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 \(U_0\), virtual pair transfer produces quartet hopping at order \(K^2/U_0\), whereas electron hopping alone contributes only at order \(t^4/U_0^3\). 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 \(d\geq2\), 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.
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 (U_0), virtual pair transfer produces quartet hopping at order (K^2/U_0), whereas electron hopping alone contributes only at order (t^4/U_0^3). 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\geq2), 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.
The condensation of electron quartets, known as charge-4e superconductivity (SC), represents a novel quantum state of matter beyond the standard paradigm of Cooper pairing. However, concrete microscopic models realizing this phase in two dimensions remain a central challenge. Here, we introduce a non-engineered and sign-problem-free model, unambiguously demonstrating the emergence of a robust and high-temperature charge-4e SC phase using unbiased quantum Monte Carlo simulations. At zero temperature, the phase diagram reveals that charge-4e SC is the primary ground state in the strong-coupling regime. At finite temperature in the absence of charge-2e SC, we identify charge-4e SC through a Berezinskii-Kosterlitz-Thouless transition, marked by a universal jump in the superfluid stiffness consistent with a condensate of charge 4e. Remarkably, the transition temperature Tc increases nearly linearly with interaction strength, providing a robust mechanism for high-Tc quartet superconductivity. Furthermore, spectral analysis reveals a prominent pseudogap above Tc arising from strong phase fluctuations. Our results establish a canonical and numerically exact model system for charge-4e superconductivity, offering crucial guidance for its realization in experimental platforms such as moiré materials and ultracold atomic systems.
Unlike conventional charge-2e superconductors, a charge-4e superconductor exhibits long-range coherence of electron quartets rather than Cooper pairs. Clear zero-temperature realizations of charge-4e superconductivity remain rare. Here, we investigate the zero-temperature phase diagram of the attractive SU(4) Hubbard model with numerically exact, large-scale quantum Monte Carlo (QMC) simulations overcoming major technical hurdles. We identify both charge-2e and charge-4e superconducting phases. Upon increasing interaction, charge-2e correlations are suppressed and eventually vanish, while the charge-4e correlations remain robust and converge with system size, signaling the onset of a quartet-condensed phase. Interestingly, across the charge-2e--charge-4e transition, single electrons remain gapped, while charge-2e correlations exhibit a scaling behavior inconsistent with a conventional Landau description. These features are naturally captured by a fractionalized framework in which the physical charge-2e order parameter is a composite field coupled to an emergent non-Abelian gauge structure. We formulate an Sp(4) gauge-Higgs theory that realizes deconfined quantum pseudocriticality between the Higgs (charge-2e) phase and the confined (charge-4e) phase. The Sp(4) gauge-Higgs theory yields pseudocriticality through a fixed-point collision, and its one-loop collision-point exponents quantitatively track the QMC results. Our results establish charge-4e superconductivity as a bona fide zero-temperature phase, provide a simple model for future studies in a numerically exact framework, and reveal an unconventional route to superconducting criticality.
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum q=0 and q=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.
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.
We study a microscopic model for four spinless fermions on the square lattice which exhibits a quartet bound state in the strong coupling regime. The four-particle quantum states are analyzed using symmetry arguments and by introducing a zoo of relevant lattice animals. These considerations, as well as variational and exact diagonalization calculations, demonstrate the existence of a narrow quartet band at small hopping and a first-order transition to delocalized fermions at a critical hopping parameter, in qualitative contrast to, e.g., the BCS-BEC crossover in the attractive Hubbard model. In the case of pure attraction, an intermediate phase is found, in which a more extended and presumably more mobile hybrid quartet dominates the ground state. We comment on the relevance of the spin degree of freedom and on the reasons why electron quartetting is rarely observed in real materials.
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.