Discrete Shift and Polarization from Response to Symmetry Defects in Interacting Topological Phases

TL;DR

The study investigates how crystalline symmetry protected topological (SPT) invariants manifest in interacting 2D lattice systems by analyzing the Hofstadter model with symmetry defects. Using DMRG on an interacting Hofstadter lattice with disclinations and dislocations, it extracts defect-bound charges to obtain the topological invariants $\mathscr{S}_{\text{o}}$ and $\vec{\mathscr{P}}_{\text{o}}$, observing quantization in both the integer quantum Hall (IQH) and charge density wave (CDW) regimes. The results show $\mathscr{S}_{\text{o}}$ and $\vec{\mathscr{P}}_{\text{o}}$ are robust to interactions and finite-size effects, confirming the crystalline SPT classification persists in strongly correlated phases and enabling defect-based probes via matrix product state methods. This work lays the groundwork for applying MPS and related approaches to study crystalline defects in 2D interacting lattices, with potential extensions to cold-atom and photonic platforms and to other symmetry-protected or fractionalized phases.

Abstract

We extend the previous study of extracting crystalline symmetry-protected topological invariants to the correlated regime. We construct the interacting Hofstadter model defined on square lattice with the rotation and translation symmetry defects: disclination and dislocation. The model realizes Chern insulator and the charge density wave state as one tunes interactions. Employing the density matrix renormalization group (DMRG) method, we calculate the excess charge around the defects and find that the topological invariants remain quantized in both phases, with the topological quantity extracted to great precision. This study paves the way for utilizing matrix product state, and potentially other quantum many-body computation methods, to efficiently study crystalline symmetry defects on 2D interacting lattice systems.

Figures (11)

• Figure 1: The disclination (a,c) with $\Omega = \frac{\pi}{2}$ and dislocation (b,d) with $\vec{b} = (0,-1)$ constructed using a cut-and-glue procedure (see SI suppl for details). The green solid lines denote the hopping of electrons while the gray dashed lines illustrate the "snake" of the one-dimensional matrix product state. In the case of dislocation, we apply periodic boundary condition in $y$ to eliminate boundary effects.
• Figure 2: (a) The excess charge measured around the disclination of different bond dimensions and interaction strengths. (b) The excess charge measured in regions with different sizes and different interaction strengths. The red dashed lines in (a) and (b) highlight the $1/8$ quantization of $\delta Q$. (c) The charge pumping by threading an infinitesimal flux at the disclination. The $C = \frac{dn}{d\phi}$ is calculated by measuring the change of the total charge inside the selected area, which gives the Chern number of the system according to the Streda formula. The red dotted lines highlight the integer quantization (d) The density plot at $V = 0.3$. We only show the lattice away from the boundary. The shaded region $W$ with $R=2$ is where we choose to compute the excess charge $\delta Q$. The color on lattice sites labels the charge density according to the color bar.
• Figure 3: With periodic boundary condition in $y$, the dislocation is put in a cylinder. We shift the data by $\Delta$ to show its convergence in (a) (b), and (c). (a) The excess charge measured around the dislocation of different bond dimensions for different interaction strengths. (b) The size scaling of excess charge for three different interaction strengths $V= 0,0.3,0.9$. (c) The charge pumping by threading an infinitesimal flux into the cylinder. The Chern number $C = \frac{dn}{d\phi}$ is calculated by measuring the change of the total charge of the left boundary. (d) The density plot near the dislocation with $V = 0.3$. We choose the shaded region to compute the excess charge $\delta Q$. The color on lattice sites labels the charge density.
• Figure 4: Transition from IQH to CDW as the interaction strength increases close to $\nu = \frac{1}{2}$. The IQH becomes unstable as the $V$ approaches the band gap. Panel (a) displays the evolution of charge occupation polarization $n_1 - n_2$ on 2 sites and the density-density correlation between them $\langle n_1 n_2\rangle - \langle n_1 \rangle \langle n_2 \rangle$ as we tune $V$. Panel (b) is the excess charge $\delta Q$ along this path. The inset shows the CDW pattern, where the two sites encircled denote the position of $n_1$ and $n_2$ used in (a). The two red dashed lines highlight the quantization of $1/8$ and ${1/4}$.
• Figure 5: Cut-and-glue procedure for constructing the disclination with the center $O$, where $k'$ and $l'$ are the $C_4$ counterpart of $k$ and $l$. The sites inside the shaded area are removed and the red bonds are connected.