Coupled Gap Equations Research Articles

Going beyond the Chandrasekhar-Clogston limit in a flatband superconductor

The Chandrasekhar-Clogston limit normally places stringent conditions on the magnitude of the magnetic field that can coexist with spin-singlet superconductivity, restricting the critical induced Zeeman shift to a fraction of the superconducting gap. Here, we consider a model system where the spin-singlet Cooper pairing in a dispersive band crossing the Fermi level is boosted by an additional flat-band located away from the Fermi level. The boosting of the pairing in the dispersive band allows for nontrivial solutions to the coupled gap equations for spin-splitting fields considerably larger than the superconducting gaps at zero field. Further, the additional Cooper pairing in the flat-band, away from the Fermi level, increases the superconducting condensation energy without affecting the paramagnetic susceptibility of the system, making the free energy favor the superconducting state. This opens up the possibility for spin-singlet superconductivity beyond the standard Chandrasekhar-Clogston limit.

Two-loop HTL-resummed thermodynamics for mathcal{N} = 4 supersymmetric Yang-Mills theory

We compute the two-loop hard-thermal-loop (HTL) resummed thermodynamic potential for mathcal{N} = 4 supersymmetric Yang-Mills (SYM). Our final result is manifestly gauge-invariant and was renormalized using only simple vacuum energy, gluon mass, scalar mass, and quark mass counter terms. The HTL mass parameters mD, MD, and mq are then determined self-consistently using a variational prescription which results in a set of coupled gap equations. Based on this, we obtain the two-loop HTL-resummed thermodynamic functions of mathcal{N} = 4 SYM. We compare our final result with known results obtained in the weak- and strong-coupling limits. We also compare to previously obtained approximately self-consistent HTL resummations and Padé approximants. We find that the two-loop HTL resummed results for the scaled entropy density is a quantitatively reliable approximation to the scaled entropy density for 0 ≤ λ ≲ 2 and is in agreement with previous approximately self-consistent HTL resummation results for λ ≲ 6.

The p-wave superconductivity in the presence of Rashba interaction in 2DEG.

We investigate the effect of the Rashba interaction on two dimensional superconductivity. The presence of the Rashba interaction lifts the spin degeneracy and gives rise to the spectrum of two bands. There are intraband and interband pairs scattering which result in the coupled gap equations. We find that there are isotropic and anisotropic components in the gap function. The latter has the form of cos φk where . The former is suppressed because the intraband and the interband scatterings nearly cancel each other. Hence, −the system should exhibit the p-wave superconductivity. We perform a detailed study of electron-phonon interaction for 2DEG and find that, if only normal processes are considered, the effective coupling strength constant of this new superconductivity is about one-half of the s-wave case in the ordinary 2DEG because of the angular average of the additional in the anisotropic gap function. By taking into account of Umklapp processes, we find they are the major contribution in the electron-phonon coupling in superconductivity and enhance the transition temperature Tc.

Superconducting phase diagram of itinerant antiferromagnets

We study the phase diagram of the Hubbard model in the weak-coupling limit for coexisting spin-density-wave order and spin-fluctuation-mediated superconductivity. Both longitudinal and transverse spin fluctuations contribute significantly to the effective interaction potential, which creates Cooper pairs of the quasi-particles of the antiferromagnetic metallic state. We find a dominant $d_{x^2-y^2}$-wave solution in both electron- and hole-doped cases. In the quasi-spin triplet channel, the longitudinal fluctuations give rise to an effective attraction supporting a $p$-wave gap, but are overcome by repulsive contributions from the transverse fluctuations which disfavor $p$-wave pairing compared to $d_{x^2-y^2}$. The sub-leading pair instability is found to be in the $g$-wave channel, but complex admixtures of $d$ and $g$ are not energetically favored since their nodal structures coincide. Inclusion of interband pairing, in which each fermion in the Cooper pair belongs to a different spin-density-wave band, is considered for a range of electron dopings in the regime of well-developed magnetic order. We demonstrate that these interband pairing gaps, which are non-zero in the magnetic state, must have the same parity under inversion as the normal intraband gaps. The self-consistent solution to the full system of five coupled gap equations give intraband and interband pairing gaps of $d_{x^2-y^2}$ structure and similar gap magnitude. In conclusion, the $d_{x^2-y^2}$ gap dominates for both hole and electron doping inside the spin-density-wave phase.

