Phase separation in the Edwards model

The nature of charge transport within a correlated background medium can be described by spinless fermions coupled to bosons in the model introduced by Edwards. Combining numerical density matrix renormalization group and analytical projector-based renormalization methods, we explore the ground-state phase diagram of the Edwards model in one dimension. Below a critical boson frequency, any long-range order disappears and the system becomes metallic. If the charge carriers are coupled to slow quantum bosons, the Tomonaga-Luttinger liquid is attractive and finally makes room for a phase separated state, just as in the $t$-$J$ model. The phase boundary separating the repulsive from the attractive Tomonaga-Luttinger liquid is determined from long-wavelength charge correlations, whereas fermion segregation is indicated by a vanishing inverse compressibility. On approaching phase separation, the photoemission spectra develop strong anomalies.

Figure 1

Finite-size scaling of the TLL parameter $K_{\rho }$ (left panel) and inverse compressibility ${\kappa }^{-1}$ (right panel) for the 1D Edwards model with $\lambda =0.2$ at $n=1/8$. Insets show the variation with ${\omega }_0$ of the $L\to \infty$ extrapolated values. Results are obtained by DMRG for a lattice with $L$ sites and OBC. In all DMRG runs, we take into account up to $n_b=4$ bosonic pseudosites and determine $n_b$ by the requirement that the local boson density of the last pseudosite is less than ${10}^{-6}$ for all $j$. Using selectively $n_b=5$, we controlled that convergence is truly achieved. Furthermore, we keep up to $m=1200$ density-matrix eigenstates in the renormalization steps to ensure that the discarded weight is smaller than ${10}^{-8}$.

Figure 2

Ground-state phase diagram of the 1D Edwards model, showing repulsive TLL ($K_{\rho }<1$), attractive TLL ($K_{\rho }>1$), and PS (${\kappa }^{-1}=0$) regions. The phase boundaries were obtained by DMRG (filled symbols) in the course of a careful finite-size scaling analysis and for the infinite system directly by PRM (open symbols). The upper panel displays the phase diagram as a function of $n$, varying ${\omega }_0$ at fixed $\lambda =0.2$; the lower panel gives the phase diagram in the $\lambda$-${\omega }_0$ plane for fixed density $n=1/8$.

Figure 3

Local particle (filled circles) and boson (open squares) densities for the 1D Edwards model with $n=1/8$, calculated for a 48-site system with OBC at various $\lambda$ and ${\omega }_0$.

Figure 4

Line-shape of the single-particle spectral function $A(k,\omega )$ for the 1D Edwards model with $\lambda =0.2$ at $n=1/8$. Results are obtained by DDMRG for a 16-site chain with quasimomenta $k=\pi s/(L+1)$, using a broadening $\eta =0.2$.

Figure 5

PRM (inverse) photoemission spectrum [$A^{+}(k,\omega )$] $A^{-}(k,\omega )$ for the Edwards model with $\lambda =0.2$, $n=3/10$. The bold line marks the signal at the Fermi momentum $k_F=0.3\pi$.