Extending the supercurrent-fluctuation theory of the cuprate pseudogap to interlayer Josephson coupling and the regime near the superconducting transition

In underdoped cuprate superconductors, A. A. Abrikosov (Phys. Rev. B 64, 104521 (2001)) calculated pseudogap-phase observables — the photoemission spectral function, the inelastic neutron-scattering response, and the Knight shift — by averaging Doppler-shifted quasiparticle quantities over a Gaussian distribution of superfluid velocities carried by small superconducting domains with uncorrelated supercurrents, anchored to the extended saddle points of the cuprate band structure. The construction is strictly two-dimensional and phase-gradient-only, and the work itself defers the extension needed near the superconducting transition: close to the true $T_c$, interplane phase locking and Josephson energies become important and the supercurrent-only formalism breaks down.

Background

The cuprate high-temperature superconductors, when hole-doped below optimal doping (the underdoped regime), display a pseudogap: above the superconducting transition temperature $T_c$, and up to a much higher crossover temperature $T^*$, low-energy electronic spectral weight is suppressed, as seen in angle-resolved photoemission spectroscopy (ARPES, which measures how electron removal energies depend on momentum), in the NMR Knight shift (the conduction-electron contribution to the nuclear magnetic resonance frequency, which tracks the electron spin susceptibility), and in a broadened maximum in magnetic neutron scattering. One long-standing interpretation is the phase-fluctuation scenario: Emery and Kivelson (1995) argued that in superconductors with small superfluid density — equivalently small phase stiffness, the energy cost of spatially varying the order-parameter phase — phase fluctuations destroy long-range phase coherence at $T_c$ while the pairing amplitude survives to higher temperature, so the loss of phase coherence by itself can produce gap-like spectroscopic signatures. Franz and Millis (1998) made this concrete for spectroscopy: spatially varying phase gradients act as local supercurrents, and the resulting Doppler shifts of quasiparticle energies broaden the single-particle spectrum.

Abrikosov (Phys. Rev. B 64, 104521 (2001)) developed this into a quantitative, band-structure-specific model. He described the pseudogap phase as small superconducting domains carrying uncorrelated supercurrents (the spatially inhomogeneous Franz–Millis picture), with the electronic structure dominated by extended saddle points of the dispersion near $(\pi,0)$ and $(0,\pi)$ — 'hot' regions where the Fermi velocity is strongly reduced and the density of states enhanced. From a two-dimensional phase-only Ginzburg–Landau free energy restricted to these regions he derived a Gaussian distribution of superfluid velocities, $W[\mathbf{v}_s]$; averaging the Doppler-shifted quasiparticle Green's function over this distribution (the Doppler shift of a quasiparticle of momentum $\mathbf{p}$ is $\eta=\mathbf{p}\cdot\mathbf{v}_s$) yielded the ARPES spectral function; the corresponding two-particle average, building on his earlier extended-saddle-point spin susceptibility (Phys. Rev. B 57, 8656 (1998)), gave the neutron-scattering lineshape; and averaging the spin susceptibility gave the Knight shift. Local normal regions were introduced through the Landau critical velocity $v_{sc}=\Delta_a/(m v_1)$ beyond which superflow destroys superconductivity ($\Delta_a$ is the gap in the saddle regions and $v_1$ the Fermi velocity there), and the pseudogap onset $T^*$ was estimated from the condition that the thermally characteristic superfluid velocity equals this critical velocity. In all cases the calculated curves were qualitatively close to experiment.

The construction is nevertheless strictly two-dimensional and contains only phase gradients (supercurrents): no interplane hopping and no Josephson coupling between the CuO$_2$ planes. Cuprates are layered superconductors whose planes are weakly Josephson coupled (neighboring planes coherently tunnel Cooper pairs, with an energy that locks their order-parameter phases together); close to the true $T_c$, interplane phase locking and the interlayer Josephson energy become important, and a supercurrent-only two-dimensional description ceases to be valid. In addition, the model treats neither fluctuations below $T_c$ — relevant because the neutron resonance energy empirically scales with the real $T_c$ (approximately $E_r\approx 5.8\,k_BT_c$ across cuprate families) even though the single-particle gap increases with underdoping — nor the small, slowly fluctuating magnetic moments observed in underdoped samples, and it was not quantitatively fitted to data. The mean-field pairing scale $T^{**}$ of the model — the temperature at which the saddle-region pairing gap $\Delta_a$ opens, distinct from the velocity-criterion pseudogap onset $T^*$ — is also distinct from the observed $T_c$, and precisely how phase locking converts $T^{**}$ into $T_c$ was left outside the theory.

Problem Statement

Extend the supercurrent-fluctuation (phase-gradient-only, two-dimensional, saddle-point-anchored) description of the pseudogap phase in underdoped cuprate superconductors into a fuller theory that remains valid through the superconducting transition. Specifically:

1. (Chief extension) Incorporate interplane hopping and Josephson coupling between the superconducting CuO$_2$ planes - a Josephson energy depending on the order-parameter phases of different planes and not only on their gradients, i.e. interlayer phase locking and interlayer phase fluctuations - so that the theory describes temperatures close to the true $T_c$, where the supercurrent-only two-dimensional formalism breaks down, and explains how the mean-field pairing scale is converted into the observed three-dimensional $T_c$.

2. Within the same extension, address superconducting fluctuations below $T_c$ and their effect on the neutron-scattering response. The source's own motivating example is that in underdoped samples the energy of the neutron maximum varies proportional to the real $T_c$ (decreasing with underdoping) whereas the gap measured by the maximum in the ARPES spectrum increases; the extended theory should account for this behavior rather than leave it outside its scope.

The objective is a single phase-fluctuation-based theory of the underdoped cuprates covering the pseudogap regime through the three-dimensional superconducting transition, with the same Doppler-averaged spectroscopic observables (ARPES spectral function, neutron-scattering response, Knight shift) as outputs. Two further limitations lie outside the scope of this problem: the small, time-fluctuating magnetic moments observed in underdoped samples, and quantitative parameter-constrained fitting to data.

References

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