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arXiv:cond-mat/9312062 · Physical review. B, Condensed matter

Theory of spin and charge fluctuations in the Hubbard model

Yu. M. Vil’k, Liang Chen, A.–M. S. Tremblay

Published May 1, 1994• 83 citations
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Abstract

A self-consistent theory of both spin and charge fluctuations in the Hubbard model is presented. It is in quantitative agreement with Monte Carlo data at least up to intermediate coupling (U\ensuremath{\sim}8t). It includes both short-wavelength quantum renormalization effects, and long-wavelength thermal fluctuations, which can destroy long-range order in two dimensions. The last effect leads to a small energy scale, as often observed in high-temperature superconductors. The theory is conserving, satisfies the Pauli principle, and includes three-particle correlations necessary to account for the incipient Mott transition.

DOI
10.1103/physrevb.49.13267

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A self-consistent theory of both spin and charge fluctuations in the Hubbard model is presented. It is in quantitative agreement with Monte Carlo data at least up to intermediate coupling (U\ensuremath{\sim}8t). It includes both short-wavelength quantum renormalization effects, and long-wavelength thermal fluctuations, which can destroy long-range order in two dimensions. The last effect leads to a small energy scale, as often observed in high-temperature superconductors. The theory is conserving, satisfies the Pauli principle, and includes three-particle correlations necessary to account for the incipient Mott transition.
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 ground state properties of the two-dimensional half-filled one band Hubbard model in the strong (large-UUU) to intermediate coupling limit (i.e., away from the strict Heisenberg limit) using an effective spin-only low-energy theory that includes nearest-neighbor exchange, ring exchange, and all other spin interactions to order t(t∕U)3t{(t∕U)}^{3}t(t∕U)3. We show that the operator for the staggered magnetization, transformed for use in the effective theory, differs from that for the order parameter of the spin model by a renormalization factor accounting for the increased charge fluctuations as t∕Ut∕Ut∕U is increased from the $t∕U\ensuremath{\rightarrow}0$ Heisenberg limit. These charge fluctuations lead to an increase of the quantum fluctuations over and above those for an S=1∕2S=1∕2S=1∕2 antiferromagnet. The renormalization factor ensures that the zero temperature staggered moment for the Hubbard model is a monotonously decreasing function of t∕Ut∕Ut∕U, despite the fact that the moment of the spin Hamiltonian, which depends on transverse spin fluctuations only, in an increasing function of t∕Ut∕Ut∕U. We also comment on quantitative aspects of the t∕Ut∕Ut∕U and 1∕S1∕S1∕S expansions.
The influence of spin and charge fluctuations on spectra of the two-dimensional fermionic Hubbard model is considered using the strong coupling diagram technique. Infinite sequences of diagrams containing ladder inserts, which describe the interaction of electrons with these fluctuations, are summed, and obtained equations are self-consistently solved for the ranges of Hubbard repulsions [Formula: see text], temperatures [Formula: see text] and electron concentrations [Formula: see text] with t the intersite hopping constant. For all considered U the system exhibits a transition to the long-range antiferromagnetic order at [Formula: see text]. At the same time no indication of charge ordering is observed. Obtained solutions agree satisfactorily with results of other approaches and obey moments sum rules. In the considered region of the U-T plane, the curve separating metallic solutions passes from [Formula: see text] at the highest temperatures to U = 2t at [Formula: see text] for half-filling. If only short-range fluctuations are allowed for the remaining part of this region is occupied by insulating solutions. Taking into account long-range fluctuations leads to strengthening of maxima tails, which transform a part of insulating solutions into bad-metal states. For low T, obtained results allow us to trace the gradual transition from the regime of strong correlations with the pronounced four-band structure and well-defined Mott gap for [Formula: see text] to the Slater regime of weak correlations with the spectral intensity having a dip along the boundary of the magnetic Brillouin zone due to an antiferromagnetic ordering for [Formula: see text]. For [Formula: see text] and [Formula: see text] doping leads to the occurrence of a pseudogap near the Fermi level, which is a consequence of the splitting out of a narrow band from a Hubbard subband. Obtained spectra feature waterfalls and Fermi arcs, which are similar to those observed in hole-doped cuprates.
We study the three-band Hubbard model for the cuprate superconductors using Quantum Monte Carlo simulations and the fluctuation exchange approximation including a self-consistent many-body renormalization of the single-particle propagator. The energy dispersion of the low-lying one-particle excitations is in very good agreement with spectroscopic measurements for a hole doping concentration δ = 0.25 of the CuO2 planes. At the same doping, we examine the spin susceptibility yielding a non-ordered phase with short-range incommensurate fluctuations near the antiferromagnetic wave vector. In the superconducting regime, the interaction vertex for the pairing correlation function in the extended s-wave channel shows a maximum for δ = 0.20, resembling the dependence of transition temperature Tc on doping.
Recent experimental results suggest that a particular hydrodynamic theory describes charge fluctuations at long wavelengths in the square-lattice Hubbard model. Due to the continuity equation, the correlation functions for the charge and the current are directly connected: the parameters of the effective hydrodynamic model thus determine the optical conductivity. Here we investigate the validity of the proposed hydrodynamic theory in the full range of parameters of the Hubbard model. In the noninteracting case, there is no effective hydrodynamics, and the charge fluctuations present a rich variety of nonuniversal behaviors. At weak coupling, the optical conductivity is consistent with the hydrodynamic theory: at low frequency one observes a Lorentzian-shaped Drude peak, but the high-frequency asymptotics is necessarily different; the high-temperature limit for the product of the two hydrodynamic model parameters is also in agreement with numerical data. At strong coupling, we find that a generalization of the proposed hydrodynamic law is consistent with our quantum Monte Carlo, as well as the finite-temperature Lanczos results from literature. Most importantly, the temperature dependence of the hydrodynamic parameters as well as the dc resistivity are found to be very similar in the weak- and the strong-coupling regimes.
A self-consistent method for studying quantum spin fluctuations in the Mott-Hubbard antiferromagnet is presented within a diagrammatic scheme. The treatment involves the self-consistently renormalized one-particle Green's function (including its incoherent spectral part), and the renormalized spin-wave propagator, the exact long-wavelength form of which follows from an identity related to the commutation properties of spin-angular-momentum operators. In two dimensions, for the square-lattice case, we obtain M(0)=0.62 for the zero-temperature sublattice magnetization.