Quasisymmetry Enriched Gapless Criticality at Chern Insulator Transitions

Abstract

In continuous topological phase transitions (CTPTs), the low-energy physics is governed by gap-closing subspaces, where approximate "higher" symmetries, termed quasisymmetries, may emerge. Here, we introduce the notion of quasisymmetry enrichment of these transitions. Focusing on paradigmatic normal-to-Chern insulator transitions, we identify quasisymmetries in the gapless subspaces, which subdivide CTPTs of the same universality class according to quasisymmetry charges. Gapless criticalities with nontrivial charges exhibit regulated phenomena, including intrinsic correlations between charge and pseudospin currents and continuous generalized Hall conductivities governed by the generalized Středa formula, both conventionally exclusive to gapped phases. These features arise as quasisymmetry forbids certain matrix elements, rendering the generalized Berry curvature integrable. By establishing quasisymmetry as a fundamental classifying ingredient, our work adds a new dimension for understanding the rich landscape of quantum phase transitions.

Figures (4)

• Figure 1: Quasisymmetry-enriched continuous topological phase transition (CTPT) between Chern and normal insulating phases in BHZ model. (a) Schematic of pseudospin (dipole) Hall current $\bm{j}^D=\bm{j}_T-\bm{j}_B =\sigma_{H}^D(\hat{z}\times\bm{E}_{\parallel})$, describing charge Hall counter-flows $\bm{j}_T$ and $\bm{j}_B$ on top and bottom surfaces, induced by in-plane electric field $\bm{E}_\parallel$. (b) Dipole Hall conductivity $\sigma_{H}^{D}$ as a function of $V_\perp$, the energy bias between the surfaces. The yellow, blue, and green symbols correspond to the exchange field $\Delta_2$ and velocity difference $A_2$ set as ($\Delta_2,A_2$) = (0, 0), (50 meV, 0), (0, 0.5 eV·Å$^{-1}$) respectively. (c) Schematic of the gapless criticality enriched by quasisymmetry $\Pi$, which forbids the matrix element $\langle \mathds{V},\bm{k}|j_x^D|\mathds{C},\bm{k}\rangle$ at the touching point iff quasisymmetry charge $\omega_D \neq 0$. (d) Distributions of ${\rm Im}\left[\langle \mathds{V},\bm{k}|j_x^D|\mathds{C},\bm{k}\rangle \langle \mathds{C},\bm{k}|v_y|\mathds{V},\bm{k}\rangle \right]$ along high symmetry lines in Brillouin zone, for the BHZ model under $V_\perp = V_c$ and three set of ($\Delta_2,A_2$) as in (b). This quantity determines the continuity of $\sigma_{H}^{D}$ at the critical point [c.f. Eq. (\ref{['eq:limits_of_conductivity']})].
• Figure 2: Phase diagrams of BHZ-based CI model with $\Delta_{2}=A_{2}=0$ (a) and $\Delta_{2}=50$ meV, $A_2=0$ (b), and yellow dashed curves denote the phase boundaries. Color maps the magnitude of dipole Hall conductivity $\sigma_{H}^{D}$.
• Figure 3: Generalized Středa formula linking dipole Hall conductivity and equilibrium magnetic quadrupole order in gapped phase and gapless criticality. (a) Dipole Hall effect as a magnetoelectric response. The counter-flowing Hall currents on the two surfaces and boundary tunneling currents form a current loop that correspond to an orbital magnetization longitudinal to the in-plane electric field. (b) In equilibrium, the counter-flowing edge currents on the top and bottom surfaces—an edge dipole current—can be equivalently viewed as an array of current loops (dashed-arrow circles). The resulting orbital magnetization, oriented normal to the side surfaces, realizes a magnetic quadrupole order. (c) $\partial M_{xx}/\partial \mu$ as a function of chemical potential $\mu$, counting both Fermi sea and Fermi surface contributions (c.f. Appendix \ref{['Appendix-D']}). $V_\perp$ is the energy bias between the top and bottom surfaces as the tuning parameter for the CTPT, $V_c$ being the critical value. Left: $V_\perp = 0.6 V_c$, the system is a Chern insulator and the generalized Středa formula manifests as a constant $\partial M_{xx}/\partial \mu$ in the gap. Right: $V_\perp = V_c$, the gapped formula is still applicable at the gapless criticality by the quasisymmetry protection. Dashed vertical line indicates middle of the gap, and vertical axis is in unit of $-\sigma_H^Dd/2e$, the dipole Hall conductivity in gap.
• Figure 4: Quasisymmetry-enriched CTPT in other Chern insulator systems. (a) Schematics of the Haldane model composed of $s$ orbital located at honeycomb lattice with staggered magnetic flux (marked by + and -), and FeS monolayer with $d_{xz}$ and $d_{yz}$ orbitals with ferromagnetic order (marked by arrows). Alternating on-site energy $\pm V_{\perp}/2$ are induced in both models, the $t_{2}$ denotes next nearest neighbor hopping integral. (b) Distribution of sublattice Berry curvature $\Omega_{xy}^{\rm{sub}}$ in Haldane model. (c) Sublattice Hall conductivity $\sigma_{xy}^{\rm{sub}}$ as a function of $V_{\perp}$ for Haldane model with $\varphi=\pi/4$ and FeS monolayer model. (d) Phase diagram of the Haldane model under variations of $V_{\perp}$ and $\varphi$, color coded by the value of $\sigma_{H}^{\rm{sub}}$. Dashed curves denote phase boundary.