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.

We extend the theory of dipole moments in crystalline insulators to higher multipole moments. As first formulated in Ref. 1, we show that bulk quadrupole and octupole moments can be realized in crystalline insulators. In this paper, we expand in great detail the theory presented in Ref. 1, and extend it to cover associated topological pumping phenomena, and a novel class of 3D insulator with chiral hinge states. We start by deriving the boundary properties of continuous classical dielectrics hosting only bulk dipole, quadrupole, or octupole moments. In quantum-mechanical crystalline insulators, these higher multipole bulk moments manifest themselves by the presence of boundary-localized moments of lower dimension, in exact correspondence with the electromagnetic theory of classical continuous dielectrics. In the presence of certain symmetries, these moments are quantized, and their boundary signatures are fractionalized. These multipole moments then correspond to new symmetry-protected topological phases. The topological structure of these phases is described by "nested" Wilson loops, which we define. These Wilson loops reflect the bulkboundary correspondence in a way that makes evident a hierarchical classification of the multipole moments. Just as a varying dipole generates charge pumping, a varying quadrupole generates dipole pumping, and a varying octupole generates quadrupole pumping. For non-trivial adiabatic cycles, the transport of these moments is quantized. An analysis of these interconnected phenomena leads to the conclusion that a new kind of Chern-type insulator exists, which has chiral, hinge-localized modes in 3D. We provide the minimal models for the quantized multipole moments, the non-trivial pumping processes and the hinge Chern insulator, and describe the topological invariants that protect them. Contents

In the presence of crystalline symmetries, certain topological insulators present a filling anomaly: a mismatch between the number of electrons in an energy band and the number of electrons required for charge neutrality. In this paper, we show that a filling anomaly can arise when corners are introduced in Cn-symmetric crystalline insulators with vanishing polarization, having as consequence the existence of corner-localized charges quantized in multiples of e n . We characterize the existence of this charge systematically and build topological indices that relate the symmetry representations of the occupied energy bands of a crystal to the quanta of fractional charge robustly localized at its corners. When an additional chiral symmetry is present, e 2 corner charges are accompanied by zero-energy corner-localized states. We show the application of our indices in a number of atomic and fragile topological insulators and discuss the role of fractional charges bound to disclinations as bulk probes for these crystalline phases.Topological crystalline insulators (TCIs) [1][2][3][4][5][6][7] are known to exhibit a variety of quantized electromagnetic phenomena. They host bulk dipole moments that lead to surface charge densities quantized in fractions of the electronic charge e [8][9][10][11][12][13]. Recently, it was found that TCIs can also host higher bulk multipole moments that manifest lower-order moments bound to their boundaries [14,15]. For example, a quadrupole insulator in two dimensions has edge-bound dipole moments and cornerbound charges, while an octupole insulator in three dimensions has surface-bound quadrupole moments, hingebound dipole moments, and corner-bound charges. Just as in the case of insulators with symmetry-protected dipole moments, crystalline symmetries quantize the boundary signatures in quadrupole or octupole TCIs. Indeed, TCIs with quantized multipole moments are symmetry protected topological phases of matter; their quantization is robust and can change only in discrete jumps at phase transitions [14,15], unless the protecting symmetries are broken.A salient property of TCIs with quantized higher multipole moments is that some of their protected observables at the boundary are at least two dimensions less than the protecting bulk. This property has now been extended to a broader family of TCIs, broadly referred to as higher-order topological insulators . In this paper, we focus on two-dimensional (2D) higher-order TCIs having zero-dimensional topological signatures. A number of studies have recently shown examples of such TCIs which exhibit in-gap corner-localized states [15,[21][22][23][24][25][26][27], some of which have been related to fractionally quantized corner charges [15,22,23,27]. Interestingly, many such TCIs have these corner signatures in spite of vanishing quadrupole moments, and their mechanisms of protection and associated topological invariants are still not completely elucidated.In this article we systematically study 2D second-order TCIs in class AI (spinless and ti...

The theory of electric polarization in crystals defines the dipole moment of an insulator in terms of a Berry phase (geometric phase) associated with its electronic ground state. This concept not only solves the long-standing puzzle of how to calculate dipole moments in crystals, but also explains topological band structures in insulators and superconductors, including the quantum anomalous Hall insulator and the quantum spin Hall insulator, as well as quantized adiabatic pumping processes. A recent theoretical study has extended the Berry phase framework to also account for higher electric multipole moments, revealing the existence of higher-order topological phases that have not previously been observed. Here we demonstrate experimentally a member of this predicted class of materials-a quantized quadrupole topological insulator-produced using a gigahertz-frequency reconfigurable microwave circuit. We confirm the non-trivial topological phase using spectroscopic measurements and by identifying corner states that result from the bulk topology. In addition, we test the critical prediction that these corner states are protected by the topology of the bulk, and are not due to surface artefacts, by deforming the edges of the crystal lattice from the topological to the trivial regime. Our results provide conclusive evidence of a unique form of robustness against disorder and deformation, which is characteristic of higher-order topological insulators.