Microscopic derivation of the anomalous-exponent non-Fermi-liquid spectral function underlying interlayer Josephson tunneling in layered cuprates

Chakravarty and Anderson [Phys. Rev. Lett. 72, 3859 (1994)] showed that if the single-particle spectral function of a CuO2 plane obeys a homogeneity law with a nonvanishing exponent α — a non-Fermi liquid — then interlayer single-particle tunneling is incoherent, interlayer Josephson pair tunneling reduces to an instantaneous Hamiltonian, and the Josephson critical current is governed by an electronic cutoff ω_c of order the in-plane bandwidth rather than being cutoff-independent as in the Ambegaokar–Baratoff result. The spectral function itself, however, was postulated: the paper states verbatim that its derivation from a realistic microscopic Hamiltonian 'remains an open problem'. The open problem is to derive the α-spectral function from a microscopic two-dimensional Hamiltonian of the correlated electrons; to establish self-consistently the reduction to the instantaneous interlayer pair-tunneling Hamiltonian, including corrections, bath self-consistency, and explicit spin-charge separation in the superconducting state; and to determine the Josephson scaling function f(α, Δ/ω_c) for α < 1/4, where the paper's methods fail. No work is known to close the core derivation or to evaluate f(α, Δ/ω_c) for α < 1/4; the problem is open.

Background

The high-transition-temperature (high-$T_c$) cuprate superconductors are layered oxides in which superconducting CuO2 planes are stacked along the crystallographic $c$ axis with weak electronic coupling between them. Two facts frame the problem. First, the planes derive from two-dimensional doped Mott insulators: the parent compounds are antiferromagnetic insulators in which motion is blocked by strong on-site Coulomb repulsion, and the standard theory of interacting metals — Landau Fermi-liquid theory, in which low-energy excitations are long-lived quasiparticles appearing as a pole in the single-particle Green's function — appears to break down, angle-resolved photoemission (ARPES, which measures the momentum- and frequency-resolved probability for removing an electron) finding broad non-quasiparticle spectra. Second, charge transport along the $c$ axis is weak and incoherent, and adjacent planes couple through intrinsic Josephson tunneling of Cooper pairs (bound pairs of electrons that tunnel coherently between superconductors — the Josephson effect), observed directly as stacks of intrinsic junctions in Bi2Sr2CaCu2O8+δ.

A convenient way to quantify the breakdown of Fermi-liquid theory is through the single-particle spectral function $A(\mathbf{k},\omega)$ — the imaginary part of the retarded Green's function, i.e., the density of states for adding or removing an electron at momentum $\mathbf{k}$ and frequency $\omega$. A Fermi liquid has $A\sim Z\,\delta(\omega-v_F|\mathbf{k}-\mathbf{k}_F|)$ near the Fermi surface, which under a uniform rescaling $(\mathbf{k}-\mathbf{k}_F,\omega)\to(\Lambda(\mathbf{k}-\mathbf{k}_F),\Lambda\omega)$ is homogeneous of degree $-1$. The general homogeneity law is

$$A(\Lambda(\mathbf{k}-\mathbf{k}_F),\,\Lambda\omega)=\Lambda^{-1+\alpha}A(\mathbf{k}-\mathbf{k}_F,\omega),$$

where the exponent $\alpha$ is zero for a Fermi liquid; any $\alpha\neq 0$ gives a power-law spectrum $A\sim\omega^{\alpha-1}$ at the Fermi surface with no quasiparticle pole — the many-body analogue of the x-ray orthogonality catastrophe, in which an abrupt perturbation of a Fermi sea makes the overlap of the perturbed and unperturbed ground states vanish as a power law. In one space dimension such power-law spectra are exact (Luttinger liquids — one-dimensional metals whose low-energy excitations are collective charge- and spin-density waves rather than quasiparticles), and Wen [Phys. Rev. B 42, 6623 (1990)] argued that metallic non-Fermi-liquid fixed points can exist in two and higher dimensions, constructing them from coupled chains whose interchain hopping renormalizes to zero.

