Slow quenches in a quantum Ising chain; dynamical phase transitions and topology

Shraddha Sharma, Uma Divakaran, Anatoli Polkovnikov, and Amit Dutta Department of Physics, Indian Institute of Technology, Kanpur 208 016, India UM-DAE Center for Excellence in Basic Sciences, Mumbai 400 098, India Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, USA We study the slow quenching dynamics (characterized by an inverse rate, τ−1) of a onedimensional transverse Ising chain with nearest neighbor ferromagentic interactions across the quantum critical point (QCP) and analyze the Loschmidt overlap measured using the subsequent temporal evolution of the final wave function (reached at the end of the quenching) with the final time-independent Hamiltonian. Studying the Fisher zeros of the corresponding generalized “partition function”, we probe non-analyticities manifested in the rate function of the return probability known as dynamical phase transitions (DPTs). In contrast to the sudden quenching case, we show that DPTs survive in the subsequent temporal evolution following the quenching across two critical points of the model for a sufficiently slow rate; furthermore, an interesting “lobe” structure of Fisher zeros emerge. We have also made a connection to topological aspects studying the dynamical topological order parameter (νD(t)), as a function of time (t) measured from the instant when the quenching is complete. Remarkably, the time evolution of νD(t) exhibits drastically different behavior following quenches across a single QCP and two QCPs. In the former case, νD(t) increases step-wise by unity at every DPT (i.e., ∆νD = 1). In the latter case, on the other hand, νD(t) essentially oscillates between 0 and 1 (i.e., successive DPTs occur with ∆νD = 1 and ∆νD = −1, respectively), except for instants where it shows a sudden jump by a factor of unity when two successive DPTs carry a topological charge of same sign.