Hinge solitons in three-dimensional second-order topological insulators
TL;DR: In this article, the existence of stable solitons localized on the hinges of a second-order topological insulator in three dimensions was shown by means of a systematic numerical study, and the soliton propagates along the hinge unidirectionally without changing its shape.
Abstract: Higher-order topological insulators have recently witnessed rapid progress in various fields ranging from condensed matter physics to electric circuits. A well-known higher-order state is the second-order topological insulator in three dimensions with gapless states localized on the hinges. A natural question in the context of nonlinearity is whether solitons can exist on the hinges in a second-order topological insulator. Here we theoretically demonstrate the existence of stable solitons localized on the hinges of a second-order topological insulator in three dimensions when nonlinearity is involved. By means of systematic numerical study, we find that the soliton has strong localization in real space and propagates along the hinge unidirectionally without changing its shape. We further construct an electric network to simulate the second-order topological insulator. When a nonlinear inductor is appropriately involved, we find that the system can support a bright soliton for the voltage distribution demonstrated by stable time evolution of a voltage pulse.
Citations
More filters
TL;DR: In this paper, a photonic platform enables the observation of nonlinear topological corner states and solitons in a second-order topological insulator, as shown by experiments.
Abstract: Higher-order topological insulators are a novel topological phase beyond the framework of conventional bulk–boundary correspondence1,2. In these peculiar systems, the topologically non-trivial boundary modes are characterized by a co-dimension of at least two3,4. Despite several promising preliminary considerations regarding the impact of nonlinearity in such systems5,6, the flourishing field of experimental higher-order topological insulator research has thus far been confined to the linear evolution of topological states. As such, the observation of the interplay between nonlinearity and the dynamics of higher-order topological phases in conservative systems remains elusive. Here we experimentally demonstrate nonlinear higher-order topological corner states. Our photonic platform enables us to observe nonlinear topological corner states as well as the formation of solitons in such topological structures. Our work paves the way towards the exploration of topological properties of matter in the nonlinear regime, and may herald a new class of compact devices that harnesses the intriguing features of topology in an on-demand fashion. The nonlinear properties of photonic topological insulators remain largely unexplored, as band topology is linked to linear systems. But nonlinear topological corner states and solitons can form in a second-order topological insulator, as shown by experiments.
106 citations
TL;DR: In this article, the role of strong nonlinearity on the topologically robust edge state in a one-dimensional system was examined, and the robustness of frequency and stability of nonlinear edge states against disorder was investigated.
Abstract: We examine the role of strong nonlinearity on the topologically robust edge state in a one-dimensional system. We consider a chain inspired from the Su-Schrieffer-Heeger model but with a finite-frequency edge state and the dynamics governed by second-order differential equations. We introduce a cubic onsite nonlinearity and study this nonlinear effect on the edge state's frequency and linear stability. Nonlinear continuation reveals that the edge state loses its typical shape enforced by the chiral symmetry and becomes generally unstable due to various types of instabilities that we analyze using a combination of spectral stability and Krein signature analysis. This results in an initially excited nonlinear-edge state shedding its energy into the bulk over a long time. However, the stability trends differ both qualitatively and quantitatively when softening and stiffening types of nonlinearity are considered. In the latter, we find a frequency regime where nonlinear edge states can be linearly stable. This enables high-amplitude edge states to remain spatially localized without shedding their energy, a feature that we have confirmed via long-time dynamical simulations. Finally, we examine the robustness of frequency and stability of nonlinear edge states against disorder, and find that those are more robust under a chiral disorder compared to a nonchiral disorder. Moreover, the frequency-regime where high-amplitude edge states were found to be linearly stable remains intact in the presence of a small amount of disorder of both types.
41 citations
19 Apr 2021
TL;DR: In this paper, a generic topological tight-binding model for topolectric circuits was proposed by exploiting the Clifford algebra of Hermitian matrices, which can be seen as a generalization of tight-bounding models.
Abstract: This paper shows how to engineer generic topological tight-binding models on classical topolectric circuits by exploiting the Clifford algebra of Hermitian matrices.
