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arXiv:0812.3824 · Physical Review B

Efficient parametrization of the vertex function, Ω scheme, and the t,t′ Hubbard model at van Hove filling

Christoph Husemann, Manfred Salmhofer B

Published May 26, 2009• 186 citations
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Abstract

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 $N$-patch schemes, this parametrization also reduces the ambiguities in introducing boson fields.

DOI
10.1103/physrevb.79.195125

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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 \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ 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 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 derive a differential equation for the one-particle-irreducible vertex functions of interacting fermions as a function of the temperature. Formally, these equations correspond to a Wilsonian renormalization group (RG) scheme that uses the temperature as an explicit scale parameter. Our method allows us to analyze the competition between superconducting and various magnetic Fermi surface instabilities in the one-loop approximation. In particular, this includes ferromagnetic fluctuations, which are difficult to treat on an equal footing in conventional Wilsonian momentum space techniques. Applying the scheme to the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model we investigate the RG flow of the interactions at the van Hove filling with varying next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ Starting at ${t}^{\ensuremath{'}}=0$ we describe the evolution of the flow to strong coupling from an antiferromagnetic nesting regime over a d-wave regime at moderate ${t}^{\ensuremath{'}}$ to a ferromagnetic region at larger absolute values of ${t}^{\ensuremath{'}}.$ Upon increasing the particle density in the latter regime the ferromagnetic tendencies are cut off and the leading instability occurs in the triplet superconducting pairing channel.
We use a novel temperature-flow renormalization group technique to analyze magnetic and superconducting instabilities in the two-dimensional ${t\ensuremath{-}t}^{\ensuremath{'}}$ Hubbard model for particle densities close to the Van Hove filling as a function of the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. In the one-loop flow at the Van Hove filling, the characteristic temperature for the flow to strong coupling is suppressed drastically around ${t}_{c}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.33t$, suggesting a quantum critical point between ddd-wave pairing at moderate ${t}^{\ensuremath{'}}>{t}_{c}^{\ensuremath{'}}$ and ferromagnetism for ${t}^{\ensuremath{'}}<{t}_{c}^{\ensuremath{'}}$. Upon increasing the particle density in the latter regime, the leading instability occurs in the triplet pairing channel.
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.
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 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.