The superconducting instability in a non-Fermi liquid in dimensions d > 1 is considered. For a particular form of the single particle spectral function with homogeneous scaling A(λk, λ ω) = λα A(k, ω) it is shown that the pair susceptibility is also a scaling function of temperature with power defined by α. We find three different regimes depending on the scaling constant. Bardeen-Cooper-Schrieffer result is recovered for α = − 1 and it corresponds to a marginal scaling of the coupling constant. For α > − 1 the superconducting transition happens above some critical coupling. In the opposite case of α < − 1 for any fixed coupling the system undergoes a transition at low temperatures. Possible implications for theories of high-T c with a superconducting transition driven by interlayer Josephson tunnelling are discussed.
The normal-state single particle spectral function of the high temperature superconducting cuprates, measured by the angle-resolved photoelectron spectroscopy (ARPES), has been considered both anomalous and crucial to understand. Here, we report an unprecedented success of the new extremely correlated Fermi liquid theory by one of us [B. S. Shastry, Phys. Rev. Lett. 107, 056403 (2011)] to describe both laser and conventional synchrotron ARPES data (nodal cut at optimal doping) on Bi(2)Sr(2)CaCu(2)O(8+δ) and synchrotron data on La(1.85)Sr(0.15)CuO(4). It fits all data sets with the same physical parameter values, satisfies the particle sum rule and successfully addresses two widely discussed kink anomalies in the dispersion.
Near a quantum-critical point in a metal a strong fermion-fermion interaction, mediated by a soft boson, destroys fermionic coherence and also gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other, and in a class of large
N models, where the tendency to incoherence is parametrically stronger, one would naively expect an incoherent (non-Fermi liquid) normal state behavior to persist down to
T=0. However, this is not the case for quantum-critical systems, described by Eliashberg theory. In such systems, the part of the fermionic self-energy $\mathrm{\ensuremath{\Sigma}}({\ensuremath{\omega}}_{m})$, relevant for spin-singlet pairing, is large for a generic Matsubara frequency ${\ensuremath{\omega}}_{m}=\ensuremath{\pi}T(2m+1)$ but vanishes for fermions with ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$, while the pairing interaction between fermions with these two frequencies remains strong. It has been shown [Y. Wang et al. Phys. Rev. Lett. 117, 157001 (2016)] that due to this peculiarity, the onset temperature for the pairing
Tp is finite even at large
N, when the scaling analysis predicts a non-Fermi liquid normal state. We consider the system behavior below
Tp and contrast the conventional case, when ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$ are not special, and the case when the pairing is induced by fermions with ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$. We obtain the solution of the nonlinear gap equations in Matsubara frequencies and then convert to real frequency axis and obtain the spectral function $A(k,\ensuremath{\omega})$ and the density of states $N(\ensuremath{\omega})$. In a conventional BCS-type superconductor, $A(k,\ensuremath{\omega})$ and $N(\ensuremath{\omega})$ are peaked at the gap value $\mathrm{\ensuremath{\Delta}}(T)$, and the peak position shifts to a smaller $\ensuremath{\omega}$ as temperature increases towards
Tp, i.e., the gap ``closes in.'' We show that, when the pairing is induced by fermions with ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$, the situation is qualitatively different from the standard BCS result. Namely, the peak in $N(\ensuremath{\omega})$ remains at a finite frequency even at $T={T}_{p}\ensuremath{-}0$, the gap just ``fills in'' near this
T. The spectral function $A(k,\ensuremath{\omega})$ either shows almost the same ``gap filling'' behavior as the density of states, or its peak position shifts to zero frequency already at a finite $\mathrm{\ensuremath{\Delta}}$ (``emergent Fermi arc'' behavior), depending on the position of
k on the Fermi surface. As an example, we compare our results with the data for the cuprates and argue that ``gap filling'' behavior holds in the antinodal region, while the ``emergent Fermi arc'' behavior holds in the nodal region.
Recent photoemission spectroscopy measurements (T. J. Reber et al., arXiv:1509.01611) on cuprate superconductors have inferred that over a wide range of doping, the imaginary part of the electron self-energy scales as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{''}}\ensuremath{\sim}{({\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2})}^{a}$ with
a=1 in the overdoped Fermi-liquid state and
a<0.5 in the optimal to underdoped regime. We show that this non-Fermi-liquid scaling behavior can naturally be explained by the presence of a scale-invariant state of matter known as unparticles. We evaluate analytically the electron self-energy due to interactions with fermionic unparticles. We find that, in agreement with experiments, the imaginary part of the self-energy scales with respect to temperature and energy as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{''}}\ensuremath{\sim}{T}^{2+2\ensuremath{\alpha}}$ and ${\ensuremath{\omega}}^{2+2\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ is the anomalous dimension of the unparticle propagator. In addition, the calculated occupancy and susceptibility of fermionic unparticles, unlike those of normal fermions, have significant spectral weights even at high energies. This unconventional behavior is attributed to the branch cut in the unparticle propagator which broadens the unparticle spectral function over a wide energy range and nontrivially alters the scattering phase space by enhancing (suppressing) the intrinsic susceptibility at low energies for negative (positive) $\ensuremath{\alpha}$. Our work presents evidence suggesting that unparticles might be important low-energy degrees of freedom in strongly coupled systems such as the cuprate superconductors.
In this letter we consider
D-dimensional interacting Fermi liquids, and demonstrate that an attractive interaction with a finite range
Rs that is much greater than the Fermi wavelength
λF breaks the conventional BCS theory of superconductivity. In contrast to the BCS prediction of a finite superconducting gap for all attractive contact interactions, we show that a finite-range interaction does not induce a superconducting gap. Instead, the pair susceptibility develops a power-law singularity at zero momentum and zero frequency signaling quantum critical behavior without long-range ordering. Starting from this, we show that superconductivity can be stabilized by adding a short-range attractive interaction, which is always present in real electronic systems. As an example, we consider a layered quasi-two-dimensional material with attractive electron-electron interactions mediated by optical phonons. We demonstrate a dome shape of the critical temperature
Tc versus doping, strongly suppressed isotope effect, and a weak dependence of the optimal doping and maximal
Tc∗∼0.1EF on the interaction range at
Rs≫λF,
EF is the Fermi energy. We believe that these results could be relevant to high-temperature superconductors.
In this Letter we consider D-dimensional interacting Fermi liquids, and demonstrate that an attractive interaction with a finite range R_{s} that is much greater than the Fermi wavelength λ_{F} breaks the conventional BCS theory of superconductivity. In contrast to the BCS prediction of a finite superconducting gap for all attractive contact interactions, we show that a finite-range interaction does not induce a superconducting gap. Instead, the pair susceptibility develops a power-law singularity at zero momentum and zero frequency signaling quantum critical behavior without long-range ordering. Starting from this, we show that superconductivity can be stabilized by adding a short-range attractive interaction, which is always present in real electronic systems. As an example, we consider a layered quasi-two-dimensional material with attractive electron-electron interactions mediated by optical phonons. We demonstrate a dome shape of the critical temperature T_{c} versus doping, strongly suppressed isotope effect, and a weak dependence of the optimal doping and maximal T_{c}^{*}∼0.1E_{F} on the interaction range at R_{s}≫λ_{F}, E_{F} is the Fermi energy. We believe that these results could be relevant to high-temperature superconductors.