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量子自旋系统的动力学性质是统计物理和凝聚态理论研究的热点问题. 本文利用递推关系方法, 通过计算系统的自旋关联函数及谱密度, 研究了纵场对一维量子Ising模型动力学性质的影响. 对于常数纵场的情况, 发现当自旋耦合相互作用较弱时纵场能够引起不同动力学行为之间的交跨效应, 且驱使系统出现了多种振动模式, 但较强的自旋耦合相互作用会掩盖纵场的影响. 对于随机纵场的情况, 分别讨论了双模型随机纵场和高斯型随机纵场的影响, 发现不同随机类型下的动力学结果有很大的差别, 且高度依赖于随机分布中参数的选取, 如双模分布的均值, 高斯分布的均值和偏差等. 尽管常数纵场和随机纵场下的动力学结果不同, 但可以得到一个共同的结论: 当纵场所占比重较大时, 系统的中心峰值行为将得到保持. 且此结论可以推广: 系统哈密顿中非对易项的出现有利于中心峰值行为的保持.
The dynamical properties of quantum spin systems are a hot topic of research in statistical and condensed matter physics. In this paper, the dynamics of one-dimensional quantum Ising model with both transverse and longitudinal magnetic field (LMF) is investigated by the recursion method. The time-dependent spin autocorrelation function $C\left( t \right) = \overline {\left\langle {\sigma _j^x\left( t \right)\sigma _j^x\left( 0 \right)} \right\rangle } $ and corresponding spectral density$\varPhi \left( \omega \right)$ are calculated. The Hamiltonian of the model system can be written as$H = - \dfrac{1}{2}J\displaystyle\sum\limits_i^N {\sigma _i^x\sigma _{i + 1}^x - \dfrac{1}{2}\displaystyle\sum\limits_i^N {B_i^x\sigma _i^x} } - \dfrac{1}{2}\displaystyle\sum\limits_i^N {B_i^z\sigma _i^z}$ .This work focuses mainly on the effects of LMF ( $ B_i^x $ ) on spin dynamics of the Ising system, and both uniform LMF and random LMF are considered respectively. Without loss of generality, the transverse magnetic field$ B_i^z = 1 $ is set in the numerical calculation, which fixes the energy scale.The results show that the uniform LMF can induce crossovers between different dynamical behaviors (e.g. independent spins precessing, collective-mode behavior or central-peak behavior) and drive multiple vibrational modes (multiple-peaked behavior) when spin interaction ( $ J $ ) is weak. However, the effect of uniform LMF is not obvious when spin interaction is strong. For the case of random LMF, the effects of bimodal-type and Gaussian-type random LMF are investigated, respectively. The dynamical results under the two types of random LMFs are quite different and highly dependent on many factors, such as the mean values ($ {B_1} $ ,$ {B_2} $ and$ {B_x} $ ) or the standard deviation ($ \sigma $ ) of random distributions. The nonsymmetric bimodal-type random LMF ($ {B_1} \ne {B_2} $ ) may induce new vibrational modes easily. The dynamical behaviors under the Gaussian-type random LMF are more abundant than under the bimodal-type random LMF. When$ \sigma $ is small, the system undergoes two crossovers: from a collective-mode behavior to a double-peaked behavior, and then to a central-peak behavior as the mean value$ {B_x} $ increases. However, when$ \sigma $ is large, the system presents only a central-peak behavior.For both cases of uniform LMF and random LMF, it is found that the central-peak behavior of the system is maintained when the proportion of LMF is large. This conclusion can be generalized that the emergence of noncommutative terms (noncommutative with the transverse-field term $\displaystyle\sum\nolimits_i^N {B_i^z\sigma _i^z}$ ) in Hamiltonian will enhance the central peak behavior. Therefore, noncommutative terms, such as next-nearest-neighbor spin interactions, Dzyaloshinskii-Moryia interactions, impurities, four-spin interactions, etc., can be added to the system Hamiltonian to modulate the dynamical properties. This provides a new direction for the future study of spin dynamics.-
Keywords:
- Ising model /
- longitudinal magnetic field /
- spin autocorrelation function /
- spectral density
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图 1 横场取值
$B_i^z = 1$ , 纵场取值$ B_i^x = 0 $ , 0.5, 1.0, 1.5和2.0, (a)−(d)分别对应自旋耦合相互作用参数J = 0.1, 0.5, 1.0和1.5时的自旋关联函数Fig. 1. Take the transverse magnetic field
$B_i^z = 1$ and the longitudinal magnetic field$ B_i^x = 0 $ , 0.5, 1.0, 1.5 and 2.0, respectively. Spin autocorrelation functions$C\left( t \right)$ for different values of spin interactions (e.g., J = 0.1, 0.5, 1.0 and 1.5) are given in (a)−(d), respectively.