ON d+id DENSITY WAVE AND SUPERCONDUCTING ORDERINGS IN HOLE-DOPED CUPRATES

The d+id-density wave (chiral DDW) order, at the anti-ferromagnetic wave vector Q = (π, π), is assumed to represent the pseudo-gap (PG) state of a hole-doped cuprate superconductor. The pairing interaction U(k, k′) required for d+id ordering comprises of (Ux2-y2(k, k′), Uxy(k, k′)), where [Formula: see text] and [Formula: see text] with U1 > U2. The d-wave superconductivity (DSC), driven by an assumed attractive interaction of the form [Formula: see text] where V1 is a model parameter, is discussed within the mean-field framework together with the d+id ordering. The single-particle excitation spectrum in the CDDW + DSC state is characterized by the Bogoluibov quasi-particle bands — a characteristic feature of SC state. The coupled gap equations are solved self-consistently together with the equation to determine the chemical potential (μ). With the pinning of the van Hove-singularities close to μ, one is able to calculate the thermodynamic and transport properties of the under-doped cuprates in a consistent manner. The electron specific heat displays non-Fermi liquid feature in the CDDW state. The CDDW and DSC are found to represent two competing orders as the former brings about a depletion of the spectral weight (and Raman response function density) available for pairing in the anti-nodal region of momentum space. It is also shown that the depletion of the spectral weight below Tc at energies larger than the gap amplitude occurs. This is an indication of the strong-coupling superconductivity in cuprates. The calculation of the ratio of the quasi-particle thermal conductivity αxx and temperature in the superconducting phase is found to be constant in the limit of near-zero quasi-particle scattering rate.

Relation between nodes and2Δ/Tcon the hole Fermi surface in iron-based superconductors

We analyze the interplay between the absence or presence of nodes in the superconducting gap along electron Fermi surfaces (FSs) in Fe pnictides and $2\ensuremath{\Delta}/{T}_{c}$ along hole FSs ($2{\ensuremath{\Delta}}_{h}/{T}_{c}$), measured by angle-resolved photoemission spectroscopy (ARPES). We solve the set of coupled gap equations for 4- and 5-pocket models of Fe pnictides and relate the presence of the nodes to $2{\ensuremath{\Delta}}_{h}/{T}_{c}$ being below a certain threshold. Using ARPES data for $2{\ensuremath{\Delta}}_{h}/{T}_{c}$, we find that the optimally doped ${\text{Ba}}_{1\ensuremath{-}x}{\text{K}}_{x}{\text{Fe}}_{2}{\text{As}}_{2}$ and $\text{Ba}({\text{Fe}}_{1\ensuremath{-}x}{\text{Co}}_{x}){}_{2}{\text{As}}_{2}$ are likely nodeless, but isovalent ${\text{BaFe}}_{2}({\text{As}}_{1\ensuremath{-}x}{\text{P}}_{x}){}_{2}$ likely has nodes.

Superconductivity enhanced by d-density wave: A weak-coupling theory

Making a revision of mistakes in Ref. [19], we present a detailed study of the competition and interplay between the d-density wave (DDW) and d-wave superconductivity (DSC) within the fluctuation-exchange (FLEX) approximation for the two-dimensional (2D) Hubbard model. In order to stabilize the DDW state with respect to phase separation at lower dopings a small nearest-neighbor Coulomb repulsion is included within the Hartree–Fock approximation. We solve the coupled gap equations for the DDW, DSC, and π-pairing as the possible order parameters, which are caused by exchange of spin fluctuations, together with calculating the spin fluctuation pairing interaction self-consistently within the FLEX approximation. We show that even when nesting of the Fermi surface is perfect, as in a square lattice with only nearest-neighbor hopping, there is coexistence of DSC and DDW in a large region of dopings close to the quantum critical point (QCP) at which the DDW state vanishes. In particular, we find that in the presence of DDW order the superconducting transition temperature T c can be much higher compared to pure superconductivity, since the pairing interaction is strongly enhanced due to the feedback effect on spin fluctuations of the DDW gap. π-pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small. In the present work, we have developed a weak-coupling theory of the competition between DDW and DSC in 2D Hubbard model, using the static spin fluctuation obtained within FLEX approximation and ignoring the self-energy effect of spin fluctuations. For our model calculations in the weak-coupling limit we have taken U/ t=3.4, since the antiferromagnetic instability occurs for higher values of U/ t.

