Singular density fluctuations in the strange metal phase of a copper-oxide superconductor

One of the least understood phases of matter is the so-called "strange metal", whose transport properties defy any generally accepted understanding in terms of electron quasiparticles, requiring radical new ideas. It is widely believed that strange metal properties may arise from the presence a singular spectrum of fluctuations that span all momentum and energy scales. However, the existence of such fluctuations has never been established. Here, we use the new technique of momentum-resolved inelastic electron scattering (M-EELS) to measure the dynamic charge response of the strange metal Bi$_{2.1}$Sr$_{1.9}$CaCu$_2$O$_{8+x}$ (BSCCO). At optimal doping, the spectra are dominated by an electronic continuum spanning all momenta and energy up to a cutoff of 1 eV. Unlike conventional metals, the polarizability in BSCCO factors into distinct functions of energy and momentum, $\Pi^{"}(q,\omega) = f(q)\cdot g(\omega)$, indicating that collective excitations do not propagate. In overdoped materials, the polarizability also factors, but exhibits a temperature-dependent, gap-like structure, signifying an emergent energy scale. Our study implies that this strange metal resides in a scale-invariant phase in which space and time axes are decoupled.