Chakravarty and Anderson [Phys. Rev. Lett. 72, 3859 (1994)] combined this scaling form, adopted as a phenomenological input, with the interlayer tunneling hypothesis for high-$T_c$ superconductivity [Chakravarty, Sudbø, Anderson, and Strong, Science 261, 337 (1993)], in which the superconducting condensation energy comes from Josephson tunneling of pairs between adjacent layers. For two Fermi-liquid superconductors the classic Ambegaokar–Baratoff result gives $I_cR_n=\pi\Delta/2e$ at $T=0$, where $R_n$ is the normal-state resistance of the junction: the Josephson critical current is proportional to the gap $\Delta$ and independent of any high-energy cutoff — a subtle consequence of quasiparticle coherence. Chakravarty and Anderson instead generalized the superconducting (Nambu, i.e., electron–hole matrix) Green's function by endowing the normal-state inverse propagator with fractional powers, $G^{-1}\propto(\omega-\varepsilon(\mathbf{k}))^{1-\alpha}$, with an electronic cutoff $\omega_c$ of order the in-plane bandwidth restoring dimensions ($\alpha=0$ recovers BCS theory, the standard Bardeen–Cooper–Schrieffer theory of superconductivity). The resulting Green's function has branch points and poles that densely encircle $\varepsilon(\mathbf{k})$ in the complex-frequency plane (for irrational $\alpha$; rational $\alpha$ gives a discrete set of poles); selecting the physical Riemann sheet fixes the excitation spectrum $\omega(\mathbf{k})=\varepsilon(\mathbf{k})+|\Delta|$. The Josephson critical current then takes a scaling form governed by a function $f(\alpha,\Delta/\omega_c)$: for $\alpha\ge 1/4$ and $\Delta\ll\omega_c$ the dependence on $\Delta/\omega_c$ drops out, so the scale of $I_c$ is set by $\omega_c$ rather than by $\Delta$ alone — a direct manifestation of the 'cut' spectrum of the non-Fermi-liquid normal state. Single-particle interlayer tunneling was analyzed by mapping each momentum state to a two-level system (one level per layer) coupled to an Ohmic bath — a bath of harmonic oscillators whose spectral density is proportional to frequency, the standard dissipative environment of the spin-boson problem — with the bath formed by the low-energy collective modes generated by the strong in-plane correlations. With the bath exponent identified as $a'=2\alpha/(\alpha+1)$, the purely incoherent relaxation regime begins at $\alpha=1/4$ (with damped short-time oscillations surviving below it but decaying as a power law), which is also the regime in which the reduction of interlayer transport to an instantaneous pair-tunneling Hamiltonian is 'certainly justified'. An earlier attempt to derive a momentum-conserving pair-tunneling Hamiltonian by coupling single-particle tunneling to the spinon pair field [Muthukumar, Phys. Rev. B 46, 5769 (1992), in the spin-charge-separated resonating-valence-bond framework — a proposed liquid of fluctuating spin singlets in which the electron fractionalizes into a spin-carrying spinon and a charge-carrying holon] had failed, which is part of why a microscopic foundation was needed.

The paper is explicit about what it does not do: 'the derivation of this spectral density from a realistic microscopic Hamiltonian remains an open problem'; the authors are 'unable to provide a rigorous proof' that the interlayer tunneling Hamiltonian follows from non-Fermi-liquid behavior; the self-consistency problem — the same electrons create and couple to the bath — is explicitly set aside; $f(\alpha,\Delta/\omega_c)$ is not evaluated 'in a useful form' for $\alpha<1/4$; the proper non-instantaneous extension 'awaits further work'; and spin-charge separation in the superconducting state is deferred as 'more difficult'. The empirical context has shifted since 1994: incoherent $c$-axis single-particle transport is supported, while the interlayer tunneling mechanism of $T_c$ itself is contested — but the theoretical question, a microscopic foundation for non-Fermi-liquid interlayer Josephson tunneling, stands independently of that mechanism's fate.

Problem Statement

The problem, in the form left by Chakravarty and Anderson (1994), is to replace the postulated non-Fermi-liquid input of the interlayer Josephson tunneling theory with a microscopic foundation and to work out its observable consequences:

(1) Starting from a realistic microscopic Hamiltonian for the correlated electrons of a two-dimensional layer (for example, but not necessarily, the two-dimensional Hubbard model of hopping plus on-site repulsion $U$, or its large-$U$ limit the $t$–$J$ model with hopping $t$ and superexchange $J$, used for the CuO2 planes), derive the low-energy single-particle spectral function and determine whether it obeys the homogeneity law $A(\Lambda(\mathbf{k}-\mathbf{k}_F),\Lambda\omega)=\Lambda^{-1+\alpha}A(\mathbf{k}-\mathbf{k}_F,\omega)$ and, if so, with what exponent $\alpha$ (including the Fermi-liquid possibility $\alpha=0$).

(2) From that microscopic spectral function, establish — or refute — the reduction of interlayer transport used in the source paper, treating its three components as separate, independently resolvable claims: (a) that single-particle tunneling between adjacent layers is dynamically incoherent, including the self-consistent treatment of the electrons that both generate and couple to the multiparticle bath; (b) that the interlayer Josephson coupling reduces to an instantaneous pair-tunneling Hamiltonian, with the leading corrections to the instantaneous approximation given and a criterion stated for the regime in which that approximation is valid (if the reduction is instead refuted, component (b) is complete on that refutation alone, no corrections or validity criterion being required); and (c) that the superconducting state exhibits explicit spin-charge separation.

(3) Determine the Josephson critical-current scaling function $f(\alpha,\Delta/\omega_c)$ in closed or asymptotically controlled form for $\alpha<1/4$, settling how $I_c$ depends on $\Delta/\omega_c$ in that regime, where the asymptotic methods of the source paper fail.

These three items share one scientific objective: a microscopic, self-consistent theory of the non-Fermi-liquid interlayer Josephson effect. They are posed jointly, each a part of that single scientific objective. Item (3) is conditional on the outcome of item (1): if the derived exponent satisfies α < 1/4, the scaling function f(α, Δ/ω_c) must be evaluated as in (3), while if item (1) instead yields α = 0 or α ≥ 1/4, then establishing that outcome, together with the corresponding outcome of item (2), is itself a complete resolution of the problem and item (3) does not arise. Item (1) is the load-bearing core — items (2)–(3) are its consequences within the same framework and are complete only relative to it. No restriction on model class, method, or parameter regime is intended beyond what the source specifies: the two-dimensionality, the homogeneity form, and the interlayer geometry are part of the question, not a narrowing of it.

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