Self-consistent microscopic foundation of the weakly screened Coulomb interaction in extended-saddle-point layered cuprates (open issues from Abrikosov, 1994)

In a 1994 Physica C paper, A. A. Abrikosov proposed that in layered cuprate high-temperature superconductors (HTSC) the combination of extended saddle-point (flat-band, quasi-one-dimensional) spectral regions and weak electronic screening — a forward-peaked interaction modeled by the factor $K^{2}/(k^{2}+\kappa^{2})$ with a small Debye screening wavevector $\kappa$ controlled by a large dielectric constant $\epsilon_{\infty}$ — logarithmically enhances the effective phonon-mediated coupling, producing a strongly momentum-anisotropic superconducting gap, a large $2\Delta/T_{\mathrm{c}}$ ratio, a linear-in-temperature resistivity, and a small isotope effect. The paper itself left the microscopic status of this mechanism open on five linked issues: self-consistent inclusion of the bare Coulomb repulsion; Landau Fermi-liquid validity in the quasi-1D saddle regions; quantitative impurity/disorder dependence of the gap anisotropy; microscopic derivation of the screening factor and the numerical value of $\epsilon_{\infty}$; and the relation to the interlayer tunneling model. No resolution of these issues is known in the literature to date: Abrikosov's own program stopped in 1997 without closing them, later work developed forward-scattering and nonadiabatic physics in modified frameworks, the Fermi-liquid question for flat spectral regions has model-level but not material-specific answers, and the d-wave consensus together with the recent van Hove/Lifshitz-transition revival (Kordyuk, 2018) keeps the saddle-point premise alive while its microscopic justification remains missing. The problem — to decide, by a self-consistent microscopic treatment with realistic screening parameters, whether the weak-screening extended-saddle mechanism survives — remains open.

Background

The copper-oxide (cuprate) high-temperature superconductors (HTSC), discovered in 1986, lose electrical resistance at transition temperatures $T_{\mathrm{c}}$ far above those of conventional superconductors, and the microscopic mechanism of this high $T_{\mathrm{c}}$ has been contested ever since. Their defining structural feature is a stack of weakly coupled CuO$_2$ planes separated by distance $d$, making the electronic spectrum extremely anisotropic. Angle-resolved photoemission spectroscopy (ARPES) on YBa$_2$Cu$_3$O$_{6.9}$, YBa$_2$Cu$_4$O$_8$ and Bi$_2$Sr$_2$CaCu$_2$O$_8$ showed in the early 1990s that near the $(\pi,0)$ points of the Brillouin zone the band is nearly flat along one momentum direction over an extended segment of length $P_{y0}$: an "extended saddle point". A saddle point of a band produces a van Hove singularity, a non-analytic peak in the electronic density of states, and the extended flat character makes the surrounding region effectively quasi-one-dimensional: electrons there disperse in only one momentum direction, as in a 1D wire.

A. A. Abrikosov, with J. C. Campuzano and K. Gofron, proposed that these extended saddle-point singularities, lying close to the Fermi energy, are central to the physics of the cuprates. In the source paper (Physica C 222, 191 (1994)) he developed a quantitative model of this program. The ingredients are: the Debye screening wavevector $\kappa$ (in a metal, mobile electrons screen a charge over length $1/\kappa$, with $\kappa$ set by the density of states and the dielectric constant); the high-frequency dielectric constant $\epsilon_{\infty}$ entering the electronic screening of the layered, partly ionic crystal; and a model interaction in which the squared electron-phonon matrix element is multiplied by the factor $K^{2}/(k^{2}+\kappa^{2})$, where $K$ is the reciprocal-lattice period and $k$ the momentum transfer. Small $\kappa$ makes this factor strongly peaked at small momentum transfer (forward-focused scattering), i.e. weak screening. Using only the saddle-region density of states for the dispersion $\xi=p_{x}^{2}/(2m_{x})-(\mu-\epsilon_{0})$ (with $\mu-\epsilon_{0}$ the chemical potential's distance from the saddle-point energy), the paper obtains

$$\kappa^{2}(0)=\frac{2\sqrt{2}\,N_{\mathrm{s}}e^{2}m_{x}^{1/2}P_{y0}}{\pi\,\epsilon_{\infty}\,d\,(\mu-\epsilon_{0})^{1/2}},$$

