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physica status solidi (b)

On the Definition of the Current Operator Application to the Hubbard Model

J. Monecke

Published Apr 1, 1974• 5 citations
Publisher

Abstract

Abstract A disagreement existing in the literature between various calculations of the electrical conductivity y in the single‐band Hubbard model is resolved. It is replied to an article by Cabib and Kaplan who tried to disprove that there are no absorption and emission lines near ω = U due to the translational invariance. The physical reason for this result and some of its consequences are discussed.

DOI
10.1002/pssb.2220620210

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On the Definition of the Current Operator Application to the Hubbard Model

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Abstract A disagreement existing in the literature between various calculations of the electrical conductivity in the single‐band Hubbard model is resolved. The error, which occurs in a paper by Monecke, is pointed out and explained to be due to the assumption that the current operator, defined as the commutator of the Hubbard Hamiltonian with the polarization operator, is invariant under the lattice translations. The existence of ambiguity in the definition of a “good” current operator (which is translationally invariant) is pointed out and resolved in a large class of situations.

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

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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 $\ensuremath{\Delta}$, 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.

Physical review. B, Condensed matter•Jun 1, 1989•2 citations
Abstract A disagreement existing in the literature between various calculations of the electrical conductivity y in the single‐band Hubbard model is resolved. It is replied to an article by Cabib and Kaplan who tried to disprove that there are no absorption and emission lines near ω = U due to the translational invariance. The physical reason for this result and some of its consequences are discussed.
Abstract A disagreement existing in the literature between various calculations of the electrical conductivity in the single‐band Hubbard model is resolved. The error, which occurs in a paper by Monecke, is pointed out and explained to be due to the assumption that the current operator, defined as the commutator of the Hubbard Hamiltonian with the polarization operator, is invariant under the lattice translations. The existence of ambiguity in the definition of a “good” current operator (which is translationally invariant) is pointed out and resolved in a large class of situations.
Abstract The chemical potential μ and the electrical dc conductivity σ in the one‐dimensional Hubbard model in the strong‐coupling region U≫t (U intra‐atomic Coulomb energy, –t nearest‐neighbour hopping integral) are calculated as functions of temperature and electron density n. In the non‐half‐filled band case n ≠ 1 the Bari‐Kaplan approximation lim (σ/t2) is shown to yield an un magnified image physical singularity in the dc conductivity at T = 0 which is connected with the discontinuity of the zero‐temperature chemical potential at n = 1. An improved calculation for n ≠ 1 going beyond the Bari‐Kaplan approximation (i.e. taking into account the band terms in μ and σ/t2) yields a finite zero‐temperature conductivity which is proportional to the first power in t. For small deviations from the half‐filled band case the conductivity as a function of increasing temperature subsequently shows a metal‐like behaviour, a semiconductor‐like behaviour, and a repeated decrease at high temperatures.
The optical conductivity of the U=\ensuremath{\infty} one-band Hubbard model is calculated on a square lattice in leading order of a 1/N expansion. To achieve this, the X-operator formulation and a Baym-Kadanoff expansion are used. For N\ensuremath{\rightarrow}\ensuremath{\infty} the optical conductivity is given by a Drude weighted \ensuremath{\delta} function. The calculated Drude weight as a function of the density is in excellent agreement with recent results of an exact diagonalization study. This supports the limit N\ensuremath{\rightarrow}\ensuremath{\infty} as being a good starting point for practical calculations. \textcopyright{} 1996 The American Physical Society.
We consider a Hubbard model in which electrons are locally coupled to the lattice and form polarons. The frequency-dependent polaron conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{pot}}$(\ensuremath{\omega}) is calculated in the narrow-band limit as a function of electron density. There are two contributions: a band term that arises from coherent motion of dressed electrons and a hopping term that describes the incoherent process by which electrons ``shake off'' their polaron distortion. The possible relevance of these results to the structure in \ensuremath{\sigma}(\ensuremath{\omega}) observed in high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors is briefly discussed.
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-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.