Spin–orbit coupling in a hexagonal ring of pendula

Abstract

We consider the mechanical motion of a system of six macroscopic pendula which are connected with springs and arranged in a hexagonal geometry. When the springs are pre-tensioned, the coupling between neighbouring pendula along the longitudinal (L) and the transverse (T) directions are different: identifying the motion along the L and T directions as the two components of a spin-like degree of freedom, we theoretically and experimentally verify that the pre-tensioned springs result in a tunable spin-orbit coupling. We elucidate the structure of such a spin-orbit coupling in the extended two-dimensional honeycomb lattice, making connections to physics of graphene. The experimental frequencies and the oscillation patterns of the eigenmodes for the hexagonal ring of pendula are extracted from a spectral analysis of the motion of the pendula in response to an external excitation and are found to be in good agreement with our theoretical predictions. We anticipate that extending this classical analogue of quantum mechanical spin-orbit coupling to two-dimensional lattices will lead to exciting new topological phenomena in classical mechanics. of a spin-orbit coupling for photons [26][27][28][29][30][31]. Spin-orbit coupling in such photonic systems has been predicted to induce various topological phenomena, such as topological phase transitions [31,32]. A detailed theoretical and experimental study of such a spin-orbit coupling for a hexagonal ring of exciton-polariton microcavities was reported in [30], where two polarisation states provided the pseudospin degrees of freedom.In the present work, we show that similar physics can be observed also in systems governed by Newtonian classical mechanics. Towards this goal, we theoretically and experimentally investigate an analogous mechanical model consisting of six pendula arranged in a hexagonal ring structure and coupled by springs. In particular, we observe that the frequency spectrum strongly depends on the ratio between the rest length of the springs and the equilibrium distance between neighbouring pendula. A mismatch between these two quantities results in a finite pre-tensioning of the springs which, as first anticipated by [33], is responsible for different effective spring constants along the longitudinal (L) and the transverse (T) directions with respect to the spring axis. This is analogous to how the coupling between neighbouring sites depends on the two polarisation states in the polariton lattice of [30]. On this basis, the mechanical system can also be interpreted as being subject to an effective spin-orbit coupling, as we show in the following.This article is organised as follows. In section 2 we introduce the mechanical system under consideration and we theoretically review the origin of the spin-orbit coupling term, first in an infinite honeycomb lattice, then in the hexagonal ring of pendula. For this latter case, we discuss in detail the oscillation eigenfrequencies and eigenmodes and we classify them in terms of the symmetry of the oscillation pattern...

Artificial gauge fields in materials and engineered systems

“…In these systems a spin-orbit coupling can be engineered by encoding an effective spin in two acoustic modes brought close to degeneracy. The experimental systems studied so far consist in discrete lattices of simple elementary mechanical systems, such as pendula [191,469] or gyroscopes [192], as well as acoustic crystals, made of a continuous fluid flowing in an engineered lattice geometry [206,470,471,472,473].…”

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Edge solitons in a nonlinear mechanical topological insulator

“…The asymptotic analysis shows that these nonlinear edge solitons are governed by the classical 1D NLS equation and are topologically protected in the same way as linear edge waves. The derivation can be readily generalized to other mechanical TIs with discrete elements [17,52], possibly with dissipation [53] and forcing [30] included. Further extensions may include TIs in continuous media that can exhibit significant nonlinearity, such as recent proposals based on magnetic solitons [54] and water waves [55].…”