3-Band theory of Fe pnictide superconductors

For the Fe pnictide superconductors, band features important for superconductivity (SC) in the neighborhood of εF are mimicked by tight-binding bands with bases of Fe dxz, dyz and dxy orbitals. In terms of this 3-band model, effective pair transfer processes originating from Coulombic integrals are shown to be largely enhanced due to the nesting between electron and hole pockets by explicitly summing up ladder diagrams. Obtained coupled gap equations lead to s±-type SC. This framework are argued to provide a scheme for discussing important elements working for the high Tc SC, although its validity is restricted to the case where Dirac cones are away from εF. Experimental results on Fe pnictides are discussed in terms of the present results. The pair transfer process explicitly due to the exchange-like integral is suggested to be considerably important for the high Tc in MFeAsO1−xFx to occur.

THE CJT CALCULATION IN STUDYING NUCLEAR MATTER BEYOND MEAN FIELD APPROXIMATION

We have introduced the Cornwall–Jackiw–Tomboulis (CJT) resummation scheme in studying nuclear matter. Based on the CJT formalism and using Walecka model, we have derived a set of coupled Dyson equations of nucleons and mesons. Neglecting the medium effects of the mesons, the usual mean field theory (MFT) results can be obtained. The beyond MFT calculations have been performed by thermodynamic consistently determining the meson effective masses and solving the coupled gap equations for nucleons and mesons together. The numerical results for the nucleon and meson effective masses at finite temperature and chemical potential in nuclear matter are discussed.

Quantum transport in noncentrosymmetric superconductors and thermodynamics of ferromagnetic superconductors

Motivated by recent findings of unconventional superconductors exhibiting multiple broken symmetries, we consider a general Hamiltonian describing coexistence of itinerant ferromagnetism, spin-orbit coupling, and mixed spin-singlet and spin-triplet superconducting pairing in the context of mean-field theory. The Hamiltonian is diagonalized and exact eigenvalues are obtained, thus allowing us to write down the coupled gap equations for the different order parameters. Our results may then be applied to any model describing coexistence of any combination of these three phenomena. As a specific application of our results, we consider tunneling between a normal metal and a noncentrosymmetric superconductor with mixed singlet and triplet gaps. The conductance spectrum reveals information about these gaps in addition to how the influence of spin-orbit coupling is manifested. Explicitly, we find well-pronounced peaks and bumps in the spectrum at voltages corresponding to the sum and the difference of the magnitude of the singlet and triplet components. Our results may thus be helpful in determining the relative sizes of the singlet and triplet gaps in noncentrosymmetric superconductors. We also consider the coexistence of itinerant ferromagnetism and triplet superconductivity as a model for recently discovered ferromagnetic superconductors. The coupled gap equations are solved self-consistently, and we study the conditions necessary to obtain the coexistent regime of ferromagnetism and superconductivity. Analytical expressions are presented for the order parameters, and we provide an analysis of the free energy to identify the preferred system state. It is found that the uniform coexistence of ferromagnetism and superconductivity is energetically favored compared to both the purely ferromagnetic state and the unitary superconducting state with zero magnetization. Moreover, we make specific predictions concerning the heat capacity for a ferromagnetic superconductor. In particular, we report a nonuniversal relative jump in the specific heat, depending on the magnetization of the system, at the uppermost superconducting phase transition. We propose that this may be exploited to obtain information about the superconducting pairing symmetry realized in ferromagnetic superconductors, in addition to the magnitude of the exchange splitting between majority- and minority-spin bands.

Color superconductivity with determinant interaction in strange quark matter

We investigate the effect of six fermion determinant interaction on color superconductivity as well as on chiral symmetry breaking. Coupled mass gap equations and the superconducting gap equation are derived through the minimisation of the thermodynamic potential. The effect of nonzero quark -- antiquark condensates on the superconducting gap is derived. This becomes particularly relevant for the case of 2-flavor superconducting matter with unpaired strange quarks in the diquark channel. While the effect of six fermion interaction leads to an enhancement of u-d superconductivity, due to nonvanishing strange quark--antiquark condensates, such an enhancement will be absent at higher densities for u-s or d-s superconductivity due to early (almost) vanishing of light quark-- antiquark condensates.

Two-band theory of specific heat and thermal conductivity in the mixed state ofMgB2

We solve the coupled gap equations for the $\sigma$- and $\pi$-bands of MgB$_2$ in the vortex state and calculate the resulting field dependencies of the specific heat coefficient $\gamma$ and the thermal conductivity $\kappa $. The crucial parameters of the theory are the interband pairing interaction $\lambda_{\pi\sigma}$ and the ratio $s=\xi_{\sigma}/\xi_{\pi}$ of the coherence lengths. For reasonably small $\lambda_{\pi\sigma}$ and s, the small gap $\Delta_{\pi}$ decreases with increasing magnetic field $H$ much faster than the large gap $\Delta_{\sigma} $. This gives rise to the observed rapid increase of $\gamma_{\pi}$ and $\kappa_{\pi}$ for small fields while $\gamma_{\sigma}$ and $\kappa_{\sigma}$ exhibit conventional field dependencies. Inclusion of intraband impurity scattering yields fairly good agreement with experiments for applied fields along the c axis.

