Dynamics of the geometric phase in inhomogeneous quantum spin chains

The dynamics of the geometric phase are studied in inhomogeneous quantum spin chains after a quench. Analytic expressions of the Pancharatnam geometric phase (PGP) are derived for both the period-2 quantum Ising chain (QIC) and the disordered QIC. In the period-2 QIC, due to the periodic modulation, the phase difference between the boundaries of the half Brillouin zone (0, pi] changes with time, and consequently, the winding number nu(D)(t) of the PGP is not quantized and thus is not topological anymore. Nevertheless, the PGP and its winding number show nonanalytic singularities at the critical times of the dynamical quantum phase transitions (DQPTs). This relation between the PGP and the DQPT is further confirmed in the disordered QIC. It is found that the critical time of the DQPT inherited from the homogeneous system and the additional one induced by weak disorder are also accompanied by the nonanalytic singularity of the PGP by decomposing the PGP into each quasiparticle mode. The connection between the nonanalytic behavior of the PGP at the critical time and the DQPT, regardless of whether the winding number is topological, can be explained by the fact that they both arise when the Loschmidt amplitude vanishes.