Completing the spin-charge-separated (spinon–holon) theory of the cuprate normal and superconducting states: $2k_F$ holon kinematics, anomalous exponent, spin and charge gaps, statistics, Hall/Raman/NMR responses, and the interlayer-tunneling estimate of $T_c$

In 'Theory of the Normal State of Cuprate Superconductors' (MRS Proceedings 169, 3 (1989)), P. W. Anderson and Yong Ren proposed that the normal (non-superconducting) state of the CuO$_2$ planes of high-$T_c$ superconductors is a two-dimensional spin-charge-separated 'tomographic Luttinger liquid': neutral spin-1/2 spinons ($Z=0$ fermions) fill the interior of a conventional Fermi surface, spinless charged holons carry their low-energy spectral weight near the $2k_F$ spanning vectors, and the physical electron is a spinon–holon composite whose spectrum fits the anomalous angle-resolved photoemission spectroscopy (ARPES) lineshapes. The paper derived predictions (pair susceptibility $\chi_p(T)\simeq N(0)(\ln J/T)^3$, a heuristic $T_c\sim t_\perp^2/t_\parallel$ from interlayer pair tunneling, an enhanced specific-heat jump, two-component NMR relaxation, linear-in-$T$ resistivity) while leaving outstanding a list of analyses: the kinematic derivation of the $2k_F$ holon weight and the anomalous exponent $\alpha$; a quantitative Hall-effect theory from the opposite spinon/holon precession; the origin and momentum structure of the superconducting gap and whether spin and charge gaps differ; possible semionic statistics once a spin gap appears; the low-frequency Raman $\omega$-dependence; the Knight shift and $1/T_1$ below $T_c$; and the microscopic justification and numerical values behind $T_c\sim t_\perp^2/t_\parallel$. Subsequent literature has partly addressed that agenda without settling it: the Hall response was treated within the program (Anderson 1991, refined 2009–2011) on the assumption of spin-charge separation, the exponent was re-derived in the later hidden-Fermi-liquid reformulation via a different mechanism rather than from the source's $2k_F$ kinematics, and the interlayer-tunneling relation was developed but remains contested, while the core premise — spin-charge separation in the two-dimensional doped Mott insulator — and the derivations of it from the microscopic model remain unproven and open.

Background

High-temperature superconductivity arises in layered copper oxides ('cuprates') whose CuO$_2$ planes, when hole-doped away from an antiferromagnetic Mott insulator (an insulator caused by strong electron–electron repulsion rather than by filled energy bands), become metallic. The 'normal' state above the superconducting transition temperature $T_c$ violates the standard theory of metals (the Landau Fermi liquid of long-lived electron-like quasiparticles): resistivity is linear in temperature over a wide range, the Hall effect is strongly temperature dependent, NMR relaxation violates the Korringa law (the linear proportionality between the nuclear spin-lattice relaxation rate $1/T_1$ and temperature in an ordinary metal), and angle-resolved photoemission spectroscopy (ARPES — the experiment measuring the probability of removing an electron of in-plane momentum $k$ and energy $\omega$, encoded in the spectral function $\rho(k,\omega)$) shows broad cusp-like lineshapes instead of sharp quasiparticle peaks with finite quasiparticle residue $Z$ at the Fermi surface. A competing phenomenology of the same anomalies is the marginal Fermi liquid of Varma, Littlewood, Schmitt-Rink, Abrahams and Ruckenstein, which retains electrons but replaces the quasiparticle pole by scale-invariant scattering. Anderson's resonating-valence-bond (RVB) program — the proposal that the parent insulator is a quantum superposition of many ways of binding the electron spins into singlet pairs — takes a different route: because the parent Mott insulator's electrons are Gutzwiller-projected (the projected wavefunction suppresses double occupancy of Cu sites), the doped state fractionalizes. In one dimension the analogous Hubbard model (electrons on a lattice with on-site repulsion) is solved exactly by the Bethe ansatz and exhibits a Luttinger liquid: a correlated metal with no quasiparticle poles in which spin and charge degrees of freedom propagate as separate collective modes ('spin-charge separation') with different velocities and anomalous power-law exponents. Anderson and Ren proposed the two-dimensional counterpart, the 'tomographic Luttinger liquid' (so called because the Fermi surface is decomposed into a continuum of one-dimensional patches, each treated as a 1D Luttinger liquid): 'spinons' are neutral spin-1/2 fermions with $Z=0$ occupying the interior of a large Fermi surface of conventional shape; 'holons' are spinless charged excitations whose low-energy spectral weight lies near the $2k_F$ 'spanning vectors' — momenta connecting opposite points of the spinon Fermi surface ($k_F$ the Fermi wavevector) — so a physical electron at momentum $k$ is a composite of one spinon near $k_F$ and one holon near $2k_F$. From the resulting kinematics they computed a superconducting pair susceptibility $\chi_p(T)\simeq N(0)(\ln J/T)^3$ — the linear response to the formation of charge-$2e$ pairs, whose growth signals the pairing instability — with $N(0)$ the density of states at the Fermi surface and $J$ the antiferromagnetic superexchange energy between neighboring Cu spins (of order $10^3$ K, acting as the high-energy cutoff); an interlayer pair-tunneling interaction $V\sim t_\perp^2/J$ ($t_\perp$, $t_\parallel$ the interlayer and in-plane hopping amplitudes), and hence the heuristic $T_c\sim t_\perp^2/t_\parallel$; an enhanced superconducting specific-heat jump $\Delta C/\gamma T_c$ ($\Delta C$ the specific-heat jump at $T_c$, $\gamma$ the normal-state electronic specific-heat coefficient) of 1.5–3 times the Bardeen–Cooper–Schrieffer value; a two-component NMR relaxation $1/T_1$ (spinon Korringa term plus a holon $T(\ln J/T)^2$ term); and a linear-in-$T$ resistivity. Transport in such a state requires composition rules (pioneered by Ioffe and Larkin) that combine spinon and holon contributions into the measured resistivity and Hall response; the interlayer tunneling mechanism itself (Wheatley, Hsu and Anderson) attributes the superconducting condensation energy to coherent tunneling of Cooper pairs between the CuO$_2$ planes, giving $V\sim t_\perp^2/J$ because single-electron interlayer hopping is incoherent in the RVB state while pair tunneling is not. The present problem is the set of quantitative completions of this framework left outstanding by the 1989 source and its program, which taken together constitute the program's demand for a microscopic derivation and experimental validation of two-dimensional spin-charge separation in the cuprates.

