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Physical review. B, Condensed matter

ac conductivity of the strongly correlated Hubbard model in the low-temperature limit

Minette M. Mohan

Published Jun 1, 1989• 2 citations
Publisher

Abstract

The ac conductivity of the strongly correlated single-band Hubbard model, in the limit of small carrier concentration, is calculated over the entire frequency spectrum for temperatures much less than the bandwidth $\ensuremath{\Delta}$, using the retraceable path approximation of Brinkman and Rice. The small-frequency ($\ensuremath{\omega}$) limit of the optical conductivity can be fitted to the first terms of the low-frequency expansion $\ensuremath{\sigma}\ensuremath{\simeq}{\ensuremath{\sigma}}_{0}[1\ensuremath{-}{(\ensuremath{\omega}\ensuremath{\tau})}^{2}]$ of the Drude conductivity. In this limit, $\ensuremath{\tau}\ensuremath{\simeq}\frac{(0.35\ifmmode\pm\else\textpm\fi{}0.06)}{T}$, in units where Boltzmann's and Planck's constant are equal to unity. This crosses over for larger frequencies to non-Drude-like behavior $\ensuremath{\sim}1\sqrt{\ensuremath{\omega}}$, and vanishes as ${(1\ensuremath{-}\frac{\ensuremath{\omega}}{\ensuremath{\Delta}})}^{2}$ asymptotically close to the band edge. This calculation should be of qualitative interest to experiments on the normal-state, infrared, and optical regions of the new high-${T}_{c}$ superconductors over the entire range of frequencies and for dilute hole concentrations in the light of the current debate in the literature over strongly correlated versus itinerant behavior.

