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arXiv:0801.1773 · Physical Review B

Generation of d -wave coupling in the two-dimensional Hubbard model from functional renormalization

H. C. Krahl, Jens Andreas Müller, CHRISTOF WETTERICH

Published Mar 30, 2009• 19 citations
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Abstract

Within the two-dimensional repulsive $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model, an attractive coupling in the $d$-wave-pairing channel is induced by antiferromagnetic fluctuations. We investigate this coupling using functional renormalization-group equations. The momentum-dependent $d$-wave coupling can be bosonized by the use of scale-dependent field transformations. We propose an effective coarse-grained model for the Hubbard model which is based on the exchange of antiferromagnetic and $d$-wave collective bosons.

DOI
10.1103/physrevb.79.094526

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Within the two-dimensional repulsive t\ensuremath−t\ensuremath′t\ensuremath{-}{t}^{\ensuremath{'}}t\ensuremath−t\ensuremath′ Hubbard model, an attractive coupling in the ddd-wave-pairing channel is induced by antiferromagnetic fluctuations. We investigate this coupling using functional renormalization-group equations. The momentum-dependent ddd-wave coupling can be bosonized by the use of scale-dependent field transformations. We propose an effective coarse-grained model for the Hubbard model which is based on the exchange of antiferromagnetic and ddd-wave collective bosons.
We analyze the competition between antiferromagnetism and superconductivity in the two-dimensional Hubbard model by combining a functional renormalization group flow with a mean-field theory for spontaneous symmetry breaking. Effective interactions are computed by integrating out states above a scale ${\ensuremath{\Lambda}}_{\mathrm{MF}}$ in one-loop approximation, which captures in particular the generation of an attraction in the ddd-wave Cooper channel from fluctuations in the particle-hole channel. These effective interactions are then used as an input for a mean-field treatment of the remaining low-energy states, with antiferromagnetism, singlet superconductivity, and triplet $\ensuremath{\pi}$ pairing as the possible order parameters. Antiferromagnetism and superconductivity suppress each other, leaving only a small region in parameter space where both orders can coexist with a sizable order parameter for each. Triplet $\ensuremath{\pi}$ pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small.
Despite its apparent simplicity, the two-dimensional Hubbard model for locally interacting fermions on a square lattice is widely considered as a promising approach for the understanding of Cooper pair formation in the quasi two-dimensional high-Tc cuprate materials. In the present work this model is investigated by means of the functional renormalization group, based on an exact flow equation for the effective average action. In addition to the fermionic degrees of freedom of the Hubbard Hamiltonian, bosonic fields are introduced which correspond to the different possible collective orders of the system, for example magnetism and superconductivity. The interactions between bosons and fermions are determined by means of the method of “rebosonization” (or “flowing bosonization”), which can be described as a continuous, scale-dependent Hubbard-Stratonovich transformation. This method allows an efficient parameterization of the momentumdependent effective two-particle interaction between fermions (four-point vertex), and it makes it possible to follow the flow of the running couplings into the regimes exhibiting spontaneous symmetry breaking, where bosonic fluctuations determine the types of order which are present on large length scales. Numerical results for the phase diagram are presented, which include the mutual influence of different, competing types of order.
Using the dynamical mean-field theory (DMFT) as a ``booster rocket'', the functional renormalization group (fRG) can be upgraded from a weak-coupling method to a powerful computation tool for strongly interacting fermion systems. The strong local correlations are treated nonperturbatively by the DMFT, while the fRG flow can be formulated such that it is driven exclusively by nonlocal correlations, which are more amenable to approximations. We show that the full frequency dependence of the two-particle vertex needs to be taken into account in this approach, and demonstrate that this is actually possible---in spite of the singular frequency dependence of the vertex at strong coupling. We are thus able to present results obtained from the DMFT-boosted fRG for the two-dimensional Hubbard model in the strongly interacting regime. We find strong antiferromagnetic correlations from half filling to 18% hole doping and at the lowest temperature we can access, a sizable ddd-wave pairing interaction driven by magnetic correlations at the edge of the antiferromagnetic regime.
We analyze effective ddd-wave interactions in the two-dimensional extended Hubbard model at weak coupling and small to moderate doping. The interactions are computed from a renormalization group flow. Attractive ddd-wave interactions are generated via antiferromagnetic spin fluctuations in the pairing and charge channels. Above Van Hove filling, the ddd-wave charge interaction is maximal at incommensurate diagonal wave vectors, corresponding to nematic fluctuations with a diagonal modulation. Below Van Hove filling a modulation along the crystal axes can be favored. The nematic fluctuations are enhanced by the nearest-neighbor interaction in the extended Hubbard model, but they always remain smaller than the dominant antiferromagnetic, pairing, or charge density wave fluctuations.
The phases with spontaneously broken symmetries corresponding to antiferromagnetic and ddd-wave superconducting order in the two-dimensional $t\ensuremath{-}{t}^{'}$-Hubbard model are investigated by means of the functional renormalization group. The introduction of composite boson fields in the magnetic, charge density, and superconducting channels allows an efficient parametrization of the four-fermion vertex and the study of regimes where either the antiferromagnetic or superconducting order parameter, or both, are nonzero. We compute the phase diagram and the temperature dependence of the order parameter below the critical temperature, where antiferromagnetic and superconducting order show a tendency of mutual exclusion.
Using the recently introduced multiloop extension of the functional renormalization group, we compute the magnetic, density, and superconducting susceptibilities of the two-dimensional Hubbard model at weak coupling and present a detailed analysis of their evolution with temperature, interaction strength, and loop order. By breaking down the susceptibilities into contributions from the bare susceptibility and the individual channels, we investigate their relative importance as well as the channel interplay. In particular, we trace the influence of antiferromagnetic fluctuations on the ddd-wave superconductivity and provide an analytical understanding for the observed behavior.