Cooper Condensation and Pair Wave Functions in Strongly Correlated Electrons

TL;DR

The paper addresses how to identify and characterize superconducting states in strongly correlated electrons without prior assumptions. It develops a 2RDM-based Penrose-Onsager framework, using symmetry projections to resolve condensate structure and detect both simple and fragmented condensates. Applying this approach to the 2D Hubbard model with AFQMC and DMRG, it uncovers conventional s-wave pairing, FFLO states under spin imbalance, d-wave stripe order with a significant triplet component, and a fragmented supersolid state, while revealing detailed Cooper-pair internal structure and scaling properties. The framework provides an unbiased, broadly applicable tool for diagnosing superconducting order in correlated quantum matter, compatible with multiple numerical methods and extendable to other platforms and models.

Abstract

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.

Figures (12)

• Figure 1: Spectral properties of the $2$RDM and the leading eigenvector for the two-dimensional attractive Hubbard model at density $n=0.5$, for $U/t=-4$ (a,c,e) and $U/t=-10$ (b,d,f) obtained using AFQMC. (a, b) Spectrum of the $2$RDM. For both $U/t=-4$ and $U/t=-10$ we see a dominant eigenvalue growing linearly in the number of electrons $N_e$. (c, d) The condensate wave function in momentum space for $L=20$. For $U/t=-4$, the condensate wave function peaks around the non-interacting Fermi surface, indicated by the blue line. For $U/t=-10$, the wave function peaks slightly inside of the Fermi surface. (e, f) The condensate wave function in real space for $L=20$ is invariant under $C_4$ rotations. These observations establish the formation of a Cooper condensate with uniform $s$-wave pairing.
• Figure 2: Condensation and localization properties of Cooper pairs in the two-dimensional attractive Hubbard model. We compare results from AFQMC and BCS mean-field theory for quarter-filling $n=1/2$ in the upper row (a-d) and half-filling $n=1$ in the lower row (e-h). The orange lines for $L=\infty$ are obtained using AFQMC by extrapolation in $L$. (a,e) Condensate fraction $\varepsilon_0 / N_e$, (b,f) localization length $\lambda$, (c,g) inverse participation ratio IPR, as a function of $U/t$. (d,h) Scaling of the localization length with the condensate fraction. We observe a transition from delocalized Cooper pairs at small values of $U/t$ to fully localized Cooper pairs for larger values of $U/t$.
• Figure 3: Condensation and localization properties of Cooper pairs in the two-dimensional attractive Hubbard model on a cylinder of width $W=4$. We show results from DMRG, for quarter-filling $n=1/2$ and compare with the results from AFQMC (orange) on $L\times L$ lattices, extrapolated to $L=\infty$. The red lines for $L=\infty$ are obtained using DMRG by extrapolation in $L^\prime$. (a) Condensate fraction $\varepsilon_0/N_e^\nu$, (b) localization length $\lambda_{\text{cyl}}$, (c) Inverse participation ratio $\text{IPR}_{\text{cyl}}$ as a function of $U/t$. (d) Scaling of the localization length with condensate fraction.
• Figure 4: Comparison between the exponents $\nu$ and $1-K_{sc}$ as functions of $U$, where $K_{sc}$ is the superconducting Luttinger parameter. The inset displays the dominant eigenvalue $\varepsilon_0$ as a function of the number of electrons, $N_e$, for a range of values of interaction strength $U$, and shows the exponent $\nu$ with which it grows. We observe a good agreement between $\nu$ and $1-K_{sc}$, and see that the dominant eigenvalues grows algebraically, but sub-linearly, in $N_e$.
• Figure 5: The condensate wave function $\tilde{\psi}_0(k_1, k_2)$ as function of momenta, $k_1$ and $k_2$, in the $x$-direction for the two electrons, at $U/t=-10$, quarter-filling $n=1/2$ and at magnetization (a) $m=0$ and (b) $m=1/32$. For $m=0$, the condensate wave function peaks at zero total momentum, but when introducing a spin imbalance, $m=1/32$, it peaks at non-zero momentum, indicating the existence of an FFLO state of LO type.