  We propose a twisted Szegedy walk for estimating the limit behavior of a
discrete-time quantum walk on a crystal lattice, an infinite abelian covering
graph, whose notion was introduced by [14]. First, we show that the spectrum of
the twisted Szegedy walk on the quotient graph can be expressed by mapping the
spectrum of a twisted random walk onto the unit circle. Secondly, we show that
the spatial Fourier transform of the twisted Szegedy walk on a finite graph
with appropriate parameters becomes the Grover walk on its infinite abelian
covering graph. Finally, as an application, we show that if the Betti number of
the quotient graph is strictly greater than one, then localization is ensured
with some appropriated initial state. We also compute the limit density
function for the Grover walk on $\mathbb{Z}^d$ with flip flop shift, which
implies the coexistence of linear spreading and localization. We partially
obtain the abstractive shape of the limit density function: the support is
within the $d$-dimensional sphere of radius $1/\sqrt{d}$, and $2^d$ singular
points reside on the sphere's surface.
