Quasisymmetry Enrichment Advances Gapless Criticality at Chern Insulator Transitions

Researchers have uncovered a surprising link between symmetry and critical behaviour in materials undergoing transitions to become Chern insulators. Jiayu Li, Feng-Ren Fan from Soochow University, and Wang Yao from The University of Hong Kong, alongside their colleagues, demonstrate how ‘quasisymmetries’ , approximate symmetries emerging in specific energy ranges , fundamentally reshape these transitions. Their work identifies that these quasisymmetries not only classify different types of transitions within the same category, but also give rise to previously unseen phenomena, such as correlated charge and pseudospin currents and continuous Hall conductivities normally associated with insulating states. By establishing quasisymmetry as a key organising principle, this research promises a deeper understanding of the complex behaviour exhibited at the boundaries between different quantum states of matter.

Quasisymmetry Enrichment in Continuous Phase Transitions reveals novel

The research introduces the concept of quasisymmetry enrichment, focusing on normal-to-Chern insulator transitions to identify quasisymmetries within gapless subspaces. The team achieved a classification of CTPTs by establishing quasisymmetry as a fundamental ingredient, moving beyond traditional topological and energetic approaches. Traditionally, transitions are classified by jumps in topological invariants or by the presence of an energy gap, but this work highlights the importance of “higher” symmetries, quasisymmetries, that emerge within the gap-closing subspaces at criticality. This research establishes that gapless criticalities with nontrivial quasisymmetry charges exhibit unique behaviours, effectively bridging the gap between gapped and gapless phases.
The study unveils that these charges can host phenomena typically associated with gapped phases, such as protected boundary modes, directly at the quantum critical point. This breakthrough reveals that quasisymmetry acts as a classifying ingredient, allowing for a more nuanced understanding of the rich landscape of CTPTs. Focusing on the paradigmatic Bernevig-Hughes-Zhang (BHZ) model, describing magnetic topological insulator thin films, researchers demonstrated that quasisymmetries emerging within the gap-closing subspace can enrich transitions by endowing the gapless criticality with a quasisymmetry charge. Furthermore, the work establishes a continuous generalized Hall conductivity associated with a generalized Streda formula, a phenomenon conventionally exclusive to insulating phases. The study’s findings are supported by calculations based on the BHZ Hamiltonian, HBHZ(k) = ε0 (k) + A(kxsy −kysx)τz + M0(k)τx, where ε0(k) = C0 −Dk2 and M0(k) = M0 −Bk2. The parameters were calculated by embedding the model in a square lattice with magnetic point group symmetry 4m′m′ and demonstrate the unconventional nature of this CTPT through the intrinsic dipole Hall conductivity σD H, measuring the response of the pseudospin current jD to an in-plane electric field E∥.

Quasisymmetry Identification and Topological Phase Classification are crucial

Researchers investigated continuous topological phase transitions (CTPTs) by focusing on gap-closing subspaces in which quasisymmetries can emerge. In particular, they analyzed the paradigmatic transition from a normal insulator to a Chern insulator to demonstrate quasisymmetry enrichment. The calculations employed a flowing charge dipole, p, to generate an orbital magnetic moment, m, defined as m = 1/4 (p × v − v × p), where v denotes the velocity operator. To validate this framework, the researchers evaluated the integral of I₍surf,1₎ˡᵏ using the parameterization method described in Appendix B. This yielded I₍surf,1₎ˡᵏ ∝ R₀,₀ sign(∆₀) (A² − 2B∆₀ + 2D|∆₀|), where R₀,₀ is defined in Eq. (B6) and ∆₀ denotes the critical gap parameter. In contrast, for quasisymmetry-enriched CTPTs with nontrivial charge, R₀,₀ vanishes and consequently I₍surf,1₎ˡᵏ = 0. This condition is enforced by the quasisymmetry operator Π [Eq. (8)] within the Bernevig–Hughes–Zhang (BHZ) model. Detailed derivations and supporting numerical data are provided in the Supplemental Material, Section S6, which further substantiates the results and offers a comprehensive account of the methodology.

Quasisymmetry reveals new topological transition features in plasma

Experiments focused on the paradigmatic normal-to-Chern insulator transition, identifying quasisymmetries within gapless subspaces that subdivide CTPTs based on quasisymmetry charges. The team measured the dipole Hall conductivity (σD H) as a function of energy bias (V⊥) between surfaces, finding distinct behaviours at the gapless criticality. Specifically, σD H exhibited discontinuous behaviour when the exchange field (∆2) or velocity difference (A2) were zero, similar to charge Hall conductivity. These measurements were conducted on the Bernevig-Hughes-Zhang (BHZ) model, a magnetic topological insulator thin film, with parameters carefully controlled to access the rich phase diagram.

Results demonstrate that the continuity of σD H is directly linked to the integrability of the dipole Berry curvature. At the critical parameter (Vc), the valence and conduction bands touch, causing a divergent contribution to the charge Berry curvature. Data shows that the dipole Hall conductivity (σD H) can reach values of 1.0 e²/h, while the product of matrix elements, crucial for determining the continuity of σD H, varies significantly along high symmetry lines in the Brillouin zone. For instance, at (Δ2, A2) = (0, 0), the distribution of Im ⟨V, k|jD x |C, k⟩⟨C, k|vy|V, k⟩ exhibits a distinct pattern compared to (50 meV, 0) and (0, 0.5 eV·Å⁻¹). The research establishes quasisymmetry as a fundamental classifying ingredient, adding a new layer to the understanding of CTPTs and potentially enabling the design of novel topological materials with tailored current correlations.

Quasisymmetry defines continuous topological phase classes in magnetic

Scientists have uncovered a novel understanding of continuous topological phases of matter, demonstrating that these phases can be classified not only by their fundamental topological properties but also by “quasisymmetries” present within their gapless subspaces. Focusing on the transition between normal and Chern insulator states, researchers identified that these quasisymmetries subdivide continuous topological phases into distinct universality classes, categorised by specific quasisymmetry charges. The authors acknowledge a limitation in their analysis: the current framework relies on linear band crossings and may not fully apply to systems with more complex band dispersions, such as those exhibiting parabolic or cubic behaviour. Future research could extend this framework to other topological invariants and crystalline insulators, potentially revealing further insights into the behaviour of topological materials and expanding our knowledge of quantum matter.