Multiloop functional renormalization group for the two-dimensional Hubbard model: Loop convergence of the response functions

Abstract

We present a functional renormalization group (fRG) study of the two dimensional Hubbard model, performed with an algorithmic implementation which lifts some of the common approximations made in fRG calculations. In particular, in our fRG flow; (i) we take explicitly into account the momentum and the frequency dependence of the vertex functions; (ii) we include the feedback effect of the self-energy; (iii) we implement the recently introduced multiloop extension which allows us to sum up all the diagrams of the parquet approximation with their exact weight. Due to its iterative structure based on successive one-loop computations, the loop convergence of the fRG results can be obtained with an affordable numerical effort. In particular, focusing on the analysis of the physical response functions, we show that the results become independent from the chosen cutoff scheme and from the way the fRG susceptibilities are computed, i.e., either through flowing couplings to external fields, or through a "post-processing" contraction of the interaction vertex at the end of the flow. The presented substantial refinement of fRG-based computation schemes paves a promising route towards future quantitative fRG analyses of more challenging systems and/or parameter regimes. G Performance of the code 45References 46Note that the index x = {σ, k} combines the spin index σ and the fermionic quadrivector k = (iν l , k) (here we disregard additional quantum dependencies, e.g., orbital), while y = {n, q} refers to the momentum structure of the coupling to the bilinears, n, and to the bosonic quadrivector q = {iω l , q} . In Eq. (13) the first term on the r.h.s. represents the expansion of the effective action in absence of external field (see Section 3), while the functional derivatives in the following terms represent the Λ-dependent susceptibility and the fermion-boson vertex in the different channels. Taking the derivative with respect to the scale parameter Λ (see Appendix B), yields the following flow equations for the susceptibility and fermion-boson vertex (assuming SU(2) symmetry and momentum-frequency as well as spin conservation)

Antiferromagnetic and d -wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group

“…29 Alternatively, one may also recover the Mermin-Wagner theorem in a purely fermionic flow via the recently developed multi-loop truncation of the flow equation hierarchy. 38,39 Extending such refinements to the DMF 2 RG is one of the most promising future directions. The temperature range in which the present implementation of the DMF 2 RG breaks down due to the divergent magnetic fluctuation term would then become accessible.…”

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

“…Using the SU(2) spin symmetry [21], we can restrict ourselves to one spin component V = F ↑↑↓↓ . From the vertex, the susceptibilities can be computed by contracting the two-particle vertex at the end of the flow (postprocessed) or alternatively via the flow of the response vertices [57] (see Appendix A for a more detailed discussion).…”

“…For a detailed description and their implementation in the multiloop extension we refer to Refs. [57,70], where we also provide the expression of the employed smooth frequency cutoff. Here we further use a refined momentum grid to resolve the peak at q = (π, π ) in the AF susceptibility at half filling.…”

“…Notably, the truncated unity fRG methodology is useful in another diagrammatic approach, the parquet scheme, whose performance can be boosted significantly in the truncated unity PA [55,56]. A major step forward was achieved by including the frequency dependence and self-energy corrections into the truncated unity fRG framework [57] and related schemes [58,59]. Supplemented by multiloop corrections [57], this paved the way for (a) internal convergence of the fRG as a function of the number of loops and (b) internal consistency in the fRG, as different ways to compute response functions (by the flow of external-field couplings or by postprocessing of the final interaction vertex) and flows with different cutoff schemes are found to agree.…”

“…Going beyond the preliminary results of Ref. [57], restricted to the calculation of static response functions, we address here the computation of the self-energy in detail. In particular, investigating the form of the multiloop flow equation for the self-energy, we find that a form-factor expansion of the two-particle vertex prevents the full reconstruction of the Schwinger-Dyson equation (SDE) [60,61] at loop convergence.…”