Gross-Neveu-Heisenberg criticality from $2+\epsilon$ expansion

The Gross-Neveu-Heisenberg universality class describes a continuous quantum phase transition between a Dirac semimetal and an antiferromagnetic insulator. Such quantum critical points have originally been discussed in the context of Hubbard models on $\pi$-flux and honeycomb lattices, but more recently also in Bernal-stacked bilayer models, of potential relevance for bilayer graphene. Here, we demonstrate how the critical behavior of this fermionic universality class can be computed within an $\epsilon$ expansion around the lower critical space-time dimension of two. This approach is complementary to the previously studied expansion around the upper critical dimension of four. The crucial technical difference near the lower critical dimension is the presence of different four-fermion interaction channels at the critical point, which we take into account in a Fierz-complete way. By interpolating between the lower and upper critical dimensions, we obtain improved estimates for the critical exponents in $2+1$ space-time dimensions. For the situation relevant to single-layer graphene, we find an unusually small leading-correction-to-scaling exponent, arising from the competition between different interaction channels. This suggests that corrections to scaling may need to be taken into account when comparing analytical estimates with numerical data from finite-size extrapolations.

Figure 1

Evolution of Gross-Neveu-Heisenberg fixed-point couplings ${\mathit{g}}_{\text{GNH}}^{★}=[h_1\left(N_{\mathrm{f}}\right),h_2\left(N_{\mathrm{f}}\right),0,0,0,h_6\left(N_{\mathrm{f}}\right)]\epsilon$ as function of flavor number $N_{\mathrm{f}}$.

Figure 2

RG flow in the plane spanned by $g_1$ and $g_6$ for fixed $g_2=h_2\left(N_{\mathrm{f}}\right)$ through the Gross-Neveu-Heisenberg fixed point for (a) $N_{\mathrm{f}}=2$, (b) $N_{\mathrm{f}}=4$, and (c) $N_{\mathrm{f}}=8$. The red dot denotes the position of the Gross-Neveu-Heisenberg fixed point $\mathcal{H}$. Gray dots labeled by ${\mathcal{O}}^{\prime }, {\mathcal{B}}^{\prime }$, and ${\mathcal{I}}^{\prime }$ indicate points in parameter space in which the flow is perpendicular to the plane $g_2=h_2\left(N_{\mathrm{f}}\right)$, which become projections of the Gaussian fixed point $\mathcal{O}$, the critical fixed point $\mathcal{B}$, and the bicritical fixed point $\mathcal{I}$ in the large-$N_{\mathrm{f}}$ limit. The gray line indicates the projection of the RG invariant subspace spanned by the fixed points $\mathcal{A}, \mathcal{B}$, and $\mathcal{C}$.

Figure 3

(a) Corrections-to-scaling exponent $\omega$ for the Gross-Neveu-Ising fixed point as a function of $N_{\mathrm{f}}$. (b) Same as (a), but for the Gross-Neveu-Heisenberg fixed point, featuring a distinct minimum near $N_{\mathrm{f}}=2$, corresponding to a slow flow towards the fixed point. The inset shows the angle $\phi$ between ${\mathit{g}}_{\text{GNH}}^{★}$ and the surface normal $\mathit{n}=(1,0,1)/\sqrt{2}$ of the invariant subspace spanned by the fixed points $\mathcal{A}, \mathcal{B}$, and $\mathcal{C}$, as function of $N_{\mathrm{f}}$.

Figure 4

Correlation-length exponent $1/\nu$ (left column), order-parameter anomalous dimension ${\eta }_{\varphi }$ (center column), and fermion anomalous dimension ${\eta }_{\psi }$ (right column) of the Gross-Neveu-Heisenberg universality class as function of space-time dimension $2<D<4$ for $N_{\mathrm{f}}=2$ (first row), $N_{\mathrm{f}}=4$ (second row), and $N_{\mathrm{f}}=8$ (last row). The different curves in each panel correspond to different Padé approximants $[m/n]$, which interpolate between the series expansions around the lower and upper critical dimensions $D=2$ and $D=4$, respectively. Data points at $D=3$ refer to literature results from $4-\varepsilon$ expansion ($▿$) [13], $1/N_{\mathrm{f}}$ expansion ($▵$) [39, 40, 42], functional RG ($\circ$) [16, 38, 40], as well as determinantal quantum Monte Carlo ($\diamond$) [27, 28, 29, 30, 31] and hybrid Monte Carlo ($\square$) [33, 34, 35, 36] simulations.