Cooper Condensation and Pair Wave Functions in Strongly Correlated Electrons

Identifying superconducting states of matter without prior assumptions is a central challenge in strongly correlated electron systems. We introduce a canonical framework for diagnosing the formation of Cooper pair condensates based on the Penrose-Onsager criterion, in which superconducting order is encoded in the spectral properties of the two-particle reduced density matrix (2RDM). Within this formulation, the symmetry and structure of the condensate are obtained by projecting the 2RDM onto irreducible representations of the underlying symmetry group, enabling an unbiased identification of both conventional and exotic superconducting states. We demonstrate the power and versatility of the approach through applications to the two-dimensional Hubbard model, using both auxiliary-field quantum Monte Carlo (AFQMC) and the density matrix renormalization group (DMRG). For attractive interactions without a magnetic field, we reveal a clear finite-size scaling of the condensate fraction on square lattices of size up to $20\times 20$. The framework further provides direct access to the internal structure and extent of Cooper pairs, which we track across the BCS-BEC crossover. Moreover, it enables a clean diagnosis of the finite-momentum Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in a magnetic field. Finally, we apply the approach to a supersolid phase in the repulsive Hubbard model with an additional next-nearest neighbor hopping $t^\prime$, where a charge-density wave coexists with a superconductor. We confirm the fragmented nature of the condensate and uncover substantial pairing correlations in the triplet channel with $p$-wave spatial symmetry in addition to the dominant singlet $d$-wave pairing. Our results establish the 2RDM-based Penrose-Onsager framework as a broadly applicable and unbiased tool for characterizing superconducting order in correlated quantum matter.