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arXiv:0708.0012 · Physical Review B

Quantum vs classical aspects of one dimensional electron-phonon systems revisited by the renormalization group method

H. Bakrim, C. Bourbonnais

Published Nov 16, 2007• 28 citations
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Abstract

An extension of the renormalization group method that includes the effect of retardation for the interactions of a fermion gas is used to reexamine the quantum and classical properties of Peierls-type states in one dimension. For models of spinless and spin-$\frac{1}{2}$ fermions interacting with either intra- or intermolecular phonons, the quantum corrections to the Peierls gap at half-filling are determined at arbitrary phonon frequency. The nature of quantum-classical transitions is clarified in weak coupling.

DOI
10.1103/physrevb.76.195115

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An extension of the renormalization group method that includes the effect of retardation for the interactions of a fermion gas is used to reexamine the quantum and classical properties of Peierls-type states in one dimension. For models of spinless and spin-12\frac{1}{2}21​ fermions interacting with either intra- or intermolecular phonons, the quantum corrections to the Peierls gap at half-filling are determined at arbitrary phonon frequency. The nature of quantum-classical transitions is clarified in weak coupling.
Using a multistep renormalization group method, we study the low-temperature phases of the interacting one-dimensional (1D) electron gas coupled to phonons. We obtain analytic expressions for the weak-coupling quantum phase boundaries of the 1D extended Holstein-Hubbard model and the 1D extended Peierls-Hubbard model for general band-filling and phonon frequency. Away from half-filling, the phase diagrams are characterized by a delicate competition between spin density wave, charge density wave, and superconducting orders. We study the dependence of the ground state on the electron-phonon (el-ph) and electron-electron (el-el) coupling strengths, the screening length, electron bandwidth, phonon frequency, doping, and type of phonon. Unlike the case in Fermi liquids, in 1D the el-ph coupling is strongly renormalized, often to stronger values. Even when the bare phonon-induced attraction is weak compared to the bare el-el repulsion, a small amount of retardation can cause the renormalized el-ph interaction to dominate the problem. We find cases in which a repulsive el-el interaction enhances the superconducting susceptibility in the presence of a retarded el-ph interaction. The spin gap and superconducting susceptibility are found to be strongly dependent on the deviation from half-filling (doping). In some cases, the superconducting susceptibility varies nonmonotonically with doping and exhibits a maximum at a particular doping. For a quasi-1D array of weakly coupled, fluctuating 1D chains, the superconducting transition temperature Tc{T}_{c}Tc​ also exhibits a maximum as a function of doping. The effect of changing the ion mass (isotope effect) on Tc{T}_{c}Tc​ is found to be largest near half-filling and to decrease rapidly with doping.
"We have studied the extended Hubbard and the Peierls-extended Hubbard models in one dimension using primarily computational methods. We discuss the derivation of the models and the computational methods used to study them, focusing primarily on our extensions over existing numerical methods. In particular, we present the construction of ""operator-loop"" updates for spin-1/2 fermions and the inclusion of electron-phonon interactions in the Stochastic Series Expansion Quantum Monte Carlo method and the use of Quantum Parallel Tempering to improve the quality of the numerical data. For the extended Hubbard model at half-filling, we have confirmed the existence of a recently reported bond-order wave phase for the ground state in a small region of parameter space between the well-known charge- and spin-density wave phases. We have studied in detail the nature of the associated quantum phase transitions between all the phases. Upon doping away from half-filling, the doubly-degenerate, spin and charge gapped long-range-ordered ground states of the model are shown to evolve to a Luther-Emery liquid with no charge gap but a finite spin gap. For the Peierls-extended Hubbard model, which includes both electron-phonon and electron-electron interactions, we have studied the dimerization transition of the half-filled system in the presence of fully quantum mechanical phonons. We show that the transition to a dimerized state occurs only when the electron-phonon coupling exceeds a finite critical value that depends on the bare phonon frequency. This is in contrast to previous studies in similar models with adiabatic phonons that had predicted a dimerized state for arbitrarily small values of the electron-phonon coupling."
