Energy Gap Equation Research Articles

Mass spectra and decay of mesons under strong external magnetic field * *Supported by the National Natural Science Foundation of China (11735007, 11890711, 11890710)

We study the mass spectra and decay process of σ and mesons under a strong external magnetic field. To achieve this goal, we deduce the thermodynamic potential in a two-flavor, hot and magnetized Nambu–Jona-Lasinio model. We calculate the energy gap equation through the random phase approximation (RPA). Then we use the Ritus method to calculate the decay triangle diagram and self-energy in the presence of a constant magnetic field B. Our results indicate that the magnetic field has little influence on the mass of at low temperatures. However, for quarks and σ mesons, their mass clearly changes, which reflects the influence of magnetic catalysis (MC). The presence of a magnetic field accelerates the decay of the meson while the presence of a chemical potential will decrease the decay process.

Effect of Multiple Orders on Superconducting Transition Temperature of Hole Doped Cuprates

Hole doped high-Tc cuprate superconductors are strongly correlated electronic systems. In these materials, various electronic orders are often found, but whether they support or compete with superconducting order is not unambiguous. Superconductivity normally manifests itself by a superconducting gap in the electronic density of states (EDOS). In cuprates, a gap appears even in the normal state called the pseudogap (PG). For certain doping range, spin density wave and charge density wave coexist with superconductivity by inducing corresponding additional gaps in the EDOS. In this study, we have tried to obtain expression for superconducting transition temperature, Tc by solving the BCS (Bardeen-Cooper-Schrieffer) energy gap equation in the presence of depleted EDOS of various origins and types. We have been successful to solve the weak-coupling BCS integral equation analytically in some special cases and also in the general case by using numerical integration. We have found that depending on conditions these non-pairing gaps/orders can enhance as well as reduce Tc.

Functional integrals and 1/h expansion in the boson–fermion model

The effective action of boson–fermion model is derived by means of the functional integrals method and Popov–Faddeev canonical transformations. The energy gap equation and excitation spectrum equation are obtained from first order and second order perturbation expansions of functional determinant. In the long wave approximation, some analytical expressions of excitation spectrum are calculated by using the 1/h expansion technique, the results showed that analytical calculation is in good agreement with the numerical calculation. Moreover, the Nambu sum rules of Higgs bosons are analyzed and discussed.

On the Bardeen-Cooper-Schrieffer(BCS) Hamiltonian

We present a detailed diagonalization calculation of the Bardeen-Cooper-Schrieffer (BCS) Hamiltonian via the unitary transformation method which allows for the Bogolyubov-Valatin transformation to be arrived at naturally. The energy gap equation is derived and solved, and averages of some representative thermodynamic quantities are obtained.KEYWORDS: Hamiltonian, unitary matrix, superconductivity, interaction, ground state.

SUPERCONDUCTING PAIRING MECHANISM OF RARE-EARTH-NICKEL-BOROCARBIDES: EFFECTS OF ELECTRON–ELECTRON AND ELECTRON–PHONON INTERACTIONS

Upon considering the three interactions namely, the electron–acoustic phonon, the electron–optical phonon and the Coulomb, the analytical solutions for the energy gap equation allows one to determine the electronic structure parameters to discuss the behavior of superconducting transition temperature (Tc) and isotope effect coefficient (α) for layered structure YNi2 B2C. Tc of 17 K is estimated for YNi2B2C with electron–acoustic phonon (λac) = 0.31, electron–optical phonon (λop) = 0.1 and Coulomb screening parameter (μ*) = 0.126 indicating that the YNi2B2C superconductor is in the intermediate coupling regime. To correlate the Tc with various coupling strengths as λac, λop and μ*, we present curves of Tc with them. The present approach also explains the conditions for the Boron and Carbon isotope effect. The negative pressure coefficient of Tc in this layered material is attributed to the contraction along c-axis under hydrostatic pressure. We suggest from these results that both the acoustic and optical phonons within the framework of a three-square well scheme consistently explains the effective electron–electron interaction leading to superconductivity in layered structure YNi2B2C.

BCS and BEC p-wave pairing in Bose-Fermi gases

The pairing of fermionic atoms in a mixture of atomic fermion and boson gases at zero temperature is investigated. The attractive interaction between fermions, that can be induced by density fluctuations of the bosonic background, can give rise to a superfluid phase in the Fermi component of the mixture. The atoms of both species are assumed to be in only one internal state, so that the pairing of fermions is effective only in odd-l channels. No assumption about the value of the ratio between the Fermi velocity and the sound velocity in the Bose gas is made in the derivation of the energy gap equation. The gap equation is solved without any particular ansatz for the pairing field or the effective interaction. The p-wave superfluidity is studied in detail. By increasing the strength and/or decreasing the range of the effective interaction a transition of the fermion pairing regime, from the Bardeen-Cooper-Schrieffer state to a system of tightly bound couples can be realized. These composite bosons behave as a weakly-interacting Bose-Einstein condensate.

