BCS Bose-Einstein Condensation Research Articles

Resonant multi-gap superconductivity at room temperature near a Lifshitz topological transition in sulfur hydrides

The maximum critical temperature for superconductivity in pressurized hydrides appears at the top of superconducting domes in Tc vs pressure curves at a particular pressure, which is not predicted by standard superconductivity theories. The high-order anisotropic Van Hove singularity near the Fermi level observed in band-structure calculations of pressurized sulfur hydride, typical of a supermetal, has been associated with the array of metallic hydrogen wire modules forming a nanoscale heterostructure at an atomic limit called the superstripe phase. Here, we propose that pressurized sulfur hydrides behave as a heterostructure made of a nanoscale superlattice of interacting quantum wires with a multicomponent electronic structure. We present first-principles quantum calculation of a universal superconducting dome where Tc amplification in multi-gap superconductivity is driven by the Fano–Feshbach resonance due to a configuration interaction between open and closed pairing channels, i.e., between multiple gaps in the BCS regime, resonating with a single gap in the BCS–Bose–Einstein condensation crossover regime. In the proposed three dimensional phase diagram, the critical temperature shows a superconducting dome where Tc is a function of two variables: (i) the Lifshitz parameter (η) measuring the separation of the chemical potential from the Lifshitz transition normalized by the inter-wire coupling and (ii) the effective electron–phonon coupling (g) in the appearing new Fermi surface including phonon softening. The results will be of help for material design of room-temperature superconductors at ambient pressure.

Screening masses of photons interacting with a two-species fermionic system in relativistic BCS–BEC crossover

We address the behavior of Debye and Meissner masses of photons in a condensate of fermion pairs in the presence of number density asymmetry. Our formalism applies to a two-species fermionic system with number density asymmetry in BCS–Bose–Einstein condensation (BEC)–relativistic BEC crossover and with variable rapidity. Our results recover the known results of the photon self-energy in the ultrarelativistic limit and the superfluid density in the nonrelativistic limit. We further consider the electromagnetic stability of the condensate and show that the Meissner mass squared can become negative in the weakly coupling BCS regime and the strongly coupling relativistic BEC regime. The electromagnetic instability is compared to the mechanical stability discussed in previous works.

Establishing gauge invariant linear response of fermionic superfluids with pair fluctuations: A diagrammatic approach

We present a fully gauge-invariant linear response theory for ultra-cold Fermi gases undergoing BCS-Bose–Einstein Condensation (BEC) crossover with pair fluctuation effects included, especially in the superfluid phase. The gauge invariance is verified by introducing an effective external electromagnetic (EM) field. For pure BCS-type superfluids, the gauge invariance of the linear response theory can be maintained by constructing a full external EM vertex. By consistently including fluctuations of the order parameters as the self-energy effect is included in the quasi-particles, the Ward identity (WI) is satisfied as a consequence of gauge invariance. The construction of a gauge invariant vertex is non-trivial since in the broken symmetry phase the Nambu–Goldstone mode intertwines with the pairing fluctuation. We find that, under a suitable diagrammatic representation, the construction of such a vertex can be greatly simplified and allows us to build a vertex with pseudogap effects and consistent with the WI in the superfluid phase. We focus on a particular approach known as the G0Gt-matrix of pairing fluctuations, but our formalism also works for the G0G0t-matrix theory as well.

Effect of the particle-hole channel on BCS-Bose-Einstein condensation crossover in atomic Fermi gases.

BCS–Bose-Einstein condensation (BEC) crossover is effected by increasing pairing strength between fermions from weak to strong in the particle-particle channel, and has attracted a lot of attention since the experimental realization of quantum degenerate atomic Fermi gases. Here we study the effect of the (often dropped) particle-hole channel on the zero T gap Δ(0), superfluid transition temperature Tc, the pseudogap at Tc, and the mean-field ratio 2Δ(0)/, from BCS through BEC regimes, using a pairing fluctuation theory which includes self-consistently the contributions of finite-momentum pairs and features a pseudogap in single particle excitation spectrum. Summing over the infinite particle-hole ladder diagrams, we find a complex dynamical structure for the particle-hole susceptibility χph, and conclude that neglecting the self-energy feedback causes a serious over-estimate of χph. While our result in the BCS limit agrees with Gor’kov et al., the particle-hole channel effect becomes more complex and pronounced in the crossover regime, where χph is reduced by both a smaller Fermi surface and a big (pseudo)gap. Deep in the BEC regime, the particle-hole channel contributions drop to zero. We predict a density dependence of the magnetic field at the Feshbach resonance, which can be used to quantify χph and test different theories.