Summary: The interplay of charge, spin and lattice degrees of freedom is studied for quasi-one-dimensional electron and spin systems coupled to quantum phonons. Special emphasis is put on the influence of the lattice dynamics on the Peierls transition. Using exact diagonalization techniques the ground-state and spectral properties of the Holstein model of spinless fermions and of a frustrated Heisenberg model with magneto-elastic coupling are analyzed on finite chains. In the non-adiabatic regime a (T = 0) quantum phase transition from a gapless Luttinger-liquid/spin-fluid state to a gapped dimerized phase occurs at a nonzero critical value of the electron/spin-phonon interaction. To study the nature of the spin-Peierls transition at finite temperatures for the infinite system, an alternative Green’s function approach is applied to the magnetostrictive XY model. With increasing phonon frequency the structure factor shows a remarkable crossover from soft-mode to central-peak behaviour. The results are discussed in relation to recent experiments on CuGeO3. 1
We study the nature of the ground state of the Su-Schrieffer-Heeger model for electron-phonon interactions in one dimension in the half-filled-band case. We consider the cases of spinless electrons (n=1n=1n=1) and spin-\textonehalf{} electrons (n=2n=2n=2), and discuss the stability of the Peierls-dimerized ground state as a function of the ionic mass and electron-phonon coupling constant. We first consider the zero-mass limit of the theory and extend our results to finite mass using renormalization-group arguments. For spinless electrons, it is found that quantum fluctuations destroy the long-range dimerization order for the small electron-phonon coupling constant if the ionic mass is finite. For spin-\textonehalf{} electrons, the system is dimerized for an arbitrary coupling constant and phonon frequency. Renormalization-group trajectories show that the low-energy behavior of the system is governed by the zero-mass limit of the theory, an nnn-component Gross-Neveu model. Monte Carlo simulations are performed for the model at finite phonon frequencies, and the results are compared with the static limit result. We study in particular the set of parameters appropriate for polyacetylene and find a 15% reduction in the phonon order parameter due to fluctuations of the phonon field. A finite-size scaling analysis of the numerical data for the cases n=1and2n=1 \mathrm{and} 2n=1and2 is performed, which confirms the results obtained from the renormalization-group analysis.
We study the nature of the ground state of a one-dimensional electron-phonon model for molecular crystals: The phonons are assumed dispersionless and couple to the local electronic density. We consider the half-filled-band sector and discuss the stability of the Peierls-dimerized ground state as a function of the phonon frequency ($\ensuremath{\omega}$), electron-phonon coupling constant ($\ensuremath{\lambda}$), and number of components of the electron spin (nnn). First, we discuss the properties of the model in the limiting cases of zero frequency and infinite frequency. We then perform a strong coupling expansion which maps the system onto a spinless fermion model with nearest-neighbor repulsion for both spinless and spin-\textonehalf{} electrons, but with different parameters in both cases. Finally, we perform a numerical study of the system using a Monte Carlo technique. We study the behavior of the order parameter and of correlation functions for various points in parameter space. We also perform a finite-size---scaling analysis of the numerical data. The conclusions of our study are the following: For the case of spinless electrons, there exists always a disordered phase for small coupling constant, and the system undergoes an infinite-order transition to a Peierls-dimerized state as the coupling constant increases beyond a critical value. The phase diagram is divided between a disordered and an ordered region by a line that connects the points ($\ensuremath{\omega}=0$, $\ensuremath{\lambda}=0$) and ($\ensuremath{\omega}=\ensuremath{\infty}$, $\ensuremath{\lambda}=\ensuremath{\infty}$). For the case of spin-\textonehalf{} electrons, the system is dimerized for arbitrary $\ensuremath{\lambda}$ and $\ensuremath{\omega}$ except in the limit $\ensuremath{\omega}=\ensuremath{\infty}$.
We investigate the influence of the electron-phonon coupling in the one-dimensional spinless Holstein model at half-filling using both a recently developed projector-based renormalization method (PRM) and an refined exact diagonalization technique in combination with the kernel polynomial method. At finite phonon frequencies the system shows a metal-insulator transition accompanied by the appearance of a Peierls distorted state at a finite critical electron-phonon coupling. We analyze the opening of a gap in terms of the (inverse) photoemission spectral functions which are evaluated in both approaches. Moreover, the PRM approach reveals the softening of a phonon at the Brillouin-zone boundary which can be understood as precursor effect of the gap formation.