Quantitative functional renormalization group description of the two-dimensional Hubbard model

Abstract

Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model. In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for twodimensional lattice problems. The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence. The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy. We quantitatively compare our fRG results for the self-energy and momentumdependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths. The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.

Quasi-particle functional Renormalisation Group calculations in the two-dimensional half-filled Hubbard model at finite temperatures

“…However, in those cases where these features do appear, they only show up very close to a mean-field-like critical point, such that the discrepancies are restricted to a very narrow region in the phase diagram. Parallel efforts to improve quantitative aspects of fRG calculations have also been conducted very recently in [17,48]. The scheme presented here shows, given the approximations made, very reasonable quantitative agreement with those works regarding the pseudo-critical scales and shares the tendencies toward pseudogap opening, while on the other hand, it gives a complementary perspective from the real-frequency axis.…”

“…It delivers Fermi-surface-resolved quasi-particle selfenergies directly on the real-frequency axis. Quantitatively, where comparable, the data for the 2D Hubbard model below agrees quite well with other advanced state-of-the-art fRG approaches [14] that in turn tie in well with results of other numerical approaches to the Hubbard model [17]. Hence, we believe that the scheme presented here is a useful addition and can, due to its conceptual simplicity, provide useful clarifications in this field.…”

“…for t = 0 and µ = 0, as a function of finite temperatures T and the bare coupling U. For this case, various data from prior works are available to which we can compare, be it from other fRG schemes (for very recent samples, see [14,17]) or other numerical methods (e.g. [30,31]).…”

“…in [13,14], yet a complete numerical treatment which fully reflects the exact flow equation remains unfeasible. A lot of progress has been made with the explicit inclusion of frequency dependencies in one-and two-particle quantities [13][14][15][16][17] and in terms of a reduction in the necessary amount of spatial degrees of freedom [15,[18][19][20]. Furthermore, various other suggestions have been made, for example to improve on the inner consistency of the method [21] or to make explicit contact to other advanced methods such as the parquet formalism [14].…”

Evolution between two orbital-selective Mott phases driven by interorbital hopping

“…Several theoretical approaches have been used to build the impurity solver of the DMFT procedure, such as quantum Monte Carlo simulations (QMC) [13][14][15][16] , renormalization-group theory [17][18][19] , and slave-variable representations [20][21][22] , etc. For the multi-orbital extensions combined with the DMFT algorithm, each solver has its limitations.…”