Low-energy effective theory and symmetry classification of flux phases on Kagome lattice

Abstract: Motivated by recent experiments on AV$_3$Sb$_5$ (A=K,Rb,Cs), the chiral flux phase has been proposed to explain time-reversal symmetry breaking. To fully understand the physics behind the chiral flux phase, we construct a low-energy effective theory based on the van-Hove points around the Fermi surface. The possible symmetry-breaking states and their classifications of the low-energy effective theory are completely studied, especially the flux phases on Kagome lattice. In addition, we discuss the relations between the low-energy symmetry breaking orders, the chiral flux and charge bond orders. We find all possible 183 flux phases on Kagome lattice within 2*2 unit cell by brute-force approach and classify them by point group symmetry. Among the 183 phases, we find 3 classes in 1*1 unit cell, 8 classes in 1*2 unit cell and 18 classes in 2*2 unit cell, respectively. These results provide a full picture of the time-reversal symmetry breaking in Kagome lattices.

• The paper identifies 183 flux phases by developing a low-energy effective model that leverages SU(3) Gell-Mann matrices for classifying symmetry breaking in Kagome lattices.
• It utilizes a single-orbital tight-binding approach at van-Hove points to capture electronic scattering and resulting chiral flux states.
• The findings provide a comprehensive framework with practical implications for understanding time-reversal symmetry breaking in materials like AV₃Sb₅.

Overview of Flux Phases on the Kagome Lattice

The paper "Low-energy effective theory and symmetry classification of flux phases on Kagome lattice" (2106.04395) explores the intricate physics underpinning chiral flux phases leading to time-reversal symmetry breaking in Kagome lattice systems, particularly focusing on materials like AV$_3$Sb$_5$. It formulates a low-energy effective theory by analyzing dominant scattering at van-Hove (vH) points near the Fermi surface and explores symmetry-breaking states alongside their classifications.

Low-Energy Effective Theory

The authors develop a low-energy effective model by considering the electronic states at three vH points in the Brillouin zone of the Kagome lattice. Utilizing a single-orbital tight-binding model, the study identifies how interactions at these points lead to symmetry-breaking states, including charge density waves (CDWs), charge bond orders (CBOs), and flux phases. The resulting effective Hamiltonian is systematically mapped using SU(3) Gell-Mann matrices, which help reveal the classifications of symmetry breaking orders, categorized into charge differences, bond orders, and flux phases.

The emphasis is on constructing a comprehensive framework linking the low-energy electronic instabilities to real-space order parameters, establishing connections to complex hopping terms tied to chiral flux phases.

Real-Space Patterns and Symmetry Classifications

A significant contribution of the paper is the identification and classification of flux states within a 2x2 unit cell of the Kagome lattice. By ensuring the charge continuity equation is satisfied (i.e., no charge sink or source at nodes), the researchers employ a brute-force approach to determine symmetrically unique flux configurations.

The paper identifies 183 flux phases, with varying unit cell configurations (1x1, 1x2, 2x2), and provides their symmetry classifications:

• 1x1 Unit Cell: Classified into Nagaosa, Flow-a, and Flow-b classes based on magnetic group symmetries.
• 2x2 Unit Cell: A detailed enumeration yields 18 classes, with varying symmetries from time-reversal preserving ($D_{6h}^{*}$) to those breaking both time and point symmetry (e.g., $C_{2v}$).
• 1x2 Unit Cell: The symmetries include $D_{2h}^{*}$ and $C_{2v}^{*}$, with 17 phases identified.

Practical Implications

This comprehensive framework enlightens the emerging field of time-reversal symmetry breaking phenomena in Kagome lattices. Notably, materials like AV$_3$Sb$_5$ serve as practical platforms for such studies, revealing fundamental insights into correlated topological matter. By systematically classifying and understanding these flux states, the study informs potential experimental validation and synthesis of materials aiming to harness unique electronic properties like the quantum anomalous Hall effect in these compounds.

Conclusion

The research presents a meticulous examination of flux phases in the Kagome lattice within the context of a low-energy effective theory. Highlighting 183 possible flux configurations, the paper bridges theoretical constructs with possible experimental realizations, thus setting the stage for future investigations into chiral anomalies and symmetry-breaking phenomena in quantum materials. These findings not only advance theoretical understanding but also provide a foundation for exploring novel electronic phases in condensed matter systems.