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This review reports a series of theoretical and experimental progress on researches of the transverse field Ising chain (TFIC) and the quantum E8 integrable model. For the TFIC, on one hand, a unique exotic quantum critical behavior of Grüneisen ratio (a ratio from magnetic or thermal expansion coefficient to specific heat) is theoretically established; on the other hand microscopic models can accommodate the TFIC universality class are substantially expanded. These progresses successfully promote a series of experiments collaborations to first-time realize the TFIC universality class in quasi one-dimensional anti-ferromagnetic material BaCo2V2O8 and SrCo2V2O8. For the quantum E8 integrable model, the low temperature local dynamics and the dynamical structure factor with zero transfer momentum of this system are analytically determined, where a cascade of edge singularities with power-law divergences are obtained in the continuum region of the dynamical structure factor. After combining with detailed quantum critical scaling behaviors analysis and large scale iTEBD calculation, it successfully facilitates a series of experiments, including THz spectrum measurements, inelastic neutron scattering and NMR experiments, to realize the quantum E8 integrable model in BaCo2V2O8 for the first time. The experimental realization of the quantum E8 integrable model substantially extends the frontiers of studying quantum integrable models in real materials. The series of progress and developments on the TFIC and the quantum E8 integrable model lay down a concrete ground to go beyond quantum integrability, and can inspire studies in condensed matter systems, cold atom systems, statistical field theory and conformal field theory.
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Keywords:
- integrable systems /
- quantum magnetism /
- universality class /
- spin dynamics
1. 引 言
量子相变指在零温下通过调节参数, 强关联多体系统发生的相变行为[1]. 在量子临界点 (quantum critical point, QCP) 附近, 强烈的量子涨落往往会涌现出奇异的量子临界行为[2-7]. 对这些量子临界行为的深入理解将有助于解答长期悬而未决的物理问题[8-16], 包括非常规高温超导体、非费米液体等[17-19], 并有助于在材料中实现量子自旋液体[20,21], 因此吸引着理论和实验的持续研究. 量子临界行为能被临界指数所刻画, 每一组量子临界指数对应于一种普适类, 并往往伴随相应的低能有效量子场论[1]. 即使微观起源不同, 属于同一普适类的模型在QCP附近也会展示出非常一致的物理, 如标度规律、关联行为及动力学激发等[1]. 因此, 对普适类的深入研究可以使人们同时理解QCP附近大量具有该普适类的微观模型, 而这意味着将实际系统调节向QCP附近是实现并研究普适类之必要. 然而, 由于真实材料内在的复杂性, 只有极少数的QCP可以被实验实现, 即使一些形式上并不复杂的微观模型, 如横场伊辛链模型[22] (transverse field Ising chain model, TFIC模型, 见方程(1)), 实验上也一直难以找到候选材料实现并研究其对应的普适类.
在QCP处对体系的微扰可能会进一步涌现出丰富的物理, 如涌现对称性和涌现可积性(涌现对称性指量子系统在低能物理中的对称性大于完整的哈密顿量的对称性. 涌现可积性指不可积系统在某些特定的参数区间成为可积系统, 产生了可积性; 或可积系统在恰当扰动下又呈现出新的可积性)[3-6,23-26]. 自1989年起Zamolodchikov等[23,24,27-29]发展了微扰共形场论(perturbed conformal field theory, pCFT). 他们通过在不同普适类的临界点处施加微扰, 发现了一系列能够由例外李代数(exceptional Lie algebra)描述的可积模型[24]. 比如, QCP处的TFIC模型在纵向磁场的微扰下会涌现出量子
E8 可积模型[23,24,27-31]. 这类涌现现象同样具有普适性, 一方面不依赖于微观模型的具体细节, 另一方面拓展了量子可积模型范围, 有利于推动实验在实际材料中对可积模型进行研究.近年来, 吴建达及其合作者在理论和实验的共同努力下, 在强关联材料BaCo2V2O8 (BCVO)和SrCo2V2O8 (SCVO) 中首次成功实现TFIC模型的普适类[22,32-35], 并在BCVO材料中首次实现量子
E8 可积模型[30,36,37]. 这一系列进展不仅是量子可积磁性理论和实验上的突破[2], 也展示了对量子临界行为、普适类、涌现对称性和涌现可积性研究的重要意义, 更开拓了于真实材料中实现并研究量子可积模型的重要方向. 本文按照历史的脉络, 从理论和实验两个方面综述了这一系列的进展. 全文组织如下: 第2节分别回顾TFIC模型和横场海森伯-伊辛链的量子临界行为及其普适类; 第3节介绍证实BCVO材料和SCVO材料中TFIC模型普适类的实验进展; 第4节概括量子E8 模型的理论背景和进展; 第5节叙述量子E8 模型在材料中的实验实现; 第6节为总结与展望.2. 伊辛普适类
2.1 横场伊辛链
HI=−J∑i(σziσzi+1+gσxi), (1) 其中
J 及gJ 分别为铁磁及横场与自旋间的相互作用强度.Sαi=σαi/2,α=x,y,z 表示在格点i 处的自旋1/2 算符,σαi 是泡利矩阵. 该模型是一精确可解模型, 利用约当–魏格纳变换 (Jordan-Wigner transformation)和波格留波夫 (Bogoliubov)变换可以对角化该哈密顿量并求解出其激发谱[1,22,32]:HI=∑kεk(γ†kγk−12), (2) 其中
εk=2J√1+g2−2gcos(ka) 为单粒子激发谱,a 为晶格常数,γ†k 和γk 分别为波格留波夫变换后的费米子产生湮灭算符. 显然, 当g≠1 时, 系统存在能隙, 该能隙在g=gc=1 时消失. 零温时, 该模型存在一个横场驱动的铁磁相和顺磁相之间的量子相变(图1)[1,22], 对应的QCP位于gc .
图 1 TFIC模型相图.g gc=1 Fig. 1. A phase diagram of TFIC, whereg gc=1 1970年, Pfeuty[22]首次研究了TFIC模型的量子相变. 1984年, Zamolodchikov等[3,30,38]证明了当
g=gc 时TFIC模型的低能物理可以被中心荷1/2 的共形场论 (central charge 1/2 conformal field theory, CFTc=1/2 )所描述. 在标度极限下[30], TFIC模型中的自旋算符σzi 及σxi 分别对应于CFTc=1/2 中自旋密度算符σ(x) 及能量密度算符ε(x) . 当|x|→0 时, 其两点关联函数具有标度行为[31,38]⟨σ(x)σ(0)⟩∼1/|x|1/4,⟨ε(x)ε(0)⟩∼1/|x|2, (3) σ(x) 和ε(x) 对应的标度维度分别是2Δσ=1/8 和2Δε=1 [30,31,38], 它们刻画了TFIC模型的普适类, 同时也刻画了二维经典伊辛模型的普适类[39] (以下简称该普适类为伊辛普适类). 然而, 自TFIC模型提出后的近半个世纪里, 具有伊辛普适类的强关联材料却一直没有被确认存在, 这是因为一方面实验上受限于实验精度, 另一方面缺乏有效的理论指导去分辨并证实观测到的量子临界行为是否属于伊辛普适类. 比如, 方程(3)所提及的标度行为是等时短程关联, 实验上不具有可操作性, 无法进行测量. 2018年, 吴建达与合作者[32]关于伊辛普适类格林艾森比率 (Grüneisen ratio)量子临界标度行为的理论突破为实验提供了一个可靠的判定标准. 这里先简要归纳伊辛普适类在QCP附近的标度规律, 这些标度规律皆可作为判断伊辛普适类的依据[1,22,32]. 首先, QCP附近系统的磁化Mz 满足Mz≈|g−gc|δ , 其中δ=1/8 . 其次, 在QCP附近考虑自旋关联函数Cij=⟨σziσzj⟩−⟨σzi⟩⟨σzj⟩, (4) 该关联随格点间距呈指数衰减, 即
