Topological Instabilities in ac-Driven Bosonic Systems

Abstract

Under non-equilibrium conditions, bosonic modes can become dynamically unstable with an exponentially growing occupation. On the other hand, topological band structures give rise to symmetry protected midgap states. In this letter, we investigate the interplay of instability and topology. Thereby, we establish a general relation between topology and instability under ac-driving. We apply our findings to create dynamical instabilities which are strongly localized at the boundaries of a finite-size system. As these localized instabilities are protected by symmetry, they can be considered as topological instabilities.PACS numbers: 05.30. Jp, 63.20.Pw, 42.50.Dv Introduction. Due to their underlying symplectic structure, bosonic systems can exhibit so-called dynamical instabilities [1][2][3]. This effect can occur in the presence of non-particle conserving terms which appear, e.g., in the Bogoliubov excitations of Bose-Einstein condensates. Thereby, the bosonic occupation of a mode grows exponentially in time due to a non-equilibrium state of the system. On the other hand, the theory of topological band structures predicts symmetry protected midgap states [4][5][6][7][8][9][10][11][12][13]. A priory, dynamical instability and topology are independent phenomena.In this letter, we formalize a relation between instabilities and topology under ac driving. More precisely, we demonstrate that different topological phases are always separated by regions of instability. Using this relation, we demonstrate how to employ topology to systematically engineer topologically protected dynamical instabilities. Thereby, spatially localized midgap modes are rendered dynamically unstable with exponentially growing bosonic occupation as has been exemplary proposed for Bose-Einstein condensates in Refs. [11,14,15], and for photonic systems [16]. Here, we suggest a very flexible tool in the form of ac-fields in order to engineer topological instabilities governed by corresponding artificial, effective Hamiltonians [17]. This simultaneously provides the possibility to detect the midgap states as their occupation increases exponentially in time. Topological instabilities are an effect with no direct analogue in fermionic topological insulators. This stresses the need for a more intensive investigation of topological effects in bosonic systems.In fermionic systems, ac-driving has been applied to control topological phases [18][19][20][21][22][23][24]. In particular, the topology of a band can change if there is a degeneracy of the form i = i + Ω, where Ω denotes the driving frequency. As this is a single-particle effect, it can also appear in bosonic systems [25,26]. The main challenge, however, is that dynamical instabilities appearing in bosonic systems generated by an ac-driving con-

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