Dynamical quantum phase transitions in transverse field <i>XY</i> chain after a double quench
Authors: Fu Hao , Cao Kai-Yuan , Zhong Ming , Tong Pei-Qing ,   School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China,   Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, China
Published: August 2021

Nonequilibrium dynamics of quantum many-body systems have achieved rapid progress from both theoretical and experimental perspectives. Recently, dynamical quantum phase transitions (DQPTs), which describe the nonanalytic behaviors of physical quantities during the time evolution, have attracted a lot of interest. The most studied protocol to drive the system out of equilibrium is via a quantum quench. Recently, the DQPTs in the Ising chain and ANNNI chain after double quench are studied. Double quench means that the Hamiltonian of the system is abruptly changed from <inline-formula><tex-math id="M16">\begin{document}$H_{0}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M16.png"/></alternatives></inline-formula> to <inline-formula><tex-math id="M17">\begin{document}$H_{1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M17.png"/></alternatives></inline-formula>, and then abruptly changed from <inline-formula><tex-math id="M18">\begin{document}$H_{1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M18.png"/></alternatives></inline-formula> to <inline-formula><tex-math id="M19">\begin{document}$H_{2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M19.png"/></alternatives></inline-formula> after a evolutionary time <i>T</i>. One can control at will whether or not DQPTs appear after the second quench by varying <i>T</i>. In this paper, we study the DQPTs arising from a double quench in the anisotropic <inline-formula><tex-math id="M20">\begin{document}$XY$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M20.png"/></alternatives></inline-formula> chain in a transverse field. The anisotropic <inline-formula><tex-math id="M21">\begin{document}$XY$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M21.png"/></alternatives></inline-formula> chain in a transverse field has two kinds of quantum phase transitions (Ising transition and anisotropic transition). We discuss mainly the effects of quench paths on the DQPTs of the transverse field <inline-formula><tex-math id="M22">\begin{document}$XY$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M22.png"/></alternatives></inline-formula> chain. By calculating the rate function of the Loschmidt echo and Fisher zeros, we find that there are three typical types of the critical times of DQPTs in the plane of the <i>T</i> and the evolution time <i>t</i>. Type I of critical times, which occurs only in a certain range of <i>T</i>, is related to the protocol of the Hamiltonian abruptly changed from <inline-formula><tex-math id="M23">\begin{document}$H_{0}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M23.png"/></alternatives></inline-formula> to <inline-formula><tex-math id="M24">\begin{document}$H_{1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M24.png"/></alternatives></inline-formula>. Type II of critical times, which occurs all the time, is related to the protocol of the Hamiltonian abruptly changed from <inline-formula><tex-math id="M25">\begin{document}$H_{0}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M25.png"/></alternatives></inline-formula> to <inline-formula><tex-math id="M26">\begin{document}$H_{2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M26.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M26.png"/></alternatives></inline-formula>. Type III of critical times, which occurs all the time, is related to the protocols of the Hamiltonian abruptly changed from <inline-formula><tex-math id="M27">\begin{document}$H_{0}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M27.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M27.png"/></alternatives></inline-formula> to <inline-formula><tex-math id="M28">\begin{document}$H_{1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M28.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M28.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M29">\begin{document}$H_{0}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M29.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M29.png"/></alternatives></inline-formula> to <inline-formula><tex-math id="M30">\begin{document}$H_{2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M30.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210728_M30.png"/></alternatives></inline-formula>. When the double quench paths pass through the same kind of transition point, only the critical times corresponding to Type I will appear after the second quench. When the double quench paths pass through different kinds of transition points, the critical times after the second quench will appear any two of the above three types, which depend on the choice of quench path.