Various BCS-like pairing theories are still invariably used to describe the unusual superconducting/superfluid states and properties of condensed matters without clarifying the fermionic or bosonic nature of Cooper pairs and the relevant mechanisms of superconductivity and superfluidity. We show that the bosonization of Cooper pairs and the novel superconductivity and superfluidity resulting from the condensation of an attractive Bose gas of Cooper pairs into a superfluid state in high-Tc cuprates and other exotic systems occur under the condition εF≲2εA (where εF is the Fermi energy, εA is the energy of the effective attraction between fermions). We argue that the simple BCS criterion (i.e. formation of an energy gap on the Fermi surface) is insufficient for the occurrence of unconventional superconductivity (superfluidity) and the superconducting/superfluid transition in attractive Bose systems is more λ-like than the usual Bose–Einstein condensation. The new and specific criteria for unconventional superconductivity and superfluidity are formulated. Mean-field theory of the three- and two-dimensional superfluid Bose condensates of Cooper pairs describes reasonably well the novel superconducting states (i.e., two distinct superconducting phases below the bulk superconducting transition temperature Tc and a vortex-like state above Tc) and properties (such as λ-like transition at Tc, first-order phase transition and gapless bosonic excitations somewhat below Tc or even far below Tc) of high-Tc cuprates in full agreement with the experimental findings. The unusual superconducting/superfluid states and properties of heavy-fermion and organic compounds, Sr2RuO4, 3He, 4He and atomic Fermi gases are explained more clearly by this theory.
Bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-
Tc cuprates and other systems
The criteria for bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-
Tc cuprates and other pseudogap matters are formulated by using the uncertainty principle and the boson mean field theory. We argue that the BCS-type s- or d-wave superconductivity occurring in the fermionic limit of Cooper pairs (which exist in ordinary metals and heavily overdoped cuprates with large Fermi energies) is not characteristic of underdoped to overdoped cuprates with low Fermi energies. The superconducting order parameter in high-
Tc cuprates and other pseudogap matters cannot be determined as the BCS-like (s- or d-wave) gap. We show that the unconventional superconductivity/superfluidity occurring in the bosonic limit of Cooper pairs exists in low Fermi energy systems where the bosonic Cooper pairs are formed at a pseudogap temperature T* above the uperconducting/superfluid transition temperature
Tc and then such bosons condenses into a Bose superfluid at
Tc. The pair condensation of attracting bosons occurs at
Tc and then their single particle condensation sets in at
Tc∗ lower than
Tc (in three dimensions (3D)) or at T=0 (in two dimensions (2D)). The coherent single particle and pair condensates of bosons exist as two distinct superfluid phases below
Tc. By solving the mean field equations for high-
Tc cuprates, the novel superconducting states (i.e., a vortex-like state existing below the temperature
Tv=Tc2D lower than T* but higher than
Tc=Tc3D and two superconducting phases below
Tc) and their properties characterized by the boson superfluid stiffness are determined and compared with the key experimental findings. The novel Bose-liquid superconductivity and superfluidity exists also in other exotic superconductors/superfluids, including quantum liquids, atomic Fermi gases and low-density nuclear matter.
The (mean field based) BCS theory is considered one of the most successful theories in condensed matter physics. It is justified in ordinary metal superconductors the coherence length
ξ is large, with two important features: the order parameter (OP) and excitation gap (EG) are identical, and the pair formation and their Bose condensation take place at the same temperature Tc. It fails to explain the underdoped cuprate superconductivity: EG is finite at Tc and thus distinct from OP. Since these superconductors belong to a large class of small
ξ materials, this failure has the potential for widespread impact. Here we have extended BCS theory in a natural way to short
ξ superconductors, based on a BCS--BEC crossover scenario, and arrived at a simple physical picture in which incoherent, finite momentum pairs become progressively more important as the pairing interaction becomes stronger, leading to the distinction between EG and OP. The superconductivity from the fermionic perspective and BEC from the bosonic perspective are just two sides of the same coin. Our theory is capable of making verifiable quantitative predictions. We obtain a cuprate phase diagram (with one free parameter) , in (semi-)quantitative agreement with experiment. The mutually compensating contributions from fermionic quasiparticles and bosonic pair excitations provides a natural explanation for the quasi-universal behavior of the in-plane superfluid density versus T. Our bosonic pair excitations also provide an intrinsic mechanism for the long mysterious linear T terms in the specific heat. Incoherent pair contributions lead to new low T power laws, consistent with existing experiments. Finally, we demonstrated that the onset of superconducting long range order leads to sharp features in the specific heat at Tc, consistent with experiment.