with $N_{\mathrm{s}}$ the number of saddle points per Brillouin zone, so that a large $\epsilon_{\infty}$ and a large $\mu-\epsilon_{0}$ weaken screening. After reducing the 3D kernel over the quasi-1D region, the effective pairing coupling is log-enhanced, $g\pi K^{2}\ln(P_{y0}/\kappa)$, producing a gap $\Delta(\boldsymbol{p})$ that is large and nearly constant inside the singular region and decays away from it roughly as $(P_{y0}/|p_{y}|)/\ln(P_{y0}/\kappa)$ — a strongly anisotropic gap consistent with the then-reported anisotropic-gap ARPES and with tunneling reports of $2\Delta/T_{\mathrm{c}}\sim 6$–$8$ in Bi- and Hg-based cuprates. The same quasi-1D kinematics gives an electron-electron scattering rate $1/\tau_{\mathrm{ee}}\propto T$ (rather than $T^{2}$), hence a resistivity $\rho=(\pi^{2}/8)\,m_{x}dT/(\hbar P_{y0}(\mu-\epsilon_{0})\epsilon_{\infty})$ linear in temperature, dominant over electron-phonon scattering for $T\ll c\kappa$ ($c$ the sound velocity). The mechanism requires $\epsilon_{\infty}\gg 1$, and a small isotope effect is obtained if $\mu-\epsilon_{0}\ll\omega(\kappa)=c\kappa$ with $T_{\mathrm{c}}\leq\mu-\epsilon_{0}$.

Two theoretical frameworks set the context for the paper's open issues. First, the Landau Fermi-liquid theory — the standard quasiparticle description of interacting fermions in two and three dimensions — is known to fail in a strictly one-dimensional interacting system, which flows to a Luttinger-liquid fixed point with no quasiparticles and power-law response governed by logarithmic singularities; whether the transverse (non-1D) couplings present in real layered materials suppress these singularities is a quantitative question. Second, Migdal's theorem, which justifies neglecting vertex corrections in conventional phonon-mediated superconductivity, becomes questionable when the density of states is singular and the relevant phonon momenta are of order $\kappa$. A competing proposal of the time, the interlayer tunneling model (Chakravarty, Sudbø, Anderson and Strong, Science 261, 337 (1993)), attributed high $T_{\mathrm{c}}$ and the gap anisotropy to Cooper-pair tunneling between the CuO$_2$ layers rather than to any screening property.

The 1994 paper explicitly acknowledges that its mechanism rests on unverified assumptions, and it is the set of issues it left open — the Coulomb repulsion dropped from the gap equation, the Fermi-liquid description assumed for quasi-1D regions, the impurity dependence of the anisotropic gap, the phenomenological screening factor, and the relation to the interlayer tunneling model — that this problem asks the solver to resolve.

Problem Statement

Provide a self-consistent microscopic foundation for the weakly screened Coulomb interaction in extended-saddle-point layered cuprates — or a quantitative no-go within the same self-consistency — deciding whether Abrikosov's 1994 mechanism survives. Concretely, starting from a layered cuprate band structure containing quasi-one-dimensional extended saddle-point regions, the microscopic status of the mechanism turns on the following linked open issues: (1) whether the pairing and scattering equations, with the bare Coulomb repulsion retained on the same footing as the forward-peaked, weakly screened phonon interaction modeled by the factor $K^{2}/(k^{2}+\kappa^{2})$ (or a microscopically derived replacement, including the correct power of the denominator), leave the longitudinal phonon attraction compensated, partially cancelled, or protected by the extreme momentum anisotropy — the paper assumed the compensation that occurs in strongly compressed matter does not happen in layered cuprates, without a self-consistent verification; (2) whether, quantitatively for the cuprate saddle regions, the transverse (non-1D) couplings suppress the one-dimensional logarithmic (Luttinger-liquid-type) singularities, i.e. whether a Landau Fermi-liquid treatment of these regions is internally consistent; (3) how impurities and disorder quantitatively modify the predicted momentum anisotropy of the gap, distinguishing isotropic neutral-defect scattering from small-angle scattering off weakly screened charged defects, in comparison with the observation that more disordered samples show more isotropic gaps; (4) the numerical value of the high-frequency dielectric constant $\epsilon_{\infty}$ relevant to low-energy electronic screening in the cuprates, and hence the screening wavevector $\kappa$, and whether the required $\epsilon_{\infty}\gg 1$ regime actually holds; and (5) the relation of the resulting mechanism to the interlayer tunneling model of high $T_{\mathrm{c}}$ and gap anisotropy. The scientific objective is a single internally consistent account that either supports or excludes the weak-screening extended-saddle explanation of high $T_{\mathrm{c}}$, the anisotropic gap and large $2\Delta/T_{\mathrm{c}}$, the linear-in-temperature resistivity, and the small isotope effect; issues (1)–(4) are the quantitative determinations such an account must settle, while issue (5) is comparative context it should address. The account must return an explicit verdict stating whether the mechanism survives or is excluded; an intermediate outcome for issue (1) such as "partially cancelled", or an outcome that depends on the parameters, resolves the task only when accompanied by that explicit verdict. That verdict is to be given for realistic cuprate parameters: an account whose outcome depends on the parameters must report the relevant threshold values together with the verdict for the realistic regime, and the mechanism counts as excluded if the account shows it cannot reproduce one or more of the phenomena the explanation is required to account for (high $T_{\mathrm{c}}$, the anisotropic gap and large $2\Delta/T_{\mathrm{c}}$, the linear-in-temperature resistivity, and the small isotope effect).

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