Percolative theories of strongly disordered ceramic high-temperature superconductors

Optimally doped ceramic superconductors (cuprates, pnictides, etc.) exhibit transition temperatures T(c) much larger than strongly coupled metallic superconductors like Pb (T(c) = 7.2 K, E(g)/kT(c) = 4.5) and exhibit many universal features that appear to contradict the Bardeen, Cooper, and Schrieffer theory of superconductivity based on attractive electron-phonon pairing interactions. These complex materials are strongly disordered and contain several competing nanophases that cannot be described effectively by parameterized Hamiltonian models, yet their phase diagrams also exhibit many universal features in both the normal and superconductive states. Here we review the rapidly growing body of experimental results that suggest that these anomalously universal features are the result of marginal stabilities of the ceramic electronic and lattice structures. These dual marginal stabilities favor both electronic percolation of a dopant network and rigidity percolation of the deformed lattice network. This "double percolation" model has previously explained many features of the normal-state transport properties of these materials and is the only theory that has successfully predicted strict lowest upper bounds for T(c) in the cuprate and pnictide families. Here it is extended to include Coulomb correlations and percolative band narrowing, as well as an angular energy gap equation, which rationalizes angularly averaged gap/T(c) ratios, and shows that these are similar to those of conventional strongly coupled superconductors.

INTERMEDIATELY COUPLED PHONON AND CHARGE FLUCTUATION INDUCED SUPERCONDUCTIVITY IN MgB2 PROBED BY A THREE SQUARE WELL APPROACH

A three square well model is developed, which allows one to easily calculate the correlation between the coupling strength parameters and superconducting transition temperature (T c ), pressure and volume derivative of T c and isotope effect exponent for MgB 2. Upon considering the three interactions, namely, electron-phonon, electron-plasmon and Coulomb, the analytical solutions for the energy gap equation allow us to understand the relative interplay of these interactions. The values of the coupling strength and of the Coulomb interaction parameter indicate that the superconductor is in the intermediate coupling regime. The superconducting transition temperature of MgB 2 for the 2D band is estimated as 41 K for [Formula: see text] and μσ*≈0.28. We present correlation curves of T c with various coupling strengths as electron-phonon [Formula: see text], electron-plasmon [Formula: see text] and the Coulomb screening parameter (μσ*). The present approach also explains the reported boron isotope effect and the pressure derivative of T c in the test material. We suggest from these results that both the plasmons and phonons within the framework of a three square well scheme consistently explain the effective electron-electron interaction leading to superconductivity in MgB 2. The implications of the effective interactive potential with both σ and π carriers and its analysis are discussed.

Baryonic 3P2-Dominant Superfluidity under Combined Pion Condensation with Isobar. II: Properties of Pairing Interaction and Numerical Results

According to the formulation developed in I, we calculate energy gaps of the baryonic 3 P2dominant superfluidity under the combined pion condensation with ∆-mixing at moderately high density in neutron star interior. Adopting a baryon-baryon potential extended from a“ root” NN potential to be workable in theN + ∆ space, we obtain the concrete form of the pairing interaction matrix elements between the quasi-baryon pairs, which constitute a two-dimensional angular-momentum stretched state and a charge triplet. With use of OPEG-B as a “root” NN potential and an available set of the parameters representing the combined pion condensation, we study the properties of two-dimensional pairing potentials and the matrix elements of pairing interaction. We find that the strong attraction of pairing interaction for the quasi-neutron pairs is brought about by the spin-orbit potential and the spin- and isospin-dependent core terms of the central potential, whose effects are enhanced due to the pion condensation. The quasi-neutron pair plays a decisive role to bring about meaningful energy gaps, while the coupling between different quasi-baryon pairs plays no important role, as a consequence of a unique feature of the combined pion condensation we adopt. We numerically solve the energy gap equation for baryon density of (2 − 6) times the nuclear density and clarify substantial aspects of resulting superfluid energy gaps, and discuss related problems by taking into account possible change in the factors affecting the energy gaps, such as baryon-baryon potentials, some of the pion condensation parameters and an effective mass of the quasi-particle. Standing on these results, we can say that the 3 P2-dominant superfluid is realized with the critical temperatures Tc of the order of 10 9 K, equivalent to the energy gaps of the order of 0.1 MeV, under the combined pion condensation in neutron star matter. The key point of the recognition lies in the aspects that the ∆-mixing provides the additional pairing attraction of short range, although the attenuation of the pairing attraction due to the charged-pion condensation comes about in the whole interaction range. This means that the superfluid of this type can provide the meaningful suppression on the emissivity of the pion direct Urca process, which makes the pion cooling scenario of neutron stars available.