26 citations
TL;DR: In this article, the existence of a second-order topological insulator in amorphous systems with time-reversal symmetry has been shown to exist in three dimensions.
Abstract: Higher-order topological insulators are established as topological crystalline insulators protected by crystalline symmetries. One celebrated example is the second-order topological insulator in three dimensions that hosts chiral hinge modes protected by crystalline symmetries. Since amorphous solids are ubiquitous, it is important to ask whether such a second-order topological insulator can exist in an amorphous system without any spatial order. Here, we predict the existence of a second-order topological insulating phase in an amorphous system without any crystalline symmetry. Such a topological phase manifests in the winding number of the quadrupole moment, the quantized longitudinal conductance, and the hinge states. Furthermore, in stark contrast to the viewpoint that structural disorder should be detrimental to the higher-order topological phase, we remarkably find that structural disorder can induce a second-order topological insulator from a topologically trivial phase in a regular geometry. We finally demonstrate the existence of a second-order topological phase in amorphous systems with time-reversal symmetry.
24 citations
TL;DR: In this paper, the authors highlight a general theory to engineer arbitrary Hermitian tight-binding lattice models in electrical LC circuits, where the lattice sites are replaced by the electrical nodes, connected to its neighbors and to the ground by capacitors and inductors.
Abstract: We highlight a general theory to engineer arbitrary Hermitian tight-binding lattice models in electrical LC circuits, where the lattice sites are replaced by the electrical nodes, connected to its neighbors and to the ground by capacitors and inductors. In particular, by supplementing each node with $n$ subnodes, where the phases of the current and voltage are the $n$ distinct roots of \emph{unity}, one can in principle realize arbitrary hopping amplitude between the sites or nodes via the \emph{shift capacitor coupling} between them. This general principle is then implemented to construct a plethora of topological models in electrical circuits, \emph{topoelectric circuits}, where the robust zero-energy topological boundary modes manifest through a large boundary impedance, when the circuit is tuned to the resonance frequency. The simplicity of our circuit constructions is based on the fact that the existence of the boundary modes relies only on the Clifford algebra of the corresponding Hermitian matrices entering the Hamiltonian and not on their particular representation. This in turn enables to implement a wide class of topological models through rather simple topoelectric circuits with nodes consisting of only two subnodes. We anchor these outcomes from the numerical computation of the on-resonance impedance in circuit realizations of first-order ($m=1$), such as Chern and quantum spin Hall insulators, second- ($m=2$) and third- ($m=3$) order topological insulators in different dimensions, featuring sharp localization on boundaries of codimensionality $d_c=m$. Finally, we subscribe to the \emph{stacked topoelectric circuit} construction to engineer three-dimensional Weyl, nodal-loop, quadrupolar Dirac and Weyl semimetals, respectively displaying surface and hinge localized impedance.
22 citations
Cites background from "Hinge solitons in three-dimensional..."
...07931 [63] Y-L....
[...]
...In this respect, topoelectric circuits, made of rather simple capacitance and inductance elements, yield a readily available route for the realization of a plethora of topological phases [30–69]....
[...]
References
More filters
TL;DR: In this paper, the authors studied three-dimensional generalizations of the quantum spin Hall (QSH) effect and introduced a tight binding model which realized the WTI and STI phases, and discussed its relevance to real materials including bismuth.
Abstract: We study three-dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where a single ${Z}_{2}$ topological invariant governs the effect, in three dimensions there are 4 invariants distinguishing 16 phases with two general classes: weak (WTI) and strong (STI) topological insulators. The WTI are like layered 2D QSH states, but are destroyed by disorder. The STI are robust and lead to novel ``topological metal'' surface states. We introduce a tight binding model which realizes the WTI and STI phases, and we discuss its relevance to real materials, including bismuth.
3,357 citations
TL;DR: In this paper, it was shown that the parity of the occupied Bloch wave functions at the time-reversal invariant points in the Brillouin zone greatly simplifies the problem of evaluating the topological invariants.
