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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
$H_{0}$ to$H_{1}$ , and then abruptly changed from$H_{1}$ to$H_{2}$ after a evolutionary time T. One can control at will whether or not DQPTs appear after the second quench by varying T. In this paper, we study the DQPTs arising from a double quench in the anisotropic$XY$ chain in a transverse field. The anisotropic$XY$ 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$XY$ 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 T and the evolution time t. Type I of critical times, which occurs only in a certain range of T, is related to the protocol of the Hamiltonian abruptly changed from$H_{0}$ to$H_{1}$ . Type II of critical times, which occurs all the time, is related to the protocol of the Hamiltonian abruptly changed from$H_{0}$ to$H_{2}$ . Type III of critical times, which occurs all the time, is related to the protocols of the Hamiltonian abruptly changed from$H_{0}$ to$H_{1}$ and$H_{0}$ to$H_{2}$ . 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.[1] Polkovnikov A, Sengupta K, Silva A, Vengalattore M 2011 Rev. Mod. Phys. 83 863
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[10] Sharma S, Divakaran U, Polkovnikov A, Dutta A 2016 Phys. Rev. B 93 144306
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[18] Heyl M 2014 Phys. Rev. Lett. 113 205701
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[21] Yin H H, Chen S, Gao X L, Wang P 2018 Phys. Rev. A 97 033624
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[22] Yang C, Wang Y, Wang P, Gao X, Chen S 2017 Phys. Rev. B 95 184201
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[23] Žunkovič B, Heyl M, Knap M, Silva A 2018 Phys. Rev. Lett. 120 130601
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[24] Karrasch C, Schuricht D 2017 Phys. Rev. B 95 075143
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[25] Zhou L, Wang Q H, Wang H, Gong J 2018 Phys. Rev. A 98 022129
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[26] 邓天舒, 易为 2019 物理学报 68 040303
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Deng T S, Yi W 2019 Acta Phys. Sin. 68 040303
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[27] Abdi M 2019 Phys. Rev. B 100 184310
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[28] Liu T, Guo H 2019 Phys. Rev. B 99 104307
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[29] Abeling N O, Kehrein S 2016 Phys. Rev. B 93 104302
Google Scholar
[30] Vajna S, Dóra B 2015 Phys. Rev. B 91 155127
Google Scholar
[31] Wang P, Gao X L 2018 Phys. Rev. A 97 023627
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[32] Yang K, Zhou L, Ma W, Kong X, Wang P, Qin X, Rong X, Wang Y, Shi F, Gong J, Du J 2019 Phys. Rev. B 100 085308
Google Scholar
[33] Jurcevic P, Shen H, Hauke P, Maier C, Brydges T, Hempel C, Lanyon B P, Heyl M, Blatt R, Roos C F 2017 Phys. Rev. Lett. 119 080501
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[34] Fläschner N, Vogel D, Tarnowski M, Rem B S, Lühmann D S, Heyl M, Budich J C, Mathey L, Sengstock K, Weitenberg C 2018 Nat. Phys. 14 265
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[38] Wang K, Qiu X, Xiao L, Zhan X, Bian Z, Yi W, Xue P 2019 Phys. Rev. Lett. 122 020501
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[39] Tia Tn, Yang H X, Qiu L Y, Liang H Y, Yang Y B, Xu Y, Duan L M 2020 Phys. Rev. Lett. 124 043001
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Google Scholar
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[42] Hou X Y, Gao Q C, Guo H, He Y, Liu T, Chien C C 2020 Phys. Rev. B 102 104305
Google Scholar
[43] Heyl M 2015 Phys. Rev. Lett. 115 140602
Google Scholar
[44] Lang J, Frank B, Halimeh J C 2018 Phys. Rev. Lett. 121 130603
Google Scholar
[45] Hagymási I, Hubig C, Legeza Ö, Schollwöck U 2019 Phys. Rev. Lett. 122 250601
Google Scholar
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Chen S, Yang C 2019 Acta Phys. Sin. 68 220304
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[55] 贺志, 余敏, 王琼 2019 物理学 报 68 240506
He Z, Yu M, Wang Q 2019 Acta Phys. Sin. 68 240506
[56] Lieb E, Schultz T, Mattis D 1961 Ann. Phys. NY 16 407
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图 1 横场中XY链的相图, 其中的点
$A, B, \cdots, F$ 表示典型的参数值所对应的位置Figure 1. Phase diagram of the
$XY$ chain in a transverse field. The points$A, B, \cdots, F$ in the phase diagram correspond to the typical parameter values.
