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Physical review. B, Condensed matter

Finite-size effects and quantum phonon fluctuations in the optical absorption edgeof dimerized chains

Ioan Bâldea, Horst Köppel, Lorenz S. Cederbaum

Published Jan 15, 1997• 16 citations
Publisher

Abstract

We study effects of quantum phonons on the optical conductivity \ensuremath{\sigma}(\ensuremath{\omega}) for finite dimerized chains by using two numerical methods: exact numerical diagonalization and by factorizing the electron-phonon wave function (AA, adiabatic ansatz). For small chains, we compare the results of these methods and find a criterion for a valid AA. AA turns out to be: (a) a substantial improvement with respect to the adiabatic approximation usually employed in condensed-matter physics, and (b) much less affected by computer storage limitations than exact diagonalization. We use AA and get \ensuremath{\sigma}(\ensuremath{\omega}) reliably for sufficiently large chains, which could be realized experimentally. The optical gap as well as the \ensuremath{\sigma}(\ensuremath{\omega}) curve near the absorption edge possess a surprisingly strong dependence on the chain size; this could be relevant for experiments on mesoscopic systems.

DOI
10.1103/physrevb.55.1481

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We study effects of quantum phonons on the optical conductivity \ensuremath{\sigma}(\ensuremath{\omega}) for finite dimerized chains by using two numerical methods: exact numerical diagonalization and by factorizing the electron-phonon wave function (AA, adiabatic ansatz). For small chains, we compare the results of these methods and find a criterion for a valid AA. AA turns out to be: (a) a substantial improvement with respect to the adiabatic approximation usually employed in condensed-matter physics, and (b) much less affected by computer storage limitations than exact diagonalization. We use AA and get \ensuremath{\sigma}(\ensuremath{\omega}) reliably for sufficiently large chains, which could be realized experimentally. The optical gap as well as the \ensuremath{\sigma}(\ensuremath{\omega}) curve near the absorption edge possess a surprisingly strong dependence on the chain size; this could be relevant for experiments on mesoscopic systems.
The effects of quantum lattice fluctuations on the Peierls transition and the optical conductivity in the one-dimensional Holstein model of spinless fermions have been studied by developing an analytical approach based on the unitary transformation method. We show that when the electron-phonon coupling constant decreases to a finite critical value, the Peierls dimerization is destroyed by the quantum lattice fluctuations. The dimerization gap is much more reduced by the quantum lattice fluctuations than the phonon order parameter. The calculated optical conductivity does not have the inverse-square-root singularity, but it has a peak above the gap edge, and there exists a significant tail below the peak. The peak of optical-conductivity spectrum is not directly corresponding to the dimerized gap. Our results of the phase diagram and the spectral-weight function agree with those of the density-matrix renormalization group and the exact diagonalization methods.
We present exact numerical results for the effects of thermal fluctuations on the experimentally relevant thermodynamic and spectral properties of Peierls chains. To this end, a combination of classical Monte Carlo sampling and exact diagonalization is used to study adiabatic half-filled Holstein and Su-Schrieffer-Heeger models. The classical nature of the lattice displacements in combination with parallel tempering permit simulations on large system sizes and a direct calculation of spectral functions in the frequency domain. Most notably, the long-range order and the associated Peierls gap give rise to a distinct low-temperature peak in the specific heat. The closing of the gap and suppression of order by thermal fluctuations involves in-gap excitations in the form of soliton-antisoliton pairs and is also reflected in the dynamic density and bond structure factors as well as in the optical conductivity. We compare our data to the widely used mean-field approximation and highlight relations to symmetry-protected topological phases and disorder problems.
We investigate the Peierls transition in nite chains by exact (Lanczos) diagonalization and within a seminumerical method based on the factorization of the electron-phonon wave function (Adiabatic Ansatz, AA). AA can be applied for mesoscopic chains up to micrometer sizes and its reliability can be checked self-consistently. Our study demonstrates the important role played for nite systems by the tunneling in the double well potential. The chains are dimerized only if their size N exceeds a critical value Nc which increases with increasing phonon frequency. Quantum phonon fluctuations yield a broad transition region. This smooth Peierls transition contrasts not only to the sharp mean eld transition, but also with the sharp RPA soft mode instability, although RPA partially accounts for quantum phonon fluctuations. For weak coupling the dimerization disappears below micrometer sizes; therefore, this eect could be detected experimentally in mesoscopic systems.
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.
The optical conductivity, \ensuremath{\sigma}(\ensuremath{\omega}), of the two-dimensional one-band Hubbard model is calculated at finite temperature using exact diagonalization techniques on finite clusters. The in-plane dc resistivity, ${\mathrm{\ensuremath{\rho}}}_{\mathit{a}\mathit{b}}$, is also evalauted. We find that at large U/t and temperature T, ${\mathrm{\ensuremath{\rho}}}_{\mathit{a}\mathit{b}}$ is approximately linear with temperature. We also observed that \ensuremath{\sigma}(\ensuremath{\omega}) displays charge excitations, a midinfrared (MIR) band, and a Drude peak. The combination of the Drude peak and the MIR oscillator strengths leads to a conductivity that decays slower than 1/${\mathrm{\ensuremath{\omega}}}^{2}$ at energies smaller than the insulator gap near half filling.
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.