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arXiv:cond-mat/0509412 · physica status solidi (b)

Magnetic and superconducting correlations in the two‐dimensional Hubbard model

W. Metzner, J. Reiss, D. Rohe

Published Dec 2, 2005• 16 citations
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Abstract

Abstract The interplay and competition of magnetic and superconducting correlations in the weakly interacting two‐dimensional Hubbard Model is investigated by means of the functional renormalization group. At zero temperature the flow of interactions in one‐loop approximation evolves into a strong coupling regime at low energy scales, signalling the possible onset of spontaneous symmetry breaking. This is further analyzed by a mean‐field treatment of the strong renormalized interactions which takes into account magnetic and superconducting order simultaneously. The effect of strong correlations on single‐particle properties in the normal phase is studied by calculating the flow of the self‐energy. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssb.200562445

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Abstract The interplay and competition of magnetic and superconducting correlations in the weakly interacting two‐dimensional Hubbard Model is investigated by means of the functional renormalization group. At zero temperature the flow of interactions in one‐loop approximation evolves into a strong coupling regime at low energy scales, signalling the possible onset of spontaneous symmetry breaking. This is further analyzed by a mean‐field treatment of the strong renormalized interactions which takes into account magnetic and superconducting order simultaneously. The effect of strong correlations on single‐particle properties in the normal phase is studied by calculating the flow of the self‐energy. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The interplay and competition of magnetic and superconducting correlations in the weakly interacting two-dimensional Hubbard Model is investigated by means of the functional renormalization group. At zero temperature the flow of interactions in one-loop approximation evolves into a strong coupling regime at low energy scales, signalling the possible onset of spontaneous symmetry breaking. This is further analyzed by a mean-field treatment of the strong renormalized interactions which takes into account magnetic and superconducting order simultaneously. The effect of strong correlations on single-particle properties in the normal phase is studied by calculating the flow of the self-energy. 1
We analyze the interplay of antiferromagnetism and pairing in the two-dimensional Hubbard model with a moderate repulsive interaction. Coupled charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are treated by a functional renormalization group flow, while the formation of gaps and order below that scale is treated in mean-field theory. The full frequency dependences of the interaction vertices and gap functions are taken into account. We compute the magnetic and pairing gap functions as a function of doping ppp and compare with results from a static approximation. In spite of the strong frequency dependences of the effective interactions and of the pairing gap, important physical results from previous static functional renormalization group calculations are confirmed. In particular, there is a sizable doping regime with robust pairing coexisting with N\'eel or incommensurate antiferromagnetism. The critical temperature for magnetic order is interpreted as the pseudogap crossover temperature. Computing the Kosterlitz-Thouless temperature from the superfluid phase stiffness, we obtain a superconducting dome in the (p,T)(p,T)(p,T) phase diagram centered around 15% hole doping.
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.
The treatment of strongly interacting two-dimensional Fermi systems constitutes one of the most challenging problems in the field of condensed matter physics. Many theoretical works focused on the strongly coupled Hubbard model, since it is expected to capture the most important physics of cuprate superconductors. Due to our methodological improvements and understanding of the frequency dependence of the two-particle vertex function, the functional renormalization group combined with the dynamical mean field theory can now be used to study competing correlations in the strongly interacting regime. While limited to moderate interaction strengths, the functional renormalization group describes efficiently systems with a hierarchy of different energy scales and competing correlations. For instance, it provides definite evidence for d-wave superconductivity in the two-dimensional Hubbard model at moderate coupling. In a first project, we study the frequency dependence of the one-particle irreducible vertex function generated by the functional renormalization group flow. The frequency dependence, which becomes singular for strong interactions, appears to be important already for moderate couplings, and it cannot be represented by separate channels each depending on a single linear combination of frequencies. For strongly interacting systems, the dynamical mean field theory captures strong local correlation effects nonperturbatively. With this approximation, we study the impact of local correlations on the magnetic susceptibility. The local dynamics strongly affects the spin response function and its momentum dependence. In contrast to the widely used random-phase approximation with self-energy corrections that predicts Néel antiferromagnetic order, the local vertex corrections favor an incommensurate order similar to the ordering instability predicted by the Fermi surface geometry, as for weakly interacting systems. The dynamical mean field theory is also used as starting point for the functional renormalization group flow. We demonstrate that, due to our improvements for the parametrization of the vertex function, this approach is actually able to access the strong coupling physics. Moreover, we derive a flow scheme that conserves the local contributions and reduces the truncation error of the flow equations. In the strongly interacting regime, we capture strong d-wave pairing correlations driven by magnetic fluctuations with a mechanism similar to the one observed in the weakly interacting system.
Abstract A refined variational wave function for the two‐dimensional repulsive Hubbard model is studied numerically, with the aim of approaching the difficult crossover regime of intermediate values of U. The issue of a superconducting ground state with d‐wave symmetry is investigated for an average electron density n ≈ 0.8 and for U = 8t. Due to finite‐size effects a clear‐cut answer to this fundamental question has not yet been reached. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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.