A. A. Abrikosov's 1979 theory of the long-range indirect exchange (RKKY) interaction between localized spins in anisotropic metals with impurities left an open core. That theory derived the r−3r^{-3} asymptotic law for strictly convex Fermi surfaces with oscillation wavenumbers set by the 'gabarite' (directional caliper) of the Fermi surface, effective two- and one-dimensional laws for cylindrical and planar Fermi-surface pieces, and proved that weak disorder (εFτ≫1\varepsilon_F\tau\gg1) only multiplies the interaction by an exponential damping factor exp⁡(−r/lr)\exp(-r/l_{\mathbf r}) with a direction-dependent mean free path. Three regimes lie beyond the reach of that analysis and remain without a general treatment: (1) degenerate Fermi-surface geometries - nonconvex surfaces, loci of vanishing curvature, and contacts between the Fermi surface and its displaced copy along lines or areas for isolated directions of the interspin vector - where the stationary-phase extremum is degenerate and the decay law changes (exponents such as r−5/2r^{-5/2}); (2) the strong-disorder regime εFτ≫̸1\varepsilon_F\tau\not\gg1, where the estimates that make vertex corrections (diagrammatic ladder corrections to the impurity-averaged response) negligible fail and the impurity diagrams require nonperturbative treatment; (3) the quantitative crossover of quasi-one- and quasi-two-dimensional conductors at interspin separations beyond (Δp)−1(\Delta p)^{-1}, the inverse warping amplitude of the Fermi surface. Later literature has resolved special cases - nearly nested Fermi surfaces, bilayer graphene's chiral bands, hyperbolic van Hove surfaces, toroidal nodal-line surfaces, and the Anderson-localized limit (where disorder localizes the electron states) - but none of the three facets in their stated generality.

Background

The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is the indirect exchange coupling between two localized magnetic impurity spins S1\mathbf S_1 and S2\mathbf S_2 embedded in a metal, mediated by the conduction electrons through the ss-dd exchange coupling of strength JJ. In a metal with a spherical Fermi surface - the surface in momentum space separating occupied from unoccupied electron states at zero temperature - with Fermi momentum p0p_0 (the radius of that sphere), the interaction oscillates as cos⁡(2p0r)/r3\cos(2p_0 r)/r^3 at large interspin separation rr in three dimensions. Real metals, however, have strongly anisotropic, non-spherical Fermi surfaces (FS), and quasi-one- and quasi-two-dimensional conductors have FS that are nearly flat sheets or nearly open cylinders, so both the oscillation period and the decay law of the indirect exchange can depend on the direction of the interspin vector.

In 1979 A. A. Abrikosov derived the anisotropic generalization of the RKKY formula (building on the disorder formalism of Abrikosov and Ryzhkin's 1978 study of quasi-one-dimensional conductivity). For any FS that is strictly convex - meaning its Gaussian curvature (the product of the two principal curvatures of the surface) is everywhere positive - the interaction retains the r−3r^{-3} power law, with an oscillation wavenumber fixed by the 'gabarite' (extent, or caliper) of the FS in the direction r^\hat{\mathbf r} of the interspin vector: D(r^)=max⁡n 2p0(n) n⋅r^D(\hat{\mathbf r})=\max_{\mathbf n}\,2p_0(\mathbf n)\,\mathbf n\cdot\hat{\mathbf r}, where n\mathbf n runs over FS points and p0(n)p_0(\mathbf n) is the Fermi momentum in that direction; the amplitude is set by the Fermi velocity and Gaussian curvature at the extremal ('gabarite') point. Geometrically this is a stationary-phase result: the momentum integral for the static spin susceptibility is dominated by the FS points where the oscillatory phase is extremal. For an ellipsoidal FS the oscillation wavenumber is obtained in closed form. When the FS contains an extended cylindrical portion the interaction becomes effectively two-dimensional, decaying as ρ−2\rho^{-2} with the in-plane separation and confined to atomic thickness along the cylinder axis; when it contains an extended planar portion it becomes effectively one-dimensional, decaying as z−1z^{-1} along the normal. For weak disorder, quantified by εFτ≫1\varepsilon_F\tau\gg1 (εF\varepsilon_F the Fermi energy and τ\tau the elastic impurity-scattering time, so that an electron completes many de Broglie oscillations between scattering events), Abrikosov proved that impurity scattering leaves the oscillation periods intact and only multiplies the interaction by exp⁡(−r/lr)\exp(-r/l_{\mathbf r}), where the mean free path lrl_{\mathbf r} (the distance traveled between scattering events) is built from the scattering time at the gabarite point selected by the direction r^\hat{\mathbf r}. Vertex corrections - diagrammatic corrections to the impurity-averaged response beyond the damping of single-particle Green's functions (the electron propagators dressed by impurity scattering) - were shown to be parametrically small, of relative order (εFτ)−1ln⁡(εFτ)(\varepsilon_F\tau)^{-1}\ln(\varepsilon_F\tau) in three dimensions and (εFτ)−1/2(\varepsilon_F\tau)^{-1/2} in the two-dimensional (cylindrical) case. In the strictly one-dimensional geometry the diagrammatic structure changes qualitatively (vertex corrections and diagrams with intersecting impurity lines contribute at leading order), and the interaction becomes proportional to cos⁡(2p0z) e−z/l/z\cos(2p_0 z)\,e^{-z/l}/z with a composite mean free path l−1=2l1−1+l2−1l^{-1}=2l_1^{-1}+l_2^{-1} defined by the correlation lengths of the specific disorder model. For quasi-one- and quasi-two-dimensional materials - stacks of weakly coupled chains or layers, whose FS is a weakly modulated plane or cylinder with modulation (warping) amplitude Δp\Delta p - the reduced-dimensional laws were argued to hold only up to interspin separations of order (Δp)−1(\Delta p)^{-1}, beyond which the interaction crosses over to a three-dimensionally decaying form; only this scale was identified, not the crossover shape.

