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

We study the slow quenching dynamics (characterized by an inverse rate τ−1) of a one-dimensional 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 nonanalyticities 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 stepwise by unity at every DPT (i.e., ΔνD=1). In the latter case, on the other hand, νD(t) essentially oscillates between 0 and 1 (i.e., successive DPTs occur with ΔνD=1 and ΔνD=−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 the same sign.We acknowledge Jun-ichi Inoue for discussions and Sei Suzuki for extensive discussions, critical comments on the manuscript, and collaboration in related works. Special thanks to Utso Bhattacharya for his critical comments. S.S. acknowledges CSIR, India and also DST, India, and A.D. and U.D. acknowledge DST, India, for financial support. A.D. and S.S. acknowledge Abdus Salam ICTP for hospitality where the initial part of the work was done. A.P. was supported by AFOSR FA9550-13-1-0039, NSF DMR-1506340, and ARO W911NF1410540. (DST, India; FA9550-13-1-0039 - AFOSR; DMR-1506340 - NSF; W911NF1410540 - ARO)Accepted manuscrip