Unification of the Bardeen, Cooper and Schrieffer (BCS) and the Bose‐Einstein condensation (BEC) theories is surveyed in terms of a generalized BEC (GBEC) finite‐temperature statistical formalism. A vital distinction is that Cooper pairs (CPs) are true bosons that may suffer a BEC since they obey BE statistics, in contrast with BCS pairs that are “hard‐core bosons” at best. A second crucial ingredient is the explicit presence of hole‐pairs (2h) alongside the usual electron‐pairs (2e). A third critical element (particularly in 2D where ordinary BEC does not occur) is the linear dispersion relation of CPs in leading order in the center‐of‐mass momentum (CMM) power‐series expansion of the CP energy. The GBEC theory reduces in limiting cases to all five continuum (as opposed to “spin”) statistical theories of superconductivity, from BCS on one extreme to the BEC theory on the other, as well as to the BCS‐Bose “crossover” picture and the 1989 Friedberg‐Lee BEC theory. It accounts for 2e‐ and 2h‐CPs in arbitrary proportions while BCS theory can be deduced from the GBEC theory but allows only equal (50%‐50%) BE condensed‐mixtures of both kinds of CPs. As it yields the precise BCS gap equation for all temperatures as well as the precise BCS zero‐temperature condensation energy for all couplings, it suggests that the BCS condensate is a BE condensate of a ternary mixture of kinematically independent unpaired electrons coexisting with equally proportioned weakly‐bound zero‐CMM 2e‐ and 2h‐CPs. Without abandoning the electron‐phonon mechanism in moderately weak coupling, and fortuituously insensitive to the BF interactions, the GBEC theory suffices to reproduce the unusually high values of Tc (in units of the Fermi temperature TF) of 0.01–0.05 empirically found in the so‐called “exotic” superconductors of the Uemura plot, including cuprates, in contrast to the low values of Tc/TF⩽10−3 roughly reproduced by BCS theory for conventional (mostly elemental) superconductors.
For the Cooper/BCS model interaction in superconductors (SCs) it is shown: a) how BCS-Bose crossover picture transition temperatures Tc, defined self-consistently by both the gap and fermion-number equations, require unphysically large couplings to differ significantly from the Tc of ordinary BCS theory defined without the number equation since here the chemical potential is assumed equal to the Fermi energy; how although ignoring either hole- or electron-Cooper-pairs in the recent "complete boson-fermion model": b) one obtains the precise BCS gap equation for all temperatures T, but c) only half the zero-temperature BCS condensation energy emerges. Results (b) and (c) are also expected to hold for neutral-fermion superfluids (SFs)--such as liquid
3He, neutron matter and trapped ultra-cold fermion atomic gases--where the pair-forming two-fermion interaction of course differs from the Cooper/BCS one for SCs.
Using the Boson-Fermion formalism of superconductivity we calculate the condensation energy for several superconductors ranging from conventional to unconventional, or high temperature superconductors. It is calculated as the difference between the Helmholtz free energies of the superconducting and the normal state, which is a gas of N attractive electron gas, while the superconducting state is formed by the condensed Cooper pairs taken as composite bosons, coming from a fraction of the electrons inside the Debye shell, plus those electrons inside and outside the Debye shell that are unable to pair. After giving the analytic expressions for the internal energy U and the entropy S we obtain the Helmholtz free energy F = U -TS for both the superconducting and the normal states as functions of temperature. In the search for universalities, we calculate the ratio of the condensation energy at T=0 to the Sommerfeld constant (the normal state electronic specific heat over the temperature when T it's almost zero) using two different methods: the Boson-Fermion formalism developed here, as well as an analytical expression deduced from a combination of the BCS and Ginzburg-Landau theories. We find for the Boson-Fermion formalism
Econd/γ0=0.252Tc1.997, which is the same behavior described by the experimental fit of Kim et al
Econd/γ0=0.2Tc2.06 and by the one recently reported by Tallon et al for overdoped cuprate superconductors; while for the Ginzburg-Landau-BCS we get the expression
Econd/γ0=0.236Tc2, also in very good agreement with the partifirst method.
The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the
s-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or
XY universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple
s-wave picture.