Cij≈e−|i−j|/ξ , 其中ξ 被称为关联长度, 满足标度律1/ξ≈|gc−g|ν ,ν=1 . 此时系统激发的能隙满足标度律Δ≈ |gc−g|zν ,z=1 被称为动力学临界指数[1,32,35].由于临界指数也可以直接用于分析热力学量临界行为, 因此在实验方面, 可以通过测量相应热力学量的量子临界行为考察相关材料的普适类性质. 比如考虑和磁热效应相关的热力学量格林艾森比率:
ΓH=αHcH=−1T(∂M/∂T)H(∂S/∂T)H=−1T(∂S/∂H)T(∂S/∂T)H, (5) 其中
H 是可控外场;cH ,αH ,M 和S 分别为摩尔比热、磁膨胀系数、每摩尔磁化强度和熵. 在TFIC模型中, 相关物理量均可被解析地求解出来[32]. 零温极限下, 格林艾森比率在横场趋于TFIC模型的QCP时趋于发散Γcr(t→0,g)≈αcv=sgng−gc|g−gc| , 而在量子临界区内, 趋于QCP时, 格林艾森比率则趋于常数[32]. 此奇异的量子临界行为一方面表明了TFIC模型的QCP是本性奇点, 另一方面则可作为TFIC模型中伊辛普适类的独特特征, 为实验探测伊辛普适类提供有效的理论指导. 值得一提的是, 格林艾森比率的发散行为意味着熵极大值会出现在QCP附近[40], 这在相关材料的量子临界测量中得到确认, 如Sr3 Ru2 O7 和CeCu6−x Aux 等[41,42]. 于是, 基于上述理论, 在实验中可以通过测量相关材料格林艾森比率的量子临界行为, 判断该材料的普适类是否是伊辛普适类.2.2 横场海森伯-伊辛链
接下来总结横场海森伯 -伊辛链在外加横场之下的可能量子相变及其普适类特征, 首先考察外加均匀横场之下的海森伯-伊辛链:
H=∑iJ[ε(SxiSxi+1+SyiSyi+1)+SziSzi+1]+Hx∑iSxi, (6) 这里
Hx 是自旋和横场的耦合系数.ε 通常取0 — 1 之间, 当\varepsilon = 0, 1 时该模型分别对应TFIC和横场海森伯链, 于是称该模型为横场海森伯-伊辛链[43,44]. 此前已经证明了该模型在加横场后会存在伊辛普适类[44]. 2019年, 吴建达及合作者[35]利用大规模infinite time-evolving block decimation (iTEBD)数值计算证明了引入垂直于所加均匀横场的交错横场后, 该模型在量子相变点处依旧展示了伊辛普适类. 考虑一般的带有交错横场的海森伯-伊辛链哈密顿量\begin{split} {\boldsymbol{H}} =\;& \sum\limits_{i}J[\varepsilon ({\boldsymbol{S}}^{x}_{i}{\boldsymbol{S}}^{x}_{i+1}+{\boldsymbol{S}}_{i}^{y}{\boldsymbol{S}}_{i+1}^{y})+{\boldsymbol{S}}_{i}^{z}{\boldsymbol{S}}_{i+1}^{z}]\\ &-\sum\limits_{i}[g_{x}{\boldsymbol{S}}_{i}^{x}+(-1)^{i}g_{y}{\boldsymbol{S}}_{i}^{y}]. \end{split} (7) 利用iTEBD数值方法, 在
g_{x}{\text{-}}g_{y}{\text{-}}\varepsilon 三维参数空间内可以精确计算该有效模型中展现伊辛普适类物理的QCPs, 所有的QCPs组成了g_{x}{\text{-}}g_{y}{\text{-}}\varepsilon 参数空间内一量子临界曲面, 如图2[35]所示.![图 2 由iTEBD计算得出的针对模型方程(7)的量子临界曲面 (图片经文献[35]允许转载, 版权归2019 IOP Publishing Ltd 所有)\r\nFig. 2. Quantum critical surface calculated by iTEBD algorithm for Eq. (7). (Reprinted with permission from Ref. [35]. Copyright 2019 IOP Publishing Ltd.)](/fileWLXB/journal/article/wlxb/2021/23/PIC/23-20211836-2.png)
该理论工作进一步考察了带有交错横场的海森伯-伊辛链存在沿
z 方向的4周期磁场微扰项的情形, 发现这一微扰项仅轻微移动QCP的位置而不改变相应的伊辛普适类. 计算结果见图3[35].![图 3 (a)交错磁化$ M_{z}(g) $的计算结果. 蓝线和红线分别代表了带有和不带有4周期项的数据. 这两条曲线可以分别用$M_{z} = $$ 0.524(g_{{\rm{c}}1}-g)^{0.126}$和$ M_{z} = 0.530(g_{{\rm{c}}2}-g)^{0.128} $来拟合, 其中$ g_{{\rm{c}}1} = 0.1454 $, $ g_{{\rm{c}}2} = 0.1456 $, 在误差精度范围内均可得到临界指数为$ 1/8 $的结论. 内嵌图是利用对数坐标轴画出的$ M_{z} $的标度行为. (b)半对数坐标轴下纠缠熵$ S_{{\rm{EE}}}(l) $和链长$ l $的关系, 两种情况都符合伊辛普适类的$ 1/2 $中心荷. (c)当$ g = 0.1448 $时, 自旋两点关联函数和距离$ i-j $间的函数关系. 内嵌图展示了$ \ln C(i-j) $在长程时与$ i-j $成正比. (d)关联长度的倒数和$ g $的函数关系, 两种情况都符合伊辛普适类中关联长度指数$ \nu = 1 $的结论(图片经文献[35]允许转载, 版权归2019 IOP Publishing Ltd 所有)\r\nFig. 3. (a) iTEBD results for staggered magnetization $ M_{z}(g) $. The blue line and red line represent iTEBD data with and without the four periodic term, respectively. The two curves can be fitted with $ M_{z} = 0.524(g_{{\rm{c}}1}-g)^{0.126} $ and $ M_{z} = 0.530(g_{{\rm{c}}2}-g)^{0.128} $, where $ g_{{\rm{c}}1} = 0.1454 $, $ g_{{\rm{c}}2} = 0.1456 $, and critical exponent $ \delta = 1/8 $ is obtained within error bar for both cases. Inset shows scaling behavior of $ M_{z} $ in $ \log-\log $ plot. (b) Entanglement entropy $ S_{{\rm{EE}}}(l) $ versus length $ l $ in a semi-$ \log $ plot, both fall into central charge $ c = 1/2 $ of TFIC universality. (c) Spin-spin correlation function versus distance $ i-j $ at $ g = 0.1448 $. Inset shows $ \ln C(i-j) $ being proportional to $ i-j $ in long range region. (d) The inverse of correlation length in terms of $ g $, both agree with correlation length exponent $ \nu = 1 $. (Reprinted with permission from Ref. [35]. Copyright 2019 IOP Publishing Ltd.)](/fileWLXB/journal/article/wlxb/2021/23/PIC/23-20211836-3.png)
图 3 (a)交错磁化M_{z}(g) M_{z} = 0.524(g_{{\rm{c}}1}-g)^{0.126} M_{z} = 0.530(g_{{\rm{c}}2}-g)^{0.128} g_{{\rm{c}}1} = 0.1454 g_{{\rm{c}}2} = 0.1456 1/8 M_{z} S_{{\rm{EE}}}(l) l 1/2 g = 0.1448 i-j \ln C(i-j) i-j g \nu = 1 Fig. 3. (a) iTEBD results for staggered magnetizationM_{z}(g) M_{z} = 0.524(g_{{\rm{c}}1}-g)^{0.126} M_{z} = 0.530(g_{{\rm{c}}2}-g)^{0.128} g_{{\rm{c}}1} = 0.1454 g_{{\rm{c}}2} = 0.1456 \delta = 1/8 M_{z} \log-\log S_{{\rm{EE}}}(l) l \log c = 1/2 i-j g = 0.1448 \ln C(i-j) i-j g \nu = 1 综上所述, 该工作利用iTEBD数值算法证明了在带有交错横场的海森伯-伊辛链中通过合理调整所加磁场, 可以到达伊辛普适类的量子临界区, 从而实现伊辛普适类的物理. 而由于BCVO和SCVO材料均可被海森伯-伊辛链所描述[33-37], 理论上预期这两种材料在横场调节之下可以实现伊辛普适类. 第3节中将叙述在BCVO和SCVO材料中实验实现伊辛普适类的一系列工作. 实验测量所得到的BCVO材料中格林艾森比率的量子临界行为完美符合了伊辛普适类中该物理量量子临界行为的理论结果, 从而为BCVO在外加强横场之下涌现伊辛普适类提供了坚实的判定依据[33]. 而利用核磁共振(NMR)实验辅以数值算法, 通过对SCVO材料及其有效模型进行研究, 我们证实了SCVO材料在弱横场下展现了伊辛普适类[34].