Theoretical Approach to the Gauge Invariant Linear Response Theories for Ultracold Fermi Gases with Pseudogap

Recent experimental progress allows for exploring some important physical quantities of ultracold Fermi gases, such as the compressibility, spin susceptibility, viscosity, optical conductivity, and spin diffusivity. Theoretically, these quantities can be evaluated from suitable linear response theories. For BCS superfluid, it has been found that the gauge invariant linear response theories can be fully consistent with some stringent consistency constraints. When the theory is generalized to stronger than BCS regime, one may meet serious difficulties to satisfy the gauge invariance conditions. In this paper, we try to construct density and spin linear response theories which are formally gauge invariant for a Fermi gas undergoing BCS-Bose-Einstein Condensation (BEC) crossover, especially below the superfluid transition temperatureTc. We adapt a particulart-matrix approach which is close to theG0Gformalism to incorporate noncondensed pairing in the normal state. We explicitly show that the fundamental constraints imposed by the Ward identities andQ-limit Ward identity are indeed satisfied.

Temperature dependence of the pair coherence and healing lengths for a fermionic superfluid throughout the BCS-BEC crossover

We calculate the pair correlation function and the order parameter correlation function, which probe, respectively, the intrapair and interpair correlations of a Fermi gas with attractive interparticle interaction, in terms of a diagrammatic approach as a function of coupling throughout the BCS-Bose-Einstein condensation (BEC) crossover and of temperature, both in the superfluid and normal phase across the critical temperature ${T}_{c}$. Several physical quantities are obtained from this calculation, including the pair coherence and healing lengths, the Tan's contact, the crossover temperature ${T}^{*}$ below which interpair correlations begin to build up in the normal phase, and the signature for the disappearance of the underlying Fermi surface which tends to survive in spite of pairing correlations. A connection is also made with recent experimental data on the temperature dependence of the normal coherence length as extracted from the proximity effect measured in high-temperature (cuprate) superconductors.

Density and spin linear response of atomic Fermi superfluids with population imbalance in the BCS–BEC crossover

We present a theoretical study of the density and spin (representing the two components) linear response of Fermi superfluids with tunable attractive interactions and population imbalance. In both linear response theories, we find that the fluctuations of the order parameter must be treated on equal footing with the gauge transformations associated with the symmetries of the Hamiltonian so that important constraints including various sum rules can be satisfied. Both theories can be applied to the whole BCS–Bose–Einstein condensation crossover. The spin linear responses are qualitatively different with and without population imbalance because collective-mode effects from the fluctuations of the order parameter survive in the presence of population imbalance, even though the associated symmetry is not broken by the order parameter. Since a polarized superfluid becomes unstable at low temperatures in the weak and intermediate coupling regimes, we found that the density and spin susceptibilities diverge as the system approaches the unstable regime, but the emergence of phase separation preempts the divergence.

FUNDAMENTAL CONSTRAINTS ON LINEAR RESPONSE THEORIES OF FERMI SUPERFLUIDS ABOVE AND BELOW Tc

We present fundamental constraints required for a consistent linear response theory of fermionic superfluids and address temperatures both above and below the transition temperature Tc. We emphasize two independent constraints, one associated with gauge invariance (and the related Ward identity) and another associated with the compressibility sum rule, both of which are satisfied in strict BCS theory. However, we point out that it is the rare many body theory which satisfies both of these. Indeed, well studied quantum Hall systems and random-phase approximations to the electron gas are found to have difficulties with meeting these constraints. We summarize two distinct theoretical approaches which are, however, demonstrably compatible with gauge invariance and the compressibility sum rule. The first of these involves an extension of BCS theory to a mean field description of the BCS-Bose Einstein condensation crossover. The second is the simplest Nozieres Schmitt–Rink (NSR) treatment of pairing correlations in the normal state. As a point of comparison we focus on the compressibility κ of each and contrast the predictions above Tc. We note here that despite the compliance with sum rules, this NSR based scheme leads to an unphysical divergence in κ at the transition. Because of the delicacy of the various consistency requirements, the results of this paper suggest that avoiding this divergence may repair one problem while at the same time introducing others.

Theories of Linear Response in BCS Superfluids and How They Meet Fundamental Constraints

We address the importance of symmetry and symmetry breaking on linear response theories of fermionic BCS superfluids. The linear theory of a noninteracting Fermi gas is reviewed and several consistency constraints are verified. The challenge to formulate linear response theories of BCS superfluids consistent with density and spin conservation laws comes from the presence of a broken U(1)$_{\textrm{EM}}$ symmetry associated with electromagnetism (EM) and we discuss two routes for circumventing this. The first route follows Nambu's integral-equation approach for the EM vertex function, but this method is not specific for BCS superfluids. We focus on the second route based on a consistent-fluctuation-of-the order-parameter (CFOP) approach where the gauge transformation and the fluctuations of the order parameter are treated on equal footing. The CFOP approach allows one to explicitly verify several important constraints: The EM vertex satisfies not only a Ward identity which guarantees charge conservation but also a $Q$-limit Ward identity associated with the compressibility sum rule. In contrast, the spin degrees of freedom associated with another U(1)$_z$ symmetry are not affected by the Cooper-pair condensation that breaks only the U(1)$_{\textrm{EM}}$ symmetry. As a consequence the collective modes from the fluctuations of the order parameter only couple to the density response function but decouple from the spin response function, which reflects the different fates of the two U(1) symmetries in the superfluid phase. Our formulation lays the ground work for application to more general theories of BCS-Bose Einstein Condensation crossover both above and below $T_c$.

