Complex charge density waves at Van Hove singularity on hexagonal lattices: Haldane-model phase diagram and potential realization in kagome metals $\text{AV}_3\text{Sb}_5$ (2104.02725v3)

Abstract: We investigate how the real and imaginary charge density waves interplay at the Van Hove singularity on the hexagonal lattices. A phenomenological analysis indicates the formation of $3Q$ complex orders at all three nesting momenta. Under a total phase condition, unequal phases at the three momenta break the rotation symmetry generally. The $3Q$ complex orders constitute a rich Haldane-model phase diagram. When effective time-reversal symmetries arise under 1-site translations, the Dirac semimetals are protected. The breakdown of these symmetries gaps the Dirac points and leads to the trivial and Chern insulator phases. These phases are deformations of purely real and imaginary orders, which exhibit trivial site and/or bond density and chiral flux orders, respectively. The exotic single-Dirac-point semimetals also appear along the gapless phase boundary. We further show that the theoretical model offers transparent interpretations of experimental observations in the kagome metals $\text{AV}_3\text{Sb}_5$ with $\text{A}=\text{K},\text{Rb},\text{Cs}$. The topological charge density waves may be identified with the complex orders in the Chern insulator phase. Meanwhile, the lower-temperature symmetry-breaking phenomena may be interpreted as the secondary orders from the complex order ground states. Our work sheds light on the nature of the topological charge density waves in the kagome metals $\text{AV}_3\text{Sb}_5$ and may offer useful indications to the experimentally observed charge orders in the future experiments.

Overview of Complex Charge Density Waves in Kagome Metals and Beyond

The research paper presents an in-depth investigation into the emergence and interplay of complex charge density waves (CDWs) at the Van Hove singularity on hexagonal lattices, with a particular focus on their manifestation in kagome metals such as $\text{AV}_3\text{Sb}_5$ (where $\text{A}=\text{K},\text{Rb},\text{Cs}$). This paper explores the theoretical framework for understanding how real and imaginary components of CDWs interact to form intricate ground states characterized by rich symmetry-breaking phenomena and topological properties.

Framework and Methodology

The paper employs a phenomenological approach grounded in Ginzburg-Landau theory to explore the dynamically generated $3Q$ complex orders originating from the interactions at the Van Hove singularity. The focus is on hexagonal lattices, including triangular, honeycomb, and kagome lattices, as they share common features in their Fermiology at the Van Hove singularity. The potential instabilities at these singularities are examined concerning both real and imaginary CDWs, where real orders manifest through site/bond density modulations and imaginary orders through staggered currents.

The theoretical model incorporates a comprehensive mean-field analysis based on irreducible pairing channels defined by specific wave symmetry orders ($s$-wave, $d_R$-wave for real, and $d_I$-wave for imaginary components) while evaluating the resultant band structures in a $2\times2$ reduced Brillouin zone. The paper determines the phase diagram by computing energy structures and Chern numbers, revealing topological insulator phases inherent to the complex orders.

Numerical Results and Lattice Implications

The numerical analysis indicates that the $3Q$ complex orders map onto a Haldane-model phase diagram, encompassing trivial insulators, Chern insulators, and critical semimetallic phases characterized by Dirac and single-Dirac-point semimetals. These findings are tied to symmetry considerations, such as the presence or absence of time-reversal and effective time-reversal symmetries. For instance, complex imaginary orders lead to intrinsic staggered and chiral flux patterns corresponding to Chern insulator phases with non-zero Chern numbers.

From a lattice perspective, the real and imaginary order components induce distinct modulations in site/bond densities and currents on triangular, honeycomb, and kagome lattices. These perturbations elucidate the link between order parameters and residual lattice symmetries, significant for understanding the microscopic nature of emerged electronic phases.

Implications for Kagome Metals $\text{AV}_3\text{Sb}_5$

In practical terms, the theoretical insights have profound implications for experimentally observed charge orders in kagome metal compounds $\text{AV}_3\text{Sb}_5$. The giant anomalous Hall effects observed in these materials may be underpinned by the topological Chern insulator states forecasted in the proposed model. The observed CDWs with unconventional charge modulations align closely with the theoretically predicted $3Q$ complex orders, suggesting these states may account for both the time-reversal breaking phenomena and the rotational symmetry-breaking signatures reported experimentally.

Moreover, the predicted coexistence and transitions between ground states (trivial and Chern insulators) offer a framework for interpreting temperature-dependent phase transitions observed in these systems, such as the emergence of secondary $1Q$ orders at lower temperatures.

Future Directions

The paper opens several avenues for future research, particularly in extending these models to other hexagonal lattice systems and exploring potential higher-order topological phenomena within complex CDWs. Moreover, the research suggests further exploration of incommensurate phases and the effects of band deformation and nonperfect Fermi surface nesting. These insights into the fascinating interplay between CDW components can aid the pursuit of new quantum materials with novel electronic properties and highlight the theoretical model's utility in guiding future experimental efforts.