Mechanism, microscopic formalism, and d-wave quasiparticle spectrum of the superconducting transition in the charge-spin separated scenario for the cuprates (Anderson, 2000)

This question originates in P. W. Anderson's Science Perspective 'Sources of Quantum Protection in High-$T_c$ Superconductivity' (2000). Anderson proposed that the anomalous normal and superconducting states of the cuprates form a 'quantum protectorate' rooted in spin-charge separation — fractionalization of the electron into neutral spinons and charged holons — and made a 'radical proposal' for the superconducting phase: the charge excitations condense with s-wave symmetry and thereby endow the spinons with quasiparticle character. The paper itself left open, and later work has not closed: the precise mechanism of the superconducting transition (with Tc conjectured to be set by the release of Gutzwiller-frustrated kinetic energy, no explicit mechanism proposed); the formal microscopic theory of a charge-spin separated superconductor (then 'rudimentary'); and the derivation of the experimentally observed $d_{x^2-y^2}$ quasiparticle gap within this scenario. Later slave-boson/gauge-theory and Gutzwiller-projected formulations give qualitative, mean-field-level accounts of the superconducting state, but a controlled microscopic theory with an explicit transition mechanism does not exist; the question remains open and actively pursued.

Background

This question originates in P. W. Anderson's Science Perspective 'Sources of Quantum Protection in High-$T_c$ Superconductivity' (Science 288, 480 (2000)), which addresses the layered copper-oxide superconductors ('cuprates'), the material class with the highest superconducting transition temperatures $T_c$ at ambient pressure (up to about 135 K; about 160 K under pressure). The parent cuprates are antiferromagnetic Mott insulators — insulating not because bands are filled but because strong on-site Coulomb repulsion $U$ forbids two electrons on the same site; doping holes into the CuO$_2$ planes produces a phase diagram with a 'strange metal' near optimal doping, a pseudogap regime (a partial gap in the single-electron spectrum along the antinodal, momentum-axis, directions while charge transport stays metallic), the superconducting phase, and the Mott insulator. Several properties contradict textbook theory: the normal-state scattering rate is linear in frequency with no phonon signatures, resistivity does not saturate near the Mott-Ioffe-Regel limit (where the mean free path approaches the lattice spacing), and although the superconducting gap has $d_{x^2-y^2}$ symmetry (it changes sign under a 90-degree rotation and vanishes along nodal directions on the Brillouin-zone diagonals, where the low-energy quasiparticles behave as massless Dirac fermions), $T_c$ is strikingly insensitive to the percent-level charged impurities (for example the off-stoichiometry of optimal YBa$_2$Cu$_3$O$_{6.93}$) that should be pair-breaking for a conventional $d$-wave superconductor.

Following Laughlin and Pines' (2000) notion of a 'quantum protectorate' — a collective quantum state, like the quantum Hall effect, whose low-energy properties are insensitive to imperfections — Anderson argued that these anomalies follow from spin-charge separation (also called charge-spin separation): the electron fractionalizes into a spinon, a neutral excitation carrying only spin-1/2, and a holon, a charged soliton carrying only charge, with independent excitation spectra. Spin-charge separation is exact in solvable one-dimensional models but unproven, and experimentally unverified, in two dimensions. In this picture, perturbations that couple to charge (impurities, phonons) hardly disturb the spin sector, which accounts for the protection; the physical electron's Green's function becomes a convolution of spinon and holon Green's functions with a power-law density of states.

For the superconducting phase the paper made what it called a 'radical proposal': the charge (holon) excitations condense with $s$-wave symmetry — hence their insensitivity to scattering — and the condensate 'automatically gives the spinons quasiparticle character', so that the quasiparticle spectrum can carry the experimentally observed $d_{x^2-y^2}$ gap. The relevant kinetic energy is 'frustrated' by the Gutzwiller projection — the enforcement of no double occupancy that defines the physics of a doped Mott insulator, encoded in the $t$-$J$ model (the large-$U$ limit of the Hubbard model of correlated electrons, with hopping $t$ and superexchange $J$). The conjecture was that $T_c$ is determined by the need to reduce this frustrated kinetic energy.

