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

Vilardi, Demetrio; Taranto, Ciro; Metzner, Walter

doi:10.1103/physrevb.99.104501

Cited by 69 publications

(62 citation statements)

References 47 publications

(78 reference statements)

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“…17(b)] and at incommensurate wave vectors for larger values of the doping [Figs. 17(c) and 17(d)], consistent with previous fRG findings [59,90]. In particular, we do not find a pairing instability for any doping at a temperature as high as 1/T = 5.…”

Section: Results Away From Half Fillingsupporting

confidence: 92%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Hille

Kugler

Eckhardt

et al. 2020

Phys. Rev. Research

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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.

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Section: Results Away From Half Fillingsupporting

confidence: 92%

Section: Introductionmentioning

confidence: 99%

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

Hille

Kugler

Eckhardt

et al. 2020

Phys. Rev. Research

View full text Add to dashboard Cite

show abstract

“…This adds an new important degree of quantitative control to the fRG, at least in the weak to intermediate coupling regime where the PA can be considered accurate. At stronger coupling, where low-frequency vertex corrections beyond the PA might appear [31,49,[88][89][90], the mfRG could provide a much better [14] setup for the proposed combination with the DMFT [22,26,91]. The loop convergence of our fRG results is also reflected in the progressive reduction of the dependence of our fRG results on the chosen cutoff scheme, which appears completely suppressed at the 8 level.…”

Section: Discussionmentioning

confidence: 72%

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

et al. 2019

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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)

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“…However, this may change for stronger interactions, where the phase stiffness may decrease due to a larger magnetic (or Mott) gap, while the pairing gap is expected to increase. A calculation with an fRG flow starting from the dynamical mean-field solution of the Hubbard model 28,29 could clarify the behavior at strong coupling.…”

Section: Discussionmentioning

confidence: 99%

Phase stiffness in an antiferromagnetic superconductor

Metzner

Yamase

2019

Phys. Rev. B

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We analyze the suppression of the phase stiffness in a superconductor by antiferromagnetic order. The analysis is based on a general expression for the phase stiffness in a mean-field state with coexisting spin-singlet superconductivity and spiral magnetism. Néel order is included as a special case. Close to half-filling, where the pairing gap is much smaller than the magnetic gap, a simple formula for the phase stiffness in terms of magnetic quasi-particle bands is derived. The phase stiffness is determined by charge carriers in small electron or hole pockets in this regime. The general analysis is complemented by a numerical calculation for the two-dimensional Hubbard model with nearest and next-to-nearest neighbor hopping amplitudes at a moderate interaction strength. The resulting phase stiffness exhibits a striking electron-hole asymmetry. In the ground state, it is larger than the pairing gap on the hole-doped side, and smaller for electron doping. Hence, in the holedoped regime near half-filling the ground state pairing gap sets the scale for the Kosterlitz-Thouless temperature T KT c , while in the slightly electron-doped regime T KT c is determined essentially by the ground state phase stiffness.

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

Cited by 69 publications

References 47 publications

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

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

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

Phase stiffness in an antiferromagnetic superconductor

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