Figure 3.

Machine learning of the phase transition from a semi-metal to an antiferromagnetic insulator in the spinful Hubbard model (4) on a honeycomb lattice using the Green’s function approach (see main text). Visualized in the side panels are representative samples of the Green’s function (calculated from the auxiliary field) for a 2 · 9 × 9 system in the two respective phases. The complex entries of these matrices are color-converted by interpreting their absolute value as the hue of the color while their angle is chosen as the saturation (HSV coloring scheme⁴¹). The main panel shows the output of the discriminating function F obtained from a CNN trained for parameters in the two fermionic phases (indicated by the red dots). Data for different system sizes 2 · L × L are shown where the colors were selected to highlight an apparent even-odd effect in the linear system size. The vertical solid line indicates the position of the phase transition in the thermodynamic limit³⁷, while the dashed line marks the position at which the antiferromagnetic order breaks down⁴² for the finite system sizes of the current study.