The theoretical understanding of conventional superconductivity as the phonon-assisted formation and condensation of two-electron Cooper pairs is a significant triumph in condensed matter physics. Here, we propose many-electron characterizations of higher-charge superconductivity with Cooper pairs consisting of more than two electrons, by implementing translation symmetrization on parent pair-density-wave-ordered states. In particular, we demonstrate many-electron constructions with vanishing charge- 2 e sectors, but with sharp signatures in charge- 4 e or charge- 6 e expectation values instead. Such characterizations are consistent with previous phenomenology of vestigial order and Ginzburg-Landau theory. Furthermore, we demonstrate that at the microscopic level, momentum conservation alone may be vital and sufficient for the robust emergence of higher-charge superconductivity. Our study thus offers a general and microscopic route to understand and characterize higher-charge superconductivity, including nontrivial experimental signatures such as fractional magnetic flux and period in interferometry, as well as localized Cooper pairs at lattice topological defects.
The theoretical understanding of conventional superconductivity as the phonon-assisted formation and condensation of two-electron Cooper pairs is a significant triumph in condensed matter physics. Here, we propose many-electron characterizations of higher-charge superconductivity with Cooper pairs consisting of more than two electrons, by implementing translation symmetrization on parent pair-density-wave-ordered states. In particular, we demonstrate many-electron constructions with vanishing charge- 2 e sectors, but with sharp signatures in charge- 4 e or charge- 6 e expectation values instead. Such characterizations are consistent with previous phenomenology of vestigial order and Ginzburg-Landau theory. Furthermore, we demonstrate that at the microscopic level, momentum conservation alone may be vital and sufficient for the robust emergence of higher-charge superconductivity. Our study thus offers a general and microscopic route to understand and characterize higher-charge superconductivity, including nontrivial experimental signatures such as fractional magnetic flux and period in interferometry, as well as localized Cooper pairs at lattice topological defects.
A direct charge-4e superconductor exhibits coherent four-electron order without condensation in any charge-2e pairing channel. We establish three complementary results. First, a particle-number-conserving Hamiltonian with electron hopping and two-body interactions moves bound quartets efficiently: for electron- and pair-hopping amplitudes t and K and local gap U0, virtual pair transfer produces quartet hopping at order K2/U0, whereas electron hopping alone contributes only at order t4/U03. Second, the microscopic theory identifies an exactly solvable parent Hamiltonian at the phase-separation boundary. Its ground state combines quartet off-diagonal long-range order and finite phase stiffness without pair condensation, while rigorous bounds establish quartet order beyond the parent in a family with explicit quartet transfer. Third, a single convergent strong-coupling expansion simultaneously controls quartet kinetics and thermodynamic stability and rules out condensation in every pair channel of the microscopic model. On bipartite hypercubic lattices at half quartet filling, it yields a parametrically controlled homogeneous phase for d≥2, with quartet order, a linear collective mode, and gapped charge-2e excitations. Virtual pair transfer thus enables a direct charge-4e superconducting ground state without pair condensation.
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 × 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 ′ , where a charge-density wave coexists with superconductivity. 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.
A phase of matter in which fermion quartets form a superconducting condensate, rather than the paradigmatic Cooper pairs, is a recurrent subject of experimental and theoretical studies. However, a comprehensive microscopic understanding of charge-4e superconductivity as a quantum phase is lacking. Here, we study a two-orbital tight-binding model with attractive Hubbard-type interactions. Such a model naturally provides the Bose-Einstein condensate as a limit for electron quartets and supports charge-4e superconductivity, as we show by mapping it to a spin-1/2 chain in this perturbative limit. Using density matrix renormalization group calculations for the one-dimensional case, we further establish that the ground state is indeed a superfluid phase of 4e charge carriers and that this phase can be stabilized well beyond the perturbative regime. Importantly, we demonstrate that 4e condensation dominates over 2e condensation even for nearly decoupled orbitals, which is a more likely scenario in electronic materials. Our model paves the way for both experimental and theoretical exploration of 4e superconductivity and provides a natural starting point for future studies beyond one dimension or more intricate 4e states.
The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite Tcdw in two dimensions. Quantum Monte Carlo calculations have determined Tcdw for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in d=2 is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine Tsc for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, where t is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher T. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.
A charge-4e superconductor forms due to the condensation of quartets of electrons. While in previous works, the mechanism for the formation of charge-4e superconductivity has been analyzed in terms of the binding of Cooper pairs in unconventional superconductors, its properties in the fermionic sector have not been studied systematically due to its inherently interacting nature even at the mean-field level. Here we propose a solvable model for a charge-4e superconductor---a spinful version of the Sachdev-Ye-Kitaev model with an anomalous quartic term. We show that the ground state is gapless and resembles a heavy Fermi liquid. We analytically solve for the superfluid density and show that it is perturbative in the strength of the charge-4e order parameter, in sharp contrast with a regular (charge-2e) superconductor. Upon lowering the temperature, we show that the correlation between charge-4e order and regular interaction terms can drive a first-order phase transition to a charge-2e superconducting state.
Superconductivity is usually understood as a phase in which charge-2e Cooper pairs are condensed. Charge-4e superconductivity has largely been discussed as a vestigial order at finite temperature emerging from charge-2e states. Primary charge-4e superconducting phases at zero temperature remain scarce in both experiments and microscopic models. Here we argue that a doped featureless Mott insulator with SU(4) symmetry provides a natural platform for primary charge-4e superconductivity, based on perturbative renormalization group arguments and group theoretic considerations. As a concrete realization, we construct a bilayer Hubbard model with tunable onsite SU(4) and Sp(4) symmetries that exhibits a featureless Mott insulating phase at half filling. Its low energy physics is captured by a generalized ESD model, featuring an effective Hamiltonian that is purely kinetic within the constrained Hilbert space. Using density matrix renormalization group (DMRG) simulations, we find a primary charge-4e superconducting phase in the SU(4) ESD model and a conventional primary charge-2e phase in the Sp(4) case. We further characterize the corresponding normal states and discuss the resulting finite temperature phase diagram.