Effects of two-loop contributions in the pseudofermion functional renormalization group method for quantum spin systems

Abstract: We implement an extension of the pseudofermion functional renormalization group (PFFRG) method for quantum spin systems that takes into account two-loop diagrammatic contributions. An efficient numerical treatment of the additional terms is achieved within a nested graph construction which recombines different one-loop interaction channels. In order to be fully self consistent with respect to self-energy corrections we also include certain three-loop terms of Katanin type. We first apply this formalism to the antiferromagnetic $J_1$-$J_2$ Heisenberg model on the square lattice and benchmark our results against the previous one-loop plus Katanin approach. Even though the RG equations undergo significant modifications when including the two-loop terms, the magnetic phase diagram -- comprising N\'eel ordered and collinear ordered phases separated by a magnetically disordered regime -- remains remarkably unchanged. Only the boundary position between the disordered and the collinear phases is found to be moderately affected by two-loop terms. On the other hand, critical RG scales, which we associate with critical temperatures $T_\text{c}$, are reduced by a factor of $\sim2$ indicating that the two-loop diagrams play a significant role in enforcing the Mermin-Wagner theorem. Improved estimates for critical temperatures are also obtained for the Heisenberg ferromagnet on the 3D simple cubic lattice where errors in $T_\text{c}$ are reduced by $\sim34\%$.