Critical behavior of Dirac fermions from perturbative renormalization

Gapless Dirac fermions appear as quasiparticle excitations in various condensed-matter systems. They feature quantum critical points with critical behavior in the 2+1-dimensional Gross-Neveu universality class. The precise determination of their critical exponents defines a prime benchmark for complementary theoretical approaches, such as lattice simulations, the renormalization group, and the conformal bootstrap. Despite promising recent developments in each of these methods, however, no satisfactory consensus on the fermionic critical exponents has been achieved, so far. Here, we perform a comprehensive analysis of the Ising Gross-Neveu universality classes based on the recently achieved four-loop perturbative calculations. We combine the perturbative series in $4-\epsilon$ spacetime dimensions with the one for the purely fermionic Gross-Neveu model in $2+\epsilon$ dimensions by employing polynomial interpolation as well as two-sided Padé approximants. Further, we provide predictions for the critical exponents exploring various resummation techniques following the strategies developed for the three-dimensional scalar $O\left(n\right)$ universality classes. We give an exhaustive appraisal of the current situation of Gross-Neveu universality by comparison to other methods. For a large enough number of spinor components $N\ge 8$ as well as for the case of emergent supersymmetry $N=1$, we find our renormalization group estimates to be in excellent agreement with the conformal bootstrap, building a strong case for the validity of these values. For intermediate $N$ as well as in comparison with recent Monte Carlo results, deviations are found and critically discussed.

Figure 1

Chiral Ising universality in $D=3$. Overview plots for the correlation-length exponent ${\nu }^{-1}$ (left panel), the boson anomalous dimension ${\eta }_{\varphi }$ (middle panel), and the fermion anomalous dimension ${\eta }_{\psi }$ (right panel) for different numbers of spinor components $N\in [1,20]$. For comparison, values from Monte Carlo (MC) calculations, the functional renormalization group (FRG), and the conformal bootstrap (cBS) are also shown. For the large-$N$ results, we applied Padé-Borel resummation.

Figure 2

Order-by-order study of two-sided Padé approximants of the three critical exponents ${\nu }^{-1}$ (left panel), ${\eta }_{\varphi }$ (middle panel), and ${\eta }_{\psi }$ (right panel) between $D\in (2,4)$ at $N=8$. We restrict ourselves to approximants, which include the same number of loop orders in the respective critical dimensions $D=2$ and $D=4$. Further, we choose approximants with $m\approx n$.

Figure 3

Order-by-order study of polynomial interpolation of the three critical exponents ${\nu }^{-1}$ (left panel), ${\eta }_{\varphi }$ (middle panel), and ${\eta }_{\psi }$ (right panel) between $D\in (2,4)$ at $N=8$. We restrict ourselves to interpolations, which include the same number of loop orders in the respective critical dimensions $D=2$ and $D=4$.

Figure 4

Interpolation of the three critical exponents ${\nu }^{-1}$ (inverse correlation-length exponent, left panel), ${\eta }_{\varphi }$ (boson anomalous dimension, middle panel), and ${\eta }_{\psi }$ (fermion anomalous dimension, right panel) between $D\in (2,4)$ at $N=8$. The shown interpolations polynomial (red line) and two-sided Padé (dark-red dashed line) are fixed by the two epsilon expansions at $D=2$ and $D=4$ in the first four derivatives. As a result, the asymptotic behavior is suppressed even far from these expansion points at the physical dimension $D=3$ and plausible values can be read off. It should also be noted that both complementary approaches for the interpolation are very close to each other and comparable to conformal bootstrap (cBS) [46], lattice Monte Carlo [9], and functional renormalization group (FRG) [51] calculations. Also note that no point for the fermionic anomalous dimension has been plotted for MC, since the point with ${\eta }_{\psi }\approx 0.38\left(1\right)$ is far above the value range of the comparable methods.

Figure 5

Chiral Ising universality for $D=3,N=8$. Sensitivity of the Borel resummed inverse correlation-length exponent ${\nu }^{-1}$ to a variation of one of the three large-order parameters $b,\lambda$, and $q$. In the first row we show the change to variation of $b$ and $\lambda$ at the optimal $q=0.16$, i.e., at the global minimum of the error estimate $\overline{E}\equiv E_4(31.5,0.74,0.16)\approx 0.009$. In this case we find plateaus with increasing spread in the loop order. For comparison, we also show the same plots at $q=0.4$ away from the optimum where these plateaus are lost. In the last plot we present the variation with respect to $q$ which has no plateaus at all and is only fixed by the intersection point. For a list of the minima of various $N$ configurations in the Borel resummation, see Table 6 in Appendix pp3.

Figure 6

Chiral Ising universality in $D=3$. Error estimates for the inverse correlation-length exponent ${\nu }^{-1}$ at $N=8$. The error axis is normalized to the smallest error in the parameters $\overline{E}\equiv E_4(31.5,0.74,0.16)\approx 0.009$. The global minimum of this parameter space (left panel) is located in an area around the value ${\nu }^{-1}\approx 0.994$, which we have zoomed in on in the right panel. The weighted mean of this subspace in parameter space has ${\nu }^{-1}\approx 0.993$ (dashed line).

Figure 7

Chiral Ising universality in $D=3$. Overview plot of the three examined critical exponents as the correlation-length exponent ${\nu }^{-1}$ (left panel), the boson anomalous dimension ${\eta }_{\varphi }$ (middle panel), and the fermion anomalous dimension ${\eta }_{\psi }$ (right panel) for different spinor component numbers $N\in [1,20]$. For comparison, the values from Monte Carlo (MC) calculations and the functional renormalization group (FRG), as well as the conformal bootstrap (cBS), were also plotted. On top of that we applied a Padé-Borel resummation on the large-$N$ expansions as in Ref. [78].