The analysis excluded, and thereby left open, three regimes. First, Fermi-surface geometries beyond the strictly convex case: nonconvex or more exotic FS for which, for isolated directions of the interspin vector, the contact between the FS and its displaced copy (the tangency geometry between the FS and the same surface shifted by the momentum transfer, which controls the singularity structure of the static susceptibility) extends along a line or over a two-dimensional area, or equivalently the stationary-phase extremum becomes degenerate because the Gaussian curvature vanishes there; a footnote of the paper observes that contacts along special lines can produce modified power laws such as r−5/2r^{-5/2} in one direction. Second, the regime of strong disorder, εFτ≫̸1\varepsilon_F\tau\not\gg1, where the estimates that render vertex corrections negligible fail and a nonperturbative treatment of the impurity diagrams may be required. Third, the quantitative form of the quasi-low-dimensional crossover at separations comparable to or larger than (Δp)−1(\Delta p)^{-1}.

Later literature has treated a number of special degenerate geometries exactly: nearly nested FS (FS with nearly parallel opposite portions mapped onto each other by a single wavevector) in layered metals (Aristov and Maleyev 1997); the chiral parabolic low-energy bands of bilayer graphene, whose degenerate spectrum strongly enhances the Kohn anomaly (the non-analyticity of the electron response at wavevectors connecting opposite points of the Fermi surface) and qualitatively changes the Friedel oscillation (the oscillatory spatial modulation of the electron density around a static impurity) and the RKKY interaction (Hwang and Das Sarma 2008; Klier, Shallcross and Pankratov 2014); a hyperbolic FS near a van Hove singularity (a saddle point of the electronic dispersion where the FS has locally negative curvature and the density of states is singular), for which the density response decays as R−2R^{-2} instead of R−3R^{-3} (Lu 2016); the toroidal FS of doped nodal-line semimetals (semimetals whose valence and conduction bands touch along a closed line in momentum space), which produces beating and two-dimensional-like decay inside a three-dimensional bulk (Wang, Liu and Zou 2022); and anisotropic-mass two-dimensional systems, where the RKKY periodicity depends on direction (Ahn and Das Sarma 2021). On the disorder side, the strongly localized (insulating) limit has been treated nonperturbatively (Sobota, Tanasković and Dobrosavljević 2007). No general classification of the asymptotic laws for degenerate FS geometries, no treatment of the metallic εFτ∼1\varepsilon_F\tau\sim1 regime with anisotropic FS, and no computation of the (Δp)−1(\Delta p)^{-1} crossover functions is known.

Problem Statement

Complete the asymptotic theory of the indirect exchange (RKKY) interaction between two localized spins in an anisotropic metal with impurities - the program initiated by Abrikosov (1979) - by covering the three regimes that analysis excluded. The single scientific objective is an asymptotically exact, direction-resolved description of the long-range spin-spin interaction for real anisotropic metals; the three deliverables below are jointly needed for that objective and are linked by the same stationary-phase and disorder-diagrammatic framework.

(1) Degenerate Fermi-surface geometry. Classify the large-rr asymptotic form of the interaction for Fermi surfaces beyond the strictly convex case: nonconvex or multiply connected surfaces, surfaces with loci of vanishing Gaussian curvature, and geometries in which, for isolated directions of the interspin vector, the contact between the Fermi surface and its displaced copy (the same surface shifted by the momentum transfer of the oscillatory phase, whose tangency controls the singularity structure of the static susceptibility) extends along a line or over a two-dimensional area. For each generic degeneracy class (a class stable under small smooth deformations of the Fermi surface), derive the power-law decay and the oscillatory structure of the interaction - including modified exponents, softer than the generic r−3r^{-3}, arising for contacts along special lines - as explicit asymptotic formulas in terms of the local Fermi-surface geometry, and specify the interspin directions to which each degenerate law applies.

