Correlations in interacting electron liquids: Many-body statistics and hyperuniformity

Disordered hyperuniform many-body systems are exotic states of matter with novel optical, transport, and mechanical properties. These systems are characterized by an anomalous suppression of large-scale density fluctuations compared to ordinary liquids. The structure factor of disordered hyperuniform systems often obeys the scaling relation $S(k)\ensuremath{\sim}\mathcal{B}{k}^{\ensuremath{\alpha}}$ with $\mathcal{B},\ensuremath{\alpha}>0$ in the limit $k\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}0$. Ground states of $d$-dimensional free fermionic gases, which are fundamental models for many metals and semiconductors, are key examples of quantum disordered hyperuniform states with important connections to random matrix theory. However, the effects of electron-electron interactions as well as the polarization of the electron liquid on hyperuniformity have not been explored thus far. In this paper, we systematically address these questions by deriving the analytical small-$k$ behaviors (and, associated, $\ensuremath{\alpha}$ and $\mathcal{B}$) of the total and spin-resolved structure factors of quasi-one-dimensional, two-dimensional, and three-dimensional electron liquids for varying polarizations and interaction parameters. We validate that these equilibrium disordered ground states are hyperuniform, as dictated by the fluctuation-compressibility relation. Interestingly, free fermions, partially polarized interacting fermions, and fully polarized interacting fermions are characterized by different values of the small-$k$ scaling exponent $\ensuremath{\alpha}$ and coefficient $\mathcal{B}$. In particular, partially polarized fermionic liquids exhibit a unique form of multihyperuniformity, in which the net configuration exhibits a stronger form of hyperuniformity (i.e., larger $\ensuremath{\alpha}$) than each individual spin component. The detailed theoretical analysis of such small-$k$ behaviors enables the construction of corresponding equilibrium classical systems under effective one- and two-body interactions that mimic the pair statistics of quantum electron liquids. Our paper thus reveals that highly unusual hyperuniform and multihyperuniform states can be achieved in simple fermionic systems and paves the way for harnessing unique hyperuniform scaling relations for applications, such as the construction of accurate density functionals.