Topological Waves in Fluids with Odd Viscosity

Abstract

Fluids in which both time-reversal and parity are broken can display a dissipationless viscosity that is odd under each of these symmetries. Here, we show how this odd viscosity has a dramatic effect on topological sound waves in fluids, including the number and spatial profile of topological edge modes. Odd viscosity provides a short-distance cutoff that allows us to define a bulk topological invariant on a compact momentum space. As the sign of odd viscosity changes, a topological phase transition occurs without closing the bulk gap. Instead, at the transition point, the topological invariant becomes ill-defined because momentum space cannot be compactified. This mechanism is unique to continuum models and can describe fluids ranging from electronic to chiral active systems.In ordinary fluids, acoustic waves with sufficiently large wavelength have arbitrarily low frequency due to Galilean invariance [1]. When either a global rotation or an external magnetic field is present, Galilean invariance is explicitly broken by either Coriolis or Lorentz forces within the fluid, respectively. Hence, the spectrum of acoustic waves becomes gapped in the bulk. Yet, a peculiar phenomenon can occur at edges or interfaces: chiral edge modes propagate robustly irrespective of interface geometry. This phenomenon analogous to edge states in the quantum Hall effect [2][3][4] was unveiled in the context of equatorial waves [5] and explored in out-of-equilibrium and active fluids [6,7]. Similar phenomena occur in lattices of circulators [8,9], polar active fluids under confinement [10] and coupled mechanical oscillators [11][12][13], including gyroscopes [14,15] and oscillators subject to Coriolis forces [16,17].In addition to Coriolis or Lorentz body forces, fluids in which time-reversal and parity are broken generically exhibit a dissipationless viscosity that is odd under each of these symmetries [18,19]. The viscosity tensor η ijkl relates the strain rate v kl ≡ ∂ k v l to the viscous part of the stress tensor σ ij = η ijkl v kl . Odd viscosity refers to the antisymmetric part of the viscosity tensor η o ijkl = −η o klij [18,19]. In an isotropic two-dimensional fluid, odd viscosity is specified by a single pseudoscalar η o , see Supplementary Information (SI) for details [19]. Odd viscosity changes sign under either time-reversal or parity, and hence must vanish when at least one of these symmetries is present. Conversely, odd viscosity is generically non-vanishing as soon as both time-reversal and parity are broken [20][21][22]. For instance, microscopic Coriolis or Lorentz forces are sufficient to induce a non-zero odd viscosity [23,24], in addition to the corresponding body forces. Odd viscosity has been studied theoretically in various systems (see SI for a partial review) including polyatomic gases [25], magnetized plasmas [24,26], flu-ids of vortices [27][28][29][30], chiral active fluids [31], quantum Hall states and chiral superfluids/superconductors [32][33][34][35][36][37][38][39][40][41][42]. Its presence has been exp...

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