On gap equations and color-flavor locking in cold dense QCD with three massless flavors

The superconductivity in cold dense QCD with three massless flavors is analyzed in the framework of the Schwinger-Dyson equation. The set of two coupled gap equations for the color antitriplet, flavor antitriplet ( 3 ̄ , 3 ̄ ) and the color sextet, flavor sextet (6,6) order parameters is derived. It is shown that the antitriplet-antitriplet gives the dominant contribution to the color-flavor locked order parameter, while the sextet-sextet is small but nonzero.

Coupled gap equations for the screening masses in the $SU(2)$ Higgs model

The complete set of static screening masses is determined for the SU(2) Higgs model from one-loop coupled gap equations. Results from the version, containing scalar fields both in the fundamental and adjoint representations are compared with the model arising when the integration over the adjoint scalar field is performed. A non-perturbative and non-linear mapping between the couplings of the two models is proposed, which exhibits perfect decoupling of the heavy adjoint scalar field. Also the alternative of a gauge invariant mass resummation is investigated in the high temperature phase.

Possible Superconductivity in the Doped Ladder CompoundLa1−xSrxCuO2.5

LaCuO_2.5 is a system of coupled, two-chain, cuprate ladders which may be doped systematically by Sr substitution. Motivated by the recent synthesis of single crystals, we investigate theoretically the possibility of superconductivity in this compound. We use a model of spin fluctuation-mediated superconductivity, where the pairing potential is strongly peaked at \pi in the ladder direction. We solve the coupled gap equations on the bonding and antibonding ladder bands to find superconducting solutions across the range of doping, and discuss their relevance to the real material.

Two-band superconductivity in the doped spin ladder La 1− xSr xCuO 2.5

LaCuO 2.5 is a system of coupled, two-chain, cuprate spin ladders which may be doped systematically by Sr substitution. Motivated by the recent synthesis of single crystals, we conduct a theoretical analysis to seek superconductivity in this compound. We test this possibility within a model of spin fluctuation-mediated superconductivity, where the pairing potential is strongly peaked at π in the ladder direction. We solve the coupled gap equations on the bonding and antibonding ladder bands to find superconducting solutions across the range of doping, and discuss their relevance to the real material.

Analyzing chiral symmetry breaking in supersymmetric gauge theories

We compare gap equation predictions for the spontaneous breaking of global symmetries in supersymmetric Yang-Mills theory to nonperturbative results from holomorphic effective action techniques. In the theory without matter fields, both approaches describe the formation of a gluino condensate. With $N_f$ flavors of quark and squark fields, and with $N_f$ below a certain critical value, the coupled gap equations have a solution for quark and gluino condensate formation, corresponding to breaking of global symmetries and of supersymmetry. This appears to disagree with the newer nonperturbative techniques, but the reliability of gap equations in this context and whether the solution represents the ground state remain unclear.

Coupled gap equations for the screening masses in hot SU(N) gauge theory

Coupled 1-loop gap equations are studied numerically for non-Abelian electric and magnetic screening in various versions of the three-dimensional effective gauge models. Corrections due to higher dimensional and non-local operators are assessed quantitatively. Comparison with numerical Monte-Carlo investigations suggests that quantitative understanding beyond the qualitative features can be achieved only by going beyond the present treatment.

Coupled gap equations for the screening masses in hot SU(N) gauge theory

Coupled 1-loop gap equations are studied numerically for non-Abelian electric and magnetic screening in various versions of the three-dimensional effective gauge models. Corrections due to higher dimensional and non-local operators are assessed quantitatively. Comparison with numerical Monte-Carlo investigations suggests that quantitative understanding beyond the qualitative features can be achieved only by going beyond the present treatment.

Mixed (s+id)-wave order parameters in the Van Hove scenario

In the Van Hove scenario including orthorhombic distortion effect, we develop a pair of coupled gap equations for the mixed (s+id)-wave order parameter. It is found that a mixed s+id symmetry state is realized in a certain range of relative strength of the s and d interactions, and there are two second-order transitions between the mixed and the pure symmetry states. Particular attention is paid to the temperature dependence of two components in the mixed order parameter as well as their evolution from a pure s to a pure d symmetry state.