Spectral signatures of the Luttinger liquid to the charge-density-wave transition

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.

Figure 1

DMRG ground-state phase diagram of the 1D half-filled HMSF (cf. Refs. 2, 9). In the adiabatic limit ${\omega }_0\to 0$ the critical dimensionless coupling constant ${\lambda }_{\mathrm{c}}$ converges to zero. For ${\omega }_0>0$, due to quantum phonon fluctuations, there exists a finite critical coupling $g_{\mathrm{c}}\left({\omega }_0\right)$. For ${\omega }_0\to \infty$, the model exhibits a Kosterlitz-Thouless phase transition near $g_{\mathrm{c}}^{*}=g_{\mathrm{c}}({\omega }_0\to \infty )$.[2] Crosses indicate parameter sets used below.

Figure 2

(Color online) Left panel [(a), (c)]: Spectral density for photoemission [$A^{-}(k,\omega )$; red dashed lines] and inverse photoemission [$A^{+}(k,\omega )$; black solid lines] from CPT in the metallic LL phase. Energies are measured relative to the Fermi level $E_{\mathrm{F}}$. Right panel [(b), (d)]: Corresponding exact phonon spectral function $B(Q,\omega )$ for the allowed wave numbers $Q$ of $N$-site clusters. Electron and phonon spectra have been calculated in the adiabatic [attractive LL; (a), (b)] and antiadiabatic [repulsive LL; (c), (d)] weak-to-intermediate EP coupling regimes. Dashed vertical lines in (b), (d) indicate the bare phonon frequency.

Figure 3

(Color online) As in Fig. 2, but for parameters in the insulating CDW phase, and for the case of a band insulator [(a), (b)] and a polaronic superlattice [(c), (d)].

Figure 4

(Color online) Density plots of the (CPT) phonon spectral function $B(q,\omega )$, where panels (a), (b), (c), and (d) correspond to the attractive LL (${\omega }_0∕t=0.4$, $g^2=2.5$), band insulator (${\omega }_0∕t=0.4$, $g^2=5$), repulsive LL (${\omega }_0∕t=4$, $g^2=0.5$), and polaron superlattice regimes (${\omega }_0∕t=4$, $g^2=5$), respectively.