Quantitative test of the extended-saddle-point Fermi-liquid mechanism for the fine structure of the neutron spin resonance in high-temperature superconducting cuprates

In hole-doped cuprate superconductors such as YBa$_2$Cu$_3$O$_{6+x}$ (YBCO), inelastic neutron scattering shows a sharp magnetic “resonance” at the antiferromagnetic wave vector $(\pi,\pi)$, whose momentum-space fine structure — an incommensurate splitting of the peak that evolves with neutron energy loss — was discovered in 1998–2000. A. A. Abrikosov [Phys. Rev. B 62, 15156 (2000)] derived a closed-form expression for the imaginary part of the spin susceptibility near this resonance from BCS quasiparticles near extended saddle-point (van Hove) singularities of the electronic bands with a slight transverse bending, and showed qualitative agreement with the neutron data then available at two energies. He explicitly left open the quantitative validation: a fit to multi-energy neutron maps, determination of the band parameters from photoemission, the microscopic origin of a phenomenological relaxation constant, orthorhombic corrections, and whether interaction (random-phase-approximation) corrections are needed for the absolute lineshape. Although the experimental prerequisite is long satisfied and the same phenomenology is quantitatively described within the competing spin-exciton framework, the specific quantitative test of the minimal saddle-point model — and in particular the question whether the bare Fermi-liquid response suffices or an interaction-induced pole is required — has never been carried out; the one direct engagement (Voo, Chen, and Wu, 2002) explicitly found no clear evidence favoring either scenario. The problem remains open.

Background

The cuprate superconductors conduct electricity in CuO$_2$ planes, and their most characteristic magnetic signature is measured by inelastic neutron scattering (INS): an incoming neutron transfers momentum $\mathbf{q}$ and energy $\Omega$ to the sample, and the measured scattering intensity is proportional to the imaginary part of the dynamical spin susceptibility, $\chi''(\mathbf{q},\Omega)$, the spectrum of spin fluctuations. In the superconducting state of YBa$_2$Cu$_3$O$_{6+x}$ (YBCO), $\chi''$ develops a sharp maximum — the “resonance” — at the antiferromagnetic wave vector $\mathbf{Q}=(\pi,\pi)$ (in units of inverse lattice spacing; this is the momentum connecting a copper site to its diagonally opposite neighbor, at which the antiferromagnetically ordered parent compounds order), at an energy of about 41 meV in optimally doped samples (the hole concentration at which $T_c$ is largest); the resonance intensity appears only below the superconducting transition temperature $T_c$, and its energy scales with $T_c$ across the doping range. In 1998–2000 it was discovered that when the neutron energy loss is fixed slightly above or below the resonance energy, the peak in momentum space splits into “incommensurate” components displaced from $(\pi,\pi)$ (Mook et al., Nature 395, 580 (1998); Arai et al., Phys. Rev. Lett. 83, 608 (1999); Bourges et al., Science 288, 1234 (2000)). The origin of this fine structure was intensely contested: proposed mechanisms included “dynamic stripes” (slowly fluctuating quasi-one-dimensional charge and spin order), nesting of the Fermi surface (nearly parallel segments of the surface separating occupied from empty states, which enhance scattering at particular wave vectors), and collective spin modes.

A. A. Abrikosov developed, in Phys. Rev. B 57, 8656 (1998) and in the source paper Phys. Rev. B 62, 15156 (2000), an analytic account based on “extended saddle-point singularities”: angle-resolved photoemission spectroscopy (ARPES) had shown that in several cuprates the electronic bands near $(\pi,0)$ and $(0,\pi)$ are extremely flat over an extended momentum region along one direction, forming a van Hove singularity — a maximum of the electronic density of states at a saddle point of the dispersion — with strongly anisotropic effective masses, a heavy mass $M$ along the flat direction and a light mass $m$ transverse to it ($M\gg m$), located close to the Fermi level. Modeling the two saddle vicinities $a$ (near $(\pi,0)$) and $b$ (near $(0,-\pi)$) by quadratic dispersions $\xi_a=p_x^2/2m-(p_y+q_y)^2/2M-\mu_1$ and $\xi_b=p_y^2/2m-(p_x-q_x)^2/2M-\mu_1$, where $\mu_1$ is the chemical potential measured from the saddle-point energy and $\mathbf{q}$ is the deviation of the neutron momentum from $(\pi,\pi)$, and treating the superconducting state as a BCS (Bardeen–Cooper–Schrieffer) Fermi liquid — the standard mean-field theory of superconductivity, applied here to weakly interacting quasiparticles — with a $d$-wave gap $\Delta$ (a superconducting energy gap whose order parameter changes sign under a 90° rotation of the CuO$_2$ plane, vanishing along the plane diagonals and being largest along the Cu–O bond directions) that is maximal at the saddle points, the source paper computed the resonance as transitions between maxima of the quasiparticle density of states on opposite sides of the Fermi level. In the exactly flat-saddle limit of the predecessor paper this yields a resonance at energy $\Omega_0=\Delta+\sqrt{\Delta^2+\mu_1^2}$ but no momentum dependence; retaining the small transverse bending $\beta=m/M\ll1$ produces a closed-form expression for $\chi''$ as a sum of four logarithms in the scaled momenta $s_{x,y}=q_{x,y}/\sqrt{2m\mu_1}$ and the reduced detuning $\nu_1=(\Omega_{o1}-\Omega)/\Omega_{o1}$, where $\Omega_{o1}$ is the resonance energy including the small $\beta$ correction, whose arguments vanish along lines in the $(s_x,s_y)$ plane that move as the energy is detuned from resonance — so the central peak splits into incommensurate peaks, qualitatively matching the measured energy slices at 34 and 24.5 meV of Mook et al. for illustrative parameters $\gamma=\mu_1/\Delta=0.5$, $\beta=0.2$, and a phenomenological relaxation $\alpha=0.001$ that regularizes the logarithmic divergences. The paper argued that no random-phase-approximation (RPA) resummation — the standard ladder-type enhancement $\chi=\chi_0/(1-J\chi_0)$ of the bare “bubble” susceptibility $\chi_0$ by an effective spin interaction $J$ — and no stripes or exotic collective scenarios are needed even for a qualitative explanation of the fine structure, and that after a unit conversion the calculated absolute magnitude is of the right order compared with the measured maximum $\chi''$.

