In recent experiments with ultracold atoms, both two-dimensional (2d) Chern insulators and one-dimensional (1d) topological charge pumps have been realized. Without interactions, both systems can be described by the same Hamiltonian, when some variables are being reinterpreted. In this paper, we study the relation of both models when Hubbard interactions are added, using the density-matrix renormalization-group algorithm. To this end, we express the fermionic Hofstadter model in a hybrid-space representation, and define a family of interactions, which connects 1d Hubbard charge pumps to 2d Hubbard Chern insulators. We study a three-band model at particle density $\rho =2/3$, where the topological quantization of the 1d charge pump changes from Chern number $C=2$ to $C=-1$ as the interaction strength increases. We find that the $C=-1$ phase is robust when varying the interaction terms on narrow-width cylinders. However, this phase does not extend to the limit of the 2d Hofstadter-Hubbard model, which remains in the $C=2$ phase. We discuss the existence of both topological phases for the largest cylinder circumferences that we can access numerically. We note the appearance of a ferromagnetic ground state between the strongly interacting 1d and 2d models. For this ferromagnetic state, one can understand the $C=-1$ phase from a bandstructure argument. Our method for measuring the Hall conductivity could similarly be realized in experiments: We compute the current response to a weak, linear potential, which is applied adiabatically. The Hall conductivity converges to integer-quantized values for large system sizes, corresponding to the system’s Chern number.

中文翻译：

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混合空间阶梯上费米-霍夫施塔特-哈伯德模型中的拓扑阶段

在最近的超冷原子实验中，已经实现了二维（2d）Chern绝缘子和一维（1d）拓扑电荷泵。如果没有交互作用，则在重新解释某些变量时，两个系统可以用相同的哈密顿量描述。在本文中，我们使用密度矩阵重归一化组算法研究了添加哈伯德相互作用时两个模型的关系。为此，我们用混合空间表示法表示费米离子的Hofstadter模型，并定义了一系列相互作用，将1d Hubbard电荷泵连接到2d Hubbard Chern绝缘子。我们研究了粒子密度的三波段模型$\rho =2/3$，其中一维电荷泵的拓扑量化从陈恩数变化 $C=2$ 至 $C=-\mathrm{1个}$随着交互强度的增加。我们发现$C=-\mathrm{1个}$当更改窄宽度圆柱体上的相互作用项时，相位稳定。但是，此阶段并未扩展到2d Hofstadter-Hubbard模型的极限，后者仍保留在$C=2$相。我们讨论了可以通过数值访问的最大圆柱周长的两个拓扑阶段的存在。我们注意到在强相互作用的1d和2d模型之间出现了铁磁基态。对于这种铁磁状态，人们可以理解$C=-\mathrm{1个}$带结构参数的相位。我们测量霍尔电导率的方法可以类似地在实验中实现：我们计算对绝热施加的弱线性电位的电流响应。对于大系统尺寸，霍尔电导率收敛到整数量化值，对应于系统的Chern数。