Information propagation in one-dimensional XY-$\Gamma$ chains

The bond-dependent Kitaev model offers a playground in which one can search for quantum spin liquids. In these Kitaev materials, a symmetric off-diagonal $\Gamma$ term emerges, hosting a number of remarkable features, which has been particularly challenging to fully understand. One primary question that arises after recognizing a new phase is how information will spread in it. Out-of-time-ordered commutators and entanglement entropy describe processes whereby information about the initial condition of a unitarily evolving system propagates over the system. A possible way to investigate dynamics in such systems is by considering one-dimensional models. We investigate here the one-dimensional spin-1/2 XY model in a transverse field with a $\Gamma$ interaction with periodic boundary conditions imposed. We will show that the $\Gamma$ interaction constructs an asymmetric "light-cone" with different butterfly velocities. In addition, it leads to faster information propagation in the spiral phase and slower propagation in the ferromagnetic and paramagnetic phases. Interestingly, we observe a pronounced effect in the entanglement entropy, explicitly showing up as a two-stage linear growth in time as fast/slow then slow/fast for quenches originating from the spiral phase. We hope our work paves the way for studying more about the spreading of information in one-dimensional Kitaev materials, which can in turn help to discover unknown aspects of higher-dimensional models.