Problem Statement

Complete and test the quantitative spin-charge-separated (spinon–holon, 'tomographic Luttinger liquid') theory of the CuO$_2$-plane normal and superconducting states proposed by Anderson and Ren (1989), by supplying the specific quantitative analyses that the framework left incomplete, each within (or explicitly against) that framework:

1. Kinematics and exponent. Derive from the two-dimensional doped Mott-insulator problem (the $t$-$J$ or Hubbard model with Gutzwiller projection; the $t$-$J$ model is the strong-repulsion limit of the Hubbard model, retaining hole hopping $t$ and antiferromagnetic exchange $J$) why the holon's low-energy spectral weight concentrates at the $2k_F$ spanning vectors of the spinon Fermi surface, and derive formally the anomalous exponent $\alpha$ of the composite electron spectral function $\rho(k,\omega)$, fixing its doping dependence and its effect on the ARPES lineshape fits and on the response functions treated in items 2, 5 and 6.
2. Hall effect. Turn the qualitative observation that spinons and holons give opposite Larmor precession into a quantitative theory of the temperature-dependent Hall coefficient and Hall angle ($\theta_H$, the angle between the current and the induced transverse electric field, with $\tan\theta_H=\rho_{xy}/\rho_{xx}$ the ratio of Hall to longitudinal resistivity; e.g. the observed $\cot\theta_H\propto T^2$ with linear-in-$T$ resistivity), including the composition rule by which spinon and holon transport combine.
3. Superconducting-state gap structure. Determine the microscopic origin and momentum structure of the superconducting gap, in particular whether the spin and charge (spinon and holon) channels develop the same gap, and whether the spin gap opens more slowly than the charge gap below $T_c$.
4. Statistics. Establish or exclude the emergence of semionic statistics of the fractionalized excitations once a spin gap appears — in two spatial dimensions, identical particles can in principle carry anyonic exchange phases intermediate between the bosonic ($0$) and fermionic ($\pi$), and a semion is the case of exchange phase $\pi/2$ — and determine its observable consequences.
5. Raman response. Compute the low-frequency $\omega$-dependence of the electronic Raman response (the inelastic light-scattering spectrum) and compare with cuprate Raman data.
6. NMR below $T_c$. Analyze the Knight shift (the shift of the nuclear magnetic resonance frequency caused by the spin polarization of the charge carriers, which measures the electron spin susceptibility) and the spin-lattice relaxation rate $1/T_1$ below $T_c$ for the spinon and holon channels, and compare with experiment.
7. $T_c$ estimate. Supply the microscopic justification and numerical values ($t_\perp$, $t_\parallel$, $N(0)$, $J$, and the precise form of the interlayer interaction $V$) behind the heuristic pair-tunneling relation $T_c\sim t_\perp^2/t_\parallel$ (with $V\chi_p(T_c)=1$ and $\chi_p(T)\simeq N(0)(\ln J/T)^3$), and test it against the interlayer-coherence and transition-temperature systematics of the cuprate families.

The common objective binding these items is to determine whether the cuprate CuO$_2$ planes realize a single, quantitatively verified spin-charge-separated ('tomographic Luttinger liquid') theory — by deriving the fractionalized state from the microscopic model and confronting the resulting response functions with experiment, or by rigorously excluding that framework. The seven analyses are independent sub-questions: a cross-reference from one item to another does not make their resolutions depend on one another. Reproducing the qualitative trends of the relevant cuprate data, together with the order of magnitude of any numerical values the framework supplies, is evidence for the framework, but not proof of it.

References

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