3. BCVO和SCVO材料中的伊辛普适类
利用上述伊辛普适类的相关理论, 吴建达及合作者[32-37]对BCVO和SCVO材料展开测量并证实了这两类准一维材料在外加横场之下能够实现伊辛普适类. BCVO和SCVO是能被海森伯-伊辛链有效描述的准一维反铁磁材料, 在外加磁场下一般性的有效哈密顿量为[43]
\begin{split} {\boldsymbol{H}} =\;& J\sum\limits_{i}[\varepsilon ({\boldsymbol{S}}^{x}_{i}{\boldsymbol{S}}^{x}_{i+1}+{\boldsymbol{S}}_{i}^{y}{\boldsymbol{S}}_{i+1}^{y})+{\boldsymbol{S}}_{i}^{z}{\boldsymbol{S}}_{i+1}^{z}]\\ &-\mu_{{\rm{B}}}\sum\limits_{i}{\boldsymbol{S}}_{i}\widetilde{{\boldsymbol{g}}}{{\boldsymbol{H}}_{{0}}}, \end{split} (8) 其中
\mu_{{\rm{B}}} 是玻尔磁子,{{\boldsymbol{H}}_{\bf{0}}} 为外加磁场,\widetilde{{\boldsymbol{g}}} 代表{\boldsymbol{ g}} 张量:\widetilde{{\boldsymbol{g}}} = \left( \begin{array}{ccc} g_{xx} & g_{xy} & g_{xz} \\ g_{yx} & g_{yy} & g_{yz} \\ g_{zx} & g_{zy} & g_{zz} \end{array} \right). (9) 考虑加横向磁场
{{\boldsymbol{H}}_{\bf{0}}} = H_{0}{\boldsymbol{e}}_{x} , 由于{\boldsymbol{g}} 张量的非对角元, 此时在Y轴和Z轴会诱导出垂直于该磁场的有效磁场, 因此有[45]\begin{split} &\mu_{{\rm{B}}}\sum\limits_{i}{\boldsymbol{S}}_{i}\widetilde{{\boldsymbol{g}}}{{\boldsymbol{H}}_{{0}}} \\ =\;& \mu_{{\rm{B}}}\sum\limits_{i}[{\boldsymbol{S}}_{i}^{x}g_{xx}+ {\boldsymbol{S}}_{i}^{y}g_{xy}+{\boldsymbol{S}}_{i}^{z}g_{xz}]{\boldsymbol{H}}_{0}, \end{split} (10) 其中
\begin{split} &g_{xx} = (g_{\xi}\cos^{2}\theta + g_{\zeta}\sin^{2}\theta) \cos^{2} \phi + g_{\psi}\sin^{2}\phi, \end{split} (11) \begin{split} g_{xy} =\;& \dfrac12 (g_{\xi}\cos^{2}\theta - g_{\psi} + g_{\zeta} \sin^{2}\theta) \sin(2\phi), \end{split} (12) \begin{split} g_{xz} =\;& (g_{\zeta}-g_{\xi}) \sin\theta\cos\theta\cos\phi,\\ \phi=\;& \phi_{1}+\frac{\pi}{2}(i-1). \end{split} (13) 而
\xi 轴、\psi 轴和\zeta 轴代表将实验室的XYZ坐标轴绕Y轴转\theta 角, 绕Z轴转\phi 角后得到的材料中的局域坐标轴. CoO_{6} 的晶格结构、螺旋链结构及坐标轴间的关系如图4所示.
图 4 (a) CoO6八面体的结构; (b) CoO6四周期螺旋链的单元结构及局域坐标系\xi\psi\zeta \xi\psi\zeta Fig. 4. (a) Structure of CoO6 octahedron; (b) a unit structure of CoO6 four-period screw chain and the local coordinate\xi\psi\zeta \xi\psi\zeta 对于外加磁场沿[110]和[100]两种情况,
\phi_{1} 分别取0^{\circ} 和45^{\circ} . 以BCVO为例, 若对该材料施加沿[110]的磁场, 则g_{xy} = g_{xz} = 0 , 可以避免诱导出有效交错横场. 在低温下对该材料进行磁热效应测量, 可以得到随温度和横场变化的格林艾森比率数据. 对该数据的标度行为分析完全符合理论上对TFIC模型格林艾森比率量子临界行为的预期, 从而确认该材料QCP在B_{\perp}^{{\rm{c}}} = 40 T附近, 并在该点附近涌现出伊辛普适类(图5)[33].![图 5 (a) BCVO材料在加[110]方向磁场后的相图, 利用从不同起始温度出发的绝热磁热测量得出. 红色区域内直到$ 20 $T处标记了一有限温的三维序. 一维QCP出现在$ B_{\perp}^{{\rm{c}}} = 40 $T附近, 上方黄色区域为量子临界区. (b), (c)以磁场和温度作为自变量的格林艾森比率实验数据. 在图(b)中, 大于QCP时不同温度的数据展现了$ \varGamma_{B}\approx (B-B_{\perp}^{{\rm{c}}})^{-1} $的发散行为. 相对应地, 当场强在临界场强附近时, 低温下格林艾森比率$ \varGamma_{B}(T) $趋于收敛(图片经文献[33]允许转载, 版权归2018 American Physical Society 所有)\r\nFig. 5. (a) Phase diagram of BCVO material with a transverse field along [110], obtained from adiabatic magnetocaloric-effect measurements starting from different temperatures. Red area labels a finite-temperature three dimensional (3D) order until 20 T. The one dimensional (1D) QCP appears around $ B_{\perp}^{{\rm{c}}} = 40 $T, while the yellow area above labels quantum critical region. (b), (c) Experimental data of Grüneisen ratio in terms of magnetic field and temperature. In panel (b) data of different temperature above the QCP shows divergent behavior of $ \varGamma_{B}\approx (B-B_{\perp}^{{\rm{c}}})^{-1} $. Correspondingly, with the field being around the critical field, Grüneisen ratio converges in low temperature. (Reprinted with permission from Ref. [33]. Copyright 2018 American Physical Society)](/fileWLXB/journal/article/wlxb/2021/23/PIC/23-20211836-5.png)
图 5 (a) BCVO材料在加[110]方向磁场后的相图, 利用从不同起始温度出发的绝热磁热测量得出. 红色区域内直到20 B_{\perp}^{{\rm{c}}} = 40 \varGamma_{B}\approx (B-B_{\perp}^{{\rm{c}}})^{-1} \varGamma_{B}(T) Fig. 5. (a) Phase diagram of BCVO material with a transverse field along [110], obtained from adiabatic magnetocaloric-effect measurements starting from different temperatures. Red area labels a finite-temperature three dimensional (3D) order until 20 T. The one dimensional (1D) QCP appears aroundB_{\perp}^{{\rm{c}}} = 40 \varGamma_{B}\approx (B-B_{\perp}^{{\rm{c}}})^{-1} 然而, 40 T左右的量子临界磁场过于昂贵, 难以从此出发继续深入研究与伊辛普适类相关的物理. 此后不久, 吴建达及合作者[34-37]在对SCVO和BCVO的进一步研究中发现, 沿[100] (或[010])方向施加一横向磁场所诱导出的沿[010] (或[100])的有效交错横场能极大降低达到QCP所需要的磁场强度, 相关的数值分析已在2.2节中提及(图2)[35]. 这里介绍吴建达及合作者对施加沿[100]方向磁场的SCVO材料的伊辛普适类QCP的实验进展. 此时, 该材料的有效哈密顿量为[34]
\begin{split} {\boldsymbol{H}} =\;& J\sum\limits_{i}[\varepsilon({\boldsymbol{S}}^{x}_{i}{\boldsymbol{S}}^{x}_{i+1}+{\boldsymbol{S}}^{y}_{i}{\boldsymbol{S}}^{y}_{i+1})+ {\boldsymbol{S}}^{z}_{i}{\boldsymbol{S}}^{z}_{i+1}]\\ &-\mu_{{\rm{B}}}g_{x}\sum\limits_{i}\big\{ {\boldsymbol{HS}}_{i}^{x}+{\boldsymbol{H}}_{y}(-1)^{i}{\boldsymbol{S}}_{i}^{y}\\ &+{\boldsymbol{H}}_{z}{\boldsymbol{S}}^{z}_{i}\cos[\pi(2i-1)/4]\big\}, \end{split} (14) 其中
