Slow quench dynamics of the Kitaev model: Anisotropic critical point and effect of disorder

We study the nonequilibrium slow dynamics for the Kitaev model both in the presence and the absence of disorder. For the case without disorder, we demonstrate, via an exact solution, that the model provides an example of a system with an anisotropic critical point and exhibits unusual scaling of defect density n and residual energy Q for a slow linear quench. We provide a general expression for the scaling of n(Q)generated during a slow power-law dynamics, characterized by a rate tau(-1) and exponent alpha, from a gapped phase to an anisotropic quantum critical point in d dimensions, for which the energy gap Delta((k) over right arrow) similar to k(i)(z) for m momentum components (i= 1, ... , m) and similar to k(i)(z') for the rest d-m components (i=m+ 1, ... , d) with z <= z': n similar to t(-[m+(d-m)z/z']v alpha/(zv alpha+1))(Q similar to tau(-[(m+z)+(d-m)z/z']v alpha/(zv alpha+1))). These general expressions reproduce both the corresponding results for the Kitaev model as a special case for d=z'=2 and m=z=v=1 and the well- known scaling laws of n and Q for isotropic critical points for z=z'. We also present an exact computation of all nonzero, independent, multispin correlation functions of the Kitaev model for such a quench and discuss their spatial dependence. For the disordered Kitaev model, where the disorder is introduced via random choice of the link variables D-n in the model's fermionic representation, we find that n similar to tau(-1/2) and Q similar to tau(-1)(Q similar to tau(-1/2))for a slow linear quench ending in the gapless (gapped) phase. We provide a qualitative explanation of such scaling.