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基于U(1)对称的无限矩阵乘积态张量网络算法提取Luttinger液体参数K

王秀娟 李生好

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基于U(1)对称的无限矩阵乘积态张量网络算法提取Luttinger液体参数K

王秀娟, 李生好

Extracting Luttinger liquid parameter K based on U(1) symmetric infinite matrix product states

Wang Xiu-Juan, Li Sheng-Hao
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  • 本文数值研究了自旋$ S=1/2,1,2 $的各向异性量子XXZD模型的Luttinger液体参数K. 首先, 利用$ U(1) $对称的无限矩阵乘积态算法(iMPS)得到在Luttinger液体相中的基态波函数. 通过二分量子涨落F和有限纠缠标度指数$ \kappa $的关系可以提取出Luttinger液体参数K. 对于自旋$ S=1/2, D=0 $的量子XXZD模型, 本文利用$ U(1) $对称的iMPS的算法得到的数值结果与精确解符合得很好. 在参数$ D\leqslant -2 $的区域, 自旋$ S=1 $的XXZD模型的哈密顿量可以被映射到一个自旋$ S=1/2 $的有效XXZ模型, 本文计算了在这个区域内的Luttinger液体参数K与精确解基本是一致的, 相对误差小于$ 1\% $. 此外, 在参数$ \varDelta=-0.5, D=0 $处, 本文数值计算的Luttinger液体参数与密度矩阵重整化群(DMRG)的结果也是一致的. 这些研究结果表明: 当系统具有$ U(1) $对称性时, 利用$ U(1) $对称的iMPS的方法可以提取无能隙相中的Luttinger液体参数. 本文利用此方法还研究了自旋$ S=1 $的XXZD模型在其他参数下的Luttinger液体参数, 以及自旋$ S=2 $的XXZD模型的Luttinger液体参数.
    We numerically calculate Luttinger liquid parameter K in the anisotropic spin XXZD models with spin $s = 1/2$, 1, and 2. In order to obtain groundstate wavefunctions in Luttinger liquid phases, we employ the $U(1)$ symmetric infinite matrix product states algorithm (iMPS). By using relation between the bipartite quantum fluctuations F and the so-called finite-entanglement scaling exponents $\kappa$, the Luttinger liquid parameter K can be extracted. For $s = 1/2$ and $D=0$, the numerically extracted Luttinger liquid parameter K is shown to be good agreement with the exact value. On using the fact that the spin-1 XXZD Hamiltonian with $ D \leqslant - 2$ can be mapped to an effective spin-1/2 XXZ model, we calculate the Luttinger liquid parameter for the region of $ D \leqslant - 2$. It is shown that our numerical value of the Luttinger liquid parameter agree well with the exact values, here, the relative error less than $1\%$. Also, our Luttinger liquid parameter at $\Delta = - 0.5$ and $ D = 0$ is shown to be consistent with the result form the density matrix renormalization group (DMRG) method. These results suggest that the $U(1)$ symmetric iMPS method can be applicable to calculate Luttinger liquid parameters if any system has a $U(1)$ symmetry for gapless phases. For instance, we present our Luttinger liquid parameters for the first time for the spin-1 XXZD model under the other parameters and the spin-2 XXZD model with $D = 1.5$.
      通信作者: 李生好, shanshui510@163.com
    • 基金项目: 国家自然科学基金(批准号: 11104362)、重庆市基础科学与前沿技术研究专项(批准号: cstc2018jcyjAX0812)、重庆市教委科学技术研究项目(批准号: KJQN201801212)和陕西省自然科学基金(批准号: 2019JM-017)资助的课题.
      Corresponding author: Li Sheng-Hao, shanshui510@163.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11104362), the Research Program of Basic Research and Frontier Technology of Chongqing, China (Grant No. cstc2018jcyjAX0812), the Science and Technology Research Program of Chongqing Municipal Education Commission, China (Grant No. KJQN201801212), and the Natural Science Foundation of Shanxi Province, China (Grant No. 2019JM-017).
    [1]

    Sachdev S 1999 Quantum Phase Transitions (Cambridge: Cambridge University)pp3—5

    [2]

    White S R 1992 Phys. Rev. Lett. 69 2863Google Scholar

    [3]

    Landau D P, Binder K 2011 A Guide to Monte-Carlo Simulatios in Statistical Physics (Cambridge: Cambridge University)pp70-73

    [4]

    Evenbly G, Vidal G 2009 Phys. Rev. B 79 144108.

    [5]

    Jordan J, Orus R, Vidal G, Verstraete F, Cirac J I 2008 Phys. Rev. Lett. 101 250602Google Scholar

    [6]

    Singh S, Zhou H Q, Vidal G 2010 New J. Phys. 12 033029Google Scholar

    [7]

    Jiang H C, Weng Z Y, Xiang T 2008 Phys. Rev. Lett. 101 090603Google Scholar

    [8]

    Czarnik P, Dziarmaga J 2015 Phys. Rev. B 92 035120.

