37 research outputs found
Quantized Electric Multipole Insulators
In this article we extend the celebrated Berry-phase formulation of electric
polarization in crystals to higher electric multipole moments. We determine the
necessary conditions under which, and minimal models in 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 manifest topologically protected corner
states carrying fractional charge, i.e., fractionalization at the boundary of
the boundary. To characterize these new insulating phases of matter, we
introduce a new paradigm whereby `nested' Wilson loops give rise to a large
number of new topological invariants that have been previously overlooked. We
propose three realistic experimental implementations of this new topological
behavior that can be immediately tested.Comment: Main text: 9 pages, 6 figures. Supplementary Material: 37 pages, 15
figures. Submitted on Jul 25, 201
Response to polarization and weak topology in Chern insulators
Chern insulators present a topological obstruction to a smooth gauge in their
Bloch wave functions that prevents the construction of exponentially-localized
Wannier functions - this makes the electric polarization ill-defined. Here, we
show that spatial or temporal differences in polarization within Chern
insulators are well-defined and physically meaningful because they account for
bound charges and adiabatic currents. We further show that the difference in
polarization across Chern-insulator regions can be quantized in the presence of
crystalline symmetries, leading to "weak" symmetry-protected topological
phases. These phases exhibit charge fractional quantization at the edge and
corner interfaces and with concomitant topological states. We also generalize
our findings to quantum spin-Hall insulators and 3D topological insulators. Our
work settles a long-standing question and deems the bulk polarization as the
fundamental quantity with a "bulk-boundary correspondence", regardless of
whether a Wannier representation is possible
Nonlinear breathers with crystalline symmetries
Nonlinear lattice models can support "discrete breather" excitations that
stay localized in space for all time. By contrast, the localized Wannier states
of linear lattice models are dynamically unstable. Nevertheless, symmetric and
exponentially localized Wannier states are a central tool in the classification
of band structures with crystalline symmetries. Moreover, the quantized
transport observed in nonlinear Thouless pumps relies on the fact that -- at
least in a specific model -- discrete breathers recover Wannier states in the
limit of vanishing nonlinearity. Motivated by these observations, we
investigate the correspondence between nonlinear breathers and linear Wannier
states for a family of discrete nonlinear Schr\"odinger equations with
crystalline symmetries. We develop a formalism to analytically predict the
breathers' spectrum, center of mass and symmetry representations, and apply
this to nonlinear generalizations of the Su-Schrieffer-Heeger chain and the
breathing kagome lattice.Comment: 16+4 pages, 8+1 figure