J\approx 7.5 meV, 各向异性因子\varepsilon = 0.47 .g_{x}\approx 2.79 为旋磁比率, 诱导出的有效交错横场{\boldsymbol{H}}_{y}\approx 0.4 {\boldsymbol{H }} , 四周期场{\boldsymbol{H}}_{z}\approx 0.14 {\boldsymbol{H}} . 利用NMR实验可以测量此时的QCP及其普适类, 测量结果见图6[34]. 在SCVO中, 沿[100]方向调节磁场会涌现出两个QCPs, 记为H_{{\rm{c}}1} 和H_{{\rm{c}}2} . 其中,H_{{\rm{c}}1}\approx 7.03 T代表三维序的终点, 而H_{{\rm{c}}2}\approx 7.7 T对应伊辛普适类的量子临界点[34].![图 6 外加[100]方向磁场的SCVO材料相图. 图中AFM, QCR和QD分别代表反铁磁相、量子临界区和量子无序区. 两个QCPs分别为$ H_{{\rm{c}}1}\approx 7.03\; {\rm{T}} $和$ H_{{\rm{c}}2}\approx 7.7\; {\rm{T}} $. 蓝色线可以被曲线$ T_{{\rm{cr}}}^{{\rm{L, H}}}\sim \vert H_{{\rm{c}}2}-H \vert $所拟合, 揭示了该QCP对应伊辛普适类. 内嵌图是两QCPs附近拟合曲线的放大 (图片经文献[34]允许转载, 版权归2019 American Physical Society 所有)\r\nFig. 6. Phase diagram of SCVO with transverse field along [100]. The AFM, QCR and QD in the figure corresponds to anti-ferromagnetic phase, quantum critical region and quantum disordered region, respectively. The two QCPs are $ H_{{\rm{c}}1}\approx 7.03 \;{\rm{T}} $ and $ H_{{\rm{c}}2}\approx 7.7\; {\rm{T}} $. The blue line can be fitted with $ T_{{\rm{cr}}}^{{\rm{L, H}}}\sim \vert H_{{\rm{c}}2}-H \vert $, implying the Ising universality of the 1D QCP. Inset is an enlarged figure of the fitting curves near two QCPs. (Reprinted with permission from Ref. [34]. Copyright 2019 American Physical Society)](/fileWLXB/journal/article/wlxb/2021/23/PIC/23-20211836-6.png)
图 6 外加[100]方向磁场的SCVO材料相图. 图中AFM, QCR和QD分别代表反铁磁相、量子临界区和量子无序区. 两个QCPs分别为H_{{\rm{c}}1}\approx 7.03\; {\rm{T}} H_{{\rm{c}}2}\approx 7.7\; {\rm{T}} T_{{\rm{cr}}}^{{\rm{L, H}}}\sim \vert H_{{\rm{c}}2}-H \vert Fig. 6. Phase diagram of SCVO with transverse field along [100]. The AFM, QCR and QD in the figure corresponds to anti-ferromagnetic phase, quantum critical region and quantum disordered region, respectively. The two QCPs areH_{{\rm{c}}1}\approx 7.03 \;{\rm{T}} H_{{\rm{c}}2}\approx 7.7\; {\rm{T}} T_{{\rm{cr}}}^{{\rm{L, H}}}\sim \vert H_{{\rm{c}}2}-H \vert 这篇工作表明, 对于能被海森伯-伊辛链有效描述的磁性材料而言, 若外加横向磁场时能诱导出新的平面内交错磁场, 这些诱导场能够有效压制达到临界磁场所需要的的场强大小, 从而在一个较小的磁场环境下实现伊辛普适类.
本节总结了在BCVO和SCVO中实现伊辛普适类的实验工作[33,34]. 在实现伊辛普适类之后, 笔者及合作者结合理论工作进展[30,46], 推动了量子
E_{8} 可积模型(以下简称E_{8} 模型)的物理实现. 下文将首先介绍与该模型相关的理论背景及研究进展[30,46-50], 再详细描述理论及实验共同合作促成的该奇异模型在BCVO材料中的实现.4. 量子E8可积模型
1990年前后, Zamolodchikov[23]和Fateev等[24]证明了对应于某一特定中心荷的CFT, 若它被一相关元场所微扰, 该CFT对应的哈密顿量会成为能够被例外李代数所描述的可积模型[16,27-31]. 特别地, 对于中心荷
c = 1- 6/\left[p(p+1)\right] 的CFT哈密顿量{\boldsymbol{H}}_{p} , 其被相关元场{\boldsymbol{\varPhi}}_{p} 微扰的哈密顿量为\hat{{\boldsymbol{H}}}_{p} = {\boldsymbol{H}}_{p}+\lambda \int {\boldsymbol{\varPhi}}_{p}(x){\rm{d}}x. (15) 当
p = 6, \; 4, \; 3 时, 该微扰哈密顿量可以分别被E_{6}, \; E_{7}, \; E_{8} 例外李代数所描述, 对应的系统分别包含有6, 7, 8种准粒子激发[24,51-57]. 在QCP处, TFIC模型是中心荷c = 1/2 的CFT[3], 在标度极限下, 自旋密度算符为相关元场[30,31], 在其微扰下的哈密顿量{\boldsymbol{H}} = {\boldsymbol{H}}_{c = 1/2}+h \int {\boldsymbol{\sigma}}(x){\rm{d}}x (16) 成为一个激发谱能够被
E_{8} 例外李代数所描述的量子可积模型[16,27-38], 其对应的格点模型为被一纵向磁场所微扰的QCP处(g = g_{\rm{c}} = 1 )的TFIC模型[30,36,37]:{\boldsymbol{H}} = -J\sum\limits_{i}\left( {\boldsymbol{\sigma}}_{i}^{z}{\boldsymbol{\sigma}}_{i+1}^{z} + {\boldsymbol{\sigma}}_{i}^{x} + h_{z}{\boldsymbol{\sigma}}_{i}^{z} \right), (17) 其中处在QCP处的TFIC哈密顿量
-J\displaystyle\sum\nolimits_{i}( {\boldsymbol{\sigma}}_{i}^{z}{\boldsymbol{\sigma}}_{i+1}^{z} + {\boldsymbol{\sigma}}_{i}^{x}) 对应中心荷c = 1/2 的哈密顿量{\boldsymbol{H}}_{c = 1/2} , 纵向场微扰-J\displaystyle\sum\nolimits_{i}h_{z}{\boldsymbol{\sigma}}_{i}^{z} 对应元场微扰h \displaystyle\int {\boldsymbol{\sigma}}(x){\rm{d}}x , 而h_{z}J 是纵向磁场与自旋间的耦合强度, 对应于标度极限下方程(16)中的h . 对TFIC模型而言, 标度极限由如下转换方式给出[30]:a\rightarrow 0,\; \; J\rightarrow \infty, \; \; h_{z}\rightarrow 0, \; \; aJ \to \text{\rm{finite value}}. (18) 即在标度极限下, 晶格常数
a 与纵场h_{z} 趋于0, 同时保持能隙\varDelta\approx Jh_{z}^{8/15} 和2 Ja 为有限值. 由此, 在标度极限下h \approx J^{15/8}h_{z},\; \; {\boldsymbol{\sigma}}(x)\approx J^{1/8} {\boldsymbol{\sigma}}_{j}^{z}, \; \; {\boldsymbol{\varepsilon}}(x) \approx J {\boldsymbol{\sigma}}_{j}^{x}. (19) E_{8} 模型包含有8种不同的准粒子激发. 其中最轻的准粒子质量m_{1} = Ch^{8/15} ,C = 4.4049\cdots [58]. 而次轻准粒子质量m_{2} 与m_{1} 满足黄金分割, 即m_{2} = 2\cos(\pi /5)m_{1}\approx 1.618 m_{1} . 其他准粒子的质量都可以通过m_{1} 及m_{2} 精确地得到.E_{8} 模型相图如图7所示. 该相图给出了E_{8} 准粒子激发与伊辛普适类的联系: 在TFIC模型的伊辛普适类处, QCP附近的物理可以用中心荷1/2 的共形场论描述. 此时纵向磁场的微扰提供了一个微弱的禁闭效应, 引起一系列完美共振的介子态, 从而产生8种E_{8} 模型的准粒子激发[59,60]. 因此纵向磁场较小的区间正是可以涌现出量子E_{8} 模型的区间, 如图7中蓝色区域所示.而这些准粒子间的精确质量关系亦见于图7.
