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arXiv:cond-mat/0611164 · Physical Review B

Renormalized mean-field analysis of antiferromagnetism and d -wave superconductivity in the two-dimensional Hubbard model

J. Reiss, D. Rohe, Walter Metzner

Published Feb 20, 2007• 92 citations
arXivHTMLPDF

Abstract

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 $d$-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.

DOI
10.1103/physrevb.75.075110

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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 \ensuremathΛMF{\ensuremath{\Lambda}}_{\mathrm{MF}}\ensuremathΛMF​ 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 \ensuremathπ\ensuremath{\pi}\ensuremathπ 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 \ensuremathπ\ensuremath{\pi}\ensuremathπ pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small.
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 perform a renormalized mean-field study of the two-dimensional repulsive Hubbard model, focusing on the intricate interplay and possible coexistence of magnetic, charge, and superconducting orders. We improve on conventional mean-field theory by utilizing a renormalization group framework that captures high-energy fluctuations. This method generates effective magnetic and d -wave pairing interactions, and allows for an unbiased exploration of coexisting phases at weak and moderate interaction strengths. Unrestricted mean-field calculations of the effective Hamiltonian on large finite lattices are combined with analyses in the thermodynamic limit, revealing a rich phase diagram with extensive regions of coexisting orders. We find that d -wave superconductivity coexists with Néel order on the electron-doped side. On the hole-doped side, superconductivity is found to coexist with spiral or stripe magnetic orders. Within the stripe-ordered region, the superconducting order parameter is spatially modulated, with a period that follows the charge modulation of the stripes. Below Van Hove filling, pairing provides the primary energy gain, while the stripe order yields only a small, and hence fragile, additional energy lowering.
We derive an efficient and unbiased method for computing order parameters in correlated electron systems with competing instabilities. Charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are taken into account by a functional renormalization group flow, while the formation of order below that scale is treated in mean-field theory. The method captures fluctuation-driven instabilities such as ddd-wave superconductivity. As a first application we study the competition between antiferromagnetism and superconductivity in the ground state of the two-dimensional Hubbard model.
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.
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.
We describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the complete flow from the bare action of the microscopic model to the effective low-energy action is obtained from a differential flow equation. We apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation for the effective interactions is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signalled by the diverging interactions we compute the flow of the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength and band-filling) we find antiferromagnetic order or d-wave superconductivity as leading symmetry breaking instability.