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Luttinger parameters and momentum distribution function for the half-filled spinless fermion Holstein model: A DMRG approach

We reexamine the nature of the metallic phase of the one-dimensional half-filled Holstein model of spinless fermions. To this end we determine the Tomonaga-Luttinger-liquid correlation parameter $K_ρ$ by large-scale density-matrix renormalisation-group (DMRG) calculations, exploiting (i) the leading-order scaling relations between the ground-state energy and the single-particle excitation gap and (ii) the static charge structure factor in the long-wavelength limit. While both approaches give almost identical results for intermediate-to-large phonon frequencies, we find contrasting behaviour in the adiabatic regime: (i) $K_ρ>1$ (attractive) versus (ii) $K_ρ<1$ (repulsive). The latter result for the correlation exponent is corroborated by data obtained for the momentum distribution function $n(k)$, which puts the existence of an attractive metallic state in the spinless fermion Holstein model into question. We conclude that the scaling relation must be modified in the presence of electron-phonon interactions with noticeable retardation.