Effect of quantum lattice fluctuations on the dimerized ground state of the Takayama-Lin-Liu-Maki model

Abstract

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 ${\mathit{g}}^{2}$ 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 ${\mathit{m}}_{\mathit{p}}$ 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 ${\mathit{m}}_{\mathit{p}}$(${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$=\ensuremath{\infty})/${\mathit{m}}_{\mathit{p}}$(${\mathrm{\ensuremath{\omega}}}_{2{\mathit{k}}_{\mathit{F}}}$=0) is equal to 1/cosh(\ensuremath{\pi}/2${\mathit{ng}}^{2}$) (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 ${\mathit{m}}_{\mathit{p}}$ compared with the adiabatic value. \textcopyright{} 1996 The American Physical Society.