Solutions of the energy gap equation for D‐wave paired systems

AbstractD‐wave paired isotropic Fermi system is considered in the frame of BCS approach. All admissible solutions of the energy gap equation are given for arbitrary temperatures. The geometry, value of the energy gap and the thermodynamic properties in the ground state as well in the critical temperature is presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Influence of electron-phonon interaction on superconducting state parameters of MgB2

A three-square well model is employed for the three interactions namely, electron–acoustic phonon, electron–optical phonon, and Coulomb in the calculation of superconducting transition temperature (Tc) for layered structure MgB2. The analytical solutions for the energy gap equation allow us to understand the relative interplay of these interactions. The values of the coupling strength and of the Coulomb interaction parameter indicate that the test material is in the intermediate coupling regime. The superconducting transition temperature of MgB2 is estimated as 41 K for λac ≈ 0.3, λop ≈ 0.1, and μ* ≈ 0.07. We suggest from these results that both the acoustic and optical phonons within the framework of a three-square well scheme consistently explains the effective electron–electron interaction leading to superconductivity in layered structure MgB2.

Influence of Electron–Phonon Interaction on Transition Temperature and Isotope Effect of Ba0.6K0.4BiO3

We employ a three-square-well model for the three interactions namely, electron-acoustic phonon, electron-optical phonon, and Coulomb in the calculation of superconducting transition temperature (Tc) and isotope effect coefficient (α) for cubic perovskite Ba0.6K0.4BiO3. The analytical solutions for the energy gap equation allow us to understand the relative interplay of these interactions. To correlate the Tc with various coupling strengths as electron-acoustic phonon (λac), electron-optical phonon (λop) and Coulomb (μ*), we present curves of Tc with them. The values of the coupling strength and of the Coulomb interaction parameter indicate that the superconductor is in the intermediate coupling regime. The superconducting transition temperature of optimally doped Ba—K—BiO is estimated as 29 K for λac of ≈0.3, λop of ≈0.2, and μ* of ≈0.12. The present approach also explains the reported oxygen isotope effect in the test material. We suggest from these results that both the acoustic and optical phonons within the framework of a three-square-well scheme consistently explains the effective electron–electron interaction leading to superconductivity in doped cubic perovskites.

The Possibility of Hyperon Superfluids in Neutron Star Cores

The possible existence of hyperon superfluidity in neutron star cores is studied by using several 1 S0 YY interactions from the OBE baryon-baryon potentials. It is found that not only Λ but also Σ − and Ξ − hyperons could very likely be superfluid. With the increase of the total baryon density ρ toward the center of neutron stars, hyperons (Y )begin to appear as new constituents. The subject of hyperonmixing has gathered much attention from an early stage of theoretical works on neutron stars. 2) - 11) The hyperon fraction yY increases with ρ and in the core region they are important components comparable to nucleons, interestingly affecting the properties of neutron stars. Here we address the question of whether Λ, Σ − and Ξ − admixed could be superfluid. The occurrence of hyperon superfluidity plays a key role in the rapid cooling scenario of neutron stars, i.e., the so-called “hyperon cooling”, 12) - 15) as one of the non-standard cooling scenarios to explain the unusually low surface temperatures observed for some neutron stars. In a preceding work, 16) we concentrated our attention on the case of the Λ superfluid, and we showed that Λ superfluidity exists, though in a limited density region, with the critical temperature Tc ∼ 10 8 −10 9 K. In the present work we discuss the cases of Σ − - and Ξ − -superfluids in reference to the Λ case. Unfortunately, our present knowledge of YY (and also YN )interactions is very limited as compared to the NN interaction. For this reason, we calculate the pairing energy gap by choosing several baryon-baryon (BB) potentials based on the hypothesis of SU(3)invariance and see what can be said regarding the existence or nonexistence of Y superfluids. In this choice, we pay particular attention to the compatibility with hypernuclear data. Although hyperons participate in the high-density region ( ρ> ∼ 2ρ0; where ρ0 = 0.17 nucleons /fm 3 is the nuclear density), the fractional density ρY (≡ yY ρ)is relatively small because yY is not large (10% − 30% at most), and thus the Fermi energy � FY (= ¯ 2 (3π 2 ρY ) 2/3 /2MY , with MY the hyperon mass)is rather low. Therefore, the pairing interaction responsible for Y superfluidity should be that in the 1 S0 pair state. Because there are different Fermi surfaces for every hyperon species, the pairing correlation can be restricted to that of the same Y species. Then the energy gap equation to be solved here has a well-known form of the 1 S0-type. 16)

