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Abstract A microtubule is a cylindrical biological polymer that plays key roles in cellular structure, transport, and signalling. In this work, based on studies of electronic properties of polyacetelene and mechanical properties of microtubules themselves (Spakowitz A. J.,Phys. Rev. Lett.,103(2009) 248101), we explore the possibility that microtubules could act as topological insulators that are gapped to electronic excitations in the bulk but possess robust electronic bounds states at the tube ends. Through analyses of structural and electronic properties, we model the microtubule as a cylindrical stack of Su-Schrieffer-Heeger chains (originally proposed in the context of polyacetylene) describing electron hopping between the underlying dimerized tubulin lattice sites. We postulate that the microtubule is mostly uniform, dominated purely by GDP-bound dimers, and is capped by a disordered regime due to the presence of GTP-bound dimers as well. In the uniform region, we identify the electron hopping parameter regime in which the microtubule is a topological insulator. We then show the manner in which these topological features remain robust when the hopping parameters are disordered. We briefly mention possible biological implications for these microtubules to possess topologically robust electronic bound states. 

We present a formulation for investigating quench dynamics acrossquantum phase transitions in the presence of decoherence. We formulatedecoherent dynamics induced by continuous quantum non-demolitionmeasurements of the instantaneous Hamiltonian. We generalize thewell-studied universal Kibble-Zurek behavior for linear temporal driveacross the critical point. We identify a strong decoherence regimewherein the decoherence time is shorter than the standard correlationtime, which varies as the inverse gap above the groundstate. In thisregime, we find that the freeze-out time \bar{t}\sim\tau^{{2\nu z}/({1+2\nu z})} t - ∼ τ 2 ν z / ( 1 + 2 ν z ) for when the system falls out of equilibrium and the associatedfreeze-out length \bar{\xi}\sim\tau^{\nu/({1+2\nu z})} ξ ‾ ∼ τ ν / ( 1 + 2 ν z ) show power-law scaling with respect to the quench rate 1/\tau 1 / τ ,where the exponents depend on the correlation length exponent \nu ν and the dynamical exponent z z associated with the transition. The universal exponents differ fromthose of standard Kibble-Zurek scaling. We explicitly demonstrate thisscaling behavior in the instance of a topological transition in a Cherninsulator system. We show that the freeze-out time scale can be probedfrom the relaxation of the Hall conductivity. Furthermore, onintroducing disorder to break translational invariance, we demonstratehow quenching results in regions of imbalanced excitation densitycharacterized by an emergent length scale which also shows universalscaling. We perform numerical simulations to confirm our analyticalpredictions and corroborate the scaling arguments that we postulate asuniversal to a host of systems.