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

TL;DR

This study uses an advanced mean-field approach to explore the complex coexistence of magnetic, charge, and superconducting phases in the 2D Hubbard model, revealing rich phase diagrams and spatially modulated superconductivity.

Contribution

It introduces a renormalization group-enhanced mean-field method to unbiasedly analyze coexisting orders in the Hubbard model at various doping levels.

Findings

Superconductivity coexists with Néel order on the electron-doped side.

Superconductivity coexists with spiral or stripe magnetic orders on the hole-doped side.

Stripe order induces spatial modulation of the superconducting order parameter.

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 N\'eel 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.