The uniqueness and approximation of a positive solution of the Bardeen–Cooper–Schrieffer gap equation

In this paper we study the Bardeen–Cooper–Schrieffer energy gap equation at finite temperatures. When the kernel is positive representing a phonon-dominant phase in a superconductor, the existence and uniqueness of a gap solution is established in a class which contains solutions obtainable from bounded domain approximations. The critical temperatures that characterize superconducting–normal phase transitions realized by bounded domain approximations and full space solutions are also investigated. It is shown under some sufficient conditions that these temperatures are identical. In this case the uniqueness of a full space solution follows directly. We will also present some examples for the nonuniqueness of solutions. The case of a kernel function with varying signs is also considered. It is shown that, at low temperatures, there exist nonzero gap solutions indicating a superconducting phase, while at high temperatures, the only solution is the zero solution, representing the dominance of the normal phase, which establishes again the existence of a transition temperature.

The q-Deformed Energy Gap Equation in SLq(2) Superconductive Thermodynamic Model

The q-deformed energy gap equations in SLq(2) superconductive thermodynamic model are derived by using quantum group method.

A model for high-temperature superconductors using the extended Hubbard hamiltonian

We derive a method to study the phase diagram for high-temperature superconductors (HTCS's). Our starting point is the Hubbard hamiltonian with a weak attractive interaction to obtain the formation of bound pairs. We consider this attractive potential at different positions for different compounds. We then construct a wave function of the BCS type by a variational method using the Fourier transform of this extended Hubbard potential and then derive an energy gap equation which can be applied to the s wave and d wave channel. This simple approach allows us to obtain the critical temperature as a function of the doping concentration which gives very good agreement with the experimental phase diagrams of YBaCuO and La(Sr,Ba)CuO compounds.

A Method to study phase diagrams for high temperature superconductors using the extended Hubbard Hamiltonian

We derive a method to study the phase diagram for high temperature superconductors (HTCS). Our starting point is the Hubbard Hamiltonian with a weak attractive interaction to obtain the formation of bound pairs. We consider this attractive potential at different positions for different compounds. We use a BCS type variational method to derive an energy gap equation. This simple approach allows us to obtain the critical temperature as function of the doping concentration which gives very good agreement with the experimental phase diagrams of YBaCuO and La(Sr,Ba)CuO compounds.

Energy gap equation of 1D polyconjugated π-systems in the Hubbard approximation

It is shown that the energy gap of an arbitrary homonuclear or heteronuclear alternant or nonalternant π-conjugated system, having a singlet ground state, is given by the formula Δ E σ={Δ 2 1(σ) corr+[Δ 2(σ) corr+Δ top+Δ geom] 2} 1 2 , where Δ top and Δ geom are factors depending on the molecular topology and geometry, respectively. In the general case, the correlation corrections Δ 1 (σ) and Δ 2(σ) differ for MOs with different spin (σ=α,β) and Δ E β≠Δ E β. In some particular cases Δ E α= Δ E β. The correlation correction Δ 2(σ) corr corresponds to a non-uniform charge distribution in the π-system. In the case of homo-nuclear alternant systems Δ 2(σ) corr=0. The investigations are carried out using the method of alternant molecular orbitals (AMO) in the Hubbard approximation. The band structure of the fully oxidized form of 1D poly(para-aniline)-poly(para-phenyleneimine), has been studied.

Dynamical mass generation for the gravitino in N = 1 supergravity

The Volkov-Akulov field is coupled to supergravity and it is gauged away through a field redefinition, remaining with a negative cosmological constant plus N = 1 supergravity lagrangian. Then the gravitino sector is quantized and a positive cosmological constant is obtained along with a mass-like term for the gravitino. Imposing the effective cosmological constant to be zero, consequently a genuine mass term for the gravitino is obtained. The corresponding energy-gap equation shows that this mass turns out to be of the order of the Planck mass.

A METHOD OF TEMPERATURE CORRECTION FOR ELECTRON TUNNELING SPECTRUM

When the effective phonom spectrum is calculated from the electron tunneling spe-ctrum by invertion of strongly coupled superconducting energy gap equation, the electron density of states must be measured under the condition T→OK. Otherwise the error of the electron density of states and the deviation of effective phonon spectrum from the real one will be very large. In this paper a method is given for correcting this temperature effect, and the above-mentioned difficulty can be overcome. The correct effective phonon spectrum, andλand μ* values can be obtained by this method using electron tunneling spectrum data measured even at T/Tc=0.22.