Unresolved quantitative developments in Abrikosov's extended saddle-point model of high-$T_c$ cuprate superconductors

In a 1999 paper, A. A. Abrikosov proposed a theory of layered high-$T_c$ cuprate superconductivity built on extended saddle-point (strengthened van Hove) singularities in the electronic spectrum, weak screening of the Coulomb interaction, and a phonon-mediated attraction plus a small short-range repulsion of Hund/spin-fluctuation origin. The paper itself acknowledged that the theory was quantitatively incomplete, and the present problem is the unfinished agenda the source explicitly acknowledged: (i) reconciling the observed positive low-temperature curvature of the upper critical field with the paramagnetic (Pauli-limit) effect, including a quantitative assessment of the conjectured sequence of inhomogeneous LOFF phases that the source cites from Buzdin and Brison for reduced-dimensionality systems where orbital destruction becomes impossible; (ii) extending the reduced singular-region interaction Hamiltonian so that low-energy excitations (low-temperature specific heat and kinetics) become calculable; and (iii) a quantitative theory of the negative copper isotope effect and its nonmonotonic oxygen-concentration dependence. No later work is known that completes any of these items within the extended saddle-point model, and the low-temperature $H_{c2}$ curvature and the copper isotope effect remain without consensual quantitative explanation in any framework. The problem is therefore open, with the core being the three quantitative developments the source itself acknowledged, to be confronted with the subsequent experimental record.

Background

The layered copper-oxide (cuprate) superconductors retain superconductivity at temperatures far above those of conventional superconductors, and although the gap symmetry is now established to be predominantly $d_{x^2-y^2}$ (the superconducting energy gap changes sign under a 90-degree rotation of the momentum within the CuO$_2$ planes), no microscopic mechanism is consensually accepted. One concrete proposal, developed by A. A. Abrikosov in the 1990s and summarized in his 1999 paper, is the extended saddle-point (ESP) model. Its ingredients are: (a) extended saddle-point singularities — flat, nearly dispersionless stretches of the electronic band lying close to the Fermi level in the CuO$_2$ planes, a strengthening of the usual van Hove singularity (a point where the band dispersion is stationary and the electronic density of states is enhanced); angle-resolved photoemission had found the Fermi level pinned near such features in several cuprates. (b) Weak screening — in layered, low-carrier-density materials the Coulomb interaction is screened inefficiently, so the inter-electron repulsion between states in different singular regions acquires a singular momentum dependence. (c) A conventional phonon-mediated attraction together with a small short-range repulsion of Hund's (intra-atomic exchange) or spin-fluctuation origin. Within this model Abrikosov derived closed formulas for $T_c$ and the condensation energy, gave qualitative explanations of the oxygen and copper isotope effects, predicted a division of layered cuprates into two groups according to whether the order parameter is of odd ($d$-wave-like, sign-changing) or even ($s$-wave-like) symmetry, described the roughly 41 meV peak in the inelastic neutron scattering of YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) via transitions between singular regions, and addressed the positive curvature of the upper critical field.

Several terms of art are needed to state what remains undone. The upper critical field $H_{c2}$ is the magnetic field at which superconductivity is destroyed; near $T_c$ it usually rises linearly in $1-T/T_c$, and a convex-upward ('positive curvature') shape, seen in cuprates, is anomalous. Beyond the orbital pair-breaking of the vortices, the paramagnetic (Pauli) effect — the destruction of spin-singlet Cooper pairs once the Zeeman spin polarization exceeds the pair binding — normally restores negative curvature at low temperature and large field. The LOFF (Larkin-Ovchinnikov-Fulde-Ferrell) states are spatially inhomogeneous superconducting phases with a modulated order parameter that pair electrons from partially spin-split Fermi surfaces; in systems of reduced dimensionality, where the orbital destruction becomes impossible, they are predicted to appear as a sequence of phases with increasing critical field near the Pauli limit, which Buzdin and Brison suggested could explain the persistent positive slope of $H_{c2}$ at low temperatures. The isotope effect is the shift of $T_c$ upon isotopic substitution, quantified by the exponent $\alpha=-d\ln T_c/d\ln M$; phonon-mediated BCS theory gives $\alpha\approx 0.5$, whereas in cuprates the oxygen isotope effect is anomalous and the copper isotope effect was measured to be negative ($T_c$ rising upon substituting heavier Cu) and nonmonotonic in oxygen content.

The 1999 source states that its reduced interaction Hamiltonian, which retains only the singular momentum regions, suffices for $T_c$ and the total energy but fails when low-energy excitations play the major role, as in the specific heat and kinetics at low temperatures; that accounting for the paramagnetic (Pauli) effect restores a negative curvature of $H_{c2}$ at lower temperatures and larger fields, which does not fit the experimental data; and that in a simple model one cannot hope to reproduce all the details of the copper isotope effect. Independent early criticism exists: a strong-coupling Eliashberg-type analysis (Eliashberg theory being the standard beyond-BCS treatment of electron-boson-mediated pairing) by Radtke and Norman found extended van Hove singularities alone insufficient to reach the observed $T_c$ values, and later reviews note the model is not widely accepted. The open problem below is nevertheless the source's own acknowledged agenda, retained at its original generality.

Problem Statement

Complete the quantitative development of Abrikosov's extended saddle-point (ESP) model of layered high-$T_c$ cuprates for the gaps the 1999 source itself explicitly acknowledged. The problem has three components:

1. Low-temperature upper critical field. The model reproduces the positive curvature of $H_{c2}(T)$ near $T_c$, but inclusion of the paramagnetic (Pauli) effect leads, in the source's words, to the restoration of a negative curvature at lower temperatures and larger fields, which does not fit the experimental data in which $H_{c2}$ continues to rise with positive slope. Reconcile the model with this observation — evaluating as a possible origin the conjecture the source cites from Buzdin and Brison, that in cases of reduced dimensionality (films, wires in longitudinal field), where the orbital destruction becomes impossible, the paramagnetic effect leads to a sequence of phase transitions between different inhomogeneous LOFF (Larkin-Ovchinnikov-Fulde-Ferrell) phases with increasing critical fields, or by an alternative mechanism internal to the model.

2. Low-energy excitations. The reduced interaction Hamiltonian, retaining only the singular momentum regions, is sufficient for $T_c$ and the total energy but fails where low-energy excitations play the major role, as in the specific heat and kinetics at low temperatures. Construct the treatment beyond the reduced singular-region Hamiltonian and compute the low-temperature specific heat and kinetic coefficients within it.

3. Copper isotope effect. The model's explanation of the negative copper isotope exponent is qualitative and, by the source's own statement, cannot reproduce all the details in a simple model. Carry out the more comprehensive calculation reproducing the experimental features the source cites, in particular the nonmonotonic dependence of the exponent on oxygen concentration.

The three developments are to be confronted with the experimental data the source cited — the pulsed-field and resistive $H_{c2}(T)$ curves, the low-temperature specific-heat and transport data, and the measured copper isotope exponent versus oxygen content.

References

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