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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 theU(1) symmetric infinite matrix product states algorithm (iMPS). By using relation between the bipartite quantum fluctuations F and the so-called finite-entanglement scaling exponentsκ , the Luttinger liquid parameter K can be extracted. Fors=1/2 andD=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 withD \leqslant - 2 can be mapped to an effective spin-1/2 XXZ model, we calculate the Luttinger liquid parameter for the region ofD \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 than1\% . Also, our Luttinger liquid parameter at\Delta = - 0.5 andD = 0 is shown to be consistent with the result form the density matrix renormalization group (DMRG) method. These results suggest that theU(1) symmetric iMPS method can be applicable to calculate Luttinger liquid parameters if any system has aU(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 withD = 1.5 .[1] Sachdev S 1999 Quantum Phase Transitions (Cambridge: Cambridge University)pp3—5
[2] White S R 1992 Phys. Rev. Lett. 69 2863
Google 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 250602
Google Scholar
[6] Singh S, Zhou H Q, Vidal G 2010 New J. Phys. 12 033029
Google Scholar
[7] Jiang H C, Weng Z Y, Xiang T 2008 Phys. Rev. Lett. 101 090603
Google 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 127202
Google 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 050301
Google 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 115125
Google Scholar
[15] Song H F, Rachel S, Hur K Le 2010 Phys. Rev. B 82 012405
Google Scholar
[16] Song H F, Rachel S, Flindt C,Klich I, Laflorencie N, Hur K Le 2012 Phys. Rev. B 85 035409
Google 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 35 465
Google Scholar
[19] Pollmann F, Mukerjee S, Turner A, Moore J E 2009 Phys. Rev. Lett. 102 255701
Google Scholar
[20] 苏耀恒, 陈爱民, 王洪雷, 相春环 2017 66 120301
Google Scholar
Su Y H, Chen A M, Wang H L, Xiang C H 2017 Acta Phys. Sin. 66 120301
Google Scholar
[21] Yang C N, Yang C P 1966 Phys. Rev. 150 321
Google Scholar
[22] Chen W, Hida K, Sanctuary B C 2003 Phys. Rev. B 67 104401
Google Scholar
[23] Kjall J A, Zaletel M P, Mong R S K, Bardarson J H, Pollmann F 2013 Phys. Rev. B 87 235106
Google Scholar
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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Figure 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和\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} 和\tilde{ {\lambda}}^{\rm B} 及粒子数\tilde{n_r} Figure 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和\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 Figure 3. Correlation length
\xi and fluctuation F of spinS = 1/2 XXZD model as a function of the truncation dimension\chi for various parameters\varDelta . Here, fixed parameterD = 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 Figure 4. Correlation length
\xi and fluctuation F of spinS = 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 Figure 5. Correlation length
\xi and fluctuation F of spinS = 2 XXZD model as a function of the truncation dimension\chi for various parameters\varDelta . Here, fixed parameterD = 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 
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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
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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
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[1] Sachdev S 1999 Quantum Phase Transitions (Cambridge: Cambridge University)pp3—5
[2] White S R 1992 Phys. Rev. Lett. 69 2863
Google 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 250602
Google Scholar
[6] Singh S, Zhou H Q, Vidal G 2010 New J. Phys. 12 033029
Google Scholar
[7] Jiang H C, Weng Z Y, Xiang T 2008 Phys. Rev. Lett. 101 090603
Google 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 127202
Google 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 050301
Google 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 115125
Google Scholar
[15] Song H F, Rachel S, Hur K Le 2010 Phys. Rev. B 82 012405
Google Scholar
[16] Song H F, Rachel S, Flindt C,Klich I, Laflorencie N, Hur K Le 2012 Phys. Rev. B 85 035409
Google 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 35 465
Google Scholar
[19] Pollmann F, Mukerjee S, Turner A, Moore J E 2009 Phys. Rev. Lett. 102 255701
Google Scholar
[20] 苏耀恒, 陈爱民, 王洪雷, 相春环 2017 66 120301
Google Scholar
Su Y H, Chen A M, Wang H L, Xiang C H 2017 Acta Phys. Sin. 66 120301
Google Scholar
[21] Yang C N, Yang C P 1966 Phys. Rev. 150 321
Google Scholar
[22] Chen W, Hida K, Sanctuary B C 2003 Phys. Rev. B 67 104401
Google Scholar
[23] Kjall J A, Zaletel M P, Mong R S K, Bardarson J H, Pollmann F 2013 Phys. Rev. B 87 235106
Google Scholar
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