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.