Critical behavior of the ${\mathrm{QED}}_3$-Gross-Neveu model: Duality and deconfined criticality

We study the critical properties of the ${\mathrm{QED}}_3$-Gross-Neveu model with $2N$ flavors of two-component Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For $N=1$, this theory has recently been suggested to be dual to the SU(2) noncompact ${\mathrm{CP}}^1$ model that describes the deconfined phase transition between the Néel antiferromagnet and the valence bond solid on the square lattice. For $N=2$, the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin-$1/2$ systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the ${\mathrm{QED}}_3$-Gross-Neveu model for all values of $N$. This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain ${\mathrm{QED}}_3$ (without scalar-field coupling) unstable at low values of $N$. The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter $\epsilon =4-D$. We compute the scalar-field anomalous dimension ${\eta }_{\varphi }$, the correlation-length exponent $\nu$, as well as the scaling dimension of the flavor-symmetry-breaking bilinear $\overline{\psi }{\sigma }^z\psi$ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.

Figure 1

One-loop diagrams determining the flows of the four-fermion couplings $u$ and $v$. Solid (wiggly) lines correspond to fermion (gauge) fields.

Figure 2

Diagrams that determine the gauge-field anomalous dimension ${\eta }_a$ (left) and the fermion selfenergy (middle). In the flow equation for $e^2$, the contributions from the fermion self-energy and explicit vertex correction (right) cancel due to the Ward identity associated with the U(1) gauge symmetry.

Figure 3

RG flow for $N=1$. The panels in (b)–(d) display the RG flow within different subspaces of the full theory space spanned by $u$, $v$, and $e^2$, as schematically depicted in (a). (b) $u\text{−}v$ plane for $e^2=0$. (c) $u\text{−}v$ plane for $e^2=e_{*}^2=3/4$. (d) $u\text{−}{e}^2$ plane for $v=0$. Besides the Gaussian fixed point (G), the only quantum critical point with just one relevant direction is the ${\mathrm{QED}}_3$-GN fixed point. It describes the TRS-breaking transition. There are furthermore two critical points in the uncharged sector $e^2=0$, which, however, receive a second relevant direction along the $e^2$ axis: the Gross-Neveu fixed point (GN) and the Thirring fixed point (T).

Figure 4

One-loop diagrams determining the flows of the Yukawa coupling $g^2$ and the ${\varphi }^4$ coupling $\lambda$, as well as the scalar-field anomalous dimension ${\eta }_{\varphi }$. Dashed lines correspond to the scalar field.

Figure 5

Diagrams that cancel in the flow equation for the charge $e^2$ due to the Ward identity. In the flow equation for the Yukawa coupling $g^2$, the contribution from the fermion self-energy (right) cancels with the gauge-dependent part of the vertex correction (first diagram in Fig. 4).

Figure 6

RG flow for $N=1$ in $(\lambda ,g^2)$ plane to leading order in $\epsilon =4-D$. For visualization purposes, here we have put the charge to its infrared fixed-point value $e^2=e_{*}^2=3\epsilon /4$, and we tune the system to criticality with $r\equiv 0$. The infrared stable fixed point at $g_{*}^2>0$ and ${\lambda }_{*}>0$ is the ${\mathrm{QED}}_3$-GN quantum critical point and governs the transition into the TRS-broken state with $\langle \varphi \rangle \ne 0$. G and WF at $g=0$ describe the Gaussian and Wilson-Fisher fixed points.

Figure 7

Critical exponents ${\eta }_{\varphi }$ (a) and $1/\nu$ (b), and scaling dimension $\left[\overline{\psi }{\sigma }^z\psi \right]$ (c) as function of space-time dimension $D$ in the critical ${\mathrm{QED}}_3$-GN theory for $N=1$. Solid curves: polynomial interpolation between lowest-order $2+\epsilon$ expansion result and $4-\epsilon$ expansion result. Dashed curves near lower and upper critical dimensions, respectively: plain extrapolation of $\epsilon$ expansion for comparison. For $1/\nu$ (b), the two dashed curves near $D=4$ correspond to the inverse of Eq. (32) (upper curve) and the expansion of $1/\nu$ itself (lower curve), cf. Ref. [50].