Instabilities of a Generalized Gross-Neveu Quantum Criticality

TL;DR

The paper investigates instabilities of a generalized Gross–Neveu–Yukawa quantum critical point in a solvable large-$N$ framework with a boson-to-fermion flavor ratio $\gamma$. Using conformal field theory and Bethe–Salpeter equations, it identifies two competing zero-temperature instabilities—$s$-wave superconductivity and ferromagnetism—that destabilize the conformal saddle point when fermion renormalization is strong, with the onset controlled by the fermion scaling dimension $Δ$ and the TRS-breaking parameter $α$. A key result is the condition for complex scaling dimensions: ferromagnetic instability arises when $k_s(D/2)>1$ (with $γ$-dependent thresholds), and superconductivity occurs when $|1-2α|\,k_s(D/2)>1$, yielding $α_c=\tfrac12 \pm \tfrac{1}{2k_s(D/2)}$, while $p$-wave and current-order channels remain stable. The work also finds that larger $γ$ enhances superconductivity and discusses implications for Dirac quantum criticality, potential holographic links, and unexpected superconductivity in $0+1$D under repulsive interactions.

Abstract

We study the instabilities to the conformal critical point of an exactly solvable family of Gross-Neveu models. Using conformal field theory techniques, we construct the zero-temperature phase diagram and identify the superconducting and ferromagnetic phases that destabilize the critical point. Both instabilities appear only when the fermions are strongly renormalized, above a critical anomalous dimension. A higher fermion anomalous dimension also raises the critical degree of time-reversal-symmetry breaking required to suppress superconductivity, indicating that pairing becomes more robust with stronger renormalization.

Figures (3)

• Figure 1: Zero temperature phase diagram for the generalized GNY model, as a function of $\alpha$, the degree of time-reversal symmetry $\mathcal{T}$ breaking, and the fermion scaling dimension $\Delta$. Time reversal symmetry is preserved at $\alpha = 0, 1$, and maximally broken at $\alpha = 1/2$. The fermion scaling dimension depends on the dimensionality $D$ and the ratio of bosons to fermions $\gamma$. Colored regions indicate instabilities: blue and orange denote $s$-wave superconductivity where $\psi^T D_T \psi$ and $\psi^T D_T \pmb{\gamma}_5 \psi$ respectively condense. Green denotes ferromagnetic instabilities, where $\bar{\psi} \pmb{\gamma}_5 \psi$ condense.
• Figure 2: Bethe-Salpeter equations for the three point functions with (a) the charged fermion bilinear $\mathcal{S} = \psi^T \textbf{s} \psi$, and (b) the charge neutral fermion bilinear $\mathcal{M} = \bar{\psi} \textbf{m} \psi$. The straight lines denote fermion propagators, squiggly lines, boson propagators, and the dashed, disorder averaging. A dot indicates a fully dressed three-point function.
• Figure 3: (a) $k_{s}(D/2)$ versus $\Delta$. $k_s(D/2) > 1$ for $0+1$D at $\Delta > \Delta_0 = 0.07088$, at all $\Delta$ in $1+1$D, and for $2+1$D at $\Delta > \Delta_2 = 1.07314$. $\Delta_{0,2}$ are marked by black dots. (b) The operators $\mathcal{M}^-$ (or $\mathcal{S}^\pm$ for $\alpha = 0,1$) acquire complex scaling dimensions $D/2+\mathbf{i} f$, where $k_s(D/2 + i f) = 1$. (c) The imaginary component of the scaling dimension $\rm{Im}\{h_{\mathcal{M}^-}\}$ is shown.