两次淬火下横场中<i>XY</i>链的动力学量子相变

符浩; 曹凯源; 钟鸣; 童培庆

doi:10.7498/aps.70.20210728

摘要

研究了两次淬火下横场中XY链的动力学量子相变. 两次淬火是指系统的Hamilton量先由H₀淬火到H₁, 演化一段时间T后再由H₁淬火到H₂. 由于横场中XY链存在两种不同的量子相变(Ising相变和各向异性相变), 因此主要讨论淬火路径对横场中XY链的动力学量子相变的影响, 发现第2次淬火后系统发生动力学量子相变的临界时间存在三种典型的情形. 情形I对应于临界时间在一定的T范围内出现, 它与由H₀淬火到H₁的临界时间相联系. 情形II对应于临界时间在任意T时总是出现, 它与由H₀直接淬火到H₂的临界时间相联系. 情形III对应于临界时间也在任意T时总是出现, 它同时与由H₀淬火到H₁以及由H₀直接淬火到H₂的临界时间相联系. 还发现两次淬火都经过同一类相变点时, 第2次淬火后只会出现情形I对应的临界时间. 而两次淬火经过不同类相变点时, 第2次淬火后的临界时间会出现上述三种情形中的任意两种, 它与淬火路径有关.

关键词:

Abstract

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.

Keywords:

作者及机构信息

1.
南京师范大学物理科学与技术学院, 南京　210023

2.
南京师范大学, 江苏省大规模复杂系统数值模拟重点实验室, 南京　210023

通信作者: 童培庆, pqtong@njnu.edu.cn

基金项目: 国家自然科学基金(批准号: 11975126, 11575087)资助的课题

Authors and contacts

1.
School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China

2.
Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, China

Corresponding author: Tong Pei-Qing, pqtong@njnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11975126, 11575087)

文章全文 : translate this paragraph

参考文献

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[41]	cheraghi H, Mahdavifar S 2020 Sci. Rep. 10 4407 Google Scholar
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[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
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施引文献

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

Fig. 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.

下载: 全尺寸图片幻灯片

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$1.0$

Fig. 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.

下载: 全尺寸图片幻灯片

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$T=T_{6}\approx1.3879$

Fig. 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.

下载: 全尺寸图片幻灯片

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$T=T_{7}\approx1.5210$

Fig. 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 $T=T_{7}\approx1.5210$.

下载: 全尺寸图片幻灯片

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$T=T_{9}\approx1.3879$

Fig. 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.

下载: 全尺寸图片幻灯片

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$T=T_{10}\approx0.9170$

Fig. 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 $T=T_{10}\approx0.9170$.

下载: 全尺寸图片幻灯片

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[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
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[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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两次淬火下横场中XY链的动力学量子相变