The dc resistivity of cuprate superconductors along the crystallographic cc axis (perpendicular to the CuO2_2 planes) follows a power law whose exponent varies between roughly +1+1 and −1-1 across compounds and dopings, while the optical conductivity σc(ω)\sigma_c(\omega) is non-Drude, nearly frequency independent into the mid-infrared, and below the Mott-Ioffe-Regel limit. Turlakov and Leggett proposed that both anomalies reflect a fluctuation-induced (Coulomb-blockade-like) suppression of incoherent interplane tunneling: Gaussian voltage ('detuning') fluctuations with ohmic spectrum ⟨δVω2⟩=αω\langle\delta V_\omega^2\rangle=\alpha\omega dephase the tunneling event, yielding σc(T)∝T2α\sigma_c(T)\propto T^{2\alpha} for the diffusive channel and a correspondence between voltage and frequency dependences, σc(V)↔σc(ω)\sigma_c(V)\leftrightarrow\sigma_c(\omega). The framework is, by its authors' account, qualitative/semiquantitative, and its completion defines the open problem: identify the microscopic many-body mechanism producing the empirically required α∼O(1)\alpha\sim O(1) low-frequency, long-wavelength charge fluctuations (simple Fermi-liquid/RPA estimates give α≪1\alpha\ll 1); incorporate interplane Coulomb correlations, acoustic out-of-phase plasmons, and intra-unit-cell versus inter-cell hopping into the noise spectrum; extract the fluctuation spectrum from optics/EELS/RIXS and test the predicted T2αT^{2\alpha} scaling and σc(V)↔σc(ω)\sigma_c(V)\leftrightarrow\sigma_c(\omega) mapping across cuprate families and dopings; and extend the theory to the superconducting state and the τhop∼τab\tau_{\mathrm{hop}}\sim\tau_{ab} crossover. Subsequent literature leaves these items unresolved: the anomalously large low-qq charge response is now directly measured and the anticipated interplane plasmon modes have been observed, but no work derives α\alpha microscopically or performs the proposed quantitative transport tests. The problem remains open.

Background

High-temperature (high-TcT_c) cuprate superconductors are layered oxides whose mobile holes reside primarily in two-dimensional CuO2_2 sheets separated by charge-reservoir layers. Transport perpendicular to the sheets — along the crystallographic cc axis — is anomalous in the normal (non-superconducting) state [Turlakov and Leggett 2001; Hussey et al. 2011]. Whereas the in-plane resistivity is famously linear in temperature, the out-of-plane resistivity ρc(T)\rho_c(T) follows a power law, ρc∼Tγ\rho_c\sim T^\gamma, whose exponent γ\gamma ranges from about +1+1 to −1-1 depending on compound and doping; the cc-axis optical conductivity σc(ω)\sigma_c(\omega) is nearly frequency independent from low frequencies to the mid-infrared and lies below the Mott-Ioffe-Regel limit (the minimum conductivity of a Boltzmann quasiparticle whose mean free path equals the lattice constant), a behavior called 'confinement'. The cc-axis hopping time τhop\tau_{\mathrm{hop}} estimated from σc\sigma_c is much longer than the in-plane scattering time τab∼ℏ/kT\tau_{ab}\sim\hbar/kT in most materials, becoming comparable to it only in heavily overdoped YBa2_2Cu3_3O7_7 and La2−x_{2-x}Srx_xCuO4_4, where a Drude-like σc(ω)\sigma_c(\omega) reappears. Negative cc-axis magnetoresistance in Tl2_2Ba2_2CuO6_6 [Hussey et al. 1996] indicated that in-plane momentum is not conserved during an interplane tunneling event.

Turlakov and Leggett (2001) proposed that these anomalies arise from a fluctuation-induced blockade of interplane tunneling, in analogy with the dynamical Coulomb blockade and zero-bias anomalies of mesoscopic systems (the suppression of tunneling by the environmental electromagnetic modes that the tunneling electron itself excites). In their tunneling-Hamiltonian treatment the tunneling matrix element acquires a fluctuating phase, and assuming Gaussian phase fluctuations the tunneling probability is P(δr,δt)=exp⁡[−R(δr,δt)]P(\delta r,\delta t)=\exp[-R(\delta r,\delta t)] with R=∫dω d2q ⟨δVq,ω2⟩(1−cos⁡(ωδt+q⋅δr))coth⁡(ω/2T)/ω2R=\int d\omega\, d^2q\,\langle\delta V^2_{q,\omega}\rangle(1-\cos(\omega\delta t+\mathbf{q}\cdot\delta\mathbf{r}))\coth(\omega/2T)/\omega^2, where ⟨δVq,ω2⟩\langle\delta V^2_{q,\omega}\rangle is the spectrum of the local voltage ('detuning') fluctuations between the two planes. For the ohmic long-wavelength spectrum ⟨δVω2⟩=αω\langle\delta V_\omega^2\rangle=\alpha\omega, tunneling acquires a dephasing rate 1/τϕ=2παkT1/\tau_\phi=2\pi\alpha kT and the dc cc-axis conductivity of the diffusive channel follows σc(T)∝(kT/ωc)2α/(2πα)\sigma_c(T)\propto(kT/\omega_c)^{2\alpha}/(2\pi\alpha), where ωc∼1 \omega_c\sim 1\,eV is a high-frequency cutoff set by the under-barrier traversal time; the same Gaussian structure implies that the frequency dependence of the optical conductivity equals the voltage dependence of the tunneling conductance, σc(ω)=σc(V→ω)\sigma_c(\omega)=\sigma_c(V\to\omega). The dimensionless noise coefficient α\alpha is the central microscopic input. Simple Fermi-liquid, random-phase-approximation (RPA), and hydrodynamic estimates all give α≪1\alpha\ll 1 (e.g. α=1/4π\alpha=1/4\pi from electron-hole pairs, α=1/(2πkFd)\alpha=1/(2\pi k_F d) with interplane RPA screening, dd being the interplane spacing), whereas the ohmic loss Im(−1/ϵab)=γω\mathrm{Im}(-1/\epsilon_{ab})=\gamma\omega with γ≃0.2 \gamma\simeq 0.2\,eV−1^{-1} extracted from optical and Raman data implies α≃1\alpha\simeq 1 — exactly what the phenomenology requires. The microscopic origin of the anomalously large low-frequency, long-wavelength charge fluctuations, a non-Fermi-liquid property absent from RPA, is thus the core open issue. Within the same paper, the spatial decay of the tunneling probability defines an effective tunneling area SeffS_{\mathrm{eff}} much smaller than the coherence area set by the fluctuation wave-vector cutoff qcq_c; tunneling is 'diffusive' (in-plane momentum not conserved) when τhop≫τab\tau_{\mathrm{hop}}\gg\tau_{ab}, with a 'specular' momentum-conserving channel expected near the overdoped crossover; noise in different planes is treated as uncorrelated; and in bilayer materials interplane Coulomb coupling brings in acoustic (out-of-phase) plasmons that modify the noise, an effect left uncalculated.