Competing microscopic calculations contemporary with the source paper (Norman, Phys. Rev. B 61, 14751 (2000); Kao, Si, and Levin, Phys. Rev. B 61, R11898 (2000)) reproduced the same phenomenology within the RPA framework using tight-binding bands fitted to ARPES data, and the later literature converged on describing the resonance as a “spin exciton”: a spin-1 particle-hole bound state that forms just below the pair-breaking continuum edge $2\Delta$ — the minimum energy needed to break a Cooper pair, set by twice the superconducting gap — where the RPA denominator softens. The source paper itself stated what would be needed to turn its qualitative picture into a validated quantitative theory — more neutron energies, more precise ARPES of the saddle region, a microscopic relaxation, and orthorhombic corrections — and these items, together with the disputed status of RPA corrections, define the open problem stated below.

Problem Statement

Determine, quantitatively and with model parameters justified and constrained by independent measurements (e.g., ARPES of the extended saddle-point regions) wherever available, whether the minimal extended-saddle-point Fermi-liquid mechanism reproduces the observed fine structure of the neutron spin resonance in high-$T_c$ cuprates — concretely, in YBa$_2$Cu$_3$O$_{6+x}$ (YBCO), the material to which the source paper's comparison and the available multi-energy fine-structure maps refer — or, if it does not, establish precisely which physical ingredients are responsible for the discrepancy. The scientific objective underlying all components is the microscopic character of the resonance: whether its position, momentum-space fine structure, lineshape, and absolute intensity follow from the bare BCS Fermi-liquid response of quasiparticles near extended van Hove (saddle-point) singularities, or require interaction-induced (spin-exciton) corrections. Aspects of such a quantitative test include the following; the sufficiency verdict of item (5) depends on the parameter-controlled comparison of item (1) and the ARPES-based parameter determination of item (2), while items (3) and (4) bear on that verdict:

(1) Parameter-controlled comparison with multi-energy neutron data. Fit the analytic susceptibility $\chi''(\mathbf{q},\Omega)$ — the sum-of-logarithms expression with transverse bending $\beta=m/M$ — to constant-energy neutron maps (momentum-space intensity maps at fixed neutron energy loss) measured at several energies across the resonance (the data that did not exist in 2000), assessing peak positions, incommensurate splitting wave vectors, momentum widths, and the energy evolution of the pattern, rather than comparing two illustrative energy slices at chosen parameters.

(2) Band parameters from independent measurements. Using ARPES measurements of the extended saddle-point regions (including more precise data than were available in 2000), determine whether the quadratic two-mass dispersion of the model is a faithful local description for the measured materials and doping range, and extract the values of $m$, $M$, the saddle-point offset $\mu_1$, and the gap $\Delta$, so that neutron-side adjustment is limited to what a global fit to the full multi-energy data set requires, rather than tuning to selected features.

(3) Microscopic relaxation. Replace the phenomenological regulator $\alpha$ — introduced in the source as a relaxation most likely originating from disorder — by a microscopic derivation (for example from impurity or quasiparticle damping in a $d$-wave superconductor) that fixes its origin and magnitude in the samples compared, or otherwise justify its value from independent measurements.

(4) Orthorhombicity. Extend the calculation to the orthorhombic inequivalence of YBCO (unequal masses and offsets at the two saddle vicinities) and determine how it modifies the predicted momentum-space patterns, including comparison with untwinned-crystal data (single-domain crystals, avoiding the averaging over domains with interchanged in-plane axes that occurs in twinned samples).

(5) Sufficiency of the bare mechanism. Decide, by quantitative comparison of the absolute lineshape and intensity within the same band model, whether subleading interaction corrections of the RPA type are required — the specific point disputed between the source paper and the critique by Bourges et al. — or whether the bare susceptibility with saddle-point singularities suffices; this includes confronting the tension that the bare threshold $\Omega_0=\Delta+\sqrt{\Delta^2+\mu_1^2}\ge2\Delta$ sits at or above the pair-breaking continuum edge, while the observed resonance energy lies below the $2\Delta_{\max}$ values inferred from ARPES and tunneling.

The verdict sought is sufficiency or insufficiency of the minimal mechanism, reached through the parameter-justified multi-energy YBCO neutron-map comparison of item (1) and the ARPES-based parameter determination of item (2) — including confronting the $\Omega_0\ge2\Delta$ threshold tension — together with the physical ingredients responsible for the discrepancy when the verdict is insufficiency.

References

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