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Physical review. B./Physical review. B

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

Y. H. Wang, Yin Zhong

Published Nov 21, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1103/cmj2-ssw8

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This study explores the ground-state phase diagram and topological properties of the spinless 1D Su-Schrieffer-Heeger (SSH) model with nearest-neighbor (NN) interactions at quarter filling. We analyze key physical quantities such as the local electron density distribution, correlation functions for bond-order-wave (BOW) and charge-density-wave (CDW) -- by integrating twisted boundary conditions with the Lanczos technique and employing high-precision numerical diagonalization methods, complemented by a mean-field approximation (MFA) based on bond-order and charge-density modulation analysis. This approach enables precise identification of phase transition critical points. Our results indicate that the system exhibits a topologically trivial band insulating (BI) phase for strong attractive interactions, with its upper boundary forming a downward-opening curve peaking at V/t≃−2.3V/t\simeq-2.3V/t≃−2.3 and extending to V/t≃−2.6V/t\simeq-2.6V/t≃−2.6. Within −2.6≤V/t≤−0.5-2.6 \leq V/t \leq -0.5−2.6≤V/t≤−0.5, a BOW phase emerges for ∣δt/t∣>0.45\left|δt/t\right| > 0.45∣δt/t∣>0.45, with its boundaries converging as ∣δt/t∣\left|δt/t\right|∣δt/t∣ decreases, terminating at a single point at ∣δt/t∣≃0.45\left|δt/t\right|\simeq0.45∣δt/t∣≃0.45. In other parameter regions, a CDW phase is realized. Through this analysis, we elucidate the topological properties of the interacting spinless SSH model at quarter filling, highlighting the competition among CDW, BOW, and BI phases. By tuning VVV and δtδtδt, the system exhibits diverse correlated phenomena, offering new insights into one-dimensional quantum phase transitions and the interplay between topology and order.
超冷原子系统为实现动态 \mathrmZ_2 规范场和探索强关联量子物态提供了重要平台。近年来，动态 \mathrmZ_2 规范场与物质场的耦合已在该平台中得到实现，这引发了人们对\mathrmZ_2 规范理论研究的极大关注。本文研究了具有最近邻交替跃迁的一维无自旋费米子与动态 \mathrmZ_2 规范场在半填充下的耦合，结合微扰理论与密度矩阵重正化群（density matrix renormalization group，DMRG）数值方法，探究局域 \mathrmZ_2 规范约束对体系拓扑性质和有序相行为的影响。在解禁闭极限下，系统被限制在满足局域高斯定律的物理希尔伯特空间，基态纠缠谱呈现二重简并。在强耦合极限下，文中利用微扰理论给出了描述禁闭相中硬核玻色子二聚体的有效 t_B-U_B 模型，发现系统形成无能隙的卢廷格液体（Luttinger liquid，LL）相和有能隙的二聚体密度波（dimer density wave，DDW）相。在一般禁闭下，规范场介导的线性禁闭势促使两个费米子配对形成玻色子二聚体，使原 Su－Schrieffer－Heeger（SSH）模型的单粒子拓扑边缘态描述不再适用。本文发现由于费米子最近邻交替跃迁的竞争，系统形成了两种不同的二聚体密度波（ \mathrmDDW_0 和 \mathrmDDW_1 ）相，它们可通过二聚体的位置加以区分。对二聚体密度振荡的观察表明，两相在体态均表现为玻色子二聚体激发，而在边缘处差异显著。\mathrmDDW_0 相的边缘处保留费米子激发，而DDW _1 相的边缘低能自由度为玻色子二聚体。本研究揭示了一维 \mathrmZ_2 规范SSH 模型在禁闭下诱导出二聚体密度波相的物理机制，为在超冷原子实验中观测规范场中的禁闭动力学提供了思路。
We investigate the quantum phases of a half-filled generalized interacting Su-Schrieffer-Heeger model with intracell, nearest-neighbor, and next-nearest-neighbor intercell hoppings, together with an on-site inter-sublattice interaction. In the noninteracting limit, the model hosts one topologically trivial phase and two symmetry-protected topological (SPT) phases, distinguished under periodic boundary conditions by different winding numbers and under open boundary conditions by two-fold and four-fold entanglement-spectrum degeneracies, respectively. When interactions are introduced, these free-fermion SPT phases evolve into distinct interacting topological phases that retain characteristic signatures such as entanglement-spectrum degeneracy structures, boundary modes, and nonzero string order parameters. For strong repulsive interactions, a symmetry-breaking phase with unequal but spatially uniform sublattice densities appears between the trivial and topological regimes. For strong attractive interactions, period-2 and period-4 charge-density-wave phases emerge from particle clustering. At intermediate attractive interactions, the competition between interaction-induced localization and hopping-induced delocalization gives rise to a Luttinger liquid phase, a paired Luttinger liquid phase, and a gapless symmetry-protected topological (gSPT) phase. The gSPT phase is characterized by a gapless charge mode together with symmetry-protected current-carrying edge states. We further characterize the gapless phases and the associated quantum phase transitions through central charges and critical exponents.
