The Renormalization Group Studies on Four Fermion Interaction Instabilities on Algebraic Spin Liquids

TL;DR

This work analyzes four-fermion instabilities in algebraic spin liquids by applying a systematic 1/N renormalization-group approach to three square-lattice spin-liquid states with different gauge structures. It identifies the SU(4N) flavor-preserving perturbations as generally irrelevant, while certain flavor-broken channels, notably those reducing to Sp(4N), can generate finite-coupling fixed points or drive symmetry-breaking transitions, depending on N. The authors develop a Majorana fermion formulation for the π-flux SU(2) state to make SU(2) gauge and SU(2) spin symmetries explicit, then extend to a large-N analysis that reveals potential Sp(4N) critical points and competing orders between Neel, VBS, and gauge-symmetry-breaking phases. The results illuminate possible deconfined-critical or multicritical scenarios in algebraic spin liquids and offer a framework for connecting to SO(5)-like order parameter manifolds, with broader relevance to Dirac-fermion systems such as graphene. Overall, the paper maps out the landscape of instabilities and phase boundaries driven by four-fermion interactions in Dirac spin liquids with emergent gauge dynamics.

Abstract

We study the instabilities caused by four fermion interactions on algebraic spin liquids. Renormalization group (RG) is used for three types of previously proposed spin liquids on the square lattice: the staggered flux state of SU(2) spin system, the $π-$flux state of SU(4) spin system, and the $π-$flux state of SU(2) spin system. The low energy field theories of the first two types of spin liquids are QED3 with emerged SU(4) and SU(8) flavor symmetries, the low energy theory of the $π-$flux SU(2) spin liquid is the QCD3 with SU(2) gauge field and emergent Sp(4) (SO(5)) flavor symmetry. Suitable large-N generalization of these spin liquids are discussed, and a systematic 1/N expansion is applied to the RG calculations. The most relevant four fermion perturbations are identified, and the possible phases driven by relevant perturbations are discussed.

Figures (5)

• Figure 1: Feynman diagrams contribute to the linear orders in both Eqs. \ref{['rgsun']} and \ref{['rgspn']}. The dashed lines are dressed photon propagators, and the full circles denote the trace in Dirac space.
• Figure 2: Feynman diagrams which only contribute to the linear orders in Eqs. \ref{['rgsun']}, but not in Eqs. \ref{['rgspn']}.
• Figure 3: Feynman diagrams which contribute to the quadratic order of the RG equations (\ref{['rgsun']}) and (\ref{['rgspn']}). Notice that since we only calculate to the order of unity in the quadratic terms, diagram G only contributes to equation (\ref{['rgsun']}) but not (\ref{['rgspn']}), and diagram H only contributes to equation (\ref{['rgspn']}) but not (\ref{['rgsun']}).
• Figure 4: Feynman diagrams which contribute to the difference of scaling dimensions of fermion bilinears $\bar{\psi}T^a_{\mathrm{su(4N)/sp(4N)}}\psi$ and $\bar{\psi}T^a_{\mathrm{sp(4N)}}\psi$.
• Figure 5: The RG flow when $N < N_c = \mathrm{Min}[N_{c1}, N_{c2}]$, the horizontal axis is $g_2$, and the vertical axis is $\lambda$ in (\ref{['lambda']}). At the Sp(4N) fixed point the order parameters with equally strongest correlations are $\bar{\psi}T^a_{\mathrm{sp(2N)}}\psi$ and $\bar{\psi}\mu^z\psi$, thus phase $A$ is most likely to be order $\langle \bar{\psi}T^a_{\mathrm{sp(2N)}}\psi \rangle$, and phase $B$ is most likely to be $\langle\bar{\psi}\mu^z\psi\rangle$. Notice that $\lambda$ is an RG eigenvector at the SU(4N) fixed point but not at the Sp(4N) fixed point, but phase A and B can be driven by $\lambda$ perturbation at the Sp(4N) fixed point.