arXiv subjects

Jia-Xiao Dai

Publications and source records attributed to Jia-Xiao Dai.

4 recordsLinked to original sources

arXiv2022

Takagi Topological Insulator on the Honeycomb Lattice

Recently, real topological phases protected by $PT$ symmetry have been actively investigated. In two dimensions, the corresponding topological invariant is the Stiefel-Whitney number. A recent theoretical advance is that in the presence of the sublattice symmetry, the Stiefel-Whitney number can be equivalently formulated in terms of Takagi's factorization. The topological invariant gives rise to a novel second-order topological insulator with odd $PT$-related pairs of corner zero modes. In this article, we review the elements of this novel second-order topological insulator, and demonstrate the essential physics by a simple model on the honeycomb lattice.

cond-mat.mes-hall↗

arXiv2022

Tensor Theory for Higher Dimensional Chern Insulators with Large Chern Numbers

Recent advances in topological artificial systems open the door to realizing topological states in dimensions higher than the usual three-dimensional space. Here, we present a "tensor product" theory, which offers a method to construct Chern insulators with arbitrarily high dimensions and Chern numbers. Particularly, we show that the tensor product of a $d_A$D Chern insulator $\langle \mathcal{H}_A^{(κ_{A})}, C_A\rangle$ with a $d_B$D Chern insulator $\langle \mathcal{H}_B^{(κ_B)}, C_B\rangle$ leads to a $(d_A+d_B)$D Chern insulator $\langle \mathcal{H}_{A B}^{(κ_A\star κ_B)},-2C_AC_B\rangle $, where in the brackets, $\mathcal{H}^{(κ)}$ is the $d$D Hamiltonian with $d$ even, $C$ is the corresponding $(d/2)$th Chern number, and $κ$ labels the five non-chiral Altland-Zirnbauer symmetry classes A, AI, D, AII and C. The four real classes AI, D, AII and C form a Klein four-group under the multiplication `$\star$' with class AI the identity, and class A is the zero element. Our theory leads to novel higher-dimensional topological physics. (i) The construction can generate large higher-order Chern numbers, e.g., for some cases the resultant classification is $8\mathbb{Z}$. (ii) Fascinatingly, the boundary states feature flat nodal hypersurfaces with nontrivial Chern charges. For the constructed $(d_A+d_B)$D Chern insulator, a boundary perpendicular to a direction of $\mathcal{H}_A$ generically hosts $|C_A|$ $d_B$D nodal hypersurfaces, each of which has topological charge $\pm 2C_B$. Under perturbations, each nodal hypersurface bursts into stable unit nodal points, with the total Chern charge conserved. Examples are given to demonstrate our theory, which can be experimentally realized in artificial systems such as acoustic crystals, electric circuit arrays, ultracold atoms, or mechanical networks.

cond-mat.mes-hall↗

arXiv2021

Takagi topological insulator with odd $\mathcal P\mathcal T$ pairs of corner states

We present a novel class of topological insulators, termed the Takagi topological insulators (TTIs), which is protected by the sublattice symmetry and spacetime inversion ($\mathcal P\mathcal T$) symmetry. The required symmetries for the TTIs can be realized on any bipartite lattice where the inversion exchanges sublattices. The protecting symmetries lead to the classifying space of Hamiltonians being unitary symmetric matrices, and therefore Takagi's factorization can be performed. Particularly, the global Takagi's factorization can (cannot) be done on a $3$D ($2$D) sphere. In 3D, there is a $\mathbb{Z}_2$ topological invariant corresponding to the parity of the winding number of Takagi's unitary-matrix factor over the entire Brillouin zone, where the $\mathbb Z_2$ nature comes from the $O(N)$ gauge degrees of freedom in Takagi's factorization. In 2D, the obstruction for a global Takagi's factorization is characterized by another $\mathbb{Z}_2$ topological invariant, equivalent to the second Stiefel-Whitney number. For the third-order topological phases, the $3$D TTIs feature a parity condition for corner zero-modes, i.e., there always exist odd $\mathcal P\mathcal T$ pairs of corners with zero-modes. Moreover, for any $\mathcal P\mathcal T$ invariant sample geometry, all configurations of corner zero-modes satisfying the parity condition can exist with the same nontrivial bulk topological invariant. Actually, without closing the bulk gap, the boundary phase diagram have a cellular structure, where each topological boundary phase associated with a particular (cross-order) boundary-mode pattern corresponds to a contractible cell with certain dimension in the parameter space.

cond-mat.mes-hall↗

arXiv2020

Boundary criticality of $PT$-invariant topology and second-order nodal-line semimetals

For conventional topological phases, the boundary gapless modes are determined by bulk topological invariants. Based on developing an analytic method to solve higher-order boundary modes, we present $PT$-invariant $2$D topological insulators and $3$D topological semimetals that go beyond this bulk-boundary correspondence framework. With unchanged bulk topological invariant, their first-order boundaries undergo transitions separating different phases with second-order-boundary zero-modes. For the $2$D topological insulator, the helical edge modes appear at the transition point for two second-order topological insulator phases with diagonal and off-diagonal corner zero-modes, respectively. Accordingly, for the $3$D topological semimetal, the criticality corresponds to surface helical Fermi arcs of a Dirac semimetal phase. Interestingly, we find that the $3$D system generically belongs to a novel second-order nodal-line semimetal phase, possessing gapped surfaces but a pair of diagonal or off-diagonal hinge Fermi arcs.

cond-mat.mes-hall↗