Quantum Fisher Information Reveals UV-IR Mixing in the Strange Metal

The density-density response in optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ has recently been shown to exhibit conformal symmetry. Using, the experimentally inferred conformal dynamic susceptibility, we compute the resultant quantum Fisher information (QFI), a witness to multi-partite entanglement. For a Fermi liquid, we find that the QFI grows quadratically as the temperature increases, consistent then with the phase space available for scattering in the standard theory of metals. By contrast, the QFI in a strange metal increases as a power law at as the temperature decreases, but ultimately extrapolates to a constant at $T=0$. The constant is of the form, $ω_g^{2Δ}$, where $Δ$ is the conformal dimension and $ω_g$ is the UV cutoff which is on the order of the pseudogap. As this constant {depends on both UV and IR properties}, it illustrates that multipartite entanglement in a strange metal exhibits UV-IR mixing, a benchmark feature of doped Mott insulators as exemplified by dynamical spectral weight transfer. We conclude with a discussion of the implication of our results for low-energy reductions of the Hubbard model.