Ising and Gross-Neveu model in next-to-leading order

We study scalar and chiral fermionic models in next-to-leading order with the help of the functional renormalization group. Their critical behavior is of special interest in condensed-matter systems, in particular graphene. To derive the $\beta$ functions, we make extensive use of computer algebra. The resulting flow equations were solved with pseudospectral methods to guarantee high accuracy. New estimates of critical quantities for both the Ising and the Gross-Neveu model are provided. For the Ising model, the estimates agree with earlier renormalization-group studies of the same level of approximation. By contrast, the approximation for the Gross-Neveu model retains many more operators than all earlier studies. For two Dirac fermions, the results agree with both lattice and large-$N_f$ calculations, but for a single flavor, different methods disagree quantitatively, and further studies are necessary.

Figure 1

Fixed-point solution to the Ising model in NLO with the Litim regulator. The derivative of the effective potential is plotted in the left panel, whereas the wave-function renormalization is shown on the right. The blue solid lines correspond to the solution where we fixed $Z_{\chi }\left({\rho }_0\right)=1$, whereas the orange dashed lines have $Z_{\chi }\left(0\right)=1$. One can see that the potentials start to deviate already for small values of the field. On the other hand, the two wave-function renormalizations differ mainly by a shift.

Figure 2

Dependence of the first critical exponent ${\theta }_1$ (blue dots) and the anomalous dimension $\eta$ (orange squares) of the Ising model on the exponential regulator parameter $a$. An interpolation helps to guide the eye.

Figure 3

Fixed-point solution to the Gross-Neveu model in three dimensions for two fermion flavors. The regulator parameter is $a=2$. Importantly, the function $X_1$ is positive, as it contributes to the denominator of propagator functions and can be seen as the second-order derivative analog of the usual Yukawa coupling $g_{\chi }$.

Figure 4

Regulator dependence of physical quantities of the Gross-Neveu model for $N_f=2$. In the left panel, the first critical exponent is shown. It does not display an extremum, and the principle of minimum sensitivity cannot be applied here. In the right panel, the bosonic (blue dots) and fermionic (orange squares) anomalous dimensions are plotted. Interpolations help to guide the eye.

Figure 5

Regulator dependence of physical quantities of the Gross-Neveu model for $N_f=1$. In the left panel, the first critical exponent is shown. Similar to the case of two fermion flavors, it shows no extremum, and the principle of minimum sensitivity cannot be applied here. In the right panel, the bosonic (blue dots) and fermionic (orange squares) anomalous dimensions are plotted. Interpolations help to guide the eye.