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arXiv:0807.2341 · Physical Review B

Spin, charge, and single-particle spectral functions of the one-dimensional quarter filled Holstein model

Fakher F. Assaad

Published Oct 24, 2008• 25 citations
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Abstract

We use a recently developed extension of the weak-coupling diagrammatic determinantal quantum Monte Carlo method to investigate the spin, charge, and single-particle spectral functions of the one-dimensional quarter filled Holstein model with phonon frequency ${\ensuremath{\omega}}_{0}=0.1t$. As a function of the dimensionless electron-phonon coupling we observe a transition from a Luttinger to a Luther-Emery liquid with dominant $2{k}_{f}$ charge fluctuations. Emphasis is placed on the temperature dependence of the single-particle spectral function. At high temperatures and in both phases it is well accounted for within a self-consistent Born approximation. In the low-temperature Luttinger liquid phase we observe features that compare favorably with a bosonization approach retaining only forward scattering. In the Luther-Emery phase, the spectral function at low temperatures shows a quasiparticle gap that matches half the spin gap, whereas at temperatures above which this quasiparticle gap closes characteristic features of the Luttinger liquid model are apparent. Our results are based on lattice simulations on chains up to $L=20$ for two-particle properties and on cluster dynamical mean-field theory calculations with clusters up to 12 sites for the single-particle spectral function.

DOI
10.1103/physrevb.78.155124

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We use a recently developed extension of the weak-coupling diagrammatic determinantal quantum Monte Carlo method to investigate the spin, charge, and single-particle spectral functions of the one-dimensional quarter filled Holstein model with phonon frequency \ensuremathω0=0.1t{\ensuremath{\omega}}_{0}=0.1t\ensuremathω0​=0.1t. As a function of the dimensionless electron-phonon coupling we observe a transition from a Luttinger to a Luther-Emery liquid with dominant 2kf2{k}_{f}2kf​ charge fluctuations. Emphasis is placed on the temperature dependence of the single-particle spectral function. At high temperatures and in both phases it is well accounted for within a self-consistent Born approximation. In the low-temperature Luttinger liquid phase we observe features that compare favorably with a bosonization approach retaining only forward scattering. In the Luther-Emery phase, the spectral function at low temperatures shows a quasiparticle gap that matches half the spin gap, whereas at temperatures above which this quasiparticle gap closes characteristic features of the Luttinger liquid model are apparent. Our results are based on lattice simulations on chains up to L=20L=20L=20 for two-particle properties and on cluster dynamical mean-field theory calculations with clusters up to 12 sites for the single-particle spectral function.
Single- and two-particle excitation spectra of the one-dimensional, half-filled Holstein-Hubbard model are calculated using the continuous-time quantum Monte Carlo method. In the metallic phase, the results are consistent with a Luther-Emery liquid that has gapped spin and single-particle excitations but a gapless charge mode. However, given the initially exponential dependence of the spin gap on the backscattering matrix element, the numerical excitation spectra appear gapless in the weak-coupling regime, and therefore resemble those of a Luttinger liquid. The Mott phase has the expected charge gap and gapless spin excitations. The Peierls state shows a charge, spin, and single-particle gap, a soft phonon mode, backfolded shadow bands, and soliton excitations. Arguments and numerical evidence for the existence of a nonzero spin gap throughout the metallic phase are provided in terms of equal-time spin and charge correlation functions.
Electron- and phonon spectral functions of the one-dimensional, spinless-fermion Holstein model at half filling are calculated in the four distinct regimes of the phase diagram, corresponding to an attractive or repulsive Luttinger liquid at weak electron-phonon coupling, and a band- or polaronic insulator at strong coupling. The results obtained by means of kernel polynomial and systematic cluster approaches reveal substantially different physics in these regimes and further indicate that the size of the phonon frequency significantly affects the nature of the quantum Peierls phase transition, the latter being either of the soft-mode or central-peak type. The generic features observed are relevant to several classes of low-dimensional materials.
The quantum phase transition between a repulsive Luttinger liquid and an insulating Peierls state is studied in the framework of the one-dimensional spinless Holstein model. We focus on the adiabatic regime but include the full quantum dynamics of the phonons. Using continuous-time quantum Monte Carlo simulations, we track, in particular, the dynamic charge structure factor and the single-particle spectrum across the transition. With increasing electron-phonon coupling, the dynamic charge structure factor reveals the emergence of a charge gap and a clear signature of phonon softening at the zone boundary. The single-particle spectral function evolves continuously across the transition. Hybridization of the charge and phonon modes of the Luttinger liquid description leads to two modes, one of which corresponds to the coherent polaron band. This band acquires a gap upon entering the Peierls phase, whereas the other mode constitutes the incoherent, high-energy spectrum with backfolded shadow bands. Coherent polaronic motion is a direct consequence of quantum lattice fluctuations. In the strong-coupling regime, the spectrum is described by the static, mean-field limit. Importantly, whereas finite electron density in general leads to screening of polaron effects, the latter reappear at half-filling due to charge ordering and lattice dimerization.
The one-dimensional Holstein model and its generalizations have been studied extensively to understand the effects of electron-phonon interaction. The half-filled case is of particular interest, as it describes a transition from a metallic phase with a spin gap due to attractive backscattering to a Peierls insulator with charge-density-wave order. Our quantum Monte Carlo results support the existence of a metallic phase with dominant power-law charge correlations, as described by the Luther-Emery fixed point. We demonstrate that for Holstein and also for purely fermionic models the spin gap significantly complicates finite-size numerical studies, and explains inconsistent previous results for Luttinger parameters and phase boundaries. On the other hand, no such complications arise in spinless models. The correct low-energy theory of the spinful Holstein model is argued to be that of singlet bipolarons with a repulsive, mutual interaction. This picture naturally explains the existence of a metallic phase, but also implies that gapless Luttinger liquid theory is not applicable.
We compute the finite-temperature single-particle spectral function of a one-dimensional Luttinger liquid coupled to an optical phonon band. The calculation is performed exactly for the case in which electron-phonon coupling is purely forward scattering. We extend the results to include backward scattering with a renormalization group treatment. The dispersion contains a change in velocity at the phonon energy, qualitatively similar to the case of electron-phonon coupling in a Fermi liquid. If the backward scattering part of the the electron-phonon interaction is not too strong compared to the forward scattering part, coupling to phonons also produces a pronounced peak in the spectral function at low energies. The calculated spectral function is remarkably similar to the angle-resolved photoemission spectra of the high-temperature superconductors, including the apparent presence of ``nodal quasiparticles,'' the presence of a ``kink'' in the dispersion, and the non-Fermi-liquid frequency and temperature dependencies. Although a microscopic justification has not been established for treating the electronic dynamics of the cuprates as quasi-one-dimensional, at the very least we take the quality of the comparison as evidence of the non-Fermi-liquid character of the measured spectra.
We study the dynamical properties of the two-dimensional square lattice Holstein Hamiltonian using numerically exact determinant quantum Monte Carlo simulations. In particular, we report systematic calculations of the model's single-particle spectral function and optical conductivity over a range of phonon energies, electron-phonon coupling strengths, carrier concentrations, and temperatures. In doing so, we map the evolution of the system from a dressed metallic phase to a (bi)polaron liquid/insulator to a charge-density-wave insulator as the carrier concentration is tuned from dilute values to half-filling. We corroborate these results using several equal-time correlation measurements and related proxy quantities reconstructed from time-displaced correlation measurements. This paper is also accompanied by an extensive open data set, covering over 16,640 unique parameter values.