Semi-Lorentz invariance, unitarity, and critical exponents of symplectic fermion models

TL;DR

This paper investigates a two-derivative, N-component symplectic fermion model with Sp(2N) symmetry that yields spin-1/2 without full Lorentz invariance. It demonstrates unitarity via a C-pseudo-Hermitian Hamiltonian and analyzes the renormalization group, revealing a low-energy fermionic Wilson-Fisher fixed point with exponents computed up to two loops. The authors establish an analytic continuation to the O(M) Wilson-Fisher fixed point with M=-2N and compute exponents for composite bilinears, highlighting potential applications to 2+1D quantum critical spin liquids and deconfined quantum criticality, albeit with cautions about model-to-lattice mappings. The work provides a framework for describing spinons in condensed matter scenarios using semi-Lorentz invariant, unitary field theory and offers concrete predictions for critical behavior in 3D contexts.

Abstract

We study a model of N-component complex fermions with a kinetic term that is second order in derivatives. This symplectic fermion model has an Sp(2N) symmetry, which for any N contains an SO(3) subgroup that can be identified with rotational spin of spin-1/2 particles. Since the spin-1/2 representation is not promoted to a representation of the Lorentz group, the model is not fully Lorentz invariant, although it has a relativistic dispersion relation. The hamiltonian is pseudo-hermitian, H^\dagger = C H C, which implies it has a unitary time evolution. Renormalization-group analysis shows the model has a low-energy fixed point that is a fermionic version of the Wilson-Fisher fixed points. The critical exponents are computed to two-loop order. Possible applications to condensed matter physics in 3 space-time dimensions are discussed.

Figures (5)

• Figure 1: Four-fermion vertex function relevant to the calculation of the $\beta$-function.
• Figure 2: Tree-level contribution to the vertex function.
• Figure 3: One-loop contributions to the vertex function. The three graphs give rise to the same loop topology, shown on the right. The group-theory factors of the individual diagrams are $-(N-2)$, 1, 1, where the minus sign of the first graph results from the closed fermion loop.
• Figure 4: Two-loop topologies for the vertex function. Each topology receives contributions from several diagrams. The combined group-theory factors are listed below each graph.
• Figure 5: One- and two-loop diagrams contributing to the fermion self-energy.