The interplay and competition of topology and electron--electron interactions have fascinated researchers since the discovery of topological insulators. The Su--Schrieffer--Heeger--Hubbard (SSH-Hubbard) model is a prototypical model which includes both nontrivial topology and interactions. Due to its simplicity, there are several artificial quantum systems which can realize such a model to a good approximation. Here we focus on the quarter-filled case, where interactions and dimerization open a charge gap. In particular, we study the single-particle spectral function for the extended SSH-Hubbard model with magnetic field and explore several parameter limits where effective model descriptions arise. In the strongly dimerized limit, we show that the low-energy excitations of the spectral function resemble a half-filled Hubbard model with effective dimer sites and renormalized couplings. For strong magnetic field and interactions, we find physics akin to the spinless Su--Schrieffer--Heeger model at half filling, featuring a noninteracting topological phase transition. Moreover, in light of the recent realization of this model in quantum dot simulation, we provide evidence for the stability of the topological phase towards moderate nonlocal interactions in the experimentally expected parameter range.
The density matrix renormalization group and quantum Monte Carlo methods are used to describe coupled trimer chains in a magnetic field hhh. The Hamiltonian contains exchange terms involving the intratrimer coupling J1{J}_{1}J1​ (taken as the unit of energy) and the intertrimer coupling J2{J}_{2}J2​, plus the Zeeman interaction for a magnetic field hhh along the zzz direction. Results for the magnetization per trimer mmm are calculated in regimes of positive and negative values of the ratio J=J2/J1J={J}_{2}/{J}_{1}J=J2​/J1​, from which the rich field-induced ground state phase diagram hhh versus JJJ is derived, with the presence of a Luttinger liquid, the 1/3 plateau (m=1/2m=1/2m=1/2), and the one of fully polarized magnetization (m=3/2m=3/2m=3/2). Also, zero-field Lanczos calculation of the spin-wave dispersion from the 1/3 plateau for Sz=1{S}^{z}=1Sz=1 is shown at the previous regimes of JJJ values. In addition, we also report on the decay of correlation functions of trimers along open chains, as well as the average two-magnon distribution. The ground state is ferrimagnetic for $0<J\ensuremath{\le}1$, and is a singlet for $\ensuremath{-}1\ensuremath{\le}J<0$. In the singlet phase, the spin correlation functions along the legs present an antiferromagnetic power-law decay, similar to the spin-1/2 linear chain, thus suggesting that the ground state is made of three coupled antiferromagnetically oriented chains. In the singlet phase, the dimensionless thermal magnetic susceptibility per site normalized by 1/∣J∣1/|J|1/∣J∣ gets closer to $1/{\ensuremath{\pi}}^{2}$ as the temperature $T\ensuremath{\rightarrow}0$. For the ferrimagnetic phase, we fit the susceptibility to the experimental data for the compound Pb3Cu3(PO4)4{\mathrm{Pb}}_{3}{\mathrm{Cu}}_{3}{({\mathrm{PO}}_{4})}_{4}Pb3​Cu3​(PO4​)4​ and estimate the model exchange couplings: J1=74.8K{J}_{1}=74.8\phantom{\rule{4pt}{0ex}}\text{K}J1​=74.8K and J=0.4J=0.4J=0.4. These values imply a range of energies for the magnon excitations that are in accord with the data from neutron scattering experiments on Pb3Cu3(PO4)4{\mathrm{Pb}}_{3}{\mathrm{Cu}}_{3}{({\mathrm{PO}}_{4})}_{4}Pb3​Cu3​(PO4​)4​ for two excitation modes. The 1/3 plateau closes only at 1/∣J∣=01/|J|=01/∣J∣=0 with J<0J<0J<0.
We explore the phase diagram of the extended attractive SU(3)\mathrm{SU}(3)SU(3) Hubbard chain with two-body hopping and nearest-neighbor attraction at half-filling. In the large on-site attraction limit, we identify three different phases: phase separation (PS); Tomonaga-Luttinger liquid (TLL); and charge density wave (CDW). Our analysis reveals that the $\ensuremath{\eta}$-clustering state, a three-component generalization of the $\ensuremath{\eta}$-pairing state, becomes the ground state at the boundary between the PS and TLL phases. On an open chain, this state exhibits an edge-edge correlation, which we call boundary off-diagonal long-range order (bODLRO). Using the density matrix renormalization group (DMRG) method, we numerically study the phase diagram of the model with large but finite on-site interactions and find that the numerical results align with those obtained in the strong coupling limit.