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arXiv:0902.1651

Efficient Fermionic One-Loop RG for the 2D Hubbard Model at Van Hove Filling

Christoph Husemann, Manfred Salmhofer B

Published Feb 10, 2009• 0 citations
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Abstract

We propose a novel parametrization of the four-point vertex function in the one-loop one-particle irreducible renormalization group (RG) scheme for fermions. It is based on a decomposition of the effective two-fermion interaction into fermion bilinears that interact via exchange bosons. Besides being more efficient than previous N-patch schemes, this parametrization also reduces the ambiguity of introducing boson fields. We apply this parametrization to the two-dimensional (t,t')-Hubbard model using a novel $Ω$-frequency regularization.

DOI
10.48550/arxiv.0902.1651

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We propose a novel parametrization of the four-point vertex function in the one-loop one-particle irreducible renormalization group (RG) scheme for fermions. It is based on a decomposition of the effective two-fermion interaction into fermion bilinears that interact via exchange bosons. Besides being more efficient than previous N-patch schemes, this parametrization also reduces the ambiguity of introducing boson fields. We apply this parametrization to the two-dimensional (t,t')-Hubbard model using a novel ΩΩΩ-frequency regularization.
We propose a parametrization of the four-point vertex function in the one-loop one-particle irreducible renormalization group (RG) scheme for fermions. It is based on a decomposition of the effective two-fermion interaction into fermion bilinears that interact via exchange bosons. The numerical computation of the RG flow of the boson propagators reproduces the leading weak-coupling instabilities of the two-dimensional Hubbard model at van Hove filling as they were obtained in a temperature RG scheme [C. Honerkamp and M. Salmhofer, Phys. Rev. B 64, 184516 (2001)]. Instead of regularizing with temperature, we here use a soft frequency $\ensuremath{\Omega}$ regularization that likewise does not artificially suppress ferromagnetism. Besides being more efficient than previous NNN-patch schemes, this parametrization also reduces the ambiguities in introducing boson fields.
We consider the application of the two-loop functional renormalization group (fRG) approach to study the low-dimensional Hubbard model. This approach accounts for both the universal and nonuniversal contributions to the RG flow. While the universal contributions were studied previously within the field-theoretical RG for the one-dimensional Hubbard model with linearized electronic dispersion and the two-dimensional Hubbard model with flat Fermi surface, the nonuniversal contributions to the flow of the vertices and susceptibilities appear to be important at large momenta scales. The two-loop fRG approach is also applied to the two-dimensional Hubbard model with a curved Fermi surface and the van Hove singularities near the Fermi level. The vertices and susceptibilities in the end of the flow of the two-loop approach are suppressed in comparison with both the one-loop approach with vertex projection and the modified one-loop approach with corrected vertex projection errors. The results of the two-loop approach are closer to the results of one-loop approach with the projected vertices, the difference of the results of one- and two-loop fRG approaches decreases when going away from the van Hove band filling. The quasiparticle weight remains finite in two dimensions at not too low temperatures above the paramagnetic ground state.
We derive an efficient method for treating renormalization contributions at two-loop level within the functional renormalization group in the one-particle irreducible formalism for fermions. It is based on a decomposition of the two-particle vertex in charge, magnetic and pairing channels. The method treats single-particle and collective excitations in all channels on equal footing, allows for the description of symmetry breaking and captures collective mode fluctuation physics in the infrared. As a first application, we study the superfluid ground state of the two-dimensional attractive Hubbard model. We obtain superfluid gaps that are reduced by fluctuations in comparison to the one-loop approximation and demonstrate that the method captures the renormalization of the amplitude mode by long-range phase fluctuations.
We complement previous functional renormalization group (fRG) studies of the two-dimensional Hubbard model by mean-field calculations. The focus falls on Van Hove filling and the hopping amplitude ${t}^{\ensuremath{'}}/t=0.341$. The fRG data suggest a quantum critical point (QCP) in this region and in its vicinity a singular fermionic self-energy $\mathrm{Im}\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})/\ensuremath{\omega}\ensuremath{\sim}\ensuremath{-}{\left|\ensuremath{\omega}\right|}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\approx}0.26$ [K-U. Giering and M. Salmhofer, Self-energy flows in the two-dimensional repulsive Hubbard model, Phys. Rev. B 86, 245122 (2012)]. Here we start a more detailed investigation of this QCP using a bosonic formulation for the effective action, where the bosons couple to the order parameter fields. To this end, we use the channel decomposition of the fermionic effective action developed in C. Husemann and M. Salmhofer, Efficient Parametrization of the Vertex Function, Omega-Scheme, and the (t, t')-Hubbard Model at Van Hove Filling, Phys. Rev. B 79, 195125 (2009), which allows us to perform Hubbard-Stratonovich transformations for all relevant order parameter fields at any given energy scale $\mathrm{\ensuremath{\Omega}}$. We stop the flow at a scale $\mathrm{\ensuremath{\Omega}}$ where the correlations of the order parameter field are already pronounced, but the flow is still regular, and derive the effective boson theory. It contains ddd-wave superconducting, magnetic, and density-density interactions. We analyze the resulting phase diagram in the mean-field approximation. We show that the singular fermionic self-energy suppresses gap formation both in the superconducting and magnetic channel already at the mean-field level, thus rounding a first-order transition (without self-energy) to a quantum phase transition (with self-energy). We give a simple effective model that shows the generality of this effect. In the two-dimensional Hubbard model, the effective density-density interaction is peaked at a nonzero frequency, so that solving the mean-field equations already involves a functional equation instead of simply a matrix equation (on a technical level, similar to incommensurate phases). Within a certain approximation, we show that such an interaction leads to a short quasiparticle lifetime.
Abstract: We illustrate the algorithmic advantages of the recently introduced single-boson exchange (SBE) formulation for the one-loop functional renormalization group (fRG), by applying it to the two-dimensional Hubbard model on a square lattice. We present a detailed analysis of the fermion-boson Yukawa couplings and of the corresponding physical susceptibilities by studying their evolution with temperature and interaction strength, both at half filling and finite doping. The comparison with the conventional fermionic fRG decomposition shows that the rest functions of the SBE algorithm, which describe correlation effects beyond the SBE processes, play a negligible role in the weak-coupling regime above the pseudo-critical temperature, in contrast to the rest functions of the conventional fRG. Remarkably, they remain finite also at the pseudo-critical transition, whereas the corresponding rest functions of the conventional fRG implementation diverge. As a result, the SBE formulation of the fRG flow allows for a substantial reduction of the numerical effort in the treatment of the two-particle vertex function, paving a promising route for future multiboson and multiloop extensions.
We present a functional renormalization group calculation for the two-dimensional $(t,{t}^{\ensuremath{'}})$-Hubbard model at Van Hove filling. Using a channel decomposition we describe the momentum and frequency dependence of the vertex function in the normal phase. Compared to previous studies that neglect frequency dependencies we find higher pseudocritical scales and a smaller region of ddd-wave superconductivity. A large contribution to the effective interaction is given by a forward-scattering process with finite frequency exchange. We test different frequency parametrizations and in a second calculation include the frequency dependence of the imaginary self-energy. We also generalize the channel decomposition to frequency-dependent fermion-boson vertex functions.