Interlayer-induced spinon pairing and the microscopic origin of the normal-state spin gap in underdoped cuprate superconductors

The normal state of underdoped cuprate superconductors exhibits a spin gap: below a temperature $T^*$ well above the superconducting $T_c$, the low-energy magnetic response is progressively suppressed without any phase transition. Anderson (1996) proposed a microscopic mechanism within the interlayer tunneling program: because coherent single-particle hopping between CuO2 layers is negligible, the dominant interlayer couplings are pair tunneling and interlayer superexchange, both conserving in-plane momentum, and these pair the spinons (spin-1/2 excitations of a spin-charge-separated normal state) in the even linear combination of the two layers of a YBa2Cu3O6+x bilayer, leaving the odd combination unpaired. An exact 16-state-per-k diagonalization yielded a temperature-independent gap, a computed $\chi(T)$ that fits susceptibility data, and a predicted even/odd neutron selection rule for the spin-gap excitations, while its assumptions (BCS-reduced interaction, equal charge and spin velocities, Fermi-liquid eigenoperators) were left unjustified. Channel-resolved neutron scattering has since measured the even/odd phenomenology (hard even-channel gap, gapless odd channel carrying the superconducting-state resonance), but the theoretical interpretation converged on the itinerant spin-exciton picture, and no controlled derivation or refutation of the interlayer spinon-pairing mechanism exists. This record poses the problem: determine the microscopic origin of the spin gap, in particular establish or exclude interlayer-induced spinon pairing with a controlled treatment of the Luttinger-liquid normal state, confront it with the channel-resolved, doping- and temperature-dependent data, and identify the signatures of the unpaired odd combination.

Background

High-temperature superconductivity arises in layered copper-oxide compounds (cuprates) whose structural unit is a CuO2 plane doped with holes; 'underdoped' means the hole density is below the value at which the superconducting transition temperature $T_c$ is maximal. In the underdoped regime the metallic state above $T_c$ is anomalous, and its most robust signature is the spin gap: the uniform spin susceptibility, measured by the Knight shift of nuclear magnetic resonance (the shift of the nuclear resonance frequency proportional to the electronic spin susceptibility) and by the NMR relaxation rate, together with the low-energy inelastic neutron scattering, are progressively suppressed below a temperature $T^*$ well above $T_c$, without any phase transition or broken symmetry. The spin gap is one facet of the broader pseudogap, a partial suppression of electronic spectral weight in the normal state also seen in photoemission and infrared conductivity. Millis and Monien showed quantitatively that in the bilayer compound YBa2Cu3O6+x (two closely spaced CuO2 planes per unit cell, with x controlling the hole density) the NMR rates on different nuclei can only be reconciled if the bilayer structure and an interplane exchange coupling are included.

The source of this record, P. W. Anderson's 1996 paper, proposed a microscopic mechanism within his interlayer tunneling program. Its premises: (i) the isolated CuO2 plane is a 'normal' Fermi-surface state which in the general case is a Luttinger liquid, a metal whose spin and charge excitations are separate with different velocities $v_s$ and $v_c$; the spin excitations (spinons) are neutral spin-1/2 quasiparticles which, in the self-conjugate Majorana-like form emphasized by Talstra, Strong, and Anderson, are BCS-like objects even in the normal state; (ii) coherent single-particle hopping between layers is negligible in the cuprates, so the only coherent interlayer couplings are second-order processes: pair tunneling (coherent transfer of an electron pair between the layers, amplitude $\lambda_J$) and interlayer superexchange (transfer of a spin pair with exchange of partners between the layers, amplitude $\lambda_S$), both conserving the in-plane momentum $\mathbf{k}$. Because two different points of the two-dimensional Fermi surface are never coupled, the gap opens independently at each $k$ as a change of spectrum rather than a phase transition, matching the gradual observed onset.

The calculational core restricts to the BCS reduced interaction (a momentum state $k$ pairs only with $-k$) and to a single velocity ($v_c$ and $v_s$ treated as equal, a Fermi-liquid simplification, not a Luttinger-liquid one). The two-layer Hamiltonian for each $k$ then acts on a 16-dimensional space and is diagonalized exactly; the resulting level splittings are temperature independent, and the computed uniform susceptibility $\chi(T)$ fits measured spin-gap curves with interlayer couplings $\lambda_J \simeq \lambda_S$ of a few meV. Physically, the interlayer interaction rewrites itself in terms of even and odd linear combinations of the two layers' electron operators, $c_{ke} = (c_{k1}+c_{k2})/\sqrt{2}$ and $c_{ko} = (c_{k1}-c_{k2})/\sqrt{2}$: spinons in the even combination are paired and acquire the gap, while the odd combination remains unpaired. For neutrons the bilayer structure factor, proportional to $1 \pm \cos(\mathbf{Q}\cdot\mathbf{d})$ with $\mathbf{d}$ the interplanar separation, selects excitations even or odd under interchange of the two planes, so the two channels are separately measurable. The theory predicts that the spin-gap excitations appear in the even channel and obey an even/odd sum rule analogous to the one Keimer's group had established for the superconducting-state resonance, the intense sharply peaked magnetic mode at about 41 meV in fully doped YBa2Cu3O7 that appears only below $T_c$ in the odd channel. The author emphasized the preliminary character of the account: the derivation's assumptions have no demonstrated validity in a full Luttinger-liquid treatment, the temperature independence of the gap and its damping by single-particle broadening when $k_B T$ exceeds the gap are asserted rather than derived, and the experimental signatures of the unpaired odd combination were unexplored. In the intervening decades the channel-resolved phenomenology has been measured in detail and interpreted in other frameworks, notably the spin-exciton picture in which the resonant modes are particle-hole bound states below the pair-breaking continuum of a superconducting state, split and weighted by interlayer exchange, while the question the source paper left open (whether the normal-state spin gap is produced by interlayer-induced spinon pairing, and whether that can be derived in a controlled way) has remained unanswered.

Problem Statement

Determine the microscopic origin of the normal-state spin gap in underdoped cuprate superconductors, in particular whether interlayer coupling produces it by pairing spinons as the source theory claims. Specifically: (1) whether, within a controlled treatment of the spin-charge-separated (Luttinger-liquid) normal state that replaces the unproven assumptions of the source derivation (a BCS-reduced interaction pairing $k$ only with $-k$; a single velocity, i.e. equal charge and spin velocities; electron operators treated as eigenoperators of the kinetic energy), the interlayer pair-tunneling and superexchange interactions open a gap in the even bilayer channel's spin spectrum while leaving the odd channel unpaired; (2) whether the temperature dependence of the gap, derived rather than asserted, is the behavior claimed by the source: intrinsic temperature independence at low temperature and damping when $k_B T$ exceeds the gap; (3) whether the derived result is consistent with the measured channel-resolved phenomenology of YBa2Cu3O6+x (a hard even-channel spin gap of about 55 meV at x near 0.5 and about 40 meV at x near 0.7, decreasing with doping; a gapless odd channel with strong low-energy suppression; the superconductivity-induced resonance confined to the odd channel) and with the doping and temperature dependence of the uniform susceptibility; and (4) what observable signatures, if any, the unpaired odd combination produces. An affirmative resolution is a controlled derivation consistent with the data; a negative one is a demonstration that interlayer-induced spinon pairing cannot account for the observed spin gap. Either way the mechanism question must be settled at the microscopic level; fitting phenomenology alone does not resolve it.

References

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