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arXiv:2604.18777 · Physical review. B./Physical review. B

Competition and coexistence of superconductivity and nematic order in a two-dimensional electron gas with quadrupolar interactions

Nei Lopes, Guilherme da Silva do Vale, Daniel Gustavo Barci

Published Aug 18, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate the interplay between superconductivity and nematic order in a two-dimensional electron gas with competing pairing and quadrupolar forward-scattering interactions. The model includes both $s$-wave and $d$-wave superconducting channels. We compute the mean-field free energy density and determine the phase diagrams as functions of interaction strengths and temperature by solving a set of coupled self-consistent equations. At zero-temperature, we find that the nematic order competes strongly with $d$-wave superconductivity, leading to a direct first-order phase transition, while its interplay with $s$-wave pairing allows for a coexistence phase characterized by an anisotropic Fermi surface with a uniform superconducting gap. At finite-temperatures, quadrupolar interactions promote the emergence of additional superconducting components, giving rise to regimes where $s$-wave, $d$-wave, and nematic orders coexist. Our results highlight the role of symmetry and interaction strength in shaping the phase structure and provide a minimal framework to describe intertwined nematic and superconducting phases in correlated electron systems.

DOI
10.1103/93xz-rklx

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We study the interplay of nematic and superconducting order in the two-dimensional Hubbard model and show that they can coexist, especially when superconductivity is not the energetically dominant phase. Due to a breaking of the C4{C}_{4}C4​ symmetry, the coexisting phase inherently contains admixture of the sss-wave pairing components. As a result, the superconducting gap exhibits nonstandard features including changed nodal directions. Our results also show that in the optimally doped regime the pure superconducting phase is typically unstable towards developing nematicity (breaking of the C4{C}_{4}C4​ symmetry). This has implications for the cuprate high-Tc{T}_{c}Tc​ superconductors, for which in this regime the so-called intertwined orders have recently been observed. Namely, the coexisting phase may be viewed as a precursor to such more involved patterns of symmetry breaking.
We analyze a mean-field model of electrons on a square lattice with two types of interaction: forward scattering favoring a ddd-wave Pomeranchuk instability and a BCS pairing interaction driving ddd-wave superconductivity. By tuning the interaction parameters a rich variety of phase diagrams is obtained. If the BCS interaction is not too strong, Fermi surface symmetry breaking is stabilized around van Hove filling, and coexists with superconductivity at low temperatures. For pure forward scattering Fermi surface symmetry breaking occurs typically via a first order transition at low temperatures. The presence of superconductivity reduces the first order character of this transition and, if strong enough, can turn it into a continuous one. This gives rise to a quantum critical point within the superconducting phase. The superconducting gap tends to suppress Fermi surface symmetry breaking. For a relatively strong BCS interaction, Fermi surface symmetry breaking can be limited to intermediate temperatures, or can be suppressed completely by pairing.
We study the competition of antiferromagnetism and ddd-wave superconductivity at zero temperature in the two-dimensional Hubbard model using cellular dynamical mean-field theory for a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette, and solve the associated cluster impurity model at zero temperature by means of exact diagonalization. The interplay between the two phases depends strongly on the strength of the correlation. At strong coupling $(U\ensuremath{\gtrsim}8t)$ the two phases do not mix, and a first-order transition takes place as a function of doping between two pure phases. At weak coupling $(U\ensuremath{\lesssim}8t)$ the two order parameters coexist within the same solution in a range of doping and the system smoothly evolves from the antiferromagnet to the superconductor. When the transition between the superconducting and the antiferromagetic phases is of the first-order, it is accompanied by a phase separation.
Using a dynamic cluster quantum Monte Carlo approximation, we study a two-dimensional Hubbard model with a small orthorhombic distortion in the nearest-neighbor hopping integrals. We find a large nematic response in the low-frequency single-particle scattering rate which develops with decreasing temperature and doping as the pseudogap region is entered. At the same time, the ddd-wave superconducting gap function develops an sss-wave component and its amplitude becomes anisotropic. The strength of the pairing correlations, however, is found to be unaffected by the strong anisotropy, indicating that ddd-wave superconductivity can coexist with strong nematicity in the system.
We apply the self-consistent renormalized perturbation theory to the Hubbard model on the square lattice at finite temperatures to study the evolution of the Fermi surface (FS) as a function of temperature and doping. Previously, a nematic phase for the same model has been reported to appear at weak coupling near a Lifshitz transition from closed to open FS at zero temperature where the self-consistent renormalized perturbation theory was shown to be sensitive to small deformations of the FS. We find that the competition with the superconducting order leads to a maximal nematic order appearing at nonzero temperature. We explicitly observe the two competing phases near the onset of nematic instability, and by comparing the grand canonical potentials, we find that the transitions are first order. We explain the origin of the interaction-driven spontaneous symmetry breaking to a nematic phase in a system with several symmetry-related Van Hove points and discuss the required conditions.
We perform a renormalized mean-field study of the two-dimensional repulsive Hubbard model, focusing on the intricate interplay and possible coexistence of magnetic, charge, and superconducting orders. We improve on conventional mean-field theory by utilizing a renormalization group framework that captures high-energy fluctuations. This method generates effective magnetic and d -wave pairing interactions, and allows for an unbiased exploration of coexisting phases at weak and moderate interaction strengths. Unrestricted mean-field calculations of the effective Hamiltonian on large finite lattices are combined with analyses in the thermodynamic limit, revealing a rich phase diagram with extensive regions of coexisting orders. We find that d -wave superconductivity coexists with Néel order on the electron-doped side. On the hole-doped side, superconductivity is found to coexist with spiral or stripe magnetic orders. Within the stripe-ordered region, the superconducting order parameter is spatially modulated, with a period that follows the charge modulation of the stripes. Below Van Hove filling, pairing provides the primary energy gain, while the stripe order yields only a small, and hence fragile, additional energy lowering.