图 2 (a)淬火路径
$A\rightarrow B\rightarrow C$ 对应的临界时间图, 图中两条虚线对应的时刻分别为$T=0.5$ 和$1.0$ ; (b1)$T=0.5$ 和(b2)$T=1.0$ 且$t>T$ 时的Fisher零点分布; (c)两次淬火过程中的率函数, 黑色和红色实线分别对应$T=0.5$ 和$1.0$ ; (d)$|B_{k}/A_{k}|$ 与k的关系, 黑色和红色实线分别对应$T=0.5$ 和$1.0$ Figure 2. (a) Location of the critical times in the
$t\text{-}T$ plane for the path$A\rightarrow B\rightarrow C$ . The dotted lines mark the times for$T=0.5$ and$1.0$ , respectively. (b) The Fisher zeros for$t>T$ with (b1)$T=0.5$ and (b2)$T=1.0$ , respectively. (c) The rate functions corresponding to$T=0.5$ and$1.0$ , respectively. (d) The relationship between$|B_{k}/A_{k}|$ and k for$T=0.5$ and$1.0$ , respectively.
图 3 (a)淬火路径
$C\rightarrow D\rightarrow E$ 对应的临界时间图, 图中虚线对应的时刻为$T=1.25$ ; (b)$T=1.25$ 且$t>T$ 时的Fisher零点分布; (c)$T=1.25$ 时两次淬火过程的率函数; (d)黑色实线是$\tan^{2}(\theta_{0, k}-\theta_{1, k})/2$ 与k的关系, 图中红色和蓝色实线为$|B_{k}/A_{k}|$ 与k的关系, 对应的时间间隔分别为$T=T_{5}\approx1.1574$ 和$T=T_{6}\approx1.3879$ Figure 3. (a) Location of the critical times in the
$t\text{-}T$ plane for the path$C\rightarrow D\rightarrow E$ . The dotted line marks the time for$T=1.25$ . (b) The Fisher zeros for$t>T$ with$T=1.25$ . (c) The rate function corresponding to$T=1.25$ . (d) The black line corresponding to the relationship between$\tan^{2}(\theta_{0, k}-\theta_{1, k})/2$ and k. The red and blue lines corresponding to the relationships between$|B_{k}/A_{k}|$ and k for$T=T_{5}\approx1.1574$ and$T=T_{6}\approx1.3879$ , respectively.
图 4 (a)淬火路径
$B\rightarrow C\rightarrow D$ 对应的临界时间图, 图中虚线对应的时刻为$T=1.5$ ; (b)$T=1.5$ 且$t>T$ 时的Fisher零点分布; (c)$T=1.5$ 时两次淬火过程的率函数; (d)黑色实线是$\tan^{2}(\theta_{0, k}-\theta_{1, k})/2$ 与k的关系. 红色实线为$|B_{k}/A_{k}|$ 与k的关系, 对应的时间间隔为$T=T_{7}\approx1.5210$ Figure 4. (a) Location of the critical times in the
$t\text{-}T$ plane for the path$B\rightarrow C\rightarrow D$ . The dotted line marks the time for$T=1.5$ . (b) The Fisher zeros for$t>T$ with$T=1.5$ . (c) The rate function corresponding to$T=1.5$ . (d) The black line corresponding to the relationship between$\tan^{2}(\theta_{0, k}-\theta_{1, k})/2$ and k. The red line corresponding to the relationship between$|B_{k}/A_{k}|$ and k for$T=T_{7}\approx1.5210$ .
图 5 (a)淬火路径
$D\rightarrow C\rightarrow B$ 对应的临界时间图, 图中虚线对应的时刻为$T=1.5$ ; (b)$T=1.5$ 且$t>T$ 时的Fisher零点分布; (c)$T=1.5$ 时两次淬火过程的率函数; (d)黑色实线是$\tan^{2}(\theta_{0, k}-\theta_{1, k})/2$ 与k的关系, 其中红色和蓝色实线为$|B_{k}/A_{k}|$ 与k的关系, 对应的时间间隔分别为$T=T_{8}\approx1.1574$ 和$T=T_{9}\approx1.3879$ Figure 5. (a) Location of the critical times in the
$t\text{-}T$ plane for the path$D\rightarrow C\rightarrow B$ . The dotted line marks the time for$T=1.5$ . (b) The Fisher zeros for$t>T$ with$T=1.5$ . (c) The rate function corresponding to$T=1.5$ . (d) The black line corresponding to the relationship between$\tan^{2}(\theta_{0, k}-\theta_{1, k})/2$ and k. The red and blue lines corresponding to the relationships between$|B_{k}/A_{k}|$ and k for$T=T_{8}\approx1.1574$ and$T=T_{9}\approx1.3879$ , respectively.
