Microscopic theory of electron quadrupling condensates

Electron pairing at low temperatures leads to superconductivity. A fundamental question is whether more complex states - characterized by order in four-electron composite objects, termed electron quadrupling or composite order - can exist in materials, and if so, under what conditions they emerge and what properties they exhibit. These states lie beyond the scope of Bardeen-Cooper-Schrieffer theory, and a microscopic description of them remained elusive. In the first part of the paper, we provide a general microscopic framework to describe these and the other four-fermion composite states. In the second part of the paper, we derive and solve a specific fermionic model in two and three dimensions that hosts time-reversal symmetry-breaking electron quadrupling order. The fermionic microscopic theory is used to estimate the specific heat and electron density of states.