图 2 横场取值
$B_i^z = 1$ , 纵场取值$ B_i^x = 0 $ , 0.5, 1.0, 1.5和2.0, (a)−(d)分别对应自旋耦合相互作用参数J = 0.1, 0.5, 1.0和1.5时的谱密度.Fig. 2. Take the transverse magnetic field
$B_i^z = 1$ and the longitudinal magnetic field$ B_i^x = 0 $ , 0.5, 1.0, 1.5 and 2.0, respectively. The corresponding spectral density$\varPhi \left( \omega \right)$ for different values of spin interactions (e.g., J = 0.1, 0.5, 1.0 and 1.5) are given in (a)−(d), respectively.
图 3 随机纵场满足双模分布时的自旋关联函数和谱密度 (a), (b)对应
$ {B_1} = 1.3 $ 和$ {B_2} = 0.7 $ 时的结果; (c), (d)为$ {B_1} = 1.8 $ 和$ {B_2} = 0.2 $ 时的结果Fig. 3. Spin autocorrelation functions and the corresponding spectral densities for bimodal-type random longitudinal magnetic field. The results for
$ {B_1} = 1.3 $ and$ {B_2} = 0.7 $ are given in (a) and (b), and the results for$ {B_1} = 1.8 $ and$ {B_2} = 0.2 $ are given in (c) and (d), respectively.
图 4 随机纵场满足高斯分布时的谱密度 (a)−(d)分别对应
$ \sigma = 0.3, {\text{ }}0.8, {\text{ }}1.0, {\text{ }}1.8 $ 时的结果.Fig. 4. Spectral densities for Gaussian-type random longitudinal magnetic field. The results for
$ \sigma = 0.3, {\text{ }}0.8, {\text{ }}1.0 $ and 1.8 are given in (a)−(d), respectively. -
[1] Plascak J A, Pires A S T, Sá Barreto F C 1982 Solid State Commun. 44 787
Google Scholar
[2] Plascak J A, Sá Barreto F C, Pires A S T, Goncalves L L 1983 J. Phys. C: Solid State Phys. 16 49
Google Scholar
[3] Watarai S, Matsubara T 1984 J Phys. Soc. Jpn. 53 3648
Google Scholar
[4] Levitsky R R, Zachek I R, Mits E V, Grigas J, Paprotny W 1986 Ferroelectrics 67 109
Google Scholar
[5] Wu W, Ellman B, Rosenbaum T F, Aeppli G, Reich D H 1991 Phys. Rev. Lett. 67 2076
[6] Chernodub M N, Lundgren M, Niemi A J 2011 Phys. Rev. E 83 011126
Google Scholar
[7] Storm C, Nelson P C 2003 Phys. Rev. E 67 051906
[8] Faure Q, Takayoshi S, Petit S, Simonet V, Raymond S, Regnault L P, Boehm M, White J S, Månsson M, Rüegg C, Lejay P, Canals B, Lorenz T, Furuya S C, Giamarchi T, Grenier B 2018 Nature Phys. 14 716
Google Scholar
[9] Jia X, Chakravarty S 2006 Phys. Rev. B 74 172414
Google Scholar
[10] Rønnow H M, Parthasarathy R, Jensen J, Aeppli G, Rosenbaum T F, McMorrow D F 2005 Science 308 389
[11] Fogedby H C 1978 J. Phys. C: Solid State Phys. 11 2801
Google Scholar
[12] Sen S, Mahanti S D, Cai Z X 1991 Phys. Rev. B 43 10990
Google Scholar
[13] Sen P 1997 Phys. Rev. B 55 11367
Google Scholar
[14] Osenda O, Huang Z, Kais S 2003 Phys. Rev. A 67 062321
Google Scholar
[15] Florencio J, Sá Barreto F C 1999 Phys. Rev. B 60 9555
Google Scholar
[16] Chen S X, Shen Y Y, Kong X M 2010 Phys. Rev. B 82 174404
[17] Da Conceição C M S, Maia R N P 2017 Phys. Rev. E 96 032121
[18] von Ohr S, Manssen M, Hartmann A K 2017 Phys. Rev. E 96 013315
