Strong-coupling superconductivity near Gross-Neveu quantum criticality in Dirac systems

TL;DR

This work investigates superconductivity emerging from Gross-Neveu quantum critical fluctuations in two-dimensional Dirac systems at neutrality, using a controlled SYK-inspired large-$N$, large-$M$ framework. The authors show that superconductivity appears only when the fermion anomalous dimension $ ext{η}_ ext{ψ}$ surpasses a critical threshold $ ext{η}_ ext{ψ}^c \\approx 0.14628$, with three of four bosonic channels acting as pairing glue and favoring isotropic $l=0$ gaps. A set of algebraic criteria is developed to identify admissible pairing channels in generic Dirac theories, and the analysis reveals that two-component Dirac spinors cannot sustain superconductivity near GN criticality, while multi-component spinors can, depending on the coupling channel. They further discuss how order-parameter fluctuations influence $T_c$ via BKT physics, arguing that superconductivity remains stable at low temperature despite phase fluctuations. The findings provide a concrete mechanism by which strong critical fluctuations in Dirac fluids can induce unconventional superconductivity and offer a framework to classify potential pairing states in related materials.

Abstract

We study two-dimensional massless Dirac fermions at neutrality, coupled to bosonic modes through a Yukawa interaction. We then examine the intriguing possibility that such a system, devoid of carriers at zero temperature, might nevertheless exhibit superconductivity. Remarkably, we find that superconductivity emerges in the vicinity of Gross-Neveu quantum criticality, provided the fermions cease to behave as well-defined quasiparticles, that is, once their anomalous dimension in the normal state becomes sufficiently large. In other words, well-defined fermions do not superconduct, whereas ill-defined ones do. We analyze four symmetry-distinct bosonic modes, each capable of driving normal-state criticality and, in three of the four cases, giving rise to a distinct superconducting phase. While phase fluctuations are strong in this regime, we argue that they do not destroy the superconducting state. We further characterize the resulting pairing states for a concrete Dirac model of spin-orbit coupled systems with orbitals of different parity. Our results are obtained using the SYK-inspired framework for Dirac systems introduced by Kim et al.[1], which provides a controlled approach to the strongly coupled regime of Dirac fluids near Gross-Neveu criticality.

Figures (7)

• Figure 1: Superconducting transition temperature $T_c$ as function of the anomalous dimension $\eta_\psi$ of the fermions at the Gross-Neveu mass-generating transition of two-dimensional Dirac systems within the generalized SYK approach of this paper. For system with $\eta_\psi$ larger than a threshold value $\eta_\psi^c$, fluctuations of the mass-generating boson give rise to a superconducting instability, where the largest $T_c$ occurs at intermediate values of $\eta_\psi$. $\Lambda$ is the upper cut off of the Dirac theory.
• Figure 2: Normal state single-particle spectral function $A(\boldsymbol{k}, E) = -{\rm Im}[ {\rm Tr}G(\boldsymbol{k}, i\omega\to E+i0^+)]$ vs energy $E$, showing a strong broadening due to the self energy Eq. (\ref{['eq:fullselfenergy']}) that indicates the absence of well-defined quasiparticles. For this plot we chose amplitude $A = 1$, exponent $\eta_\psi = 0.14628$, and wave-vector $|\boldsymbol{k}|= 0.25$ (red), $|\boldsymbol{k}| = 0.50$ (purple), and $|\boldsymbol{k}|=0.75$ (blue).
• Figure 3: Dependence of the anomalous exponent of the fermion spectrum on the ratio $M/N$ of the bosonic and fermionic flavors, respectively (notice, as we consider four-component Dirac spinors, we have in total $4N$ fermion flavors). Crucially, the approach used here allows for a controlled analysis of $\eta_\psi$ that is not parametrically small and reaches values up to $\eta_\psi=\tfrac{1}{2}$.
• Figure 4: Critical coupling constants $\lambda_{\rm p}^c$ for various angular momenta $l$ for the exact solution (solid lines) and approximation (dashed lines) of the single components superconducting problem that occurs for the pairing interactions $\Upsilon_1$, $\Upsilon_2$, and $\Upsilon_4$. The pairing interaction $\lambda_{\rm p}$ exceeds the critical value for pairing for states with angular momentum in space-time $l=0$ and for $\eta_\psi>0.14628$, yielding a superconducting state (SC). The inset shows $\lambda_{\rm p}$ and the critical coupling constants $\lambda_{\rm p}^c$ for higher angular momenta $l$.
• Figure 5: Same as Fig. \ref{['fig:lambda_crit']}, but for a pairing interaction $\Upsilon_3$ that leads to the $3\times3$ matrix $Q$ which occurs in the vector component $l_{V}$. The dashed-purple line corresponds to the threshold coupling obtained from the approximate solution of the differential equation, while the solid-blue line corresponds to the full solution. For this interaction the pairing interaction $\lambda_p$ (shown as black line) is always smaller than the threshold coupling for superconductivity $\lambda^c_p$, i.e. no superconductivity emerges.