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非厄米临界动力学及其在量子多体系统中的应用

张禧征 王鹏 张坤亮 杨学敏 宋智

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非厄米临界动力学及其在量子多体系统中的应用

张禧征, 王鹏, 张坤亮, 杨学敏, 宋智

Non-Hermitian critical dynamics and its application to quantum many-body systems

Zhang Xi-Zheng, Wang Peng, Zhang Kun-Liang, Yang Xue-Min, Song Zhi
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  • 近些年来, 非厄米与强关联两种元素开始融合并形成物理学中的一个重要研究领域, 相关理论与实验的进展重塑了人们对于物质的理解. 在该领域中, 研究对象并不局限于非厄米元素对多体系统能谱以及本征态性质的影响, 研究者们更加关注对量子态的操纵. 例外点作为非厄米量子力学区别于厄米量子力学中最显著的特征得到了大家广泛的关注. 除了围绕能谱例外点的非厄米拓扑能带理论以及量子探测等最新进展外, 本文重点阐述以能谱例外点为基础的临界动力学现象及其在量子多体系统中的应用. 当系统处于能谱例外点上时, 属于例外点合并子空间中的任意初始态都将投影到体系的合并态上. 基于量子态演化的方向性, 本文回顾了近年来本课题组在量子自旋系统所发现的外场诱导的动力学磁化、横场Ising模型中的有限温相变、中心-环境系统中的量子铸模以及非厄米强关联系统中的超导态制备等几个代表性工作, 着重讨论了与例外点相关的新的非平衡量子态制备方法以及探测方案.
    In recent years, two independent research fields, i.e. non-Hermitian andstrongly correlated systems have been merged, forming an important researchfield in physics. The progress of relevant theories and experiments hasreshaped our understanding of matter. In this field, the research object isnot limited to the influence of non-Hermiticity on the energy spectrum andthe eigenstate properties of many-body systems. Researchers have paid more attentionto the manipulation of quantum states. It is universally received that the exceptional point is the most significant featurethat distinguishes non-Hermitian quantum mechanics from Hermitian quantum mechanics. In addition to the recent advances in non-Hermitian topological band theory and quantum sensing around the exceptional points, this paper concentrates on the non-Hermitian critical dynamical phenomenon and its application to the quantum many-body system. When the system has an exceptional point, an arbitrary initial state belonging to the coalescent subspace will be projected on the coalescent state. Based on the directionality of the evolved quantum state, this paper reviews our several representative researches in recent years, including local-field-induced dynamical magnetization, quantum phase transition in transverse field, Ising model at non-zero temperature, quantum mold casting in the center-environment system, as well as superconducting state preparation in the non-Hermitian strongly correlated system. We also focus on the new preparation methods and detection schemes of non-equilibrium quantum states related to exception points.
      通信作者: 宋智, songtc@nankai.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11975166, 11874225, 12047547)资助的课题
      Corresponding author: Song Zhi, songtc@nankai.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11975166, 11874225, 12047547)
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  • 图 1  自旋在(a)局域磁场、(b)各向同性全局复数磁场及(c)粒子与粒子相互作用与局域复数磁场共同作用下的动力学演化示意图. 复数磁场通过绿色阴影部分标注. 自旋与自旋的非各项同性相互作用$ J_{ij} $通过不同的颜色来区分. 根据非厄米临界动力学的理论, 图(a) 中含有一个二阶例外点, 对应两个简并态合并. 图(b)和图(c)有N个简并态合并对应N阶例外点. 从图中可以看到, 局域的复数磁场可以通过与相互作用的协作来影响系统自旋的整体取向

    Fig. 1.  Schematics of spins subjected to (a) a local complex field, (b) a global complex field, and (c) a local complex field and interaction. The complex magnetic field is shaded green. The couplings between different spins are denoted by different colors representing inhomogeneous coupling $ J_{ij} $. Two states coalesce in panel (a) and N states coalesce in panels (b) and (c). Local complex field only affects local spin without interaction, but can affect globally with interaction