    [9]

    Li W, Ran S J, Gong S S, Zhao Y, Xi B, Ye F, Su G 2011 Phys. Rev. Lett. 106 127202Google Scholar

    [10]

    Chen B B, Chen L, Chen Z Y, Li W, Weichselbaum A 2018 Phys. Rev. X 8 031082

    [11]

    Corboz P, Czarnik P, Kapteijns G, Tagliacozzo L 2018 Phys. Rev. X 8 031031

    [12]

    Singh S, Pfeifer R N C, Vidal G 2010 Phys. Rev. A 82 050301Google Scholar

    [13]

    Haghshenas R, Sheng D N 2017 arXiv: 1711.07584v1 [cond-mat.str-el]

    [14]

    Singh S, Pfeifer R N C, Vidal G 2011 Phys. Rev. B 83 115125Google Scholar

    [15]

    Song H F, Rachel S, Hur K Le 2010 Phys. Rev. B 82 012405Google Scholar

    [16]

    Song H F, Rachel S, Flindt C,Klich I, Laflorencie N, Hur K Le 2012 Phys. Rev. B 85 035409Google Scholar

    [17]

    Yang M F 2007 Phys. Rev. B 76 180403(R).

    [18]

    Boschi C D E, Ercolessi E, Ortolani F, Roncaglia M 2003 Eur. Phys. J. B 35 465Google Scholar

    [19]

    Pollmann F, Mukerjee S, Turner A, Moore J E 2009 Phys. Rev. Lett. 102 255701Google Scholar

    [20]

    苏耀恒, 陈爱民, 王洪雷, 相春环 2017 物理学报 66 120301Google Scholar

    Su Y H, Chen A M, Wang H L, Xiang C H 2017 Acta Phys. Sin. 66 120301Google Scholar

    [21]

    Yang C N, Yang C P 1966 Phys. Rev. 150 321Google Scholar

    [22]

    Chen W, Hida K, Sanctuary B C 2003 Phys. Rev. B 67 104401Google Scholar

    [23]

    Kjall J A, Zaletel M P, Mong R S K, Bardarson J H, Pollmann F 2013 Phys. Rev. B 87 235106Google Scholar

  • 图 1  (i)三指标张量${ {\varGamma}}$, 奇异值矩阵${ {\lambda}}$和粒子数n; (ii)具有$U(1)$对称的iMPS表示

    Fig. 1.  (a) Three index tensor ${ {\varGamma}}$, singular value matrix ${ {\lambda}}$ and particle number n; (b) An U(1) symmetric iMPS representation.

    图 2  更新具有$U(1)$对称的MPS的过程(a)把U门作用在具有$U(1)$对称的MPS上; (b)吸收U门缩并(a)中的张量使之成为一个两指标张量${ {\varTheta}}$;(c)对张量${ {\varTheta}}$进行奇异值分解(SVD), 得到新的张量X, Y$\tilde{{ {\lambda}}_{\rm B}}$, 同时得到新的粒子数$\tilde{n_r}$;(d)插入逆矩阵, 还原原来的原胞结构; (e)得到更新的张量$\tilde{ {\varGamma}}^{\rm A}$, $\tilde{ {\varGamma}}^{\rm B}$$\tilde{ {\lambda}}^{\rm B}$及粒子数$\tilde{n_r}$

    Fig. 2.  The process of update the U(1) symmetric MPS (a) applied gate U on the U(1) symmetric MPS, then contract the tensor network (a) into a single tensor ${ {\varTheta}}$. We compute the singular value decomposition of tensor ${ {\varTheta}}$, and get the new tensor X, Y$\tilde{{ {\lambda}}_{\rm B}}$ and particle number $\tilde{n_r}$ as in (c). (d) Insert inverse matrix and restore the original tensor structure, we obtain the new tensor $\tilde{ {\varGamma}}^{\rm A}$, $\tilde{ {\varGamma}}^{\rm B}$, $\tilde{ {\lambda}}^{\rm B}$ and particle number $\tilde{n_r}$ as in (e).

    图 3  在不同控制参量条件下, 自旋$S = 1/2$的XXZD模型的关联长度$\xi$和涨落F是截断维数$\chi$的函数. 其中, 参数$D = 0$

    Fig. 3.  Correlation length $\xi$ and fluctuation F of spin $S = 1/2$ XXZD model as a function of the truncation dimension $\chi$ for various parameters $\varDelta$. Here, fixed parameter $D = 0$

    图 4  在不同控制参量条件下, 自旋$S = 1$的XXZD模型的关联长度$\xi$和涨落F是截断维数$\chi$的函数. 其中, 各向异性参数$\varDelta = -0.5$

    Fig. 4.  Correlation length $\xi$ and fluctuation F of spin $S = 1$ XXZD model as a function of the truncation dimension $\chi$ for various parameters D. Here, fixed anisotropic parameter $\varDelta = -0.5$

    图 5  在不同控制参量条件下, 自旋$S = 2$的XXZD模型的关联长度$\xi$和涨落F是截断维数$\chi$的函数. 其中, 参数$D = 1.5$

    Fig. 5.  Correlation length $\xi$ and fluctuation F of spin $S = 2$ XXZD model as a function of the truncation dimension $\chi$ for various parameters $\varDelta$. Here, fixed parameter $D = 1.5$

    表 1  自旋S = 1/2的XXZD模型在临界区的Luttinger液体参数K, 其中参数D = 0

    Table 1.  Estimates for Luttinger liquid parameter K in the critical phase of spin S = 1/2 XXZD model with the parameter D = 0.

