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arXiv:cond-mat/9809332 · Physical review. B, Condensed matter

Dynamical properties of the two-dimensional Holstein-Hubbard model in the normal state at zero temperature: A fluctuation-based effective cumulant approach

Tugrul Hakioglu, M. Ye. Zhuravlev

Published Aug 15, 1998• 5 citations
arXivHTMLPDF

Abstract

The two-dimensional many-body Holstein-Hubbard model in the $T=0$ normal state is examined within the framework of the self-consistent coupling of charge fluctuation correlations to the vibrational ones. The parameters of our model are the adiabaticity, the electron concentration, as well as the electron-phonon and the Coulomb interaction strengths. A fluctuation-based effective cumulant approach is introduced to examine the $T=0$ normal-state fluctuations and an analytic approximation to the true dynamical entangled ground state is suggested. Our results for the effective charge-transfer amplitude, the ground state energy, the fluctuations in the phonon population, the phonon softening as well as the coupling constant renormalizations suggest that, the recent numerical calculations of de Mello and Ranninger (Ref. 5), Berger, Val\'a\ifmmode \check{s}\else \v{s}\fi{}ek, and von der Linden (Ref. 2), and Marsiglio (Refs. 4 and 8) on systems with finite degrees of freedom can be qualitatively extended to the systems with large degrees of freedom.

DOI
10.1103/physrevb.58.3777

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The two-dimensional many-body Holstein-Hubbard model in the T=0T=0T=0 normal state is examined within the framework of the self-consistent coupling of charge fluctuation correlations to the vibrational ones. The parameters of our model are the adiabaticity, the electron concentration, as well as the electron-phonon and the Coulomb interaction strengths. A fluctuation-based effective cumulant approach is introduced to examine the T=0T=0T=0 normal-state fluctuations and an analytic approximation to the true dynamical entangled ground state is suggested. Our results for the effective charge-transfer amplitude, the ground state energy, the fluctuations in the phonon population, the phonon softening as well as the coupling constant renormalizations suggest that, the recent numerical calculations of de Mello and Ranninger (Ref. 5), Berger, Val'a\ifmmode \check{s}\else \v{s}\fi{}ek, and von der Linden (Ref. 2), and Marsiglio (Refs. 4 and 8) on systems with finite degrees of freedom can be qualitatively extended to the systems with large degrees of freedom.
We present results on the dynamical correlation functions of the particle-hole symmetric Holstein-Hubbard model at zero temperature, calculated using the dynamical mean-field theory which is solved by the numerical renormalization group method. We clarify the competing influences of the electron-electron and electron-phonon interactions particularity at the different metal to insulator transitions. The Coulomb repulsion is found to dominate the behavior in large parts of the metallic regime. By suppressing charge fluctuations, it effectively decouples electrons from phonons. The phonon propagator shows a characteristic softening near the metal to bipolaronic transition but there is very little softening on the approach to the Mott transition.
[[abstract]]The accurate analytical and numerical calculations of the electronic band structure and ground state properties of strongly correlated electrons within the Hubbard model are performed by constructing convergent perturbation theory for general interaction strength and electron concentration. We test the developed perturbation approach about mean field solution in the extreme conditions of one dimensionality for entire parameter space of electron interaction U/t and electron concentration n. The many-body perturbation formalism up two second order about the generalized self-consistent field (GSCF) Hamiltonian goes beyond the range of applicability of standard perturbation theory by incorporating systematically the effect of the random-phase-type perturbation techniques and controlled expansion of the energy functional for general U/t and n. The second order perturbation correction vanishes at small and large U/t limit and performed calculations of the ground state energy show a next to the perfect numerical agreement with the Bethe-ansatz results.
A semiclassical method is presented for tunneling of self-trapped states in many-body systems with electrons and phonons. An overcomplete set of Slater determinants, lattice coordinates, and lattice momenta is used to represent a functional integral. Stationary phase equations are solved numerically without any constraint on the dynamics of electrons and phonons, i.e., without the use of the adiabatic approximation. To evaluate transition amplitudes, we integrate over small fluctuations in both electronic and phonon degrees of freedom, keeping their time order correctly. This method can be applied to general electron-phonon systems is and useful when self-trapped states have complex structures in charge or spin densities and lattice displacements. The effective hopping strength is calculated for a self-trapped kink in the commensurate charge-density-wave state in one dimension. At strong coupling in the Holstein and attractive Hubbard models, where tunneling involves effectively a single tightly bound bipolaron, this method reproduces previous analytic results. New results are obtained at intermediate coupling where the kink is extended over a couple of lattice sites and for models with both electron-electron and electron-phonon interactions.
The interplay of electron-phonon coupling and strong electronic correlations is studied in the frame of the two-dimensional Hubbard-Holstein model. Static and dynamic properties are determined by quantum Monte Carlo simulations and by Migdal-Eliashberg theory. The comparison allows us to assess the diagrammatic approach. The competition between the phonon-mediated electron-electron attraction and the local Coulomb repulsion leads to a rich phase diagram, which we study in detail for a wide range of parameters. We address the question, to which extent the systems can be described by an effective negative-U Hubbard model.
We calculate the spectral function of the one-dimensional Hubbard-Holstein model using the time-dependent density matrix renormalization group, focusing on the regime of large local Coulomb repulsion, and away from electronic half-filling. We argue that, from weak to intermediate electron-phonon coupling, phonons interact only with the electronic charge, and not with the spin degrees of freedom. For strong electron-phonon interaction, spinon and holon bands are not discernible anymore and the system is well described by a spinless polaronic liquid. In this regime, we observe multiple peaks in the spectrum with an energy separation corresponding to the energy of the lattice vibrations (i.e., phonons). We support the numerical results by introducing a well controlled analytical approach based on Ogata-Shiba's factorized wave function, showing that the spectrum can be understood as a convolution of three contributions, originating from charge, spin, and lattice sectors. We recognize and interpret these signatures in the spectral properties and discuss the experimental implications.