(2) Strong disorder. Treat the regime εFτ≫̸1\varepsilon_F\tau\not\gg1 (in particular εFτ∼1\varepsilon_F\tau\sim1), in which the weak-disorder estimates - vertex corrections of relative order (εFτ)−1ln⁡(εFτ)(\varepsilon_F\tau)^{-1}\ln(\varepsilon_F\tau) in three dimensions and (εFτ)−1/2(\varepsilon_F\tau)^{-1/2} in the two-dimensional cylindrical case - fail: for an anisotropic Fermi surface where vertex corrections and diagrams with intersecting impurity lines are no longer parametrically small, determine how the oscillation periods, the decay laws, and the exponential damping factor exp⁡(−r/lr)\exp(-r/l_{\mathbf r}) are modified through the crossover from the weak-disorder metal to the strongly disordered regime.

(3) Quasi-low-dimensional crossover. Compute the quantitative crossover functions for quasi-one- and quasi-two-dimensional conductors (weakly modulated planar or cylindrical Fermi surfaces with warping amplitude Δp\Delta p) at interspin separations r≳(Δp)−1r\gtrsim(\Delta p)^{-1}, connecting the reduced-dimensional laws (the z−1z^{-1} law along the plane normal with transverse delta-function confinement; the ρ−2\rho^{-2} law within the plane perpendicular to a cylinder axis) to the three-dimensionally decaying form.

Any answer must reproduce the known exactly solvable special cases of the same theory: the spherical Fermi surface; cylindrical and planar Fermi-surface pieces; and the strictly one-dimensional damped law. The degenerate geometries treated in the later literature (nearly nested, hyperbolic van Hove, toroidal, and chiral-band surfaces) serve as consistency checks rather than additional deliverables: the answer should not contradict those results.

References

  1. A. A. Abrikosov, Long-range interaction of spins in an anisotropic metal with impurities, Journal of Low Temperature Physics 34, 595-605 (1979). DOI: 10.1007/BF00114944, https://doi.org/10.1007/BF00114944
  2. A. A. Abrikosov and I. A. Ryzhkin, Conductivity of quasi-one-dimensional metal systems, Advances in Physics 27, 147-230 (1978). DOI: 10.1080/00018737800101364, https://doi.org/10.1080/00018737800101364
  3. A. A. Abrikosov, One-dimensional spin-glass, Journal of Low Temperature Physics 37, 111-122 (1979). DOI: 10.1007/BF00114060, https://doi.org/10.1007/BF00114060
  4. A. A. Abrikosov, Spin glass with nonmagnetic impurities, in Modern Trends in the Theory of Condensed Matter, Lecture Notes in Physics, vol. 115, Springer, pp. 251-271 (1980). DOI: 10.1007/BFB0120139, https://doi.org/10.1007/BFB0120139
  5. D. N. Aristov, Indirect RKKY interaction in any dimensionality, Physical Review B 55, 8064-8066 (1997). DOI: 10.1103/PhysRevB.55.8064, https://doi.org/10.1103/PhysRevB.55.8064
  6. D. N. Aristov and S. V. Maleyev, RKKY interaction in the nearly nested Fermi liquid, Physical Review B 56, 8841-8848 (1997). DOI: 10.1103/PhysRevB.56.8841, https://doi.org/10.1103/PhysRevB.56.8841
  7. J. A. Sobota, D. Tanasković and V. Dobrosavljević, RKKY interactions in the regime of strong localization, Physical Review B 76, 245106 (2007). DOI: 10.1103/PhysRevB.76.245106, https://doi.org/10.1103/PhysRevB.76.245106
  8. E. H. Hwang and S. Das Sarma, Screening, Kohn Anomaly, Friedel Oscillation, and RKKY Interaction in Bilayer Graphene, Physical Review Letters 101, 156802 (2008). DOI: 10.1103/PhysRevLett.101.156802, https://doi.org/10.1103/PhysRevLett.101.156802
  9. N. Klier, S. Shallcross and O. Pankratov, Asymptotic discontinuities in the RKKY interaction in the graphene Bernal bilayer, Physical Review B 90, 245118 (2014). DOI: 10.1103/PhysRevB.90.245118, https://doi.org/10.1103/PhysRevB.90.245118
  10. Chi-Ken Lu, Friedel oscillation near a van Hove singularity in two-dimensional Dirac materials, Journal of Physics: Condensed Matter 28, 065001 (2016). DOI: 10.1088/0953-8984/28/6/065001, https://doi.org/10.1088/0953-8984/28/6/065001
  11. Seongjin Ahn and S. Das Sarma, Screening, Friedel oscillations, RKKY interaction, and Drude transport in anisotropic two-dimensional systems, Physical Review B 103, 165303 (2021). DOI: 10.1103/PhysRevB.103.165303, https://doi.org/10.1103/PhysRevB.103.165303
  12. Zhong-Yi Wang, Da-Yong Liu and Liang-Jian Zou, RKKY interaction of magnetic impurities in nodal-line semimetals, Journal of Magnetism and Magnetic Materials 553, 169164 (2022). DOI: 10.1016/j.jmmm.2022.169164, https://doi.org/10.1016/j.jmmm.2022.169164