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arXiv:1010.5920 · Physical Review B

Quantized lattice dynamic effects on the Peierls transition of the extended Hubbard-Peierls model

Christopher J. Pearson, William Barford, Robert J. Bursill

Published May 4, 2011• 16 citations
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Abstract

The density matrix renormalization group method is used to investigate the Peierls transition for the extended Hubbard model coupled to quantized phonons. Following our earlier work on spin-Peierls systems, we use a phonon spectrum that interpolates between a gapped, dispersionless (Einstein) limit and a gapless, dispersive (Debye) limit to investigate the entire frequency range. A variety of theoretical probes are used to determine the quantum phase transition, including energy gap crossing, a finite-size scaling analysis, and bipartite quantum entanglement. All these probes indicate that a transition of Berezinskii-Kosterlitz-Thouless type is observed at a nonzero electron-phonon coupling ${g}_{c}$ for a nonvanishing electron-electron interaction. An extrapolation from the Einstein limit to the Debye limit is accompanied by an increase in ${g}_{c}$ for a fixed optical ($q=\ensuremath{\pi}$) phonon gap. We therefore conclude that the dimerized ground state is more unstable with respect to Debye phonons, with the introduction of phonon dispersion renormalizing the effective electron-lattice coupling for the Peierls-active mode. By varying the Coulomb interaction $U$, we observe a generalized Peierls transition, intermediate between the uncorrelated ($U=0$) and spin-Peierls ($U\ensuremath{\rightarrow}\ensuremath{\infty}$) limits, where $U$ is the Hubbard Coulomb parameter. Using the extended Hubbard model with Debye phonons, we investigate the Peierls transition in trans-polyacetylene and show that the transition is close to the critical regime.

DOI
10.1103/physrevb.83.195105

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The density matrix renormalization group method is used to investigate the Peierls transition for the extended Hubbard model coupled to quantized phonons. Following our earlier work on spin-Peierls systems, we use a phonon spectrum that interpolates between a gapped, dispersionless (Einstein) limit and a gapless, dispersive (Debye) limit to investigate the entire frequency range. A variety of theoretical probes are used to determine the quantum phase transition, including energy gap crossing, a finite-size scaling analysis, and bipartite quantum entanglement. All these probes indicate that a transition of Berezinskii-Kosterlitz-Thouless type is observed at a nonzero electron-phonon coupling gc{g}_{c}gc​ for a nonvanishing electron-electron interaction. An extrapolation from the Einstein limit to the Debye limit is accompanied by an increase in gc{g}_{c}gc​ for a fixed optical (q=\ensuremathπq=\ensuremath{\pi}q=\ensuremathπ) phonon gap. We therefore conclude that the dimerized ground state is more unstable with respect to Debye phonons, with the introduction of phonon dispersion renormalizing the effective electron-lattice coupling for the Peierls-active mode. By varying the Coulomb interaction UUU, we observe a generalized Peierls transition, intermediate between the uncorrelated (U=0U=0U=0) and spin-Peierls (U\ensuremath→\ensuremath∞U\ensuremath{\rightarrow}\ensuremath{\infty}U\ensuremath→\ensuremath∞) limits, where UUU is the Hubbard Coulomb parameter. Using the extended Hubbard model with Debye phonons, we investigate the Peierls transition in trans-polyacetylene and show that the transition is close to the critical regime.
The density-matrix renormalization-group method is used to investigate the spin-Peierls transition for Heisenberg spins coupled to quantized phonons. We use a phonon spectrum that interpolates between a gapped, dispersionless (Einstein) limit to a gapless, dispersive (Debye) limit. A variety of theoretical probes are used to determine the quantum phase transition, including energy gap crossing, a finite-size scaling analysis, bond-order autocorrelation functions, and bipartite quantum entanglement. All these probes indicate that in the antiadiabatic phonon limit a quantum phase transition of the Berezinskii-Kosterlitz-Thouless type is observed at a nonzero spin-phonon coupling, gc{g}_{\text{c}}gc​. An extrapolation from the Einstein limit to the Debye limit is accompanied by an increase in gc{g}_{\text{c}}gc​ for a fixed optical $(q=\ensuremath{\pi})$ phonon gap. We therefore conclude that the dimerized ground state is more unstable with respect to Debye phonons with the introduction of phonon-dispersion renormalizing the effective spin-lattice coupling for the Peierls-active mode. We also show that the staggered spin-spin and phonon displacement order parameters are unreliable means of determining the transition.
We use the density matrix renormalization group method to investigate the role of gapless, dispersive quantized phonons on the Peierls transition in the extended Hubbard-Peierls model. We calculate the staggered phonon order parameter as a function of the Coulomb interaction and the phonon frequency. For all parameter ranges we show that the staggered phonon order parameter scales as $\sqrt{\ensuremath{\lambda}}$ (and the dimerized bond order scales as $\ensuremath{\lambda}$) as $\ensuremath{\lambda}\ensuremath{\rightarrow}0$ (where $\ensuremath{\lambda}$ is the electron-phonon interaction). We also show that the quantum phonon predictions for the bond order follow the classical (adiabatic) predictions as a function of inverse chain size for small $\ensuremath{\lambda}$. We thus conclude that quantum fluctuations in the nuclear degrees of freedom do not destroy the Peierls state for this model.
We report quantum Monte Carlo (stochastic series expansion) results for the transition from a Mott insulator to a dimerized Peierls insulating state in a half-filled, one-dimensional extended Hubbard model coupled to optical bond phonons. Using electron-electron (e-e) interaction parameters corresponding approximately to polyacetylene, we show that the Mott-Peierls transition occurs at a finite value of the electron-phonon (e-ph) coupling. We discuss several different criteria for detecting the transition and show that they give consistent results. We calculate the critical e-ph coupling as a function of the bare phonon frequency and also investigate the sensitivity of the critical coupling to the strength of the e-e interaction. In the limit of strong e-e couplings, we map the model to a spin-Peierls chain and compare the phase boundary with previous results for the spin-Peierls transition. We point out effects of a nonlinear spin-phonon coupling neglected in the mapping to the spin-Peierls model.
"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
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.