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Journal of Physics Condensed Matter

Spin fluctuations and holon dynamics in the large-U Hubbard model

Lina Hu, Zhengming Ji, L P Shi, K. W. Yu

Published Apr 8, 1991• 1 citation
Publisher

Abstract

The spin-holon effective Hamiltonian proposed by Zou and Anderson (1988) is used to calculate the chemical potential and the self-energy correction of spinons. The authors' results show that the effect of spin-exchange interaction is to enlarge the spinon bandwidth. For low temperatures and fixed doping fraction delta , the renormalized bandwidth and the chemical potential are nearly independent of temperature, but both change rapidly with the doping fraction. In the superfluid phase, the effect of increasing the doping fraction or increasing the on-site Coulomb repulsion U is to increase the relative kinetic energy of spinons, leading to a decrease in the superfluid transition temperature T s ( delta ), which is sensitive to various choices of U and the band parameter t; the increase in U tends to shift the boundary of T s ( delta ) towards a lower doping fraction. The correction due to spin fluctuations is extremely important for small doping fractions, leading to modification of the results of Baskaran, Zou and Anderson (1987), and Ruckstein, Hirschfeld and Appel (1987). Moreover, in the region of delta <or=8t/ pi 2 U, the effect of paramagnon fluctuations renders the mean-field theory invalid. The authors also discuss the effects of holon fluctuations on the mean-field results.

