SC$^*$ Superconductivity and Spin Stiffnesses in the SU(2) Gauge Theory of the Two-Dimensional Hubbard Model

TL;DR

This work investigates SC$^*$ superconductivity within an SU(2) gauge theory for the pseudogap regime of the 2D Hubbard model, where electrons fractionalize into spinons and chargons. By deriving a non-linear sigma model for spin fluctuations and computing both spatial and temporal spin stiffnesses through a gauge-field response, the authors quantify how chargon-mediated superconductivity affects magnetic fluctuations. They find that superconductivity generally suppresses spin stiffness, especially at larger hole doping, thereby increasing quantum spin fluctuations and potentially stabilizing quantum-disordered or topologically ordered phases. The results provide a framework for understanding the competition and coexistence of magnetic order and superconductivity, with the stiffness expressions applicable to conventional systems exhibiting similar coexisting orders as well.

Abstract

We consider the SU(2) gauge theory for spin fluctuations in the two-dimensional Hubbard model, where the electron field is fractionalized in terms of spinons and chargons. In this theory, spinons are described by a non-linear sigma model, while chargons are treated as fermions at a mean-field level. We investigate the instability to a superconducting state SC*, arising from a fractionalized Fermi liquid (FL*) where pairing between chargons occurs. Consistent with previous studies, our analysis reveals a coexisting phase characterized by both magnetic and superconducting order for the chargons. The central contribution of this work is the calculation of the feedback of superconductivity on spatial and temporal spin stiffnesses, thereby quantifying its impact on spin fluctuations. Our key finding is that superconductivity significantly suppresses these spin stiffnesses, enhancing quantum spin fluctuations. This enhancement suggests that superconductivity can play a crucial role in stabilizing quantum disorder against long-range magnetic ordering.

Figures (6)

• Figure 1: Magnetic and pairing effective interactions as a function of the doping.
• Figure 2: Superconducting and magnetic gaps as functions of the doping $p=1-n$ at temperature $T=0.001t$. $\Delta_{\rm m}$ and $\Delta_{\rm p}$ in blue and orange, respectively, refer to the solution with coexisting magnetism and superconductivity. Conversely, the green data refers to the magnetic solution without the superconducting phase. Inset: incommensurability $\eta$ as a function of the doping for the magnetic solution (in green) and for the superconducting magnet one (in blue).
• Figure 3: Upper panel: derivative of the free energy shown as a function of the incommensurability factor $\eta$ for $n=0.91$. Lower panel: out-of-plane $J^{\perp}$ and in-plane $J^{{\mathsmaller{\mathsmaller{\mathsmaller{\square}}}}}$ stiffnesses as a function of $\eta$.
• Figure 4: Out-of-plane stiffnesses as a function of the doping $p=1-n$. In blue and orange the stiffnesses for the coexistence and in red and green for the purely magnetic state.
• Figure 5: In-plane stiffness as a function of the doping $p=1-n$. In blue and orange the stiffnesses for the coexistence and in red and green for the purely magnetic state.