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The quantum geometric properties of a Bloch state in momentum space are usually described by the Berry curvature and quantum metric. In realistic gapped materials where interactions and disorder render the Bloch state not a viable starting…
Quantum metrology is deeply connected to quantum geometry, through the fundamental notion of quantum Fisher information. Inspired by advances in topological matter, it was recently suggested that the Berry curvature and Chern numbers of…
We study a class of translational-invariant insulators with discrete rotational symmetry. These insulators have no spin-orbit coupling, and in some cases have no time-reversal symmetry as well, i.e., the relevant symmetries are purely…
The surface states of 3D topological insulators possess geometric structures that imprint distinctive signatures on electronic transport. A prime example is the Berry curvature, which controls, for instance, electric frequency doubling via…
Electronic topological phases of matter, characterized by robust boundary states derived from topologically nontrivial bulk states, are pivotal for next-generation electronic devices. However, understanding their complex quantum phases,…
The Chern index characterizes the topological phases of nonreciprocal photonic systems. Unlike in electronic systems, the photonic Chern number has no clear physical meaning, except that it determines the net number of unidirectional edge…
Motivated by the outstanding challenge of realizing low-temperature states of quantum matter in synthetic materials, we propose and study an experimentally feasible protocol for preparing topological states such as Chern insulators. By…
The Berry phase associated with energy bands in crystals can lead to quantized quantities, such as the quantization of electric dipole polarization in an insulator, known as a one-dimensional (1D) topological insulator (TI) phase. Recent…
Recently, it has been established that Chern insulators possess an intrinsic two-dimensional electric polarization, despite having gapless edge states and non-localizable Wannier orbitals. This polarization, $\vec{P}_{\text{o}}$, can be…
Quantum Hall (QH) states are arguably the most ubiquitous examples of nontrivial topological order, requiring no special symmetry and elegantly characterized by the first Chern number. Their higher dimension generalizations are particularly…
We study the topological phase in dipolar-coupled two-dimensional breathing square lattice of magnetic vortices. By evaluating the quantized Chern number and $\mathbb{Z}_{4}$ Berry phase, we obtain the phase diagram and identify that the…
The surface of a 3D topological insulator is a new type of two dimensional electron gas (2DEG). The exactly quantized magneto-optical or axionic response of a 3D topological insulator is a direct result of the Berry's phase protected…
Higher-order topological insulators are a new class of topological phases of matter, originally conceived for electrons in solids. It has been suggested that $\mathbb{Z}_N$ Berry phase (Berry phase quantized into $2\pi/N$) is a useful tool…
The unrelated discoveries of quasicrystals and topological insulators have in turn challenged prevailing paradigms in condensed-matter physics. We find a surprising connection between quasicrystals and topological phases of matter: (i)…
We discuss recent advances in the study of topological insulators protected by spatial symmetries by reviewing three representative, theoretical examples. In three dimensions, these states of matter are generally characterized by the…
We study three dimensional insulators with inversion symmetry, in which other point group symmetries, such as time reversal, are generically absent. Their band topology is found to be classified by the parities of occupied states at time…
Clean isotropic quantum Hall fluids in the continuum possess a host of symmetry-protected quantized invariants, such as the Hall conductivity, shift and Hall viscosity. Here we develop a theory of symmetry-protected quantized invariants for…
Topological insulators have been proposed to be best characterized as bulk magnetoelectric materials that show response functions quantized in terms of fundamental physical constants. Here we lower the chemical potential of…
Integer and fractional Chern insulators exhibit a nonzero quantized anomalous Hall conductivity due to a spontaneous breaking of time reversal symmetry. To identify nontrivial topology in their time-reversal symmetric many-body spectra, we…
We present a theory on the quantum phase diagram of AB-stacked MoTe$_2$/WSe$_2$ using a self-consistent Hartree-Fock calculation performed in the plane-wave basis, motivated by the observation of topological states in this system. At…