    图 2  (a) 对于初始态$ \left\vert \varPsi _{\mathrm{I}}\left( 0\right) \right\rangle $在时间域上的磁滞回线; (b), (c) 对于初始态$ \left\vert \varPsi _{\mathrm{II}}\left( 0\right) \right\rangle $的磁滞回线. 这里局域的复数外场被施加到格点1上. 其强度在图(a)中为$ 0.02 $, 在图(b)和图(c)中为$ 0.1 $. 弛豫时间分别为(a), (b) $ t_{{\rm{f}}} = 2\times 10^{3}J^{-1} $及(c)$ t_{{\rm{f}}} = 3\times 10^{3}J^{-1} $

    Fig. 2.  Hysteresis loops for the initial state $ \left\vert \varPsi _{\mathrm{I}}\left( 0\right) \right\rangle $ in (a) and $ \left\vert \varPsi _{\mathrm{II}}\left( 0\right) \right\rangle $ in (b), (c). The critical local complex field $ g_{1} $ is $ 0.02 $ in panels (a) and (b), and $ 0.1 $ in panel (c). The relaxation time is $ t_{{\rm{f}}} = 2\times 10^{3}J^{-1} $ in panels (a) and (b), and $ t_{{\rm{f}}} = 3\times 10^{3}J^{-1} $ in panel (c)

    图 3  (a) 通过施密特回波所给出的相图; (b) 关联函数所给出的相图, 这里$ \beta ^{-1} $是温度, $ g_{{\rm{c}}} $是量子相变点; (c) 有限横场Ising模型低能激发谱随着参数g的变化. $ E_{{\rm{g}}} $代表基态能量. 其他系统参数为$ N = 50 $$ J = 1 $. 系统的不同相通过两种不同颜色来区分. 通过图(c)可以发现, 当$ g < 1 $时, 系统的能谱都变为二重简并

    Fig. 3.  (a) Phase diagram detected from the Loschmidt echoes in this work. (b) Phase diagram studied in term of correlation function in the work of Sachdev et al.. Here $ \beta ^{-1} $ is the temperature and $ g_{{\rm{c}}} $ is the quantum critical point. (c) Spectrum of the low-lying states for a finite quantum Ising chain as a function of g, obtained numerically through exact diagonalization. $ E_{{\rm{g}}} $ is the ground-state energy. System parameters: $ N = 50 $ and $ J = 1 $. The energy gap closes at a quasicritical point, indicated by the boundary of the two shaded areas. Notably, all energy levels become twofold degeneracy simultaneously at one point, protected by the symmetry of the quasi-zero-mode operator D.

    图 4  施密特回波随时间变化曲线. 线和点分别代表不同的温度, 即$ \beta = 5 $$ \beta = 10 $. 其他系统参数为$ N = 10 $, $\kappa = $$ 0.1$$ J = 1 $. 施密特回波在不同物质相内的动力学行为不同, 最终趋近于$ 1.0 $$ 0.5 $两个定值. 需要注意的是这个结果独立于初始热态的温度. 因为D依赖于参数g, 并且其在$ g > 1 $时发散, 所以在这里的数值模拟中, 非厄米外场只取D的主导项, 即$ {\cal{H}}=H+\kappa D_{1} $

    Fig. 4.  Loschmidt echoes of different g values. The lines and dots represent the Loschmidt echoes for $ \beta = 5 $ and $ \beta = 10 $, respectively. Other parameters: $ N = 10 $, $ \kappa = 0.1 $, and $ J = 1 $. The profiles of the Loschmidt echoes in the two regions are distinct, independent of the temperature of the initial thermal states, and converge to $ 1.0 $ and $ 0.5 $, respectively.

    图 5  量子铸模系统示意图 (a)系统由两部分组成, 中心系统$ H_{{\rm{c}}} $和源系统$ H_{{\rm{s}}} $. $ H_{{\rm{in}}} $为非厄米项, 表示$ H_{{\rm{c}}} $$ H_{{\rm{s}}} $之间的连接, 并承担从$ H_{{\rm{s}}} $$ H_{{\rm{c}}} $单向传输费米子的任务. (b)该方案的紧束缚模型, 包含三种格点A, BD. 嵌入阴影区域的晶格AB (红色和蓝色实心圆)是拓扑绝缘体, 而晶格D (黄色实心圆)是一个无跳跃的平带系统, 但具有振荡的化学势. 绿色箭头表示从D点到B点的单向跳跃. 本工作的目的是通过时间演化实现以下过程: 初始时刻D格点填充满粒子, 而AB格点无粒子; 最终末态是$ H_{{\rm{c}}} $半满填充的基态. (c)动力学过程的基本机制. 在瞬时$ t_{k} $, $ H_{{\rm{s}}} $的化学势和$ H_{{\rm{c}}} $的能级共振导致例外点. 相应的例外点动力学允许费米子在简并能级之间完全转移, 并且在长时间极限下, 这样的动力学过程发生在每个$ {\boldsymbol{k}} $子空间. $ H_{{\rm{c}}} $的能带颜色表示能带反转, 说明能带可以是拓扑绝缘带. 预计$ H_{{\rm{c}}} $的下带可以完全填充

