Metallicity in a Holstein-Hubbard Chain at Half Filling with Gaussian Anharmonicity

Abstract

The Holstein-Hubbard model with Gaussian phonon anharmonicity is studied in one-dimension at half filling using a variational method based on a series of canonical transformations. A fairly accurate phonon state is chosen to average the transformed Holstein-Hubbard Hamiltonian to obtain an effective Hubbard model which is then solved using the exact Bethe - ansatz following Lieb and Wu to obtain the ground state energy, the average lattice displacement and the renormalized parameters. The Mott-Hubbard criterion, local spin moment and the von Neumann entropy (which is a measure of quantum entanglement) are calculated to determine the ground state phase diagram which shows that the width of the metallic phase flanked by the SDW and CDW phases increases with increasing anharmonicity at low and moderate values of anharmonicity but eventually saturates when the anharmonicity becomes substantially large.

A semi exact solution for a metallic phase in a Holstein-Hubbard chain at half filling with Gaussian anharmonic phonons

“…It is clear that the lattice anharmonicity has a significant effect on the GS energy. We also show the results of the anharmonic case reported recently by Lavanya et al (LSC) 25 . The LSC results have been obtained by using three canonical transformations.…”

“…The metallic phase is flanked by the SDW phase on the left and the CDW phase on the right. The figure shows that the intermediate metallic phase appearing at the CDW-SDW cross-over region is now wider compared to that predicted by LSC 25 . It is important to emphasize that it is not important by how much the present modified variational wave function broadens the width of the intermediate region, that the improved variational calculation widens the metallic phase is itself a result of great significance.…”

“… Average lattice displacement ( ) vs. e-p interaction coefficient ( ). ‘LSC’ refers to the result of Lavanya et al 25 obtained for the anharmonic case with three canonical transformations. “Present” refers to the result for the anharmonic case obtained by including the effect of the new transformation with the generator in the present work.…”

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Figure 3 ( a ) Effective hopping parameter ( vs on-site Coulomb interaction strength ( ; ( b ) Effective e-e interaction energy vs on-site Coulomb interaction strength . “LSC” refers to the result for the anharmonic case obtained by Lavanya et al 25 using three canonical transformations. “Present” refers to the result obtained for the anharmonic case obtained by including the effect of the new transformation with generator in the present work.

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“…Konior 34 has explained the importance of Gaussian phonon anharmonicity in the context of high superconductors and have shown that in the presence of this anharmonicity, the hopping parameter reduces at a slow rate causing an enhancement in the polaron mobility and the polaron bandwidth, which is a favourable condition for the phonon mechanism to stake a claim for inducing pairing. Lavanya, Sankar and Chatterjee (LSC) 25 have recently re-examined the work of CT with Gaussian anharmonic potential by applying in succession a number of unitary transformations followed by an averaging with a general many-phonon state and the Bethe ansatz technique. This gives a wider metallic phase.…”