Charge-4e Superconducting Ground State without Pair Condensation: Exact Quartet Dynamics, Rigorous Order, and a Microscopic Route

Abstract

A direct charge-\(4e\) superconductor exhibits coherent four-electron order while every charge-\(2e\) pairing channel remains uncondensed. We establish three complementary results. First, building on the \(η\)-clustering states and bipartite parent of Yoshida and Katsura, we formulate and exactly solve a minimal two-term parent on any connected graph. Its fixed-number ground states have quartet off-diagonal long-range order (ODLRO) without charge-\(2e\) ODLRO, while an exact mapping to classical hard-core exclusion dynamics yields the full fixed-sector gap and a branch of quartet-density modes. Nonzero quartet stiffness and vanishing inverse quartet compressibility identify this \(z=2\) parent as a phase-separation boundary. Second, for a finite-range fermionic family with explicit quartet transfer and sufficiently large onsite penalty, we rigorously prove quartet ODLRO without charge-\(2e\) ODLRO at half quartet filling, both at the hypercubic XY point for \(d\geq2\) and throughout a finite XXZ interval on the square lattice. Third, we derive a strong-coupling realization using only electron hopping and two-body interactions. With local gap \(U_0\), pair hopping \(K\) generates quartet motion at order \(K^2/U_0\), whereas electron hopping \(t\) first contributes at order \(t^4/U_0^3\); charge-\(2e\) excitations remain gapped at \(O(U_0)\). On bipartite lattices, positive inverse quartet compressibility opens an asymptotically controlled homogeneous \(z=1\) regime with short-ranged pair correlations, while negative curvature drives phase separation.