Fermionic Superfluidity: From High T c Superconductors to Ultracold Fermi Gases

We present a pairing fluctuation theory which self-consistently incorporates finite momentum pair excitations in the context of BCS—Bose–Einstein condensation (BEC) crossover, and we apply this theory to high T c superconductors and ultracold Fermi gases. There are strong similarities between Fermi gases in the unitary regime and high T c superconductors. Here, we address key issues of common interest, especially the pseudogap. In the Fermi gases, we summarize recent experiments including various phase diagrams (with and without population imbalance), as well as evidence for a pseudogap in thermodynamic and other experiments.

Finite temperature effects in trapped Fermi gases with population imbalance

We study the finite temperature $T$ behavior of trapped Fermi gases as they undergo BCS-Bose Einstein condensation (BEC) crossover, in the presence of a population imbalance. Our results, in qualitative agreement with recent experiments, show how the superfluid phase transition is directly reflected in the particle density profiles. We demonstrate that at $ T \neq 0$ and in the near-BEC and unitary regimes, the polarization is excluded from the superfluid core. Nevertheless a substantial polarization fraction is carried by a normal region of the trap having strong pair correlations, which we associate with noncondensed pairs or the ``pseudogap phase''.

Applying BCS–BEC crossover theory to high-temperature superconductors and ultracold atomic Fermi gases (Review Article)

This review is written at the time of the twentieth anniversary of the discovery of high-temperature superconductors, which nearly coincides with the important discovery of the superfluid phases of ultracold trapped fermionic atoms. We show how these two subjects have much in common. Both have been addressed from the perspective of the BCS—Bose–Einstein condensation (BEC) crossover scenario, which is designed to treat short coherence length superfluids with transition temperatures which are “high” with respect to the Fermi energy. A generalized mean field treatment of BCS–BEC crossover at general temperatures T, based on the BCS–Leggett ground state, has met with remarkable success in the fermionic atomic systems. Here we summarize this success in the context of four different cold atom experiments, all of which provide indications, direct or indirect, for the existence of a pseudogap. This scenario also provides a physical picture of the pseudogap phase in the underdoped cuprates which is a central focus of high Tc research. We summarize successful applications of BCS–BEC crossover to key experiments in high Tc systems, including the phase diagram, specific heat, and vortex core STM data, along with the Nernst effect, and exciting recent data on the superfluid density in very underdoped samples.

Collective oscillations of a Fermi gas near a Feshbach resonance

A sum rule approach is used to calculate the zero temperature oscillation frequencies of a two component trapped atomic Fermi gas in the BCS-Bose Einstein condensation crossover region. These sum rules are evaluated using a local density approximation which explicitly includes Feshbach molecules. Breathing modes show non-monotonic behavior as a function of the interaction strength, while quadrupole modes are insensitive to interactions for both spherically symmetric and axially symmetric traps. Quantitative agreement is found with experiments on atomic $^6Li$ system and with other theoretical approaches.

Observation of Bose-Einstein condensation of molecules.

We have observed Bose-Einstein condensation of molecules. When a spin mixture of fermionic 6Li atoms was evaporatively cooled in an optical dipole trap near a Feshbach resonance, the atomic gas was converted into 6Li2 molecules. Below 600 nK, a Bose-Einstein condensate of up to 900 000 molecules was identified by the sudden onset of a bimodal density distribution. This condensate realizes the limit of tightly bound fermion pairs in the crossover between BCS superfluidity and Bose-Einstein condensation.

Quantum Monte Carlo study of the three-dimensional attractive Hubbard model

We study the three-dimensional (3D) attractive Hubbard model by means of the Determinant Quantum Monte Carlo method. This model is a prototype for the description of the smooth crossover between BCS superconductivity and Bose-Einstein condensation. By detailed finite-size scaling we extract the finite-temperature phase diagram of the model. In particular, we interpret the observed behavior according to a scenario of two fundamental temperature scales; T* associated with Cooper pair formation and Tc with condensation (giving rise to long-range superconducting order). Our results also indicate the presence of a recently conjectured phase transition hidden by the superconducting state. A comparison with the 2D case is briefly discussed, given its relevance for the physics of high-Tc cuprate superconductors.

