We proposed a formula for the ${\mathbb{Z}}_{2}$ invariant for topological insulators, which remains valid without translational invariance. Our formula is a local expression, in the sense that the contributions mainly come from quantities near a point. Using almost commute matrices, we proposed a method to approximate this invariant with local information. The validity of the formula and the approximation method is proved.We proposed a formula for the ${\mathbb{Z}}_{2}$ invariant for topological insulators, which remains valid without translational invariance. Our formula is a local expression, in the sense that the contributions mainly come from quantities near a point. Using almost commute matrices, we proposed a method to approximate this invariant with local information. The validity of the formula and the approximation method is proved.

arXiv:1905.12649Physical review. B./Physical review. BNov 4, 201914 citations

Topological invariants and the ground state wavefunction of topological insulators

Zhong Wang, Shengbai Zhang

Bulletin of the American Physical SocietyMar 6, 20140 citations

Topological invariants for interacting topological insulators with inversion symmetry

Zhong Wang, Xiao-Liang Qi, Shoucheng Zhang

For interacting ${Z}_{2}$ topological insulators with inversion symmetry, we propose a simple topological invariant expressed in terms of the parity eigenvalues of the interacting Green's function at time-reversal invariant momenta and zero frequency. We derive this result from our previous formula involving the integral over the frequency-momenta space. This formula greatly simplifies the explicit calculation of ${Z}_{2}$ topological invariants in inversion-symmetric insulators with strong interactions.For interacting ${Z}_{2}$ topological insulators with inversion symmetry, we propose a simple topological invariant expressed in terms of the parity eigenvalues of the interacting Green's function at time-reversal invariant momenta and zero frequency. We derive this result from our previous formula involving the integral over the frequency-momenta space. This formula greatly simplifies the explicit calculation of ${Z}_{2}$ topological invariants in inversion-symmetric insulators with strong interactions.

arXiv:1201.6431Physical Review BApr 17, 201287 citations

Topological invariants in interacting topological insulators: Success and Breakdown

Yuan-Yao He, Han‐Qing Wu, Zi Yang Meng +1 more

Bulletin of the American Physical SocietyMar 15, 20160 citations

Geometric bulk-edge correspondence for $\mathbb{Z}_2$ -topological insulatorsGeometric bulk-edge correspondence for $\mathbb{Z}_2$-topological insulators

Alexis Drouot, Jacob Shapiro, Xiaowen Zhu

Fermionic time-reversal-invariant insulators in two dimensions--class AII in the Kitaev table--come in two topological phases. These phases are characterized by a $\mathbb{Z}_2$ -valued invariant, the Fu-Kane-Mele index. We prove a geometric bulk-edge correspondence for curved interfaces: if two such insulators occupy complementary regions separated by a curved boundary, then the $\mathbb{Z}_2$ edge index of the interface system is the product, modulo two, of the difference of the two bulk $\mathbb{Z}_2$ indices and a geometric intersection number associated with the boundary and the measurement region. The argument is a $\mathbb{Z}_2$ analogue of the curved-interface connection formula proved for Hall insulators in \cite{DZ24}.Fermionic time-reversal-invariant insulators in two dimensions--class AII in the Kitaev table--come in two topological phases. These phases are characterized by a $\mathbb{Z}_2$-valued invariant, the Fu-Kane-Mele index. We prove a geometric bulk-edge correspondence for curved interfaces: if two such insulators occupy complementary regions separated by a curved boundary, then the $\mathbb{Z}_2$ edge index of the interface system is the product, modulo two, of the difference of the two bulk $\mathbb{Z}_2$ indices and a geometric intersection number associated with the boundary and the measurement region. The argument is a $\mathbb{Z}_2$ analogue of the curved-interface connection formula proved for Hall insulators in \cite{DZ24}.

math-phCross-listedarXiv:2606.27318Jun 26, 2026

General Theory of interacting Topological insulators

Shoucheng Zhang

Bulletin of the American Physical SocietyMar 25, 20114 citations

We proposed a formula for the

{\mathbb{Z}}_{2}

invariant for topological insulators, which remains valid without translational invariance. Our formula is a local expression, in the sense that the contributions mainly come from quantities near a point. Using almost commute matrices, we proposed a method to approximate this invariant with local information. The validity of the formula and the approximation method is proved.

For interacting

{Z}_{2}

topological insulators with inversion symmetry, we propose a simple topological invariant expressed in terms of the parity eigenvalues of the interacting Green's function at time-reversal invariant momenta and zero frequency. We derive this result from our previous formula involving the integral over the frequency-momenta space. This formula greatly simplifies the explicit calculation of

{Z}_{2}

topological invariants in inversion-symmetric insulators with strong interactions.

Geometric bulk-edge correspondence for

\mathbb{Z}_2

-topological insulators

Fermionic time-reversal-invariant insulators in two dimensions--class AII in the Kitaev table--come in two topological phases. These phases are characterized by a

\mathbb{Z}_2

-valued invariant, the Fu-Kane-Mele index. We prove a geometric bulk-edge correspondence for curved interfaces: if two such insulators occupy complementary regions separated by a curved boundary, then the

\mathbb{Z}_2

edge index of the interface system is the product, modulo two, of the difference of the two bulk

\mathbb{Z}_2

indices and a geometric intersection number associated with the boundary and the measurement region. The argument is a

\mathbb{Z}_2

analogue of the curved-interface connection formula proved for Hall insulators in \cite{DZ24}.