Quantized electric multipole insulators

Abstract

The Berry phase provides a modern formulation of electric polarization in crystals. We extend this concept to higher electric multipole moments and determine the necessary conditions and minimal models for which the quadrupole and octupole moments are topologically quantized electromagnetic observables. Such systems exhibit gapped boundaries that are themselves lower-dimensional topological phases. Furthermore, they host topologically protected corner states carrying fractional charge, exhibiting fractionalization at the boundary of the boundary. To characterize these insulating phases of matter, we introduce a paradigm in which "nested" Wilson loops give rise to topological invariants that have been overlooked. We propose three realistic experimental implementations of this topological behavior that can be immediately tested. Our work opens a venue for the expansion of the classification of topological phases of matter.

Reflection-Symmetric Second-Order Topological Insulators and Superconductors

“…However, crystal symmetries can be a key to ensure that a natural surface termination -i.e., a surface termination that respects the crystal symmetries -automatically leads to a nontrivial higher-order topological phase. For example, Benalcazar et al employed a combination of multiple reflection symmetries [9], whereas Schindler et al considered C 4 T symmetry, the product of a π/2 rotation and time reversal, as well as a model with reflection symmetry [7].…”

“…For example, a second-order topological insulator in two dimensions (d = 2) has zeroenergy states at corners, but a gapped bulk and no gapless edge states. Earlier examples of higher-order topological insulators and superconductors avant la lettre appeared in works by Benalcazar et al [8][9][10] (see also [11,12]), who considered insulators and superconductors with protected corner states in d = 2 and d = 3 [13]. Sitte et al showed that a threedimensional topological insulator in a magnetic field of generic direction also acquires the characteristics of a second-order topological Chern insulator, with chiral states moving along the sample edges [14].…”

Waves cornered

“…The theoretical prediction of higher order systems rests on a generalization of electric dipole moments to multipole moments that are quantized (having only specific discrete values) 5,6 . Whereas conventional topologi cal insulators are related to dipoles, higher order insulators are related to quadrupoles, octupoles, and so on.…”

“…The presence of mela nin can trigger an immune response in the infected organism 4 , but how this occurs was unknown. On page 382, Stappers et al 5 report the identification of a protein that can recog nize a type of melanin produced by the fungus Aspergillus fumigatus. Their finding illuminates the immune system response to a fungal infection that can be lethal in people who have a suppressed immune system, such as those who have undergone transplantation surgery 6 .…”

Topological Insulators Turn a Corner

“…For example, a second-order topological insulator in two dimensions (d ¼ 2) has zero-energy states at corners, but a gapped bulk and no gapless edge states. Earlier examples of higher-order topological insulators and superconductors avant la lettre appeared in works by Benalcazar et al [8][9][10] (see also Refs. [11,12]), who considered insulators and superconductors with protected corner states in d ¼ 2 and d ¼ 3 [13].…”