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arXiv:cond-mat/0008044 · Physical review. B, Condensed matter

Signatures of spin and charge energy scales in the local moment and specific heat of the half-filled two-dimensional Hubbard model

Thereza Paiva, Richard T. Scalettar, Carey Huscroft, Andrew K. McMahan

Published Mar 13, 2001• 90 citations
arXivHTMLPDF

Abstract

Local moment formation driven by the on-site repulsion U is one of the most fundamental features in the Hubbard model. At the simplest level, the temperature dependence of the local moment is expected to have a single structure at $T\ensuremath{\sim}U,$ reflecting the suppression of the double occupancy. In this paper we show low-temperature quantum Monte Carlo data for half-filling which emphasize that the local moment also has a signature at a lower energy scale which previously had been thought to characterize only the temperatures below which moments on different sites begin to correlate locally. We discuss implications of these results for the structure of the specific heat, and connections to quasiparticle resonance and pseudogap formation in the density of states.

DOI
10.1103/physrevb.63.125116

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Local moment formation driven by the on-site repulsion U is one of the most fundamental features in the Hubbard model. At the simplest level, the temperature dependence of the local moment is expected to have a single structure at T\ensuremath∼U,T\ensuremath{\sim}U,T\ensuremath∼U, reflecting the suppression of the double occupancy. In this paper we show low-temperature quantum Monte Carlo data for half-filling which emphasize that the local moment also has a signature at a lower energy scale which previously had been thought to characterize only the temperatures below which moments on different sites begin to correlate locally. We discuss implications of these results for the structure of the specific heat, and connections to quasiparticle resonance and pseudogap formation in the density of states.
We study the formation of local magnetic moments in the strongly correlated Hubbard model within dynamical mean-field theory and associate peculiarities of the temperature dependence of local charge ${\ensuremath{\chi}}_{c}$ and spin ${\ensuremath{\chi}}_{s}$ susceptibilities with different stages of local moment formation. The local maximum of the temperature dependence of the charge susceptibility ${\ensuremath{\chi}}_{c}$ is associated with the beginning of local magnetic moment formation, while the minimum of the susceptibility ${\ensuremath{\chi}}_{c}$ and double occupation, as well as the low-temperature boundary of the plateau of the effective local magnetic moment ${\ensuremath{\mu}}_{\mathrm{eff}}^{2}=T{\ensuremath{\chi}}_{s}$ temperature dependence are connected with the full formation of local moments. We also obtain the interaction dependence of the Kondo temperature TK{T}_{K}TK​, which is compared to the fingerprint criterion of Chalupa et al. [Phys. Rev. Lett. 126, 056403 (2021)]. Near the Mott transition the two criteria coincide, while further away from the Mott transition the fingerprint criterion somewhat overestimates the Kondo temperature. The relation of the observed features to the behavior of eigenvectors/eigenvalues of fermionic frequency-resolved charge susceptibility and divergences of irreducible vertices is discussed.
Quantum Monte Carlo results for the specific heat c of the two-dimensional Hubbard model are presented. At half filling it was observed that c\ensuremath{\sim}T2{\mathrm{T}}^{2}T2 at very low-temperatures. Two distinct features were also identified: a low-temperature peak related to the spin degrees of freedom and a higher-temperature broad peak related to the charge degrees of freedom. Away from half filling the spin-induced feature slowly disappears as a function of hole doping while the charge feature moves to lower temperature. A comparison with experimental results for the high-temperature cuprates is discussed.
