Ground-state phase diagram of the quarter-filled interacting Su-Schrieffer-Heeger chain

We map the ground-state phase diagram of the quarter-filled spinless Su-Schrieffer-Heeger chain with nearest-neighbor interactions by combining exact diagonalization (ED) and density-matrix renormalization group (DMRG) under twist-averaged boundary conditions to suppress shell effects. A unified scaling analysis of the equal-time charge structure factor ${S}_{\mathrm{cdw}}(q)$---including the size dependence of ${S}_{\mathrm{cdw}}({q}_{min})$, the extrapolation of ${S}_{\mathrm{cdw}}({q}_{min})/L$ versus $1/L$, the evolution of ${K}_{\mathrm{eff}}(L)=\frac{1}{2}L\phantom{\rule{0.16em}{0ex}}{S}_{\mathrm{cdw}}({q}_{min})$, and the power-law behavior of ${S}_{\mathrm{cdw}}(2{k}_{F})$---disentangles phase separation (PS), Luttinger liquids with pairing-dominated correlations ($K>1/2$), Luttinger liquids dominated by $2{k}_{F}$ bond-order-wave (BOW) correlations ($K<1/2$), and a long-range $2{k}_{F}$ charge-density-wave (CDW) phase. The phase structure, controlled by the interaction $V/t$ and dimerization $\ensuremath{\delta}t/t$, is as follows: PS for $V/t\ensuremath{\lesssim}\ensuremath{-}2$; a pairing-dominated Luttinger liquid for $\ensuremath{-}2<V/t<0$; for $V/t>0$ and moderate $|\ensuremath{\delta}t/t|$ the system realizes a $2{k}_{F}$-BOW-dominated Luttinger liquid; and for approximately $V/t\ensuremath{\gtrsim}1.5$ together with $|\ensuremath{\delta}t/t|\ensuremath{\lesssim}0.9$ it enters a long-range $2{k}_{F}$ CDW, whose domain expands toward smaller $|\ensuremath{\delta}t/t|$ with increasing $V/t$. Quantitative agreement between ED and DMRG across system sizes and boundary treatments underscores the robustness of this phase map and provides a benchmark for the competition between interactions and dimerization in one-dimensional dimerized chains.