Two-dimensional Bose-Einstein condensation in cuprate superconductors

A binary gas of noninteracting, temperature-dependent Cooper pairs in chemical/thermal equilibrium with unpaired fermions is studied in a two-dimensional (2D) boson-fermion statistical model analogous to an atom plus diatomic-molecule system. The model naturally suggests a more convenient definition for the bosonic chemical potential whereby access into the degenerate Fermi region of positive fermion chemical potential is now possible. The linear (as opposed to quadratic) dispersion relation of the pairs yields substantially higher T cs than with BCS or pure-boson Bose-Einstein condensation theories, and fall within the range of empirical T cs for quasi-2D copper oxide superconductors.

Short coherence length superconductivity: a generalization of BCS theory for the underdoped cuprates

On the basis of the observed short coherence lengths in the cuprates we argue that a BCS—Bose-Einstein condensation (BEC) crossover approach is an appropriate starting point for correcting the mean field approach of BCS and, thereby, for addressing pseudogap phenomena in these materials. Our version of the BCS-BEC approach is based on a particular Green's function decoupling scheme which should be differentiated from others in the literature, and which yields (i) the Leggett crossover ground state (for all coupling constants g, at T = 0) and (ii) BCS theory (for all T ≤ T c over a range of small g). In this paper we provide a simple physical picture of the pseudogap phase above and below T c , and review the quantitative and qualitative implications of this theory, which, for the most part have been published in a series of recent papers.

Pair excitations, collective modes, and gauge invariance in the BCS–Bose-Einstein crossover scenario

In this paper we study the BCS Bose-Einstein condensation ~BEC! crossover scenario within the superconducting state, using a T-matrix approach which yields the ground state proposed by Leggett. Here we extend this ground state analysis to finite temperatures T and interpret the resulting physics. We find two types of bosoniclike excitations of the system: long lived, incoherent pair excitations and collective modes of the superconducting order parameter, which have different dynamics. Using a gauge invariant formalism, this paper addresses their contrasting behavior as a function of T and superconducting coupling constant g .A t a more physical level, our paper emphasizes how, at finite T, BCS-BEC approaches introduce an important parameter D pg 5D 2 2D sc into the description of superconductivity. This parameter is governed by the pair excitations and is associated with particle-hole asymmetry effects that are significant for sufficiently largeg .I n the fermionic regime, D pg represents the difference between the square of the excitation gap D 2 and that of the superconducting order parameter D sc . The parameter D pg , which is necessarily zero in the BCS ~mean field! limit increases monotonically with the strength of the attractive interaction g. It follows that there is a significant physical distinction between this BCS-BEC crossover approach ~in which g is the essential variable which determines D pg) and the widely discussed phase fluctuation scenario in which the plasma frequency is the tuning parameter. Finally, we emphasize that in the strong coupling limit, there are important differences between the composite bosons that arise in crossover theories and the usual bosons of the ~interacting! Bose liquid. Because of constraints imposed on the fermionic excitation gap and chemical potential, in crossover theories, the fermionic degrees of freedom can never be fully removed from consideration.

Spectral properties of the attractive Hubbard model

We calculate the spectral functions, the band structure and the density of states of the 2D attractive Hubbard model in the intermediate coupling regime (i.e. the crossover regime between BCS theory and Bose-Einstein Condensation) using a grand canonical quantum Monte-Carlo approach and a maximum entropy procedure. The evolution of the spectral properties as a function of temperature is discussed. In particular, on lowering the temperature, we find a splitting of the single band present at high temperatures into several distinct branches, the number of which depends on the temperature.

Properties of a Fermi liquid at the superfluid transition in the crossover region between BCS superconductivity and Bose-Einstein condensation.

The self-consistent equations, which have been derived recently as a microscopic model for the crossover between BCS superconductivity and Bose-Einstein condensation in a three-dimensional interacting Fermi system [R. Haussmann, Z. Phys. B 91, 291 (1993)], are solved numerically by repeated Fourier transformation. We find a superfluid transition temperature ${\mathit{T}}_{\mathit{c}}$ which increases monotonically with increasing attractive coupling strength. Furthermore, we determine the chemical potential \ensuremath{\mu}, the fermion distribution function n(k), and the complex effective mass 2${\mathit{m}}^{\mathrm{*}}$ of the fermion pairs at T=${\mathit{T}}_{\mathit{c}}$. The bound fermion pairs cause a power-law tail \ensuremath{\sim}${\mathit{k}}^{\mathrm{\ensuremath{-}}4}$ in n(k) for large k and behave as short-living quasiparticles in the crossover region, which is indicated by a large imaginary part of 2${\mathit{m}}^{\mathrm{*}}$.