    Δ 0 0.25 0.5 0.75 1
    K[精确] 1.0 0.8614... 0.75 0.6493... 0.5
    K[数值] 0.999 0.856 0.7529 0.6457 0.5198
    相对误差 0.001 0.0063 0.0039 0.0053 0.0396
    下载: 导出CSV

    表 2  自旋$S = 1$的XXZD模型在临界区的Luttinger液体参数K, 固定参数$\varDelta = -0.5$

    Table 2.  Estimates for Luttinger liquid parameter K in the critical phase of spin $S = 1$ XXZD model with the parameter $\varDelta = -0.5$.

    D –0.3 0 0.3 0.5 0.6
    K[数值] 3.3679 3.1275 2.6834 2.4126 2.2745
    下载: 导出CSV

    表 3  自旋S = 2的XXZD模型在临界区的Luttinger液体参数K, 固定参数$D = 1.5$

    Table 3.  Estimates for Luttinger liquid parameter K in the critical phase of spin S = 2 XXZD model with the parameter $D = 1.5$.

    Δ 0.4 0.8 1 1.2 1.6
    K[数值] 1.652 2.5227 2.4096 2.3546 2.1107
    下载: 导出CSV
  • [1]

    Sachdev S 1999 Quantum Phase Transitions (Cambridge: Cambridge University)pp3—5

    [2]

    White S R 1992 Phys. Rev. Lett. 69 2863Google Scholar

    [3]

    Landau D P, Binder K 2011 A Guide to Monte-Carlo Simulatios in Statistical Physics (Cambridge: Cambridge University)pp70-73

    [4]

    Evenbly G, Vidal G 2009 Phys. Rev. B 79 144108.

    [5]

    Jordan J, Orus R, Vidal G, Verstraete F, Cirac J I 2008 Phys. Rev. Lett. 101 250602Google Scholar

    [6]

    Singh S, Zhou H Q, Vidal G 2010 New J. Phys. 12 033029Google Scholar

    [7]

    Jiang H C, Weng Z Y, Xiang T 2008 Phys. Rev. Lett. 101 090603Google Scholar

    [8]

    Czarnik P, Dziarmaga J 2015 Phys. Rev. B 92 035120.

    [9]

    Li W, Ran S J, Gong S S, Zhao Y, Xi B, Ye F, Su G 2011 Phys. Rev. Lett. 106 127202Google Scholar

    [10]

    Chen B B, Chen L, Chen Z Y, Li W, Weichselbaum A 2018 Phys. Rev. X 8 031082

    [11]

    Corboz P, Czarnik P, Kapteijns G, Tagliacozzo L 2018 Phys. Rev. X 8 031031

    [12]

    Singh S, Pfeifer R N C, Vidal G 2010 Phys. Rev. A 82 050301Google Scholar

    [13]

    Haghshenas R, Sheng D N 2017 arXiv: 1711.07584v1 [cond-mat.str-el]

    [14]

    Singh S, Pfeifer R N C, Vidal G 2011 Phys. Rev. B 83 115125Google Scholar

    [15]

    Song H F, Rachel S, Hur K Le 2010 Phys. Rev. B 82 012405Google Scholar

    [16]

    Song H F, Rachel S, Flindt C,Klich I, Laflorencie N, Hur K Le 2012 Phys. Rev. B 85 035409Google Scholar

    [17]

    Yang M F 2007 Phys. Rev. B 76 180403(R).

    [18]

    Boschi C D E, Ercolessi E, Ortolani F, Roncaglia M 2003 Eur. Phys. J. B 35 465Google Scholar

    [19]

    Pollmann F, Mukerjee S, Turner A, Moore J E 2009 Phys. Rev. Lett. 102 255701Google Scholar

    [20]

    苏耀恒, 陈爱民, 王洪雷, 相春环 2017 物理学报 66 120301Google Scholar

    Su Y H, Chen A M, Wang H L, Xiang C H 2017 Acta Phys. Sin. 66 120301Google Scholar

    [21]

    Yang C N, Yang C P 1966 Phys. Rev. 150 321Google Scholar

    [22]

    Chen W, Hida K, Sanctuary B C 2003 Phys. Rev. B 67 104401Google Scholar

    [23]

    Kjall J A, Zaletel M P, Mong R S K, Bardarson J H, Pollmann F 2013 Phys. Rev. B 87 235106Google Scholar

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出版历程
  • 收稿日期:  2019-03-17
  • 修回日期:  2019-05-31
  • 上网日期:  2019-08-01
  • 刊出日期:  2019-08-20

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