DOI
10.1088/0953-8984/3/14/014

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The spin-holon effective Hamiltonian proposed by Zou and Anderson (1988) is used to calculate the chemical potential and the self-energy correction of spinons. The authors' results show that the effect of spin-exchange interaction is to enlarge the spinon bandwidth. For low temperatures and fixed doping fraction delta , the renormalized bandwidth and the chemical potential are nearly independent of temperature, but both change rapidly with the doping fraction. In the superfluid phase, the effect of increasing the doping fraction or increasing the on-site Coulomb repulsion U is to increase the relative kinetic energy of spinons, leading to a decrease in the superfluid transition temperature T s ( delta ), which is sensitive to various choices of U and the band parameter t; the increase in U tends to shift the boundary of T s ( delta ) towards a lower doping fraction. The correction due to spin fluctuations is extremely important for small doping fractions, leading to modification of the results of Baskaran, Zou and Anderson (1987), and Ruckstein, Hirschfeld and Appel (1987). Moreover, in the region of delta <or=8t/ pi 2 U, the effect of paramagnon fluctuations renders the mean-field theory invalid. The authors also discuss the effects of holon fluctuations on the mean-field results.
We have applied mean-field theory to calculate the spin-density-wave (SDW) order parameter \ensuremath{\Delta}, the chemical potential \ensuremath{\mu}, and the ground-state energy at various band fillings for the Hubbard model. We have found that at zero temperature and in the large-U limit, the SDW order parameter and the ground-state energy can be written as \ensuremath{\Delta}=U(1-x)/2-O(J) and Eg{\mathit{E}}_{\mathit{g}}Eg​=-U(1-x)2{)}^{2})2/4-0(J). The chemical potential obtained here for x=0 and 1 agrees well with the result obtained by the exact-diagonalization calculation on a 4\ifmmode\times\else\texttimes\fi{}4 sites cluster by Dagotto et al. [Phys. Rev. B 45, 10 741 (1992)]. In the SDW phase, the mean-field theory gives an upward shift in the chemical potential as one dopes away from half-filling, which is in contrast with the exact-diagonalization calculations. This result indicates that the Mott-Hubbard gap is depressed much more rapidly in the mean-field theory than the exact-diagonalization calculation. The doping dependence of ground-state energy calculated here is in good agreement with the above numerical calculation for doping up to 0.5. Here, U is the intrasite Coulomb repulsion, x is doping concentration, J=4t2{\mathit{t}}^{2}t2/U is the superexchange interaction parameter, and t is the hopping integral for the nearest neighbors.
We construct a new U(1) slave-spin representation for the single-band Hubbard model in the large-U limit. The mean-field theory in this representation is more amenable to describe both the spin-charge-separation physics of the Mott insulator at half-filling and the strange metal behavior at finite doping. By employing a dynamical Green’s function theory for slave spins, we calculate the single-particle spectral function of electrons. The result is comparable to that in dynamical mean field theories. We then formulate a dynamical t/U expansion for the doped Hubbard model that reproduces the mean-field results at the lowest order of expansion. To the next order of expansion, it naturally yields an effective low-energy theory of a t–J model for spinons self-consistently coupled to an XXZ model for the slave spins. We show that the superexchange J is renormalized by doping, in agreement with the Gutzwiller approximation. Surprisingly, we find a new ferromagnetic channel of exchange interactions which survives in the infinite U limit, as a manifestation of the Nagaoka ferromagnetism.
We calculate the renormalized parameters for the quasiparticles and their interactions for the Hubbard model in the paramagnetic phase as deduced from the low-energy Fermi-liquid fixed point using the results of a numerical renormalization-group calculation (NRG) and dynamical mean-field theory (DMFT). Even in the low-density limit there is significant renormalization of the local quasiparticle interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}$, in agreement with estimates based on the two-particle scattering theory of J. Kanamori [Prog. Theor. Phys. 30, 275 (1963)]. On the approach to the Mott transition we find a finite ratio for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}/\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}$, where $2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}$ is the renormalized bandwidth, which is independent of whether the transition is approached by increasing the on-site interaction UUU or on increasing the density to half filling. The leading ${\ensuremath{\omega}}^{2}$ term in the self-energy and the local dynamical spin and charge susceptibilities are calculated within the renormalized perturbation theory (RPT) and compared with the results calculated directly from the NRG-DMFT. We also suggest, more generally from the DMFT, how an approximate expression for the $\mathbf{q},\ensuremath{\omega}$ spin susceptibility $\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ can be derived from repeated quasiparticle scattering with a local renormalized scattering vertex.
Away from half filling and above its Neel temperature, the three-dimensional Hubbard model is a paramagnetic metal with antiferromagnetic spin fluctuations, where controlled momentum-resolved results have been lacking. Using connected-determinant diagrammatic Monte Carlo, we compute the momentum-resolved self-energy for n = 0.4-0.95, T/t = 0.05-0.33 and U/t = 4-8, and the full-zone spin susceptibility at twelve of these points, focusing on U/t = 4. At U/t = 4 and toward half filling the spin fluctuations become strong: for n = 0.7-0.9 at T/t = 0.14 they enhance the susceptibility peak up to sevenfold and raise the momentum-dependent part of the scattering rate to four to five times its second-order value. They are driven by an effective spin interaction screened to about 2U/3 and nearly uniform over the zone. Yet the peak height is set by the slightly smaller value of this interaction at the peak wave vector, and near magnetic order this small difference changes the height by a large factor. Two-particle self-consistent theory reproduces the peak because its sum rule fixes its single constant interaction near this value, but for n = 0.7-0.9 at T/t = 0.14 its self-energy overestimates the local scattering rate and underestimates the momentum-dependent part. Though strong, the fluctuations are short ranged, with a correlation length of at most about two lattice spacings, and for n = 0.7-0.9 and T/t >= 0.14 the scattering rate near the Fermi surface varies by at most about 1% at fixed band energy, without hot spots. At U/t = 4 dynamical mean-field theory (DMFT) reproduces the local scattering rate and quasiparticle weight within a few percent, and the nonlocal part it misses is approximately a function of the band energy. Strong spin fluctuations thus coexist with nearly isotropic quasiparticles, and the results provide a benchmark for extensions of DMFT and for spin-fluctuation theories.
A real spin-charge separation scheme is found based on a saddle-point state of the t-J model. In the one-dimensional (1D) case, such a saddle-point reproduces the correct asymptotic correlations at the strong-coupling fixed point of the model. In the two-dimensional (2D) case, the transverse gauge field confining spinon and holon is shown to be gapped at finite doping so that a spin-charge deconfinement is obtained for its first time in 2D. The gap in the gauge fluctuation disappears at half-filling limit, where a long-range antiferromagnetic order is recovered at zero temperature and spinons become confined. The most interesting features of spin dynamics and transport are exhibited at finite doping where exotic residual couplings between spin and charge degrees of freedom lead to systematic anomalies with regard to a Fermi-liquid system. In spin dynamics, a commensurate antiferromagnetic fluctuation with a small, doping-dependent energy scale is found, which is characterized in momentum space by a Gaussian peak at (\ensuremath{\pi}/a,\ensuremath{\pi}/a) with a doping-dependent width (\ensuremath{\propto} \ensuremath{\surd}\ensuremath{\delta} , \ensuremath{\delta} is the doping concentration). This commensurate magnetic fluctuation contributes a non-Korringa behavior for the NMR spin-lattice relaxation rate. There also exists a characteristic temperature scale below thich a pseudogap behavior appears in the spin dynamics. Furthermore, an incommensurate magnetic fluctuation is also obtained at a finite energy regime. In the transport, a strong-range phase intereference leads to an effective holon Lagrangian which can give rise to a series of interesting phenomena including linear-T resistivity and a T2{\mathit{T}}^{2}T2 Hall angle. We discuss the striking similarities of these theoretical features with those found in the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprates and give a consistent picture for the latter. Electronic properties like Fermi surface and superconducting pairing in this framework are also discussed.