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Physical Review Letters

Effect of Quantum Fluctuations on the Peierls Instability: A Monte Carlo Study

J. E. Hirsch, Eduardo Fradkin

Published Aug 9, 1982• 115 citations
Publisher

Abstract

The effect of quantum fluctuations of the phonon field on the ground-state character of one-dimensional electron-phonon systems is studied with a numerical simulation technique. Two different electron-phonon models are studied, both for spinless and spin-\textonehalf{} electrons, in the half-filled-band case. In both models, it is found that in the spinless case quantum fluctuations destroy the long-range dimerization order for small electron-phonon coupling constant $\ensuremath{\lambda}$. For the spin-\textonehalf{} case, the results are consistent with the Peierls transition occurring at $\ensuremath{\lambda}=0$.

DOI
10.1103/physrevlett.49.402

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Physical review. B, Condensed matter•Feb 1, 1996•1 citation
The effect of quantum fluctuations of the phonon field on the ground-state character of one-dimensional electron-phonon systems is studied with a numerical simulation technique. Two different electron-phonon models are studied, both for spinless and spin-\textonehalf{} electrons, in the half-filled-band case. In both models, it is found that in the spinless case quantum fluctuations destroy the long-range dimerization order for small electron-phonon coupling constant \ensuremathλ\ensuremath{\lambda}\ensuremathλ. For the spin-\textonehalf{} case, the results are consistent with the Peierls transition occurring at \ensuremathλ=0\ensuremath{\lambda}=0\ensuremathλ=0.
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 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."
We study the one-dimensional Holstein model with spin-12\frac{1}{2}21​ electrons at half filling. Ground-state properties are calculated for long chains with great accuracy using the density-matrix renormalization-group method and extrapolated to the thermodynamic limit. We show that for small electron-phonon coupling or large phonon frequency, the insulating Peierls ground state predicted by mean-field theory is destroyed by quantum lattice fluctuations and that the system remains in a metallic phase with a nondegenerate ground state and power-law electronic and phononic correlations. When the electron-phonon coupling becomes large or the phonon frequency small, the system undergoes a transition to an insulating Peierls phase with a twofold degenerate ground state, long-range charge-density-wave order, a dimerized lattice structure, and a gap in the electronic excitation spectrum.
Using a one-dimensional antiferromagnetic Heisenberg chain model with spin-phonon interaction, the effects of quantum lattice fluctuations on the phase diagram and correlation functions in the ground state of the spin-Peierls system are investigated by developing an analytical approach based on the unitary transformation method. To study the nonadiabatic case, the Green's function method is used to implement the perturbation treatment. We show that when the spin-phonon coupling constant decreases or phonon frequency increases, the lattice dimerization and the spin dimerization gap decrease gradually. At some finite critical value, the Peierls dimerization is destroyed by the quantum lattice fluctuations and the system becomes gapless. The inverse-square-root singularity of the density of states at the gap edge in the adiabatic case disappears because of the nonadiabatic effect, which is consistent with the measurement of the optical absorption spectrum in quasi-one-dimensional spin-Peierls systems. The dimerization gap is much more reduced by the quantum lattice fluctuations than the spin dimer order parameter is. This could result in a true spin dimerization gap that is smaller than the activation energy of the optical absorption spectrum.
The effect of quantum lattice fluctuations on the dimerized ground state is studied in the half-filled-band Takayama-Lin-Liu-Maki model. The nonadiabatic effect due to finite phonon frequency ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$\ensuremath{\gtrsim}0 is treated through an energy-dependent electron-phonon scattering function \ensuremath{\delta}(${\mathit{k}}^{\ensuremath{'}}$,k) introduced by means of a unitary transformation. This leads to a weakening of the effective (adiabatic) potential stabilizing the dimerized state and results in four-fermion interaction. By decoupling of the electronic correlations we show that our approach gives a good description of the continuous variation of the dimerization as functions of the dimensionless coupling constant g2{\mathit{g}}^{2}g2 and the phonon frequency ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$ when the ratio ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$/\ensuremath{\pi}t (t is the electron hopping integral) is not large. Our results show that, at least in the weak-coupling limit, quantum lattice fluctuations change the functional dependence of the dimerization parameter mp{\mathit{m}}_{\mathit{p}}mp​ on the coupling constant even if the ratio ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$/\ensuremath{\pi}t is small (but finite). In the spin-1/2 case our result is the same as that predicted by Fradkin and Hirsch. But in the spinless case we still predict a long-range dimerization order even if the coupling is weak and ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$/\ensuremath{\pi}t is finite. In the large ${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$ (antiadiabatic) limit we show that our effective Hamiltonian becomes an n-component Gross-Neveu model and the ratio mp{\mathit{m}}_{\mathit{p}}mp​(${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$=\ensuremath{\infty})/mp{\mathit{m}}_{\mathit{p}}mp​(${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$=0) is equal to 1/cosh(\ensuremath{\pi}/2ng2{\mathit{ng}}^{2}ng2) (n=1: spinless; n=2: spin-1/2). By using the same input parameters as those of some previous authors we get a 7.2% reduction of the dimerization parameter mp{\mathit{m}}_{\mathit{p}}mp​ compared with the adiabatic value. \textcopyright{} 1996 The American Physical Society.