图 7E_{8} E_{8} g h_{z} g h_{z} E_{8} m_{1} m_{2} m_{1} Fig. 7. A phase diagram ofE_{8} E_{8} g h_{z} g h_{z} E_{8} m_{1} m_{2} m_{1} E_{8} 模型哈密顿量的本征态可以表达为一系列E_{8} 准粒子的直积态\vert A(\theta_{1})\cdots A(\theta_{n})\rangle_{a_{1}, \;\cdots,\; a_{n}} . 其中\vert A(\theta_{i})\rangle_{a_{i}} 为单粒子态, 满足正交归一化关系{}_{a_{i}}\langle A(\theta_{i})\vert A(\theta_{j})\rangle_{a_{j}} \;=\; 2\pi\delta_{a_{i}a_{j}}\delta\;(\theta_{i}-\theta_{j}) .\theta_{i} 和a_{i}\;(i = 1, \;\cdots, \;8) 分别代表该粒子的快度和8个E_8 粒子之一, 其能量及动量满足E_{i} = m_{i}\cos\theta_{i}, \; P_{i} = m_{i}\sin\theta_{i} , 这里设定系统的特征速度c = 1 及普朗克常量\hbar = 1 .E_{8} 模型的基态是不包含任何准粒子激发的真空态, 记为\vert 0\rangle . 对时空关联函数作傅里叶变换后有动力学结构因子 (dynamical structure factor, DSF)D^{\varPhi\varPhi}(\omega, q) = \displaystyle\int {\rm{d}}\,x{\rm{d}}\,t {\rm{e}}^{{\rm{i}}(\omega t-qx)}\langle{\boldsymbol{ \varPhi}}(x, t){\boldsymbol{\varPhi}}(0, 0) \rangle , 其中{\boldsymbol{ \varPhi}} 可以取自旋密度算符或能量密度算符, 分别对应于格点模型中的{\boldsymbol{\sigma}}^{z} 和{\boldsymbol{\sigma}}^{x} .E_{8} 模型中{\boldsymbol{\sigma}}^{y} 不存在对应的CFT元场, 因此无法直接计算{\boldsymbol{\sigma}}^{y} 的DSF, 但可以证明{\boldsymbol{\sigma}}^{y} 的DSFD^{yy}(\omega, q) 与D^{zz}(\omega, q) 间满足严格关系D^{yy}(\omega, q) = \omega^{2}D^{zz}(\omega, q)/(4 J^{2}) [46]. 于是, 考虑零动量的情形, 通过引入不同数目E_{8} 粒子完备基, DSF的莱曼表示 (Lehmann representation)为[30,36,37]:\begin{split} & D^{ii}(\omega,q = 0) =\\ & \sum\limits_{n = 0}^\infty\sum\limits_{\{a_i\}}\left(\prod\limits_{a_{i}}\dfrac{1}{N_{a_{i}}!}\right)\dfrac{1}{(2\pi)^{n-2}} \int_{-\infty}^{\infty} {\rm{d}}\theta_{1}\cdots {\rm{d}}\theta_{n}\\ & |F^{\sigma^{i}}_{n}(\theta_{1},\cdots,\theta_{n})|^2\delta\left(\omega-\sum\limits_{i = 1}^{n}E_{i}\right)\delta\left(\sum\limits_{i = 1}^{n}P_{i}\right), \end{split} (20) 其中
i = x, \; z , 而D^{yy}(\omega, q = 0) 由前述关系给出.F^{\sigma^{i}}_{n}(\theta_{1}, \cdots, \theta_{n}) 为对应于{\boldsymbol{\sigma}}^{i} 算符的包含n 粒子的形式因子 (Form Factor, FF)[28],F^{\sigma^{i}}_{n}(\theta_{1},\cdots,\theta_{n}) = \langle0|{\boldsymbol{\sigma}}^{i}\vert A(\theta_{1}),\cdots,A(\theta_{n})\rangle_{a_{1},\cdots,a_{n}}. (21) 关于FF的一般性讨论可参考文献[30, 31, 61-64]. 而对于
E_{8} 模型FF的求解则可见于文献[30, 47], 这里不再赘述.在方程(20)中, 通过引入不同粒子数态, 依粒子数的不同, 总体的贡献可分解为不同的通道, 即单粒子通道、两粒子通道、三粒子通道和四粒子通道[30]. 由于质量最大的
E_{8} 粒子的质量约在4.8 m_{1} 左右, 而计算表明随着能量增加谱权重急剧衰减, 因此选取DSF直至\omega = 5 m_{1} . 以D^{yy}(\omega) 为例, 其DSF谱如图8所示. 这一解析计算的结果展示了丰富的物理: 首先, 对于单粒子通道, 所有的谱均为共振峰, 位置分别对应了E_{8} 模型中8种准粒子的质量. 随着能量的增大共振峰的强度不断衰减. 这是由于在粒子质量增大的同时, 单粒子形式因子的绝对数值也在不断减小, 而单粒子通道的谱权重的峰值正比于|F|^2/m , 因此质量越大的单粒子对应的峰值越小. 其次, 对于两粒子通道, 含不同类型粒子a 和b 的通道谱存在谱边发散行为1/\sqrt{\omega-\omega_{0}} , 位于这两粒子的质量之和边界\omega_{0} = m_{a}+m_{b} 处. 这是由两粒子态密度的发散行为所导致的, 即范·霍夫奇性 (Van Hove’s singularity)[30]. 两相同类型的粒子组合却不存在这种发散行为, 这是由于它们的FF特殊结构恰好能够消除由态密度导致的发散. 最后, 三粒子和四粒子通道的谱贡献不存在上述的发散行为, 且二者行为相似, 贡献都比较小. 这是由于随着引入粒子数的增多, FF的绝对大小在不断衰减, 同时由于相空间的增大, 奇性行为被消去. 从图8的解析数据可以预期, 随着能量的不断升高, 高能区的DSF谱会不断衰减, 其行为相对来说较为平凡. 将解析计算得出的DSF谱线以实验测量的能量分辨率作展宽后[30], 便可以与实验进行对比以验证具体材料中是否实现E_{8} 模型.![图 8 $ D^{yy}(\omega) $的解析计算结果 (a)单粒子通道的谱权重贡献, 8个峰的位置依次对应$ E_{8} $模型8种粒子的质量, 其中$ m_{1} = 1 $; (b)—(d)二粒子、三粒子和四粒子通道的谱权重贡献, 图中分别标出了谱线和峰的具体贡献来源(图片经文献[30]允许转载, 版权归2021 American Physical Society 所有)\r\nFig. 8. (a) Spectrum contribution from single particle channel, the eight delta peaks one-by-one correspond to the masses of the eight particles in the quantum $ E_{8} $ model with $ m_{1} = 1 $; (b)–(d) contributions from two, three and four particle channels. The detailed contributions of different peaks and curves are highlighted in the sub-figures. (Reprinted with permission from Ref. [30]. Copyright 2021 American Physical Society)](/fileWLXB/journal/article/wlxb/2021/23/PIC/23-20211836-8.png)
图 8D^{yy}(\omega) E_{8} m_{1} = 1 Fig. 8. (a) Spectrum contribution from single particle channel, the eight delta peaks one-by-one correspond to the masses of the eight particles in the quantumE_{8} m_{1} = 1 5. BCVO材料中的E8激发
2010年, 牛津大学的Coldea实验组[65]利用非弹性中子散射技术(inelastic neutron scattering, INS)对准一维磁性材料CoNb
_{2} O_{6} 展开了自旋动力学谱的测量. 其测量结果提供了量子E_{8} 模型可在材料中实现的初步实验证据, 这体现在质量最轻的两个粒子质量比满足黄金分割. 然而, 受限于实验分辨率, 其测量结果未能得到较为精确的连续谱, 无法就连续谱区域的相关的E_{8} 物理做出有效的比对, 从而未能就该材料是否能完整实现量子E_{8} 模型给出确定性的回答. 之后, 2020 年对该材料的太赫兹 (THz) 谱测量[66]得到了详细且精确的连续谱, 实验所展示的连续谱区域特征和解析上得到的E_{8} 谱存有明显的偏差, 因此该材料是否能实现量子E_{8} 模型有待进一步深入仔细的研究[67,68]. 接下来聚焦于 BCVO 材料中实现量子E_{8} 模型的工作.前文所述的