The specific heat C of the one-dimensional half-filled Hubbard model is studied by Feynman path-integral quantum Monte Carlo simulations as a function of temperature (T). The strength of the electronic correlations is modified by a variable intramolecular two-electron repulsion U. For weak correlations, C shows a single maximum. For intermediate correlations C has two maxima in the T interval studied. For strong correlations only one maximum and one minimum in C are found in the T interval considered. The influence of spin and electronic excitations on the T dependence of C are discussed. The C data are correlated with calculated probabilities of single occupancy, electronic charge fluctuations, and the electronic chemical potential. \textcopyright{} 1996 The American Physical Society.
We report on a nonperturbative approach to the one-dimensional (1D) and two-dimensional (2D) Hubbard models that is capable of recovering both strong- $(U\ensuremath{\gg}t)$ and weak-coupling $(U\ensuremath{\ll}t)$ limits, with U the on-site Coulomb repulsion and t the kinetic energy. Dynamical corrections to the electron self-energy in the single particle Green function are explicitly included by expanding in terms of the 16 eigenstates that characterize two nearest neighbor sites. We first show that even when U is much smaller than the bandwidth, the Mott-Hubbard gap never closes at half filling in both 1D and 2D. Consequently, the Hubbard model at half filling is always in the strong-coupling nonperturbative regime. For both large and small U, we find that the population of nearest-neighbor singlet states approaches a value of order unity as $\stackrel{\ensuremath{\rightarrow}}{T}0$ as would be expected for antiferromagnetic order. We also find that the double occupancy is a smooth monotonic function of U and approaches the anticipated noninteracting limit of 1/41/41/4 as $\stackrel{\ensuremath{\rightarrow}}{U}0$ and vanishes as $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}.$ Finally, we compute the heat capacity [C(T,U)][C(T,U)][C(T,U)] for both 1D and 2D. Our results for 1D at moderate to high temperatures are in quantitative agreement with those of the exact Bethe ansatz solution, differing by no more than 11%.1 In addition, we find that in 2D, the C(T,U)C(T,U)C(T,U) curves vs T for different values of U exhibit a universal crossing point at two characteristic temperatures $T\ensuremath{\approx}1.7t\ifmmode\pm\else\textpm\fi{}0.1t$ and $T\ensuremath{\approx}0.4\ifmmode\pm\else\textpm\fi{}0.1t$ as is seen universally in Hubbard models and experimentally in a wide range of strongly correlated systems such as 3He,{}^{3}\mathrm{He},3He, UBe3,{\mathrm{UBe}}_{3},UBe3​, and ${\mathrm{CeCu}}_{6\ensuremath{-}x}{\mathrm{Al}}_{x}.$ The success of this method in recovering well-established results that stem fundamentally from the Coulomb interaction suggests that local dynamics are at the heart of the physics of strongly correlated systems.
The Feynman path-integral quantum Monte Carlo (PI QMC) approach has been employed to study the specific heat C of the one-dimensional (1D) half-filled Hubbard chain as a function of temperature. For not too strong many-body interactions C exhibits a single maximum only. With increasing correlation strength the C curve shows an intermediate minimum separating a sharp low-temperature maximum and a broad high-temperature profile. The first peak is caused by spin excitations and the high-temperature values of C are determined by charge degrees of freedom. The contribution of both quantities to the net specific heat curve has been quantified in the present theoretical setup. The specific heat data are interpreted with the aid of calculated charge fluctuations, probabilities Pi(n) to find n = 2, 1, 0 electrons at the respective lattice sites, a spin density wave (SDW) order parameter and the electronic chemical potential μ.
We present single-particle and thermodynamic properties of the half-filled single-band Hubbard model in 2D calculated in the self-consistent fluctuation exchange approximation. The low-energy excitations at moderate temperatures and small UUU are quasiparticles with a short lifetime. As the temperature is lowered, coupling to evolving spin fluctuations leads to the extinction of these quasiparticles, signaled by a weak pseudogap in the density of states and by a positive slope in $\mathrm{Re}\ensuremath{\Sigma}({\mathbf{k}}_{F},\ensuremath{\varepsilon})$ and a local maximum in $|\mathrm{Im}\ensuremath{\Sigma}({\mathbf{k}}_{F},\ensuremath{\varepsilon})|$ at $\ensuremath{\varepsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$. We explain these results using a simple spin-fluctuation model.