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arXiv:2312.13348 · Physical review. B./Physical review. B

Charge- 4 e superconductivity in a Hubbard model

Martina O. Soldini, Mark H. Fischer, Titus Neupert

Published Jun 11, 2024• 11 citations
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Abstract

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.

DOI
10.1103/physrevb.109.214509

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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-4e4e4e 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-4e4e4e 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 4e4e4e charge carriers and that this phase can be stabilized well beyond the perturbative regime. Importantly, we demonstrate that 4e4e4e condensation dominates over 2e2e2e 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 4e4e4e superconductivity and provides a natural starting point for future studies beyond one dimension or more intricate 4e4e4e states.
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.
A charge-4e4e4e superconductor forms due to the condensation of quartets of electrons. While in previous works, the mechanism for the formation of charge-4e4e4e 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-4e4e4e 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-4e4e4e order parameter, in sharp contrast with a regular (charge-2e2e2e) superconductor. Upon lowering the temperature, we show that the correlation between charge-4e4e4e order and regular interaction terms can drive a first-order phase transition to a charge-2e2e2e superconducting state.
Superconductivity is usually understood as a phase in which charge-2e2e2e Cooper pairs are condensed. Charge-4e4e4e superconductivity has largely been discussed as a vestigial order at finite temperature emerging from charge-2e2e2e states. Primary charge-4e4e4e 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)SU(4)SU(4) symmetry provides a natural platform for primary charge-4e4e4e 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)SU(4)SU(4) and Sp(4)Sp(4)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-4e4e4e superconducting phase in the SU(4)SU(4)SU(4) ESD model and a conventional primary charge-2e2e2e phase in the Sp(4)Sp(4)Sp(4) case. We further characterize the corresponding normal states and discuss the resulting finite temperature phase diagram.
A direct charge-4e4e4e superconductor exhibits coherent four-electron order without condensation in any charge-2e2e2e 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 ttt and KKK and local gap U0U_0U0​, virtual pair transfer produces quartet hopping at order K2/U0K^2/U_0K2/U0​, whereas electron hopping alone contributes only at order t4/U03t^4/U_0^3t4/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≥2d\geq2d≥2, with quartet order, a linear collective mode, and gapped charge-2e2e2e excitations. Virtual pair transfer thus enables a direct charge-4e4e4e superconducting ground state without pair condensation.
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.
Quantum design of Cooper quartets in a double quantum dot system coupled to ordinary superconducting leads is presented as a novel platform for the study of an elusive many-body state of matter, that is at the basis of the phenomenon of charge-4e superconductivity. A fundamentally novel, maximally correlated ground state, in the form of a superposition of vacuum |0⟩ and four-electron state |4e⟩, emerges as a narrow resonance and it is promoted by an attractive interdot interaction. A novel phenomenology in the dissipationless transport regime is elucidated, that yields typical flux quantization in units of h/4e and manifests in non-local multi-terminal coherence and in two-Cooper pair transport properties mediated by the quartet ground state. The results open the way to the exploration of correlation effects and non-local coherence in hybrid superconducting devices, parity-protected quantum computing schemes and more generally, the work poses the basis for the design and simulation of novel correlated states of matter starting from ordinary ingredients available in a quantum solid state lab.