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arXiv:2512.01124 · Physical Review Letters

Dynamics of Superconducting Pairs in the Two-Dimensional Hubbard Model

G. Sordi, E. M. O’Callaghan, C. Walsh, M. Charlebois, P. Sémon, A.–M. S. Tremblay

Published May 18, 2026• 0 citations
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Abstract

The frequency structure of the superconducting correlations in cuprates gives insights on the pairing mechanism. Here we present an exhaustive study of this problem in the two-dimensional Hubbard model with cellular dynamical mean-field theory. To this end, we systematically quantify the dependence on doping δ and interaction strength U of the superconducting gap, of the frequency scales where d-wave pairing occurs, and of their relative contribution to pairing. For all values of U and δ, we find pair-forming processes confined to frequencies set by the superexchange interaction and followed by pair-breaking processes, ruling out both pair-forming and pair-breaking processes on the scale of U. This suggests that at high frequencies, the effect of U is eliminated by the d-wave paring, and that at small frequencies, U generates the superexchange interaction that leads to low-frequency pair-forming processes providing the net contribution to pairing.

DOI
10.1103/22h2-jxh4

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The frequency dependence of the superconducting gap function represents the dynamics of the electrons forming the Cooper pairs. Bearing the high-temperature superconducting cuprates in mind, we calculate the superconducting gap function in the two-dimensional Hubbard model by means of a cluster extension of the dynamical mean-field theory. We scrutinize the frequency dependence and its change with temperature, doping concentration and interaction strength. The calculated superconducting gap function shows various different behaviors from the anomalous self-energy, owing to a strong frequency dependence of the normal self-energy. The calculated results also suggest that the superconductivity is stronger when the interaction strength is as large as or slightly smaller than the bare band width and is suppressed for larger interactions.
Doping a Mott insulator gives rise to unconventional superconducting correlations. Here we address the interplay between ddd-wave superconductivity and Mott physics using the two-dimensional Hubbard model with cellular dynamical mean-field theory on a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette. Our approach is to study superconducting correlations from the perspective of a cluster quantum impurity model embedded in a self-consistent bath. At the level of the cluster, we calculate the probabilities of the possible cluster electrons configurations. Upon condensation, we find an increased probability that cluster electrons occupy a four-electron singlet configuration, enabling us to identify this type of short-range spin correlation as key to superconducting pairing. The increased probability of this four-electron singlet comes at the expense of a reduced probability of a four-electron triplet with no significant probability redistribution of fluctuations of charges. This allows us to establish that superconductivity at the level of the cluster primarily involves a reorganization of short-range spin correlations rather than charge correlations. We gain information about the bath by studying the spectral weight of the hybridization function. Upon condensation, we find a transfer of spectral weight leading to the opening of a superconducting gap. We use these insights to interpret the signatures of superconducting correlations in the density of states of the system and in the zero-frequency spin susceptibility.
We study the fluctuations responsible for pairing in the d -wave superconducting state of the two-dimensional Hubbard model at intermediate coupling within a cluster dynamical mean-field theory with a numerically exact quantum impurity solver. By analyzing how momentum- and frequency-dependent fluctuations generate the d -wave superconducting state in different representations, we identify antiferromagnetic fluctuations as the pairing glue of superconductivity in both the underdoped and the overdoped regime. Nevertheless, in the intermediate coupling regime, the predominant magnetic fluctuations may differ significantly from those described by conventional spin fluctuation theory.
We analyze the interplay of antiferromagnetism and pairing in the two-dimensional Hubbard model with a moderate repulsive interaction. Coupled charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are treated by a functional renormalization group flow, while the formation of gaps and order below that scale is treated in mean-field theory. The full frequency dependences of the interaction vertices and gap functions are taken into account. We compute the magnetic and pairing gap functions as a function of doping ppp and compare with results from a static approximation. In spite of the strong frequency dependences of the effective interactions and of the pairing gap, important physical results from previous static functional renormalization group calculations are confirmed. In particular, there is a sizable doping regime with robust pairing coexisting with N\'eel or incommensurate antiferromagnetism. The critical temperature for magnetic order is interpreted as the pseudogap crossover temperature. Computing the Kosterlitz-Thouless temperature from the superfluid phase stiffness, we obtain a superconducting dome in the (p,T)(p,T)(p,T) phase diagram centered around 15% hole doping.
We analyze the influence of phonons on the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-pairing instability in the Hubbard model on the two-dimensional square lattice at weak to moderate interaction UUU, using a functional renormalization group scheme with frequency-dependent interaction vertices. As measured by the pairing scale, the B1g{B}_{1g}B1g​ buckling mode enhances the pairing, while other phonon modes decrease the pairing. When various phonon modes are included together, the net effect on the scale is small. However, in situations where ddd-wave superconductivity and other tendencies, e.g., antiferromagnetism, are closely competing, the combined effect of different phonons may be able to tip the balance towards pairing.
We present an infinite-dimensional lattice of two-by-two plaquettes, the quadruple Bethe lattice, with Hubbard interaction and solve it exactly by means of the cluster dynamical mean-field theory. It exhibits a ddd-wave superconducting phase that is related to a highly degenerate point in the phase diagram of the isolated plaquette at that the ground states of the particle number sectors N=2,3,and4N=2,\phantom{\rule{4pt}{0ex}}3,\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}4N=2,3,and4 cross. The superconducting gap is formed by the renormalized lower Slater peak of the correlated, hole-doped Mott insulator. We engineer parts of the interaction and find that pair hoppings between X/YX/YX/Y momenta are the main two-particle correlations of the superconducting phase. The suppression of the superconductivity is caused by the diminishing of pair hopping correlations in the overdoped regime and by the charge blocking in the underdoped one. The optimal doping is $\ensuremath{\sim}0.15$ at which the underlying normal state shows a Lifshitz transition. Tuning intra- and interplaquette hoppings allows to disentangle superconductivity from antiferromagnetism as the latter requires larger interplaquette hoppings. We expect our results to provide additional insight into the nature of the superconductivity in cuprates.