Abstract: Topological insulators are materials with a bulk excitation gap generated by the spin-orbit interaction that are different from conventional insulators. This distinction is characterized by ${Z}_{2}$ topological invariants, which characterize the ground state. In two dimensions, there is a single ${Z}_{2}$ invariant that distinguishes the ordinary insulator from the quantum spin-Hall phase. In three dimensions, there are four ${Z}_{2}$ invariants that distinguish the ordinary insulator from ``weak'' and ``strong'' topological insulators. These phases are characterized by the presence of gapless surface (or edge) states. In the two-dimensional quantum spin-Hall phase and the three-dimensional strong topological insulator, these states are robust and are insensitive to weak disorder and interactions. In this paper, we show that the presence of inversion symmetry greatly simplifies the problem of evaluating the ${Z}_{2}$ invariants. We show that the invariants can be determined from the knowledge of the parity of the occupied Bloch wave functions at the time-reversal invariant points in the Brillouin zone. Using this approach, we predict a number of specific materials that are strong topological insulators, including the semiconducting alloy ${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$ as well as $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Sn}$ and HgTe under uniaxial strain. This paper also includes an expanded discussion of our formulation of the topological insulators in both two and three dimensions, as well as implications for experiments.
3,349 citations
"Hinge solitons in three-dimensional..." refers background in this paper
...When 1 < |M/J | < 3, the Hamiltonian represents a strong topological insulator with odd number of Dirac points in the energy spectra of surface states; when |M/J | < 1, it describes a weak topological insulator with even number of Dirac points in the surface energy spectra [84, 85]....
[...]
Journal Article•
2,856 citations
"Hinge solitons in three-dimensional..." refers background in this paper
...3, 26 [4] Shabat A and Zakharov V 1972 Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media Sov....
[...]
...Solitons exist in various nonlinear systems, such as nonlinear optics [1–3], BoseEinstein condensates (BECs) [4–8] and Fermi superfluids [9–14]....
[...]
TL;DR: In this paper, it was shown that the fundamental time-reversal invariant (TRI) insulator exists in $4+1$ dimensions, where the effective field theory is described by the $(4 + 1)$-dimensional Chern-Simons theory and the topological properties of the electronic structure are classified by the second Chern number.
Abstract: We show that the fundamental time-reversal invariant (TRI) insulator exists in $4+1$ dimensions, where the effective-field theory is described by the $(4+1)$-dimensional Chern-Simons theory and the topological properties of the electronic structure are classified by the second Chern number. These topological properties are the natural generalizations of the time reversal-breaking quantum Hall insulator in $2+1$ dimensions. The TRI quantum spin Hall insulator in $2+1$ dimensions and the topological insulator in $3+1$ dimensions can be obtained as descendants from the fundamental TRI insulator in $4+1$ dimensions through a dimensional reduction procedure. The effective topological field theory and the ${Z}_{2}$ topological classification for the TRI insulators in $2+1$ and $3+1$ dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of measurable phenomena, the most striking of which is the topological magnetoelectric effect, where an electric field generates a topological contribution to the magnetization in the same direction, with a universal constant of proportionality quantized in odd multiples of the fine-structure constant $\ensuremath{\alpha}={e}^{2}∕\ensuremath{\hbar}c$. Finally, we present a general classification of all topological insulators in various dimensions and describe them in terms of a unified topological Chern-Simons field theory in phase space.
2,658 citations
Journal Article•
TL;DR: A tight binding model is introduced which realizes the WTI and STI phases, and its relevance to real materials, including bismuth is discussed.
Abstract: We study three-dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where a single ${Z}_{2}$ topological invariant governs the effect, in three dimensions there are 4 invariants distinguishing 16 phases with two general classes: weak (WTI) and strong (STI) topological insulators. The WTI are like layered 2D QSH states, but are destroyed by disorder. The STI are robust and lead to novel ``topological metal'' surface states. We introduce a tight binding model which realizes the WTI and STI phases, and we discuss its relevance to real materials, including bismuth.
2,325 citations
"Hinge solitons in three-dimensional..." refers background in this paper
...When 1 < |M/J | < 3, the Hamiltonian represents a strong topological insulator with odd number of Dirac points in the energy spectra of surface states; when |M/J | < 1, it describes a weak topological insulator with even number of Dirac points in the surface energy spectra [84, 85]....
[...]