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Physical review. B, Condensed matter

Renormalized perturbation theory of magnetic instabilities in the two-dimensional Hubbard model at small doping

Andrey V. Chubukov, David M. Frenkel

Published Nov 1, 1992• 115 citations
Publisher

Abstract

We consider magnetic instabilities in the two-dimensional Hubbard model at small doping. We find that the renormalization of the effective interaction prevents an immediate instability of a commensurate antiferromagnetic state upon doping. At increased doping levels, our calculations indicate the occurrence of an instability in the channel of transverse spin fluctuations. This instability is known to lead to a spiral magnetic phase. We also consider a dynamical spin susceptibility near the instability and find that conventional spin waves play no role in the transition. Instead, the incommensurate instability is governed by collective fermionic excitations coupled to the spin background.

DOI
10.1103/physrevb.46.11884

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Physical review. B, Condensed matter•Oct 1, 1991•23 citations
We consider magnetic instabilities in the two-dimensional Hubbard model at small doping. We find that the renormalization of the effective interaction prevents an immediate instability of a commensurate antiferromagnetic state upon doping. At increased doping levels, our calculations indicate the occurrence of an instability in the channel of transverse spin fluctuations. This instability is known to lead to a spiral magnetic phase. We also consider a dynamical spin susceptibility near the instability and find that conventional spin waves play no role in the transition. Instead, the incommensurate instability is governed by collective fermionic excitations coupled to the spin background.
We study the Hubbard model on a square lattice, using the dynamical vertex approximation and the parquet approximation. These methods allow us to describe the mutual interference of spin-fluctuations in the particle-hole channel and superconducting fluctuations in the cooperon channel in an unbiased way. For small dopings we find predominant commensurable antiferromagnetic spin- and d-wave superconducting fluctuations; for larger doping incommensurate antiferromagnetic spin fluctuations are concomitant to triplet s-wave superconducting fluctuations.
We analyze the density dependence of the spin stiffnesses and the stability of magnetic order with respect to quantum fluctuations in the two-dimensional Hubbard model close to half-filling. The stiffnesses are computed from the spin susceptibility obtained from a random-phase approximation in a magnetically ordered state. For a sizable next-to-nearest-neighbor hopping amplitude and a moderate Hubbard interaction, the mean-field ground state is a Néel antiferromagnet in the electron-doped regime at and above half-filling and a planar circular spiral state in the hole-doped regime below half-filling. Upon electron doping, the Néel stiffness decreases smoothly and not very steeply. By contrast, the in-plane and out-of-plane stiffnesses in the spiral state drop abruptly at half-filling. The out-of-plane stiffness even drops to zero and then increases again very slowly upon increasing hole doping. At finite temperatures, the Néel-to-spiral transition is shifted into the hole-doped regime, the stiffnesses are continuous functions of the density, and they vanish at the transition. For small hole doping, the spin stiffnesses describe the quantum fluctuations only in a small momentum range, which shrinks to zero upon approaching half-filling. Using the above results, we show that the quantum ground state of the lightly electron-doped Hubbard model remains Néel ordered, while quantum fluctuations probably destroy the spiral long-range order in the hole-doped regime, giving rise to a quantum disordered state with a spin gap.
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.
Using a path-integral formalism and decomposing the Hubbard interaction into charge and spin order parameters, we develop a 1/S loop expansion of the two-dimensional Hubbard model. At the saddle-point level, we present a detailed study of the energetics of twisted antiferromagnetic order. Doping the antiferromagnetic state leads to a rich mean-field phase diagram at zero temperature, which includes spiral phases, a column phase, and ferromagnetism. The on-site magnetization, the density of states, the plasma frequency, and the quasiparticle weights are evaluated for the twisted spin-density-wave (SDW) states as a function of doping and U/t. Doping-induced frustration remarkably leads to a closure of the Mott-Hubbard band gap for any U/t. At one-loop order, we delineate the collective spin and charge fluctuations of the mean-field SDW state. The action for the two transverse spin modes at half-filling maps to a nonlinear \ensuremath{\sigma} model, which tends to disorder upon doping. Twisted magnetic ground states exhibit a third Goldstone mode due to the complete breaking of rotational invariance. For the commensurate SDW, the charge fluctuations couple only to longitudinal spin fluctuations, whereas, for the incommensurate metallic SDW states, the charge modes are coupled nontrivially to the massless spin waves. This suggests a metallic state quite unlike traditional Fermi liquids.
In a 1/S loop expansion of the two-dimensional Hubbard model, doping leads to incommensurate twist spin-density-wave (SDW) states at the saddle point. Further doping reduces the on-site magnetization and increases the SDW pitch, conspiring to close the effective Mott-Hubbard gap at \ensuremath{\delta}\ensuremath{\sim}0.07--0.30 for U/t\ensuremath{\sim}2.5--10. At the one-loop level, the two massless fluctuations map to the nonlinear \ensuremath{\sigma} model at zero doping while for the twist SDW states, three massless spin-wave modes arise which are strongly coupled to the charge fluctuations, therefore implying nontrivial charge-spin response functions for the metal.
The homogeneous Hartree-Fock antiferromagnetic (AF) state of the doped Hubbard model is shown to be unstable for infinitesimal doping concentration. Driven by long-wavelength intraband fluctuations, this instability with respect to infinitesimal fluctuations around the mean-field state is inferred from the nature of the static spin response, which increases away from the AF wave vector (\ensuremath{\pi},\ensuremath{\pi}). The effective interaction between antiparallel-spin holes in the lower band is found to contain a short-range attractive component, with an energy scale Jx. We discuss the crossover of the characteristic length scale for intraband fluctuations from the antiferromagnetic state to the spiral state, also unstable for infinitesimal doping concentration.