Interacting Topological Phases Demonstrate Quantized Invariants with Defects on 2D Lattices

The behaviour of electrons in materials with imperfections, such as dislocations and disclinations, presents a fundamental challenge in condensed matter physics, and recent work by Lu Zhang, Min Long, Yuxuan Zhang, and colleagues at institutions including the University of Hong Kong and the University of Science and Technology of China, sheds new light on this problem. The team investigates how interactions between electrons affect the properties of a special type of material, known as a topological phase, when it contains these symmetry-breaking defects. They demonstrate that even with strong interactions, certain fundamental quantities remain precisely defined around the defects, revealing a surprising robustness of topological order, and opening up new avenues for understanding and potentially harnessing these exotic states of matter in materials science. This research establishes a powerful computational approach for studying complex, interacting systems with crystalline defects, promising significant advances in the design of novel electronic devices.

They extend previous research on crystalline symmetry-protected topological invariants to systems where electrons strongly interact, constructing interaction-induced shift and polarization vectors to characterize the topological response to defects. These vectors quantify changes in the position of Wannier centers and the induced electric polarization, providing a direct measure of topological order in strongly correlated materials. The team demonstrates that these quantities are resilient to local disturbances, serving as reliable indicators of topological phases even in complex materials.

Defect-Engineered Topology in Correlated Systems

Scientists investigated crystalline symmetry-protected topological invariants within correlated systems by constructing the interacting Hofstadter model on a square lattice. This model incorporates rotation and translation symmetry defects, specifically disclinations and dislocations, engineered through a cut-and-glue procedure to modify the lattice geometry. Through precise numerical calculations using a density matrix renormalization group method, the team demonstrated that these invariants remain quantized even when the system exhibits charge density wave behaviour, verifying their robustness in complex electronic environments. Large-scale calculations revealed how charge accumulates around the disclinations and dislocations, and the team meticulously extracted the defect-bound charge, quantifying the topological invariants of discrete shift and polarization. The researchers systematically tuned the interaction strength within the model, and remarkably, observed that even in this strongly correlated state, the quantized topological responses persisted. This finding demonstrates the robustness of these crystalline symmetry-protected invariants beyond single-particle band analysis, and establishes matrix product state methods as a controlled tool for exploring crystalline defects and their associated topological phenomena.

Robust Topological Invariants in Correlated Systems

This research successfully extends the understanding of crystalline symmetry-protected topological invariants into systems where interactions between electrons are significant. Scientists constructed a model, based on the interacting Hofstadter model with deliberately introduced defects, to investigate how these invariants behave in correlated electron systems. The study confirms the validity of classifying topological order in systems with both conventional and crystalline symmetries, and establishes a practical method for constructing symmetry defects within a matrix product state framework. Importantly, the research provides numerical validation of a definition for absolute polarization, a crucial quantity for characterizing topological phases. The findings have implications for the realization of these topological states in physical systems such as cold atoms or photonic structures, offering a pathway for experimental verification. The authors acknowledge that their current work focuses on a specific model and that further investigation is needed to explore other symmetries and more complex topological phases.

Defect-Induced Polarization in Chern Insulators

This supplemental material details the methodology and supporting evidence for research investigating the relationship between topological order, geometric defects, and measurable quantities like charge polarization. It provides a rigorous account of how disclinations and dislocations are created within a lattice model, crucial for studying their effects on topological properties. The document meticulously describes the cut-and-glue procedure, ensuring correct hopping amplitudes to preserve the underlying physics. The document details the parameters used in Density Matrix Renormalization Group calculations, including the number of states kept and maximum truncation error, ensuring the reliability of the numerical results.

System size and chemical potential are also specified to gap the bulk region of the cluster. The robustness of the results to disorder is explored by adding random interactions, demonstrating that the excess charge around the defects remains near the quantized value. The document addresses the ambiguity in defining polarization density in Chern insulators, explaining how large gauge transformations can be used to probe the bulk polarization. This supplemental material allows other researchers to reproduce the results, demonstrates methodological rigor, and provides supporting evidence for the claims made in the main paper.