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arXiv:1203.2887 · Physical Review B

Incommensurate nematic fluctuations in the two-dimensional Hubbard model

Christoph Husemann, Walter Metzner

Published Aug 10, 2012• 52 citations
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Abstract

We analyze effective $d$-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 $d$-wave interactions are generated via antiferromagnetic spin fluctuations in the pairing and charge channels. Above Van Hove filling, the $d$-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.

DOI
10.1103/physrevb.86.085113

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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.
Motivated by the recent discovery of the anomalously nearest-neighbor attraction arising from the electron-phonon coupling, we quantitatively investigate the enhancing effects of this additional attractive channel on the ddd-wave superconductivity based on dynamic cluster quantum Monte Carlo calculations of a doped two-dimensional extended Hubbard model with nearest-neighbor attraction $\ensuremath{-}V$. Focusing on the range of $0<\ensuremath{-}V/t\ensuremath{\le}2$, our simulations indicate that the dynamics of ddd-wave projected pairing interaction is attractive at all frequencies and increases with ∣V∣|V|∣V∣. Moreover, turning on $\ensuremath{-}V$ attraction enhances the $(\ensuremath{\pi},\ensuremath{\pi})$ spin fluctuations but only enhances (suppresses) the charge fluctuations for small (large) momentum transfer. Thus, at $V/t=\ensuremath{-}1$ relevant to the ``holon folding branch'', the charge fluctuations are insufficient to compete with the ddd-wave pairing interaction strengthened by enhanced spin fluctuations. Our work suggests the underlying rich interplay between the spin and charge fluctuations in giving rise to the superconducting properties.
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 study the spin and charge fluctuations of the extended Hubbard model with on-site interaction UUU and first neighbor interaction VVV on the two-dimensional square lattice in the weak to intermediate coupling regime. We propose an extension of the two-particle self-consistent approximation that includes the effect of functional derivatives of the pair-correlation functions on irreducible spin and charge vertices. These functional derivatives were ignored in our previous work. We evaluate them assuming particle-hole symmetry. The resulting theory satisfies conservation laws and the Mermin-Wagner theorem. Our current results are in much better agreement with benchmark quantum Monte Carlo results. This theory allows us to reliably determine the crossover temperatures toward renormalized-classical regimes, and hence, the dominant instability as a function of UUU and VVV. We have considered fillings n=1n=1n=1 and n=0.75n=0.75n=0.75. Either spin or charge fluctuations can dominate. The wave vector is self-determined by the approach.
We investigate the structure of the pairing potential in the stripe phase of the two-dimensional Hubbard model. Based on the random-phase approximation we discuss in detail the interactions in the charge and spin channel and compare our calculations with related considerations in commensurate antiferromagnets. Our main finding is that due to the incommensurate charge-density wave formation the exchange of collective modes in the charge channel is significantly enhanced with respect to the spin-bag approach whereas due to the inhomogeneous charge distribution the coupling to transverse spin fluctuations tends to be suppressed.
We compute the two-particle quantities relevant for superconducting correlations in the two-dimensional Hubbard model within the dynamical cluster approximation. In the normal state we identify the parameter regime in density, interaction, and second-nearest-neighbor hopping strength that maximizes the d_{x^{2}-y^{2}} superconducting transition temperature. We find in all cases that the optimal transition temperature occurs at intermediate coupling strength, and is suppressed at strong and weak interaction strengths. Similarly, superconducting fluctuations are strongest at intermediate doping and suppressed towards large doping and half filling. We find a change in sign of the vertex contributions to d_{xy} superconductivity from repulsive near half filling to attractive at large doping. p-wave superconductivity is not found at the parameters we study, and s-wave contributions are always repulsive. For negative second-nearest-neighbor hopping the optimal transition temperature shifts towards the electron-doped side in opposition to the van Hove singularity, which moves towards hole doping. We surmise that an increase of the local interaction of the electron-doped compounds would increase T_{c}.
Several new classes of compounds can be modeled in first approximation by electrons on the triangular lattice that interact through on-site repulsion UUU as well as nearest-neighbor repulsion VVV. This extended Hubbard model on a triangular lattice has been studied mostly in the strong coupling limit for only a few types of instabilities. Using the extended two-particle self-consistent approach (ETPSC), that is valid at weak to intermediate coupling, we present an unbiased study of the density and interaction dependent crossover diagram for spin- and charge-density wave instabilities of the normal state at arbitrary wave vector. When UUU dominates over VVV and electron filling is large, instabilities are chiefly in the spin sector and are controlled mostly by Fermi surface properties. Increasing VVV eventually leads to charge instabilities. In the latter case, it is mostly the wave vector dependence of the vertex that determines the wave vector of the instability rather than Fermi surface properties. At small filling, nontrivial instabilities appear only beyond the weak coupling limit. There again, charge-density wave instabilities are favored over a wide range of dopings by large VVV at wave vectors corresponding to $\sqrt{(3)}\ifmmode\times\else\texttimes\fi{}\sqrt{(3)}$ superlattice in real space. Commensurate fillings do not play a special role for this instability. Increasing UUU leads to competition with ferromagnetism. At negative values of UUU or VVV, neglecting superconducting fluctuations, one finds that charge instabilities are favored. In general, the crossover diagram presents a rich variety of instabilities. We also show that thermal charge-density wave fluctuations in the renormalized-classical regime can open a pseudogap in the single-particle spectral weight, just as spin or superconducting fluctuations.