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

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.

Figures (12)

• Figure 1: Effective interactions $U^m$ and $U^p$ obtained from the fRG flow as a function of filling $n$, for a bare interaction $U=4t$, and $t'=-0.2t$.
• Figure 2: Magnetic phase diagram for $U = 4t$ and $t' = -0.2t$. The colored pixels show the results of the real-space calculation on a $28\times 28$ lattice. The black lines have been obtained in the thermodynamic limit. The solid black line shows the divergence of the fRG flow and represents the onset of the magnetically ordered phase. Along the black dash-dotted lines, the fRG flow diverges in the pairing channel, signaling the onset of superconductivity as the leading instability. The dashed line inside the magnetically ordered regime indicates the transition from Néel order to spiral magnetic order. The dotted line indicates an instability of the spiral phase revealed by a divergence of the charge susceptibility.
• Figure 3: Average magnetization $m$ as a function of filling $n$ and temperature $T$ for the parameters from Fig. \ref{['fig: MagneticPhaseDiagram']}. The white line represents $T^*$. The magnetization is highest for low temperatures near half-filling.
• Figure 4: Average superconducting amplitude $\Delta$ in units of $t$ as a function of filling $n$ and temperature $T$ for the parameters from Fig. \ref{['fig: MagneticPhaseDiagram']}. The white line represents $T^*$.
• Figure 5: Magnetic, charge, and superconducting order pattern for the stripe state we find at $T = 0.02t$, $n = 0.84$. Only $10 \times 10$ sites are shown here for better visibility. In panel (a) one can see the collinear spin pattern with its amplitude modulated in $x$-direction. In panel (b) we see the charge pattern, which is also modulated along the $x$-axis with the lowest filling coinciding with the smallest local magnetization. In panel (c) the superconducting amplitude $\Delta_j$ on each site is shown. It is also modulated along the $x$-axis with the superconducting amplitude being maximal where the local filling is minimal.