Consistency between reflection M-EELS and optical spectroscopy measurements of the long-wavelength density response of Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+x}$

Abstract

The density fluctuation spectrum captures many fundamental properties of strange metals. Using momentum-resolved electron energy-loss spectroscopy (M-EELS), we recently showed that the density response of the strange metal Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+x}$ (Bi-2212) at large momentum, $q$, exhibits a constant-in-frequency continuum [Mitrano, PNAS $\text{115}$, 5392 (2018); Husain, PRX $\text{9}$, 041062 (2019)] reminiscent of the marginal Fermi liquid (MFL) hypothesis of the late 1980s [Varma, PRL $\text{63}$, 1996 (1989)]. However, reconciling this observation with infrared (IR) optics experiments, which show a well-defined plasmon excitation at $q\sim 0$, has been challenging. Here we report M-EELS measurements of Bi-2212 using 4$\times$ improved momentum resolution, allowing us to reach the optical limit. For momenta $q<0.04$ r.l.u., the M-EELS data show a plasmon feature that is quantitatively consistent with IR optics. For $q>0.04$ r.l.u., the spectra become incoherent with an MFL-like, constant-in-frequency form. We speculate that, at finite frequency, $\ddot{I}‰$, and nonzero $q$, some attribute of this Planckian metal randomizes the probe electron, causing it to lose information about its own momentum.

9 pages, 7 figures; copy editing, improved figure resolution