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.

Figure 1

GS energy (ε) vs. on-site e-e correlation strength U.

Figure 2

(a) 〈x _i〉 as a function of g for different values of λ _ap and γ. (b) η vs. g. (c) t _eff/t as a function of g. (d) U _eff as a function of g.

Figure 3

(a) t _eff/t vs. U; (b) dt _eff/dU vs. U.

Figure 4

(a) Phase diagram in α−U plane determined from the peaks in dt _eff/dU. MR represents metallic region (b) Peak-to-peak width from dt _eff/dU-plot as a function of λ _ap for two values of γ.

Figure 5

A 3D picture showing the behavior of |U _eff| (blue) and 4t _eff (red) with U and α.

Figure 6

(a) Variation of local spin moment L ₀ with g. (b) Variation of L ₀ with U for different values of λ _ap and γ.

Figure 7

L ₀ vs. U and α for λ _ap = 0.05, γ = 0.05 and t = 0.2 ω ₀.

Figure 8

Contour plots of L ₀ in α-U plane.

Figure 9

(a) Double occupancy ω as a function of g. (b) ω as a function of U. (c) ω as a function of α and U.

Figure 10

(a) Quantum Entanglement (E _ϑ) as a function of g. (b) E _ϑ as a function of U. (c) E _ϑ on the α-U plane with its contour map.

Figure 11

(a) Local spin moment L ₀ as a function of the strength of the Gaussian anharmonicity λ _ap. (b) Renormalized inter - site electron hopping term t_eff/t as a function of λ _ap. (c) Renormalized on-site e-e interaction strength as a function of λ _ap.