E_{8} 模型对应的格点模型为铁磁相互作用, 而若想在反铁磁相互作用下实现E_{8} 模型, 需要将方程(17)中的纵场转为交错纵场, 这在凝聚态系统实验中几乎不可能通过外加手段实现, 因此只能另辟蹊径. 考虑沿[010]方向的外加磁场, 材料的有效哈密顿量为[34-37]\begin{split} {\boldsymbol{H}} =\;& J\sum\nolimits_n {\left[ {{\boldsymbol{S}}_n^z{\boldsymbol{S}}_{n + 1}^z + \varepsilon({\boldsymbol{S}}_n^x{\boldsymbol{S}}_{n + 1}^x + {\boldsymbol{S}}_n^y{\boldsymbol{S}}_{n + 1}^y)} \right]} \\ &~~- {\mu _{\rm{B}}}{g_{yy}}{{H}}\sum\nolimits_n {\bigg[ {{\boldsymbol{S}}_n^y + {h_x}{{( - 1)}^n}{\boldsymbol{S}}_n^x} } \\ & ~~{ + {h_z}\cos \left(\pi \frac{{2n - 1}}{4}\right){\boldsymbol{S}}_n^z} \bigg] \\ & - {\mu _{\rm{B}}}{{H}}'\sum\nolimits_n {{{( - 1)}^n}} {\boldsymbol{S}}_n^z, \\[-12pt] \end{split} (22) 其中交错纵场
\mu_{{\rm{B}}}{{H}}'\displaystyle\sum\nolimits_{n}(-1)^{n}{\boldsymbol{S}}_{n}^{z} 仅在材料处于三维反铁磁序内(其三维量子相变点在10 T左右 (图9)), 利用链平均场描述链间相互作用时才会出现. 根据上文提及的SCVO材料的相关参数, 其对应于伊辛普适类的最低临界磁场值为7.7 T位于三维量子相变点7.0 T以外, 所以此材料缺乏能产生交错纵场的机制或手段[34-37], 难以实现E_{8} 模型. 而对BCVO材料来说, 相互作用J = 5.8 meV, 利用iTEBD数值计算及NMR实验, 可以确定其伊辛普适类的临界磁场位于\mu_{0}{{H}} = 4.7 T处 (图10), 处在三维序内且远离三维量子相变点, 因此存在由链间相互作用平均后大小为\mu_{{\rm{B}}}{{H}}' = 0.018 J 的有效弱交错纵向磁场, 为实现E_{8} 模型提供了可能. 确实, 在该材料三维序内, 进一步的非弹性中子实验将沿[010]方向所加横场调到4.7 T时得到的自旋动力学关联谱与解析计算得出的E_{8} 模型单粒子及多粒子激发谱及从(22)式出发的iTEBD数值计算结果高度符合, 如图11所示. 这是人们首次在真实材料中实现E_{8} 模型, 不仅明确测量到了单E_8 粒子激发, 同时确认了多E_8 粒子激发, 是强关联磁性系统研究的一次重大进展.![图 9 (a) BCVO的晶格结构. (b) 加沿[010]方向磁场后BCVO的相图, QCP为$ H_{{\rm{c}}}^{1 {\rm{D}}} $, 隐藏在三维序中, 其一维量子临界行为可在三维序外测得. 棕色圆圈代表由NMR实验测出的尼尔(Néel)温度$ T_{{\rm{N}}} $. $H_{{\rm{c}}}^{3{\rm{ D}}} = (10.4\pm 0.1)\;\text{T}, \; H_{{\rm{c}}}^{1 {\rm{D}}} = (4.7\pm 0.3)$T. $ E_{8} $模型在蓝色缎带区涌现. (c) 8种$ E_{8} $粒子的质量以$ m_{1} $为单位沿能量轴分布(图片经文献[37]允许转载, 版权归2021 American Physical Society 所有)\r\nFig. 9. (a) Crystal structure of BCVO. (b) Phase diagram of BCVO with a transverse field along [010] direction. QCP is at $ H_{{\rm{c}}}^{1{\rm{ D}}} $, hidden in the 3D order. Its quantum criticality can be measured outside the 3D ordering phase. Brown circles represent for Néel temperature $ T_{{\rm{N}}} $ measured by NMR experiments. $H_{{\rm{c}}}^{3 {\rm{D}}} = (10.4\pm 0.1)\;\rm {T}, \; H_{{\rm{c}}}^{1 {\rm{D}}} = (4.7\pm 0.3)\;\rm {T}$. $ E_{8} $ model emerges in the blue ribbon area. (c) Masses of eight $ E_{8} $ particles along energy axis in unit of $ m_{1} $. (Reprinted with permission from Ref. [37]. Copyright 2021 American Physical Society)](/fileWLXB/journal/article/wlxb/2021/23/PIC/23-20211836-9.png)
图 9 (a) BCVO的晶格结构. (b) 加沿[010]方向磁场后BCVO的相图, QCP为H_{{\rm{c}}}^{1 {\rm{D}}} T_{{\rm{N}}} H_{{\rm{c}}}^{3{\rm{ D}}} = (10.4\pm 0.1)\;\text{T}, \; H_{{\rm{c}}}^{1 {\rm{D}}} = (4.7\pm 0.3) E_{8} E_{8} m_{1} Fig. 9. (a) Crystal structure of BCVO. (b) Phase diagram of BCVO with a transverse field along [010] direction. QCP is atH_{{\rm{c}}}^{1{\rm{ D}}} T_{{\rm{N}}} H_{{\rm{c}}}^{3 {\rm{D}}} = (10.4\pm 0.1)\;\rm {T}, \; H_{{\rm{c}}}^{1 {\rm{D}}} = (4.7\pm 0.3)\;\rm {T} E_{8} E_{8} m_{1} ![图 10 三维反铁磁序外NMR实验测量结果 (a) 在不同横场下测得自旋-晶格弛豫率$ 1/T_{1} $关于温度的函数. 下箭头标记出决定$T_{{\rm{N}}}$的$ 1/T_{1} $的峰. (b)拟合$ 1/T_{1} $得到能隙, 温度区间为6—12 K. 具体细节见文献[37]. QCP位于$ (4.7\pm 0.3) $T. 内嵌图是对数坐标下放大的低温区数据. 灰色直线区域内的量子临界行为可以拟合为$ 1/T_{1}\sim T^{-0.75} $(图片经文献[37]允许转载, 版权归2021 American Physical Society 所有)\r\nFig. 10. NMR experiment result outside the 3D anti-ferromagnetic ordering phase. (a) Spin-lattice relaxation rate $ 1/T_{1} $ with different transverse fields as a function of temperature. The down arrows label $ 1/T_{1} $ peaks that determining $T_{{\rm{N}}}$. (b) Energy gap by fitting $ 1/T_{1} $, with temperature being from 6 K to 12 K. The details can be found in Ref. [37]. QCP is at $(4.7\pm 0.3)$ T. Inset: a $ \log-\log $ plot of enlarged data in low-temperature region. The quantum critical behavior around the gray line area can be fitted by $ 1/T_{1}\sim T^{-0.75} $. (Reprinted with permission from Ref. [37]. Copyright 2021 American Physical Society)](/fileWLXB/journal/article/wlxb/2021/23/PIC/23-20211836-10.png)