图 6 (a)淬火路径
$C\rightarrow B\rightarrow D$ 对应的临界时间图, 图中虚线对应的时刻为$T=1.5$ ; (b)$T=1.5$ 且$t>T$ 时的Fisher零点分布; (c)$T=1.5$ 时两次淬火过程的率函数; (d)黑色实线是$\tan^{2}(\theta_{0, k}-\theta_{1, k})/2$ 与k的关系, 其中红色实线为$|B_{k}/A_{k}|$ 与k的关系, 对应的时间间隔为$T=T_{10}\approx0.9170$ Figure 6. (a) Location of the critical times in the
$t\text{-}T$ plane for the path$C\rightarrow B\rightarrow D$ . The dotted line marks the time for$T=1.5$ . (b) The Fisher zeros for$t>T$ with$T=1.5$ . (c) The rate function corresponding to$T=1.5$ . (d) The black line corresponding to the relationship between$\tan^{2}(\theta_{0, k}-\theta_{1, k})/2$ and k. The red line corresponding to the relationship between$|B_{k}/A_{k}|$ and k for$T=T_{10}\approx0.9170$ . -
[1] Polkovnikov A, Sengupta K, Silva A, Vengalattore M 2011 Rev. Mod. Phys. 83 863
Google Scholar
[2] Dziarmaga J 2010 Adv. Phys. 59 1063
Google Scholar
[3] Eisert J, Friesdorf M, Gogolin C 2015 Nat. Phys. 11 124
Google Scholar
[4] Moessner R, Sondhi S L 2017 Nat. Phys. 13 424
Google Scholar
[5] Bloch I, Dalibard J, Zwerger W 2008 Rev. Mod. Phys. 80 885
Google Scholar
[6] Blatt R, Roos C F 2012 Nat. Phys. 8 277
Google Scholar
[7] Bloch I, Dalibard J, Nascimbne S 2012 Nat. Phys. 8 267
Google Scholar
[8] Heyl M, Polkovnikov A, Kehrein S 2013 Phys. Rev. Lett. 110 135704
Google Scholar
[9] Vajna S, Dóra B 2014 Phys. Rev. B 89 161105(R
Google Scholar
[10] Sharma S, Divakaran U, Polkovnikov A, Dutta A 2016 Phys. Rev. B 93 144306
Google Scholar
[11] Cao K Y, Li W W, Zhong M, Tong P Q 2020 Phys. Rev. B 102 014207
Google Scholar
[12] Hickey J M, Genway S, Garrahan J P 2014 Phys. Rev. B 89 054301
Google Scholar
[13] Bhattacharjee S, Dutta A 2018 Phys. Rev. B 97 134306
Google Scholar
[14] Qiu X, Deng T S, Guo G C, Yi W 2018 Phys. Rev. A 98 021601
Google Scholar
[15] Zache T V, Mueller N, Schneider J T, Jendrzejewski F, Berges J, Hauke P 2019 Phys. Rev. Lett. 122 050403
Google Scholar
[16] Ding C X 2020 Phys. Rev. B 102 060409(R
Google Scholar
[17] Schmitt M, Kehrein S 2015 Phys. Rev. B 92 075114
Google Scholar
[18] Heyl M 2014 Phys. Rev. Lett. 113 205701
Google Scholar
[19] Karrasch C, Schuricht D 2013 Phys. Rev. B 87 195104
Google Scholar
[20] Kriel J N, Karrasch C, Kehrein S 2014 Phys. Rev. B 90 125106
Google Scholar
[21] Yin H H, Chen S, Gao X L, Wang P 2018 Phys. Rev. A 97 033624
Google Scholar
[22] Yang C, Wang Y, Wang P, Gao X, Chen S 2017 Phys. Rev. B 95 184201
Google Scholar
[23] Žunkovič B, Heyl M, Knap M, Silva A 2018 Phys. Rev. Lett. 120 130601
Google Scholar
[24] Karrasch C, Schuricht D 2017 Phys. Rev. B 95 075143
Google Scholar
[25] Zhou L, Wang Q H, Wang H, Gong J 2018 Phys. Rev. A 98 022129
Google Scholar
[26] 邓天舒, 易为 2019 物理学报 68 040303
Google Scholar
Deng T S, Yi W 2019 Acta Phys. Sin. 68 040303
Google Scholar
[27] Abdi M 2019 Phys. Rev. B 100 184310