    Fig. 5.  Schematics for the system and process of quantum mold casting: (a) The system consists of two parts, central system $ H_{\rm{c}} $ and source system $ H_{\rm{s}} $. The target state is the ground state of $ H_{\rm{c}} $, which can be topologically non-trivial or not. $ H_{\rm{s}} $ is a topologically trivial system, providing the supply of fermions. Both $ H_{\rm{c}} $ and $ H_{\rm{s}} $ are Hermitian, while $ H_{\rm{in}} $ is non-Hermitian, representing the connection between $ H_{\mathrm{c}} $ and $ H_{\mathrm{s}} $, and taking the role to transport fermions unidirectionally from $ H_{\mathrm{s}} $ to $ H_{\mathrm{c}} $. (b) A tight-binding model for the scheme, which contains three sets, A, B, and D. Lattices A and B (red and blue filled circles) embedded in the shadow area is topological insulator, while lattice D (yellow filled circle) is a flat-band (hopping-free) system but with oscillating chemical potential. Green arrows represent unidirectional hopping from D to B lattices. The aim of this work is to realize the following process via time evolution. Initially, D lattice is fully filled, while A and B are empty. The final state is expected to a half-filled ground state of $ H_{\rm{c}} $. (c) The underlying mechanism of the dynamic process. At instant $ t_{{\boldsymbol{k}}} $, the chemical potential and energy levels of $ H_{\rm{c}} $ are resonant, leading to exceptional points. The corresponding (EP) dynamics allows a complete transfer of fermions between the degenerate energy levels. In the long-time limit, such dynamics occurs at each $ {\boldsymbol{k}} $ sector again and again. The band color of $ H_{\rm{c}} $ illustrates the band inversion, indicating that the energy band can be topological insulating band. It is expected the lower band of $ H_{\rm{c}} $ can be completely filled

    图 6  (a)—(d) 4格点Hubbard模型中关联函数$ |\langle \varPhi \left( t\right) \vert \eta _{i}^{+}\left\vert \varPhi \left( t\right) \right\rangle | $以及$ \langle \varPhi \left( t\right) \vert \eta _{i}^{+}\eta _{i+r}^{-}\left\vert \varPhi \left( t\right) \right\rangle $随时间的变化图. 初始态被制备在$ H_{0} $的真空态$\left\vert {V_{{\rm{vac}}}}\right\rangle$中, 其相互作用强度$U = 2J$. 随后其运动由外加局域虚数磁场来驱动. 对于图(a)和图(c), 局域虚数磁场为$g_{1}=g = 0.2J$; 对于图(b)和图(d), 系统受到一个各向同性的磁场驱动, 其强度为$g_{j}=g = 0.2J$ $ (j = 1,\cdots, N) $. 需要注意的是此时外场$ H_{{\rm{I}}} $处于例外点上. (e), (f) 稳态关联函数与相对距离之间的函数曲线. 对于图(e), 弛豫时间为$ t_{{\rm{f}}} = 400/J $, 而对于图(f), $ t_{{\rm{f}}} = 100/J $. 从图中可以看到当$ i\neq j $时, $ \langle \varPsi \left( t_{{\rm{f}}}\right) \vert \eta _{i}^{+}\eta _{j}^{-}\left\vert \varPsi \left( t_{{\rm{f}}}\right) \right\rangle = 1/4 $; 当$ i=j $时, $ \langle \varPsi \left( t_{{\rm{f}}}\right) \vert \eta _{i}^{+}\eta _{j}^{-}\left\vert \varPsi \left( t_{{\rm{f}}}\right) \right\rangle = 1/2 $, 与正文中的结论一致

