Edge states and topological pumping in stiffness-modulated elastic plates

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Edge states and topological pumping in stiffness-modulated elastic plates

Emanuele Riva, Matheus I. N. Rosa, and Massimo Ruzzene

Phys. Rev. B 101, 094307 – Published 23 March 2020

Abstract

We demonstrate that modulations of the stiffness properties of an elastic plate along a spatial dimension induce edge states spanning nontrivial gaps characterized by integer-valued Chern numbers. We also show that topological pumping is induced by smooth variations of the phase of the modulation profile along one spatial dimension, which results in adiabatic edge-to-edge transitions of the edge states. The concept is first illustrated numerically for sinusoidal stiffness modulations and then experimentally demonstrated in a plate with square-wave thickness profile. The presented numerical and experimental results show how continuous modulations of properties may be exploited in the quest for topological phases of matter. This opens new possibilities for topology-based waveguiding through slow modulations along a second dimension, spatial or temporal.

Received 19 November 2019
Revised 2 March 2020
Accepted 3 March 2020

DOI:https://doi.org/10.1103/PhysRevB.101.094307

Physics Subject Headings (PhySH)

Acoustic metamaterials Continuum mechanics Topological insulators Topological phases of matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Emanuele Riva ¹, Matheus I. N. Rosa², and Massimo Ruzzene³

¹Department of Mechanical Engineering, Politecnico di Milano, Milano 20156, Italy
²School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
³Department of Mechanical Engineering, University of Colorado Boulder, Boulder, Colorado 80309, USA

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Vol. 101, Iss. 9 — 1 March 2020

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Images

Figure 1
Stiffness modulations and plate configurations. (a) Plate (shaded gray solid) characterized by a harmonic stiffness modulation $D (x, y) = D_{0} {1 + a_{m} cos [κ_{m} x + ϕ (y)]}$ (colored surface). The schematic illustrates a linear phase change from $ϕ_{i}$ to $ϕ_{f}$ . (b) Schematic of plate with square-wave modulation of thickness $h (x, y) = h_{0} (1 + a_{m} sgn {cos [κ_{m} x + ϕ (y)]})$ . The phase also varies linearly from $ϕ_{i}$ to $ϕ_{f}$ . (c) Top view of modulation in (a) illustrating the shift of the profile characterized by a tilting angle $α$ . (d) Top view and perspective view (inset) of square wave modulated plate employed in experiments. The sample is characterized by parameters $λ_{m} = 1.6 cm, h_{0} = 4.7$ cm, $a_{m} = 0.38, L_{x} = 31.2$ cm, and $L_{y} = 43.7$ cm and phase varying linearly from $ϕ_{i} = 0.7 π$ to $ϕ_{f} = - 0.7 π$ .
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Figure 2
Dispersion properties and topological pumping for elastic plate with harmonic stiffness modulation $D (x, y) = D_{0} {1 + a_{m} cos [κ_{m} x + ϕ (y)]}$ . (a) Dispersion surfaces $Ω (μ_{x}, ϕ)$ for $μ_{y} = 0$ showing two bands separated by a gap, with information on Chern numbers and gap label. (b) Frequency spectrum of the finite plate with $L_{x} = 20 λ_{m}$ and $μ_{y} = 0$ as a function of $ϕ$ : Black straight lines corresponding to finite structure modes are superimposed to the bulk bands (shaded gray regions), while an edge mode (red curve) spans the gap. (c) Representative left-localized, bulk, and right-localized modes corresponding to the points marked in (b). (d) Variation of the finite plate spectrum in (b) as a function of $μ_{y}$ . The red surface represents the dispersion of the edge state, while shaded gray volumes are the bulk bands. The red line at $μ_{y} = 0$ highlights the transition of the edge state from left-localized (dashed) to right localized (solid) at $ϕ = π$ . (e) Steady-state response $| w (x, y) |$ of the modulated plate at frequency $Ω = 10.97$ , where the associated colormap also represents normalized displacement. Topological pumping occurs through a transition of the edge state from left localized ( $ϕ_{i} = 0.5 π$ ) to right localized ( $ϕ_{f} = 1.5 π$ ) due to the phase modulation $ϕ (y) = ϕ_{i} \to ϕ_{f}$ . (f) Cross section of dispersion diagram in (d) at frequency $Ω = 10.97$ as a function of $ϕ \in [0.5 π, 1.5 π]$ . Black (straight) lines and red curves, respectively, denote bulk and edge modes of the finite plate. The contours of the spectrogram of the displacement field $| \hat{w} (y, μ_{y}) |$ reveal that pumping occurs through a transition along the wave-number branch of the edge state.
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Figure 3
Dispersion properties and experimental observation of topological pumping in plate with square-wave modulation of the thickness $h (x, y) = h_{0} (1 + a_{m} sgn {cos [κ_{m} x + ϕ (y)]})$ . (a) Dispersion properties and edge state (red surface) for a finite plate with $L_{x} = 31.2 cm$ and free-free boundary conditions along $x$ . Bulk bands are represented by shaded gray volumes, along with Chern number and gap labels information. (b) Detail of dispersion for $ϕ \in [- 0.7 π, 0.7 π]$ , corresponding to the interval considered for the phase modulation of thickness. The experimentally measured frequency response spatially averaged over the plate surface is displayed alongside the dispersion. Each resonant peak defines a wave-number branch highlighted by dashed and solid red lines in the dispersion surface, along which topological pumping occurs at the corresponding frequency. [(c)–(e)] Experimentally measured velocity field $| \dot{w} (x, y) |$ for selected resonant peaks II, IV, and V. The transitions from localization at the left boundary to localization at the right boundary that characterize topological pumping are quantified by the spectrograms displayed in (f)–(h), which confirm that energy is concentrated around the wave-number branches of the edge states.
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Figure 4
Experimental observation of transient topological pumping in square-modulated plate. Panels (a)–(c) display snapshots of the measured velocity field for three subsequent time instants, where the transition from left-localized (a), to bulk (b), and, finally, to right-localized mode (c) as the wave propagates along $y$ can be observed.
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