which many-body states with broken Cooper pairs get populated. Therefore, we define this temperature as the characteristic temperature signalling the onset of a BCS-type pair-correlated state for electrons on a spherical surface. Such spherical electron systems are realized in multielectron bubbles in liquid helium, for whic…
The combination with superconductivity and topological nontrivial band structure provides a promising route towards novel quantum states such as topological superconductivity. Here, we report the first observation of superconductivity (4.1 K) in Ba5In4Bi5 single crystal, a noncentrosymmetric topological semimetal featuring nodal chain loops at the high-symmetry points R and X. The magnetization, resistivity, and specific heat capacity measurements reveal that Ba5In4Bi5 is a moderately coupled type-II Bardeen-Cooper-Schrieffer superconductor. Bulk superconductivity is suggested from the magneti…
We theoretically study the ground-state properties and the condensations of exciton-like Cooper pairs and biexciton-like Cooper quartets in an electron-hole system. Applying the variational approach associated based on the quartet Bardeen-Cooper-Schrieffer (BCS) model to the four-component fermionic system consisting of spin-1/2 electrons and spin-1/2 holes, we show how Cooper pairs and quartet correlations appear in the equation of state at the thermodynamic limit. The biexciton-like four-body correlations survive even at the high-density regime as a many-body BCS-like state of Cooper quartet…
We discuss the charge and the spin tunneling currents between two Bardeen-Cooper-Schrieffer (BCS) superconductors, where one density of states is spin-split. In the presence of a large temperature bias across the junction, we predict the generation of a spin-polarized thermoelectric current. This thermo-spin effect is the result of a spontaneous particle-hole symmetry breaking in the absence of a polarizing tunnel barrier. The two spin components, which move in opposite directions, generate a spin current larger than the purely polarized case when the thermo-active component dominates over the…
We extend the one-pair Cooper configuration towards Bardeen-Cooper-Schrieffer (BCS) model of superconductivity by adding one-by-one electron pairs to an energy layer where a small attraction acts. To do it, we solve Richardson's equations analytically in the dilute limit of pairs on the one-Cooper pair scale. We find, through only keeping the first order term in this expansion, that the $N$ correlated pair energy reads as the energy of $N$ isolated pairs within a $N(N-1)$ correction induced by the Pauli exclusion principle which tends to decrease the average pair binding energy when the pair n…
Cooper pairing between a conduction electron ($c$ electron) and an $f$ electron, referred to as the "$c$-$f$ pairing," is examined to explain s-wave superconductivity in heavy-fermion systems. We first apply the Schrieffer-Wolff transformation to the periodic Anderson model assuming deep $f$ level and strong Coulomb repulsion. The resulting effective Hamiltonian contains direct and spin-exchange interactions between $c$ and $f$ electrons, which are responsible for the formation of the $c$-$f$ Cooper pairs. The mean-field analysis shows that the fully gapped $c$-$f$ pairing phase with anisotrop…