Google Scholar
[19] Hadjiagapiou I A 2011 Physica A 390 2229
Google Scholar
[20] Liu Z Q, Kong X M, Chen X S 2006 Phys. Rev. B 73 224412
Google Scholar
[21] Theodorakis P E, Georgiou I, Fytas N G 2013 Phys. Rev. E 87 032119
[22] Crokidakis N, Nobre F D 2008 J. Phys.: Condens. Matter 20 145211
[23] Liu Z Q, Jiang S R, Kong X M 2014 Chin. Phys. B 23 087505
Google Scholar
[24] Kenzelmann M, Coldea R, Tennant D A, Visser D, Hofmann M, Smeibidl P, Tylczynski Z 2002 Phys. Rev. B 65 144432
Google Scholar
[25] Simon J, Bakr W S, Ma R, Tai M E, Preiss P M, Greiner M 2011 Nature 472 307
Google Scholar
[26] Senthil T 1998 Phys. Rev. B 57 8375
Google Scholar
[27] Dmitriev D V, Krivnov V Y 2004 Phys. Rev. B 70 144414
Google Scholar
[28] Neto M A, De Sousa J R 2013 Physica A 392 1
Google Scholar
[29] Corrêa Silva E V, Skea J E F, Rojas O, De Souza S M, Thomaz M T 2008 Physica A 387 5117
Google Scholar
[30] Do Nascimento D A, Neto M A, De Sousa J R, Pacobahyba J T 2012 J. Magn. Magn. Mater. 324 2429
[31] Do Nascimento D A, Pacobahyba J T, Neto M A, Salmon O D R, Plascak J A 2017 Physica A 474 224
[32] Zhao Z Y, Liu X G, He Z Z, Wang X M, Fan C, Ke W P, Li Q J, Chen L M, Zhao X, Sun X F 2012 Phys. Rev. B 85 134412
Google Scholar
[33] Kopeć T K, Usadel K D, Büttner G 1989 Phys. Rev. B 39 12418
[34] Ovchinnikov A A, Dmitriev D V, Krivnov V Y, Cheranovskii V O 2003 Phys. Rev. B 68 214406
Google Scholar
[35] Liu Z Q, Jiang S R, Kong X M, Xu Y L 2017 Physica A 473 536
[36] Viswanath V S, Müller G 1994 The Recursion Method—Application to Many-body Dynamics (Berlin: Springe-Verlag)
[37] Mezei F, Murani A P 1979 J. Magn. Magn. Mater. 14 211
Google Scholar
[38] Lee M H 1982 Phys. Rev. Lett. 49 1072
Google Scholar
[39] Lee M H 1982 Phys. Rev. B 26 2547
Google Scholar
[40] Lee M H 2000 Phys. Rev. E 62 1769
Google Scholar
[41] Florencio J, De Alcantara Bonfim O F 2020 Front. Phys. 8 557277
Google Scholar
[42] Sur A, Jasnow D, Lowe I J 1975 Phys. Rev. B 12 3845
Google Scholar
[43] Yuan X J, Kong X M, Xu Z B, Liu Z Q 2010 Physica A 389 242
Google Scholar
[44] 袁晓娟, 赵邦宇, 陈淑霞, 孔祥木 2010 59 1499
Google Scholar
Yuan X J, Zhao B Y, Chen S X, Kong X M 2010 Acta Phys. Sin. 59 1499
Google Scholar
[45] Nunes M E S, De Mello Silva É, Martins P H L, Plascak J A, Florencio J 2018 Phys. Rev. E 98 042124
[46] Li Y F, Kong X M 2013 Chin. Phys. B 22 037502
Google Scholar
[47] 李银芳, 申银阳, 孔祥木 2012 61 107501
Google Scholar
Li Y F, She Y Y, Kong X M 2012 Acta Phys. Sin. 61 107501
Google Scholar
[48] Huang X, Yang Z 2015 Solid State Commun. 204 28
Google Scholar
[49] De Souza W L, De Mello Silva É, Martins P H L 2020 Phys. Rev. E 101 042104
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