    Fig. 6.  (a)–(d) Evolution of the correlators $ |\langle \varPhi \left( t\right) \vert \eta _{i}^{+}\left\vert \varPhi \left( t\right) \right\rangle | $ and $ \langle \varPhi \left( t\right) \vert \eta _{i}^{+}\eta _{i+r}^{-}\left\vert \varPhi \left( t\right) \right\rangle $, averaged over all sites for the $ 4 $-site Hubbard model. The initial state is prepared in vacuum state $\left\vert {V_{{\rm{vac}}}}\right\rangle$ of $ H_{0} $ with interaction $U = 2J$, and then it is driven by the system with the local imaginary field $g_{1}=g = 0.2J$ for panels (a) and (c), and homogeneous dissipation $g_{j}=g = 0.2J$ $(j = 1,\cdots, N)$ for panels (b) and (d), respectively. Notice that $ H_{{\rm{I}}} $ is at EP such that $ \lambda /\gamma = 1 $. (e), (f) The correlation values of steady state for different relative distance ($ \langle \varPsi \left( t_{{\rm{f}}}\right) \vert \eta _{i}^{+}\eta _{j}^{-}\left\vert \varPsi \left( t_{{\rm{f}}}\right) \right\rangle $) at relaxation time $ t_{{\rm{f}}} = 400/J $ for panel (e) and $ t_{{\rm{f}}} = 100/J $ for panel (f). It is shown that $\langle \varPsi \left( t_{{\rm{f}}}\right) \vert \eta _{i}^{+}\eta _{j}^{-}\left\vert \varPsi \left( t_{{\rm{f}}}\right) \right\rangle = 1/4$ for $ i\neq j $ and $\langle \varPsi \left( t_{{\rm{f}}}\right) \vert \eta _{i}^{+}\eta _{j}^{-}\left\vert \varPsi \left( t_{{\rm{f}}}\right) \right\rangle = 1/2$ for $ i=j $, which confirms the understanding in the main text

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    Majorana E 2006 EJTP 3 293

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    Feshbach H 1958 Ann. Phys. 5 357Google Scholar

    [5]

    Feshbach H 1958 Ann. Phys. 19 287

    [6]

    Schrödinger E 1926 Ann. Phys. 384 489Google Scholar

    [7]

    Ashida Y, Gong Z P, Ueda M 2020 Adv. Phys. 69 249

    [8]

    Cohen-Tannoudji C, Dupnot-Roc J, Grynberg G 1998 Atom-photon Interactions: Basic Processes and Applications (Berlin: Wiley-VCH)

    [9]

    Anderson P W 1972 Science 177 393Google Scholar

    [10]

    Lee T D, Yang C N 1952 Phys. Rev. 87 410Google Scholar

    [11]

    Zhou Y H, Shen H Z, Yi X X 2018 Phys. Rev. A 97 043819Google Scholar

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    Song F, Yao S Y, Wang Z 2019 Phys. Rev. Lett. 123 170401Google Scholar

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    [14]

    Longhi S 2020 Phys. Rev. B 102 201103Google Scholar

    [15]

    Liu T, He J J, Yoshida T, Xiang Z L, Nori F 2020 Phys. Rev. B 102 235151Google Scholar

    [16]

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    [17]

    Tang L Z, Zhang G Q, Zhang L F, Zhang D W 2021 Phys. Rev. A 103 033325Google Scholar

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    Mao L, Deng T S, Zhang P F 2021 Phys. Rev. B 104 125435Google Scholar

    [19]

    Li J X, Xu L, Zhao Y H, He Z, Wang Q 2021 Laser Phys. 31 075202Google Scholar

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    Ohlsson T, Zhou S 2021 Phys. Rev. A 103 022218Google Scholar

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    Pan J S, Li L H, Gong JB 2021 Phys. Rev. B 103 205425Google Scholar

    [22]

    Xue W T, Hu Y M, Song F, Wang Z 2022 Phys. Rev. Lett. 128 120401Google Scholar

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出版历程
  • 收稿日期:  2022-05-10
  • 修回日期:  2022-07-06
  • 上网日期:  2022-08-12
  • 刊出日期:  2022-09-05

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