DOI
10.1103/physrevb.39.12395

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The ac conductivity of the strongly correlated single-band Hubbard model, in the limit of small carrier concentration, is calculated over the entire frequency spectrum for temperatures much less than the bandwidth \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ, using the retraceable path approximation of Brinkman and Rice. The small-frequency (\ensuremathω\ensuremath{\omega}\ensuremathω) limit of the optical conductivity can be fitted to the first terms of the low-frequency expansion \ensuremathσ\ensuremath≃\ensuremathσ0[1\ensuremath−(\ensuremathω\ensuremathτ)2]\ensuremath{\sigma}\ensuremath{\simeq}{\ensuremath{\sigma}}_{0}[1\ensuremath{-}{(\ensuremath{\omega}\ensuremath{\tau})}^{2}]\ensuremathσ\ensuremath≃\ensuremathσ0​[1\ensuremath−(\ensuremathω\ensuremathτ)2] of the Drude conductivity. In this limit, \ensuremathτ\ensuremath≃(0.35\ifmmode±\else\textpm\fi0.06)T\ensuremath{\tau}\ensuremath{\simeq}\frac{(0.35\ifmmode\pm\else\textpm\fi{}0.06)}{T}\ensuremathτ\ensuremath≃T(0.35\ifmmode±\else\textpm\fi0.06)​, in units where Boltzmann's and Planck's constant are equal to unity. This crosses over for larger frequencies to non-Drude-like behavior \ensuremath∼1\ensuremathω\ensuremath{\sim}1\sqrt{\ensuremath{\omega}}\ensuremath∼1\ensuremathω​, and vanishes as (1\ensuremath−\ensuremathω\ensuremathΔ)2{(1\ensuremath{-}\frac{\ensuremath{\omega}}{\ensuremath{\Delta}})}^{2}(1\ensuremath−\ensuremathΔ\ensuremathω​)2 asymptotically close to the band edge. This calculation should be of qualitative interest to experiments on the normal-state, infrared, and optical regions of the new high-Tc{T}_{c}Tc​ superconductors over the entire range of frequencies and for dilute hole concentrations in the light of the current debate in the literature over strongly correlated versus itinerant behavior.
We derive analytical expressions for the spectral moments of the dynamical response functions of the Hubbard model using the high-temperature series expansion. We consider generic dimension ddd as well as the infinite-ddd limit, arbitrary electron density nnn, and both finite and infinite repulsion UUU. We use moment-reconstruction methods to obtain the one-electron spectral function, the self-energy, and the optical conductivity. They are all smooth functions at high temperature and, at large UUU, they are featureless with characteristic widths of the order of the lattice hopping parameter ttt. In the infinite-ddd limit, we compare the series expansion results with accurate numerical renormalization group and interaction expansion quantum Monte Carlo results. We find excellent agreement down to surprisingly low temperatures, throughout most of the bad-metal regime, which applies for $T\ensuremath{\gtrsim}(1\ensuremath{-}n)D$, the Brinkman-Rice scale. The resistivity increases linearly in TTT at high temperature without saturation. This results from the 1/T1/T1/T behavior of the compressibility or kinetic energy, which play the role of the effective carrier number. In contrast, the scattering time (or diffusion constant) saturates at high TTT. We find that $\ensuremath{\sigma}(n,T)\ensuremath{\approx}(1\ensuremath{-}n)\ensuremath{\sigma}(n=0,T)$ to a very good approximation for all nnn, with $\ensuremath{\sigma}(n=0,T)\ensuremath{\propto}t/T$ at high temperatures. The saturation at small nnn occurs due to a compensation between the density dependence of the effective number of carriers and that of the scattering time. The TTT dependence of the resistivity displays a kneelike feature which signals a crossover to the intermediate-temperature regime where the diffusion constant (or scattering time) starts increasing with decreasing TTT. At high temperatures, the thermopower obeys the Heikes formula, while the Wiedemann-Franz law is violated with the Lorenz number vanishing as 1/T21/{T}^{2}1/T2. The relevance of our calculations to experiments probing high-temperature transport in materials with strong electronic correlations or ultracold atomic gases in an optical lattice is briefly discussed.
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We study the frequency-dependent electrical conductivity of a structurally disordered single-band Hubbard model near half-filling and in the strong-correlation limit. The electronic transfer integral is assumed to have a quasi-exponential form as used in the impurity-band problem of doped semiconductors. The short-time expansion of the current correlation function is calculated diagrammatically to the second order in the inverse temperature β and up to the seventh order in electronic hopping. Based on this expansion an approximate current correlation function is constructed. By means of the Kubo formula, the configurational average of the electrical conductivity is then calculated to order β2 for various atomic densities. The validity of the high-temperature expansion is examined. We also study the effect of spin polarization on the absorption line shape and the dc resistivity at different temperatures.
Quantum simulations are quickly becoming an indispensable tool for studying particle transport in correlated lattice models. One of the central topics in the study of transport is bad-metal behavior, characterized by the direct-current resistivity linear in temperature. In the fermionic Hubbard model, optical conductivity has been studied extensively, and a recent optical lattice experiment demonstrated bad-metal behavior in qualitative agreement with theory. Far less is known about transport in the bosonic Hubbard model. We investigate the conductivity in the Bose-Hubbard model and focus on the regime of strong interactions and high temperatures. We use numerically exact calculations for small lattice sizes. At weak tunneling, we find multiple peaks in the optical conductivity that stem from the Hubbard bands present in the many-body spectrum. This feature slowly washes out as the tunneling rate gets stronger. At high temperature, we identify a regime of TTT-linear resistivity, as expected. When the interactions are very strong, the leading inverse-temperature coefficient in conductivity is proportional to the tunneling amplitude. As the tunneling becomes stronger, this dependence takes quadratic form. At very strong coupling and half filling, we identify a separate linear resistivity regime at lower temperature, corresponding to the hard-core boson regime. Additionally, we unexpectedly observe that at half filling, in a big part of the phase diagram, conductivity is an increasing function of the coupling constant before it saturates at the hard-core-boson result. We explain this feature based on the analysis of the many-body energy spectrum and the contributions to conductivity of individual eigenstates of the system.
We calculate the optical conductivity \ensuremath{\sigma}(\ensuremath{\omega}) of the Hubbard and t-J models on 10- and 16-site lattices, respectively, using a Lanczos method. Results are presented for various values of the couplings U/t, J/t, and band fillings, using free- and periodic-boundary conditions. We discuss the behavior of the kinetic energy in the ground state, the distribution of spectral weight of \ensuremath{\sigma}(\ensuremath{\omega}) between low and high energies and the sum rule. Indications of a Drude-like peak at low energy in the one-hole subspace are observed in both models. Assuming that holes behave like independent (mass-renormalized) particles in the normal state of the superconductors, our results also suggest that the optical midinfrared broadband observed experimentally can be explained by hole excitations of strongly correlated systems.
Exact formulas for the optical conductivity and the Hall conductivity of the two dimensional Hubbard model are derived in terms of an effective theory description of the local moment fluctuation in the system. In this framework, the quantum Monte Carlo simulation of the electromagnetic response of such a strongly correlated electron system becomes sign-problem-free in many physically relevant cases. In particular, it is sign-problem-free when we assume the widely used Millis-Monien-Pines form for the phenomenological susceptibility in the effective action of the fluctuating local moment, even though these local moments are now subjected to Landau damping as a result of their coupling to the itinerant quasiparticle on the fermi surface. This is true more generally when a φ4φ^{4}φ4 term is included in the effective action and is thus not restricted to the Gaussian limit. Here we demonstrate the power of this framework by studying the effect of thermal fluctuation of the local moment on the optical conductivity σxx(ω)σ^{xx}(ω)σxx(ω) and the Hall conductivity σxy(ω)σ^{xy}(ω)σxy(ω) of the cuprate superconductors. Both σxx(ω)σ^{xx}(ω)σxx(ω) and σxy(ω)σ^{xy}(ω)σxy(ω) calculated are found to exhibit a two-component structure, with a Drude component at low energy and a mid-infrared component at higher energy. Depending on the relative importance of the hole pocket and the electron pocket on the reconstructed fermi surface and the coupling strength to the local moment, the Drude component in Imσxy(ω)\mathrm{Im}σ^{xy}(ω)Imσxy(ω) can be either positive or negative.(full-length abstract can be found in the main text.)