图 10 三维反铁磁序外NMR实验测量结果 (a) 在不同横场下测得自旋-晶格弛豫率1/T_{1} T_{{\rm{N}}} 1/T_{1} 1/T_{1} (4.7\pm 0.3) 1/T_{1}\sim T^{-0.75} Fig. 10. NMR experiment result outside the 3D anti-ferromagnetic ordering phase. (a) Spin-lattice relaxation rate1/T_{1} 1/T_{1} T_{{\rm{N}}} 1/T_{1} (4.7\pm 0.3) \log-\log 1/T_{1}\sim T^{-0.75} ![图 11 (a) $ Q = (002) $QCP处INS测量数据, 场强$ H = 4.7 $T, 温度$ 0.4 $K. 蓝色方块及误差棒为实验测量数据, 黑线是高斯函数的拟合. 红色竖线对应于8种$ E_{8} $模型的粒子. 其他的峰来自于多粒子通道的连续谱贡献以及布里渊区折叠效应(标记于$ F_{1} $和$ F_{2} $处). (b) 解析计算的$ D^{xx}(\omega) $, 数据经过$ 0.08 m_{1} $洛伦兹展宽. $m_{1} = 1.2\;\text{meV}$. 红色线和黑色线分别代表单粒子激发贡献及引入多粒子激发贡献的总和. (c) iTEBD数值计算有效模型在转移动量$ q = 0 $处的激发谱, 其中蓝线和黑线代表不包含及包含折叠效应的数据. (d)—(g) 单粒子到四粒子激发通道的贡献, $ ijkl $代表激发来自于$ m_{i}m_{j}m_{k}m_{l} $通道(图片经文献[37]允许转载, 版权归2021 American Physical Society 所有)\r\nFig. 11. (a) INS data obtained from $ Q = (002) $ at QCP with $ H = 4.7 $ T, temperature $ 0.4 $ K. The blue diamonds with error bars are experimental data, the black curve is fitted with Gaussian functions. The red vertical lines correspond to eight particles of $ E_{8} $ model. Other peaks are contributed from multi-particle channels and zone-folding effect (labelled at $ F_{1} $ and $ F_{2} $). (b) Analytical result of $ D^{xx}(\omega) $ with a $ 0.08 m_{1} $ Lorentzian boradening. $m_{1} = 1.2\;\text{meV}$. Red curve and black curve are single particle spectrum and total spectrum including multi-particle excitations respectively. (c) Effective model spectrum of iTEBD calculation at $ q = 0 $, where blue curve and black curve are results of without and with zone-folding effect. (d)–(g) single particle-four particle excited channels' contributions, where $ ijkl $ represents for $ m_{i}m_{j}m_{k}m_{l} $ channel. (Reprinted with permission from Ref. [37]. Copyright 2021 American Physical Society)](/fileWLXB/journal/article/wlxb/2021/23/PIC/23-20211836-11.png)
图 11 (a)Q = (002) H = 4.7 0.4 E_{8} F_{1} F_{2} D^{xx}(\omega) 0.08 m_{1} m_{1} = 1.2\;\text{meV} q = 0 ijkl m_{i}m_{j}m_{k}m_{l} Fig. 11. (a) INS data obtained fromQ = (002) H = 4.7 0.4 E_{8} F_{1} F_{2} D^{xx}(\omega) 0.08 m_{1} m_{1} = 1.2\;\text{meV} q = 0 ijkl m_{i}m_{j}m_{k}m_{l} 6. 结 论
本文综述了在BCVO和SCVO材料中首次发现伊辛普适类并在BCVO材料中首次实现
E_{8} 模型激发的研究进展. 叙述了便于判断伊辛普适类的格林艾森比率的奇异量子临界行为. 随后介绍了横场海森伯-伊辛链并叙述了数值验证其具有伊辛普适类的理论进展. 通过将数值方法和实验相结合, 确认了BCVO和SCVO材料在外加横场之下会涌现出伊辛普适类. 在此基础上, 介绍了E_{8} 模型主要的理论进展及在理论与实验的共同推动下于BCVO材料中实现E_{8} 模型的一系列工作.这些研究进展充分展示了普适类的强大力量, 其所蕴含的迷人的物理现象持续吸引着人们对量子临界物理及相关普适类物理的进一步的探索[2]. 我们期待未来通过更多的凝聚态理论和实验物理学家的合作, 能在实际材料中实现更多非平凡的普适类, 并能通过深入研究这些奇异系统中的非平凡物理激发, 最终实现物理操纵这些奇异的激发并期望能够服务于新颖量子器件的设计与开发.
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图 1 TFIC模型相图.
g 为横场参数,g_{{\rm{c}}} = 1 为QCP. 在相变点左侧, 青蓝色实线代表零温下的铁磁相; 在相变点右侧, 黄色实线代表零温下的顺磁相. 相图中的两斜虚线为低温下各个不同无序区域的过渡边界, 而上方虚线则是量子临界区域到经典区域的过渡边界Figure 1. A phase diagram of TFIC, where
g labels transverse field, andg_{{\rm{c}}} = 1 is the QCP. The cyan solid line represents for ferromagnetic phase at zero temperature on the left of the critical point, while on the right the yellow solid line stands for a paramagnetic phase at zero temperature. The two tilted dotted lines illustrate the crossovers of different disorder phases in low temperature region, while the dotted line above shows a crossover from quantum critical region to classical region.
图 3 (a)交错磁化
M_{z}(g) 的计算结果. 蓝线和红线分别代表了带有和不带有4周期项的数据. 这两条曲线可以分别用M_{z} = 0.524(g_{{\rm{c}}1}-g)^{0.126} 和M_{z} = 0.530(g_{{\rm{c}}2}-g)^{0.128} 来拟合, 其中g_{{\rm{c}}1} = 0.1454 ,g_{{\rm{c}}2} = 0.1456 , 在误差精度范围内均可得到临界指数为1/8 的结论. 内嵌图是利用对数坐标轴画出的M_{z} 的标度行为. (b)半对数坐标轴下纠缠熵S_{{\rm{EE}}}(l) 和链长l 的关系, 两种情况都符合伊辛普适类的1/2 中心荷. (c)当g = 0.1448 时, 自旋两点关联函数和距离i-j 间的函数关系. 内嵌图展示了\ln C(i-j) 在长程时与i-j 成正比. (d)关联长度的倒数和g 的函数关系, 两种情况都符合伊辛普适类中关联长度指数\nu = 1 的结论(图片经文献[35]允许转载, 版权归2019 IOP Publishing Ltd 所有)Figure 3. (a) iTEBD results for staggered magnetization
M_{z}(g) . The blue line and red line represent iTEBD data with and without the four periodic term, respectively. The two curves can be fitted withM_{z} = 0.524(g_{{\rm{c}}1}-g)^{0.126} andM_{z} = 0.530(g_{{\rm{c}}2}-g)^{0.128} , whereg_{{\rm{c}}1} = 0.1454 ,g_{{\rm{c}}2} = 0.1456 , and critical exponent\delta = 1/8 is obtained within error bar for both cases. Inset shows scaling behavior ofM_{z} in\log-\log plot. (b) Entanglement entropyS_{{\rm{EE}}}(l) versus lengthl in a semi-\log plot, both fall into central chargec = 1/2 of TFIC universality. (c) Spin-spin correlation function versus distancei-j atg = 0.1448 . Inset shows\ln C(i-j) being proportional toi-j in long range region. (d) The inverse of correlation length in terms ofg , both agree with correlation length exponent\nu = 1 . (Reprinted with permission from Ref. [35]. Copyright 2019 IOP Publishing Ltd.)
图 4 (a) CoO6八面体的结构; (b) CoO6四周期螺旋链的单元结构及局域坐标系
\xi\psi\zeta ; (c)局域坐标系\xi\psi\zeta 与实验室坐标系XYZ的关系Figure 4. (a) Structure of CoO6 octahedron; (b) a unit structure of CoO6 four-period screw chain and the local coordinate
\xi\psi\zeta ; (c) relation between the local coordinate\xi\psi\zeta and the lab coordinate XYZ.