Google Scholar
[28] Liu T, Guo H 2019 Phys. Rev. B 99 104307
Google Scholar
[29] Abeling N O, Kehrein S 2016 Phys. Rev. B 93 104302
Google Scholar
[30] Vajna S, Dóra B 2015 Phys. Rev. B 91 155127
Google Scholar
[31] Wang P, Gao X L 2018 Phys. Rev. A 97 023627
Google Scholar
[32] Yang K, Zhou L, Ma W, Kong X, Wang P, Qin X, Rong X, Wang Y, Shi F, Gong J, Du J 2019 Phys. Rev. B 100 085308
Google Scholar
[33] Jurcevic P, Shen H, Hauke P, Maier C, Brydges T, Hempel C, Lanyon B P, Heyl M, Blatt R, Roos C F 2017 Phys. Rev. Lett. 119 080501
Google Scholar
[34] Fläschner N, Vogel D, Tarnowski M, Rem B S, Lühmann D S, Heyl M, Budich J C, Mathey L, Sengstock K, Weitenberg C 2018 Nat. Phys. 14 265
Google Scholar
[35] Martinez E A, Muschik C A, Schindler P, Nigg D, Erhard A, Heyl M, Hauke P, Dalmonte M, Monz T, Zoller P, Blatt R 2016 Nature 534 516
Google Scholar
[36] Zhang J, Pagano G, Hess P W, Kyprianidis A, Becker P, Kaplan H, Gorshkov A V, Gong Z X, Monroe C 2017 Nature 551 601
Google Scholar
[37] Nie X, Wei B B, Chen X, Zhang Z, Zhao X, Qiu C, Tian Y, Ji Y, Xin T, Lu D, Li J 2020 Phys. Rev. Lett. 124 250601
Google Scholar
[38] Wang K, Qiu X, Xiao L, Zhan X, Bian Z, Yi W, Xue P 2019 Phys. Rev. Lett. 122 020501
Google Scholar
[39] Tia Tn, Yang H X, Qiu L Y, Liang H Y, Yang Y B, Xu Y, Duan L M 2020 Phys. Rev. Lett. 124 043001
Google Scholar
[40] Kennes D M, Schuricht D, Karrasch C 2018 Phys. Rev. B 97 184302
Google Scholar
[41] cheraghi H, Mahdavifar S 2020 Sci. Rep. 10 4407
Google Scholar
[42] Hou X Y, Gao Q C, Guo H, He Y, Liu T, Chien C C 2020 Phys. Rev. B 102 104305
Google Scholar
[43] Heyl M 2015 Phys. Rev. Lett. 115 140602
Google Scholar
[44] Lang J, Frank B, Halimeh J C 2018 Phys. Rev. Lett. 121 130603
Google Scholar
[45] Hagymási I, Hubig C, Legeza Ö, Schollwöck U 2019 Phys. Rev. Lett. 122 250601
Google Scholar
[46] Huang Y P, Banerjee D, Heyl M 2019 Phys. Rev. Lett. 122 250401
Google Scholar
[47] Khatun A, Bhattacharjee S M 2019 Phys. Rev. Lett. 123 160603
Google Scholar
[48] Zhou B, Yang C, Chen S 2019 Phys. Rev. B 100 184313
Google Scholar
[49] Sun G, Wei B B 2020 Phys. Rev. B 102 094302
Google Scholar
[50] Wu Y 2020 Phys. Rev. B 101 014305
Google Scholar
[51] Divakaran U, Sharma S, Dutta A 2016 Phys. Rev. E 93 052133
Google Scholar
[52] Zhang X X, Li F J, Wang K, Xue J, Huo G W, Fang A P, Li H R 2021 Chin. Phys. B 30 090504
Google Scholar
[53] Mo H L, Zhang Q L, Wan X 2020 Chin. Phys. Lett. 37 060301
Google Scholar
[54] 杨超, 陈澍 2019 物理学报 68 220304
Google Scholar
Chen S, Yang C 2019 Acta Phys. Sin. 68 220304
Google Scholar
[55] 贺志, 余敏, 王琼 2019 物理学 报 68 240506
He Z, Yu M, Wang Q 2019 Acta Phys. Sin. 68 240506
[56] Lieb E, Schultz T, Mattis D 1961 Ann. Phys. NY 16 407
Google Scholar
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