The natural microscopic language for such states is the slave-boson representation, which implements the no-double-occupancy constraint by writing the electron operator as a product of a fermionic spinon and a bosonic charge carrier; the redundancy of this representation generates an emergent gauge field, and controlling its fluctuations is the central difficulty (the confinement-deconfinement problem, analogous to confinement in lattice gauge theory). The $Z_2$ gauge theory of electron fractionalization of Senthil and Fisher (a formulation in which the gauge structure is reduced so that fractionalized excitations can remain deconfined) supplied part of the structure the source paper invoked, while the source noted it lacked impurity scattering and long-range Coulomb effects. In slave-boson mean-field theory, $d$-wave spinon pairing together with holon condensation defines a superconducting state whose physical-electron spectral function can exhibit $d$-wave quasiparticle poles; however, as the review of Lee, Nagaosa and Wen (2006) states, the gauge fluctuations are strong and no well-controlled small expansion parameter exists. In parallel, the renormalized mean-field theory of the Gutzwiller-projected $t$-$J$ model — the 'plain vanilla' version of RVB theory (resonating valence bond: the proposal that the doped Mott insulator is built from fluctuating spin singlets rather than ordered moments), developed by Anderson, Lee, Randeria, Rice, Trivedi and Zhang (2004) — reproduces much of the superconducting-state phenomenology without literal fractionalization, attributing $T_c$ on the underdoped side to phase fluctuations (the phase stiffness, i.e. the superfluid density, depleted by nodal quasiparticle excitations, with a Kosterlitz-Thouless-type vortex-unbinding transition) and the stabilization energy to kinetic-energy gain, consistent with optical sum-rule experiments (the frequency integral of the optical conductivity, which tracks the electronic kinetic energy; Molegraaf et al., 2002). What this later work of both kinds left open is precisely this question.

Problem Statement

This question originates in P. W. Anderson's Science Perspective 'Sources of Quantum Protection in High-$T_c$ Superconductivity' (Science 288, 480 (2000); DOI: 10.1126/science.288.5465.480). Its own concluding questions, as given in the author's longer version of the text (arXiv:cond-mat/0007287), are: (i) 'The precise mechanism for the final superconducting transition is of course still in question since we have made no specification of the holon dynamics; I suggest that its Tc is determined by the need to reduce the frustrated kinetic energy of the system, but do not here propose an explicit mechanism'; (ii) 'Formal theory for a charge-spin separated superconductor is even more rudimentary', the available structure (the $Z_2$ fractionalization program of Senthil and Fisher, cited in the source) lacking impurity scattering and long-range Coulomb effects; and (iii) the connection between the proposed charge-spin separation and the experimentally observed d-wave pairing symmetry of the quasiparticle spectrum — the paper proposes that an s-wave holon condensate endows spinons with quasiparticle character, but constructs no Hamiltonian or Lagrangian realizing this and calculates no resulting spectrum.

Restated as one coherent scientific objective: construct a controlled microscopic theory of the superconducting state of a charge-spin separated (spinon-holon) description of the doped cuprate planes: a theory which, starting from a specified microscopic Hamiltonian (for example the doped Hubbard or $t$-$J$ model with the no-double-occupancy constraint, long-range Coulomb interactions, and impurity scattering), derives: (a) the superconducting transition itself — its mechanism, its order, and its transition temperature $T_c$ expressed in microscopic parameters, including whether and how $T_c$ is set by the release of the Gutzwiller-frustrated kinetic energy through the holon dynamics; (b) the quasiparticle spectrum, in particular the $d_{x^2-y^2}$ gap and the nodal Dirac fermions, derived from the fractionalized degrees of freedom rather than inserted by hand; and (c) the protection properties — the insensitivity of $T_c$ and of the gap nodes to non-magnetic disorder. The three aspects (a), (b) and (c) must follow from the same specified microscopic Hamiltonian within a single treatment that is 'controlled' — meaning a systematic approximation with an identified control parameter, as opposed to an uncontrolled mean-field ansatz in which the pairing or the gap is inserted by hand.

References

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