Later work measured the relevant charge fluctuation spectra directly: momentum-resolved electron energy loss spectroscopy (M-EELS) found a featureless power-law continuum at small momentum in optimally doped Bi2_2Sr2_2CaCu2_2O8+δ_{8+\delta} [Mitrano et al. 2018] persisting across the strange-metal (non-Boltzmann, non-Fermi-liquid) regime of the phase diagram [Husain et al. 2019], and resonant inelastic X-ray scattering (RIXS) detected the anticipated interplane collective modes — acoustic plasmons — in both electron-doped [Hepting et al. 2018] and hole-doped [Nag et al. 2020] cuprates, with bilayer plasmon bands analyzed by Bejas et al. 2024. In parallel, alternative descriptions of the same ρc(T)\rho_c(T) phenomenology were developed — small-polaron and boson-assisted interlayer tunneling [Lundin and McKenzie 2003; Ho and Schofield 2005; Gutman and Maslov 2008], phenomenological scaling for incoherent layered crystals [Levin 2004], pseudogap-controlled scaling [Luo, Su and Xiang 2008], and plaquette cellular dynamical mean-field theory (a cluster extension of DMFT, the mapping of a lattice model onto self-consistently embedded finite clusters) calculations of ρc\rho_c [Sordi et al. 2013] — none of which settles the mechanism question that the blockade framework poses.

Problem Statement

Complete the fluctuation-induced blockade theory of interlayer cc-axis transport in the normal state of cuprates into a predictive microscopic theory — or refute it. The components below serve this single overarching objective: a microscopic account of the interplane voltage-noise spectrum that the framework requires, verified against the measured charge-fluctuation spectroscopy and cc-axis transport of the cuprates. (i) Microscopic origin and magnitude of the detuning-noise coefficient α\alpha: derive, from a microscopic many-body theory beyond RPA and Fermi-liquid pictures, the mechanism producing the empirically large low-frequency, small-qq charge (voltage) fluctuations — simple estimates give α≪1\alpha\ll 1 while the phenomenology of σc(ω)\sigma_c(\omega) and ρc(T)\rho_c(T) and the optical/Raman loss function require α∼O(1)\alpha\sim O(1) — including the doping and material dependence of α\alpha. (ii) Interplane correlations and layered structure: incorporate interplane Coulomb coupling (including the acoustic out-of-phase plasmons of bilayer and multilayer materials), correlations between the noise in different planes, and the distinction between intra-unit-cell and inter-unit-cell hopping into the noise spectrum ⟨δVq,ω2⟩\langle\delta V^2_{q,\omega}\rangle, the effective tunneling area SeffS_{\mathrm{eff}}, and the effective α\alpha, replacing the assumption of uncorrelated plane noise with a treatment valid for q≲1/dq\lesssim 1/d. (iii) Quantitative experimental program: extract ⟨δVq,ω2⟩\langle\delta V^2_{q,\omega}\rangle from optical/Raman/EELS/RIXS spectroscopy across cuprate families and dopings, and test the predicted mapping σc(ω)↔σc(V→ω)\sigma_c(\omega)\leftrightarrow\sigma_c(V\to\omega) and the power law σc(T)∝T2α\sigma_c(T)\propto T^{2\alpha} of the diffusive channel, with the exponent fixed by the independently extracted α\alpha rather than fitted. (iv) Crossover regimes: extend the framework to the superconducting state (where interlayer phase coherence sets in and the fluctuation-induced blockade and the diffusive character of tunneling may evolve) and to the regime τhop∼τab\tau_{\mathrm{hop}}\sim\tau_{ab} where diffusive and specular (momentum-conserving) channels coexist, specifying how the two-channel picture and the possible restoration of in-plane momentum conservation follow from the microscopic theory.

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