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Acta Physica Sinica

Bose Density Wave Phase of One-Dimensional Spinless Fermions Coupled to Z2 Gauge Field

ZHU Jun, Liang Junjun

Published Oct 29, 2026• 0 citations

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Abstract

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。

DOI: 10.7498/aps.76.20261219

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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 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.

基于我们考虑由二断开的 Luttinger 液体组成的 spinful 系统的地理论的 bosonization，它在时间 t = 被联合 0 通过 chiral 密度密度相互作用。我们借助于 bosonization 和 Bogoliubov 转变在这个 1D 系统显示出著名纺纱费用分离。评估经分解一个粒子相等时间的关联功能， Cazalilla 的方法被扩大到纺纱费用部门。在关联的行为不与平衡的与一致的长时间限制，二的相等时间的关联联合了 spinful Luttinger 液体，这被发现。以便与试验性的结果作比较，管理非平衡系统的相等时间的关联的幂定律行为的改进代表在这个工作的结束被获得。

在这篇文章，我们在二维的强烈相关的费米子系统在量阶段转变上考察我们的最近的工作。我们由用细胞的动态吝啬地的理论(CDMFT ) 和连续时间的量蒙特卡罗算法的联合计算状态，两倍占有，和费密表面进化的密度讨论这些系统的 metalinsulator 转变性质。而且，我们由定义磁性的顺序参数探索每个状态的磁性。包括 antiferromagnetic 金属，顺磁的金属， Kondo 金属，和铁磁性的绝缘体，有许多吸引人的量状态的富有的阶段图被作出对有利的裁决有强烈相关的费米子的二维的格子。我们相信我们的结果将导致真实材料的性质的更好的理解。

We investigate a quantum many-body lattice system of one-dimensional spinless fermions interacting with a dynamical Z_{2} gauge field. The gauge field mediates long-range attraction between fermions resulting in their confinement into bosonic dimers. At strong coupling we develop an exactly solvable effective theory of such dimers with emergent constraints. Even at generic coupling and fermion density, the model can be rewritten as a local spin chain. Using the density matrix renormalization group the system is shown to form a Luttinger liquid, indicating the emergence of fractionalized excitations despite the confinement of lattice fermions. In a finite chain we observe the doubling of the period of Friedel oscillations which paves the way towards experimental detection of confinement in this system. We discuss the possibility of a Mott phase at the commensurate filling 2/3.

We study one-dimensional Su-Schrieffer-Heeger (SSH) models with quantum phonons using a continuous-time quantum Monte Carlo method. Within statistical errors, we obtain identical results for the SSH model with acoustic phonons, and a related model with a coupling to an optical bond phonon mode. Based on this agreement, we first study the Peierls metal-insulator transition of the spinless SSH model, and relate it to the Kosterlitz-Thouless transition of a spinless Luttinger liquid. In the Peierls phase, the spectral functions reveal the single-particle and charge gap, and a central peak related to long-range order. For the spinful SSH model, which has a dimerized ground state for any nonzero coupling, we reveal a symmetry-related degeneracy of spin and charge excitations, and the expected spin and charge gaps as well as a central peak. Finally, we study the SSH-

UV

model with electron-phonon and electron-electron interaction. We observe a Mott phase with critical spin and bond correlations at weak electron-phonon coupling, and a Peierls phase with gapped spin excitations at strong coupling. We relate our findings to the extended Hubbard model, and discuss the physical origin of the agreement between optical and acoustic phonons.