Entanglement polarization for the topological quadrupole phase

Abstract

We propose the entanglement dipole polarization to describe the topological quadrupole phase. The quadrupole moment can be regarded as a pair of the dipole moment, in which the total dipole moment is canceled. The entanglement polarization, we propose, is useful to detect such a constituent dipole polarization. We first introduce partitions of sites in the unit cell and divide the system into two subsystems. Then, introducing an entanglement Hamiltonian by tracing out one of the subsystems partly, we compute the dipole polarization of the occupied states associated with the entanglement Hamiltonian, which is referred to as the entanglement polarization. Although the total dipole polarization is vanishing, those of the subsystems can be finite. The entanglement dipole polarization is quantized by reflection symmetries. We also introduce the entanglement polarization of the edge states, which reveals that the edge states themselves are gapped and topologically nontrivial. Therefore, such edge states yield the zero energy edge states if the system has boundaries. This is the origin of the corner states.

12 pages, 34 figures, final version

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• ${\mathbb{Z}}_Q$ Berry Phase for Higher-Order Symmetry-Protected Topological Phases
• Direct prediction of corner state configurations from edge winding numbers in 2D and 3D chiral-symmetric lattice systems
• Second-order topological corner states with ultracold atoms carrying orbital angular momentum in optical lattices
• Higher-Order Topological Phase in a Honeycomb-Lattice Model with Anti-Kekulé Distortion
• Thouless Pumps and Bulk-Boundary Correspondence in Higher-Order Symmetry-Protected Topological Phases
• Direct dynamical characterization of higher-order topological insulators with nested band inversion surfaces
• Higher-order topological insulators in a magnetic field
• Edge states, corner states, and flat bands in a two-dimensional $\mathcal{P}\mathcal{T}$ symmetric system
• Theory of inversion-${\mathbb{Z}}_4$ protected topological chiral hinge states and its applications to layered antiferromagnets
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• Boundary-obstructed topological phases of a massive Dirac fermion in a magnetic field
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• Identifying Higher Order Topology and Fractional Corner Charge Using Entanglement Spectra
• A Dirac fermion model associated with second order topological insulator
• Higher-Order Topological Mott Insulator on the Pyrochlore Lattice
• Detecting Bulk Topology of Quadrupolar Phase from Quench Dynamics
• Engineering topologically protected zero-dimensional interface end states in antiferromagnetic heterojunction graphene nanoflakes
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• Entanglement in a second order topological insulator on a square lattice
• The General Principle behind Magnetization-induced Second-Order Topological Corner States in the Kane-Mele Model
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• Many-body multipole index and bulk-boundary correspondence
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• Effects of first- and second-order topological phases on equilibrium crystal shapes
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• Gapless States Localized along a Staircase Edge in Second-Order Topological Insulators
• Higher-order topological quantum paramagnets
• Hinge modes of three-dimensional Euler insulators