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Coexisting magnetic, charge, and superconducting orders in the two-dimensional Hubbard model

Robin Scholle1, Pietro M. Bonetti1,2, Walter Metzner1, and Demetrio Vilardi1,*

  • *Contact author: d.vilardi@fkf.mpg.de
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Phys. Rev. B 114, 185107 – Published 8 September, 2026

DOI: https://doi.org/10.1103/8ck6-482h

Abstract

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.

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Phys. Rev. B 114, 185107– Published 8 September, 2026
  • Received 26 February 2026
  • Accepted 28 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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