Quantum-critical electrodynamics of Luttinger fermions

We study the quantum electrodynamics of Luttinger fermions with quadratic band-crossing dispersion in three dimensions. The model can be viewed as the low-energy effective theory of a putative U(1) quantum spin liquid with fermionic Luttinger spinons, or as an extension of the Luttinger-Abrikosov-Beneslavskii (LAB) model that accounts for transverse gauge fluctuations with finite photon velocity. Aided by a renormalization group analysis below four dimensions, we elucidate the presence and stability of quantum critical phenomena in this model. We find that the non-Fermi liquid LAB phase is stable against gauge fluctuations and can thus also be viewed as a U(1) spin liquid with gapless Luttinger spinons. We discover a multicritical point with Lifshitz scaling that corresponds to a time-reversal symmetry-breaking quantum phase transition from the LAB state to a chiral spin liquid with spinon Landau levels and birefringent emergent photons. This multicritical point is characterized by a finite fermion-photon coupling in the infrared and can be viewed as a fermionic analog of the Rokhsar-Kivelson point in three-dimensional quantum dimer models.

Figure 1

Quantum criticality of rotationally and particle-hole symmetric Luttinger fermions (blue dispersion) coupled to a dynamical U(1) gauge field (red dispersion). The photon velocity squared $c^2$ is a relevant coupling; it drives a transition between a symmetric LAB phase with instantaneous photons and a time-reversal symmetry-breaking (TRSB) phase with a spontaneously generated background magnetic field, Landau levels for Luttinger fermions, and birefringent photons. For more than one fermion flavor, the phase boundary contains a critical line governed by a Lifshitz-QED (LQED) theory with finite fermion-photon coupling and anisotropic $z>2$ scaling for both fermions and photons.

Figure 2

One-loop 1PI diagrams contributing to (a) the fermion self-energy and (b) the photon self-energy.

Figure 3

RG flows of the gauged Luttinger fermion theory with $N_f=1$ near the LAB fixed point (red dot) in (a) the $\alpha -K^{-1}$, (b) the $\rho -K^{-1}$, and (c) the $\alpha -\rho$ planes. For (a) and (b), we set $\epsilon =1$. Flows are qualitatively similar for other values of $N_f>1$.

Figure 4

Landau levels in the fermion spectrum of the time-reversal symmetry-breaking phase with spontaneously generated magnetic field $\langle \mathit{B}\rangle =B_0\widehat{\mathit{z}}$, where ${\omega }_c=e{B}_0/m$ is the cyclotron frequency, $k_z$ is the fermion momentum in the field direction, and ${\ell }_B=\sqrt{1/e{B}_0}$ is the magnetic length.

Figure 5

(a) Isofrequency surfaces of the ordinary branch (purple) and extraordinary branch (green) of the emergent photon dispersion in the broken-symmetry phase; (b) extraordinary wave dispersion in directions parallel (red) and perpendicular (blue) to the spontaneously generated magnetic field.

Figure 6

RG flows of the gauged Luttinger fermion theory in the vicinity of the Lifshitz-QED (LQED) multicritical point ${\alpha }_{*}\ne 0, {\rho }_{*}\ne 0$ (red dot) for the threshold fermion flavor number $N_f=2$, in the $\alpha -\rho$ plane. The fixed point is stable along the $\alpha$ direction but unstable along the remaining directions. RG flows are qualitatively similar for other values of $N_f⩾2$.

Figure 7

Departure of the dynamic critical exponent $z$ from its Gaussian-limit value of $z=2$ for the LAB (all $N_f$) and LQED ($N_f⩾2$) fixed points, as a function of number of fermion flavors $N_f$.

Figure 8

Feynman rules for the U(1) gauge theory of Luttinger QBC fermions. Solid line: fermion propagator $G^{\psi }$; wavy line: photon propagator $G^A$; three-point vertex: QED vertex $C^{\mu }\propto e$; four-point vertex: QED seagull vertex $V^{\mu \nu }\propto e^2$.