图 5 (a) BCVO材料在加[110]方向磁场后的相图, 利用从不同起始温度出发的绝热磁热测量得出. 红色区域内直到
20 T处标记了一有限温的三维序. 一维QCP出现在B_{\perp}^{{\rm{c}}} = 40 T附近, 上方黄色区域为量子临界区. (b), (c)以磁场和温度作为自变量的格林艾森比率实验数据. 在图(b)中, 大于QCP时不同温度的数据展现了\varGamma_{B}\approx (B-B_{\perp}^{{\rm{c}}})^{-1} 的发散行为. 相对应地, 当场强在临界场强附近时, 低温下格林艾森比率\varGamma_{B}(T) 趋于收敛(图片经文献[33]允许转载, 版权归2018 American Physical Society 所有)Figure 5. (a) Phase diagram of BCVO material with a transverse field along [110], obtained from adiabatic magnetocaloric-effect measurements starting from different temperatures. Red area labels a finite-temperature three dimensional (3D) order until 20 T. The one dimensional (1D) QCP appears around
B_{\perp}^{{\rm{c}}} = 40 T, while the yellow area above labels quantum critical region. (b), (c) Experimental data of Grüneisen ratio in terms of magnetic field and temperature. In panel (b) data of different temperature above the QCP shows divergent behavior of\varGamma_{B}\approx (B-B_{\perp}^{{\rm{c}}})^{-1} . Correspondingly, with the field being around the critical field, Grüneisen ratio converges in low temperature. (Reprinted with permission from Ref. [33]. Copyright 2018 American Physical Society)
图 6 外加[100]方向磁场的SCVO材料相图. 图中AFM, QCR和QD分别代表反铁磁相、量子临界区和量子无序区. 两个QCPs分别为
H_{{\rm{c}}1}\approx 7.03\; {\rm{T}} 和H_{{\rm{c}}2}\approx 7.7\; {\rm{T}} . 蓝色线可以被曲线T_{{\rm{cr}}}^{{\rm{L, H}}}\sim \vert H_{{\rm{c}}2}-H \vert 所拟合, 揭示了该QCP对应伊辛普适类. 内嵌图是两QCPs附近拟合曲线的放大 (图片经文献[34]允许转载, 版权归2019 American Physical Society 所有)Figure 6. Phase diagram of SCVO with transverse field along [100]. The AFM, QCR and QD in the figure corresponds to anti-ferromagnetic phase, quantum critical region and quantum disordered region, respectively. The two QCPs are
H_{{\rm{c}}1}\approx 7.03 \;{\rm{T}} andH_{{\rm{c}}2}\approx 7.7\; {\rm{T}} . The blue line can be fitted withT_{{\rm{cr}}}^{{\rm{L, H}}}\sim \vert H_{{\rm{c}}2}-H \vert , implying the Ising universality of the 1D QCP. Inset is an enlarged figure of the fitting curves near two QCPs. (Reprinted with permission from Ref. [34]. Copyright 2019 American Physical Society)
图 7
E_{8} 模型相图. 图中蓝色实线代表了E_{8} 模型涌现的区域, 其中g 和h_{z} 分别对应于横场参数g 和方程(17)中的h_{z} . 其余E_{8} 准粒子的质量和m_{1} 和m_{2} 的关系列于右上, 其中m_{1} 为最轻准粒子的质量Figure 7. A phase diagram of
E_{8} model. The blue solid line in the figure implies region ofE_{8} model emerging, whereg andh_{z} correspond to transverse parameterg andh_{z} in Eq. (17) respectively. Relations of otherE_{8} particles' masses tom_{1} andm_{2} are listed in the top right-hand corner, wherem_{1} is mass of the lightest particle.
图 8
D^{yy}(\omega) 的解析计算结果 (a)单粒子通道的谱权重贡献, 8个峰的位置依次对应E_{8} 模型8种粒子的质量, 其中m_{1} = 1 ; (b)—(d)二粒子、三粒子和四粒子通道的谱权重贡献, 图中分别标出了谱线和峰的具体贡献来源(图片经文献[30]允许转载, 版权归2021 American Physical Society 所有)Figure 8. (a) Spectrum contribution from single particle channel, the eight delta peaks one-by-one correspond to the masses of the eight particles in the quantum
E_{8} model withm_{1} = 1 ; (b)–(d) contributions from two, three and four particle channels. The detailed contributions of different peaks and curves are highlighted in the sub-figures. (Reprinted with permission from Ref. [30]. Copyright 2021 American Physical Society)
图 9 (a) BCVO的晶格结构. (b) 加沿[010]方向磁场后BCVO的相图, QCP为
H_{{\rm{c}}}^{1 {\rm{D}}} , 隐藏在三维序中, 其一维量子临界行为可在三维序外测得. 棕色圆圈代表由NMR实验测出的尼尔(Néel)温度T_{{\rm{N}}} .H_{{\rm{c}}}^{3{\rm{ D}}} = (10.4\pm 0.1)\;\text{T}, \; H_{{\rm{c}}}^{1 {\rm{D}}} = (4.7\pm 0.3) T.E_{8} 模型在蓝色缎带区涌现. (c) 8种E_{8} 粒子的质量以m_{1} 为单位沿能量轴分布(图片经文献[37]允许转载, 版权归2021 American Physical Society 所有)Figure 9. (a) Crystal structure of BCVO. (b) Phase diagram of BCVO with a transverse field along [010] direction. QCP is at
H_{{\rm{c}}}^{1{\rm{ D}}} , hidden in the 3D order. Its quantum criticality can be measured outside the 3D ordering phase. Brown circles represent for Néel temperatureT_{{\rm{N}}} measured by NMR experiments.H_{{\rm{c}}}^{3 {\rm{D}}} = (10.4\pm 0.1)\;\rm {T}, \; H_{{\rm{c}}}^{1 {\rm{D}}} = (4.7\pm 0.3)\;\rm {T} .E_{8} model emerges in the blue ribbon area. (c) Masses of eightE_{8} particles along energy axis in unit ofm_{1} . (Reprinted with permission from Ref. [37]. Copyright 2021 American Physical Society)
图 10 三维反铁磁序外NMR实验测量结果 (a) 在不同横场下测得自旋-晶格弛豫率
1/T_{1} 关于温度的函数. 下箭头标记出决定T_{{\rm{N}}} 的1/T_{1} 的峰. (b)拟合1/T_{1} 得到能隙, 温度区间为6—12 K. 具体细节见文献[37]. QCP位于(4.7\pm 0.3) T. 内嵌图是对数坐标下放大的低温区数据. 灰色直线区域内的量子临界行为可以拟合为1/T_{1}\sim T^{-0.75} (图片经文献[37]允许转载, 版权归2021 American Physical Society 所有)Figure 10. NMR experiment result outside the 3D anti-ferromagnetic ordering phase. (a) Spin-lattice relaxation rate
1/T_{1} with different transverse fields as a function of temperature. The down arrows label1/T_{1} peaks that determiningT_{{\rm{N}}} . (b) Energy gap by fitting1/T_{1} , with temperature being from 6 K to 12 K. The details can be found in Ref. [37]. QCP is at(4.7\pm 0.3) T. Inset: a\log-\log plot of enlarged data in low-temperature region. The quantum critical behavior around the gray line area can be fitted by1/T_{1}\sim T^{-0.75} . (Reprinted with permission from Ref. [37]. Copyright 2021 American Physical Society)
图 11 (a)
Q = (002) QCP处INS测量数据, 场强H = 4.7 T, 温度0.4 K. 蓝色方块及误差棒为实验测量数据, 黑线是高斯函数的拟合. 红色竖线对应于8种E_{8} 模型的粒子. 其他的峰来自于多粒子通道的连续谱贡献以及布里渊区折叠效应(标记于F_{1} 和F_{2} 处). (b) 解析计算的D^{xx}(\omega) , 数据经过0.08 m_{1} 洛伦兹展宽.m_{1} = 1.2\;\text{meV} . 红色线和黑色线分别代表单粒子激发贡献及引入多粒子激发贡献的总和. (c) iTEBD数值计算有效模型在转移动量q = 0 处的激发谱, 其中蓝线和黑线代表不包含及包含折叠效应的数据. (d)—(g) 单粒子到四粒子激发通道的贡献,ijkl 代表激发来自于m_{i}m_{j}m_{k}m_{l} 通道(图片经文献[37]允许转载, 版权归2021 American Physical Society 所有)Figure 11. (a) INS data obtained from
Q = (002) at QCP withH = 4.7 T, temperature0.4 K. The blue diamonds with error bars are experimental data, the black curve is fitted with Gaussian functions. The red vertical lines correspond to eight particles ofE_{8} model. Other peaks are contributed from multi-particle channels and zone-folding effect (labelled atF_{1} andF_{2} ). (b) Analytical result ofD^{xx}(\omega) with a0.08 m_{1} Lorentzian boradening.m_{1} = 1.2\;\text{meV} . Red curve and black curve are single particle spectrum and total spectrum including multi-particle excitations respectively. (c) Effective model spectrum of iTEBD calculation atq = 0 , where blue curve and black curve are results of without and with zone-folding effect. (d)–(g) single particle-four particle excited channels' contributions, whereijkl represents form_{i}m_{j}m_{k}m_{l} channel. (Reprinted with permission from Ref. [37]. Copyright 2021 American Physical Society) -
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