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Physical review. B, Condensed matter

Phase diagram of one-dimensional electron-phonon systems. II. The molecular-crystal model

J. E. Hirsch, Eduardo Fradkin

Published Apr 1, 1983• 236 citations
Publisher

Abstract

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 ($n$). 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 $\ensuremath{\omega}=\ensuremath{\infty}$.

DOI
10.1103/physrevb.27.4302

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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 (\ensuremathω\ensuremath{\omega}\ensuremathω), electron-phonon coupling constant (\ensuremathλ\ensuremath{\lambda}\ensuremathλ), 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 (\ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0, \ensuremathλ=0\ensuremath{\lambda}=0\ensuremathλ=0) and (\ensuremathω=\ensuremath∞\ensuremath{\omega}=\ensuremath{\infty}\ensuremathω=\ensuremath∞, \ensuremathλ=\ensuremath∞\ensuremath{\lambda}=\ensuremath{\infty}\ensuremathλ=\ensuremath∞). For the case of spin-\textonehalf{} electrons, the system is dimerized for arbitrary \ensuremathλ\ensuremath{\lambda}\ensuremathλ and \ensuremathω\ensuremath{\omega}\ensuremathω except in the limit \ensuremathω=\ensuremath∞\ensuremath{\omega}=\ensuremath{\infty}\ensuremathω=\ensuremath∞.
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 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."
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 $\ensuremath{\lambda}=0$.
The renormalization group technique is applied to one-dimensional electron-phonon Hubbard models at half filling and zero temperature. For the Holstein-Hubbard model, the results of one-loop calculations are congruent with the phase diagram obtained by quantum Monte Carlo simulations in the (U,gph)(U,{g}_{\mathrm{ph}})(U,gph​) plane for the phonon-mediated interaction gph{g}_{\mathrm{ph}}gph​ and the Coulomb interaction UUU. The incursion of an intermediate phase between a fully gapped charge-density-wave state and a Mott antiferromagnet is supported along with the growth of its size with the molecular phonon frequency ${\ensuremath{\omega}}_{0}$. We find additional phases enfolding the base boundary of the intermediate phase. A Luttinger liquid line is found below some critical ${U}^{*}\ensuremath{\approx}{g}_{\mathrm{ph}}^{*}$, followed at larger $U\ensuremath{\sim}{g}_{\mathrm{ph}}$ by a narrow region of bond-order-wave ordering which is either charge or spin gapped depending on UUU. For the Peierls-Hubbard model, the region of the (U,gph)(U,{g}_{\mathrm{ph}})(U,gph​) plane with a fully gapped Peierls-bond-order-wave state shows a growing domination over the Mott gapped antiferromagnet as the Debye frequency ${\ensuremath{\omega}}_{D}$ decreases. A power-law dependence ${g}_{\mathrm{ph}}\ensuremath{\sim}{U}^{2\ensuremath{\eta}}$ is found to map out the boundary between the two phases, whose exponent is in good agreement with the existing quantum Monte Carlo simulations performed when a finite nearest-neighbor repulsion term VVV is added to the Hubbard interaction.
The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (g2{\mathit{g}}^{2}g2\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.
The Hubbard-Holstein model describes fermions on a discrete lattice, with on-site repulsion between fermions and a coupling to phonons that are localized on sites. Generally, at half-filling, increasing the coupling ggg to the phonons drives the system towards a Peierls charge density wave state, whereas increasing the electron-electron interaction UUU drives the fermions into a Mott antiferromagnet. At low ggg and UUU, or when doped, the system is metallic. In one dimension, using quantum Monte Carlo simulations, we study the case where fermions have a long-range coupling to phonons, with characteristic range $\ensuremath{\xi}$, interpolating between the Holstein and Fr\"ohlich limits. Without electron-electron interaction, the fermions adopt a Peierls state when the coupling to the phonons is strong enough. This state is destabilized by a small coupling range $\ensuremath{\xi}$ and leads to a collapse of the fermions, and, consequently, phase separation. Increasing interaction UUU will drive any of these three phases (metallic, Peierls, phase separation) into a Mott insulator phase. The phase separation region is once again present in the $U\ensuremath{\ne}0$ case, even for small values of the coupling range.