-
It is of considerable theoretical significance to study the effects of impurity on spin dynamics of quantum spin systems. In this paper, the dynamical properties of the one-dimensional quantum Ising model with symmetric and asymmetric link-impurity are investigated by the recursion method, respectively. The autocorrelation function $C\left( t \right) = \overline {\left\langle {\sigma _j^x\left( t \right)\sigma _j^x\left( 0 \right)} \right\rangle } $ and the associated spectral density $\varPhi \left( \omega \right) = \displaystyle\int_{ - \infty }^{ + \infty } {{\rm d}t{{\rm e}^{{\rm i}\omega t}}C\left( t \right)} $ are calculated. The Hamiltonian of the Ising model with link-impurity can be written as $\qquad\qquad\qquad\qquad\qquad H = - \displaystyle\frac{1}{2}({J_{j - 1}}\sigma _{j - 1}^x\sigma _j^x + {J_j}\sigma _j^x\sigma _{j + 1}^x) - \displaystyle\frac{1}{2}J\sum\limits_{i \ne j,j - 1}^N {\sigma _i^x\sigma _{i + 1}^x} - \frac{1}{2}B\sum\limits_i^N {\sigma _i^z} . $where $J$ is the nearest-neighbor exchange coupling of the main spin chain, $ B $ denotes the external transverse magnetic field, $\sigma _i^\alpha \left( {\alpha = x,y,z} \right)$ are Pauli matrices at site $ i $. The constant 1/2 is introduced for the convenience of theoretical deduction, and N is the number of spins. The so-called link-impurity $ {J_j} $ ($ {J_{j - 1}} $) is randomly introduced, which denotes the exchange coupling between the j th spin and the (j + 1)th spin (the (j – 1)th spin). The symmetric link-impurity and asymmetric link-impurity correspond to the case of $ {J_{j - 1}} = {J_j} $ and $ {J_{j - 1}} \ne {J_j} $, respectively. The periodic boundary conditions are assumed in the theoretical calculation. After introducing the link-impurity, the original competition between $ B $ and $J$ in the pure Ising model is broken. The dynamic behavior of the system depends on synergistic effect of multiple factors, such as the mean spin coupling $ \bar J $ between $J$ and the link-impurity, the asymmetry degree between $ {J_{j - 1}} $ and $ {J_j} $, and the strength of the external magnetic field. In calculation, the exchange couplings of the main spin chain are set to $J \equiv 1$ to fix the energy scale. We first consider the effects of symmetric link-impurity. The reference values can be set to $ {J_{j - 1}} = {J_j} \lt J $ (e.g. 0.4, 0.6 or 0.8) or $ {J_{j - 1}} = {J_j} \gt J $ (e.g. 1.2, 1.6, 2.0), which are called weak or strong impurity coupling. When the magnetic field $ B \geqslant J $ (e.g., $ B = 1 $, 1.5 or 2.0), it is found that the dynamic behavior of the system exhibits a crossover from a collective-mode behavior to a central-peak behavior as the impurity strength $ {J_{j - 1}} = {J_j} $ increases. Interestingly, for $ B \lt J $ (e.g. $ B = 0.4 $ or 0.7), there are two crossovers that are a collective-mode-like behavior to a double-peak behavior, then to a central-peak behavior as $ {J_{j - 1}} = {J_j} $ increases. For the case of asymmetric link-impurity, the impurity configuration is more complex. Using the cooperation between $ {J_{j - 1}} $ and $ {J_j} $, more freedoms of regulation can be provided and the dynamical properties are more abundant. For the case of $ B \leqslant J $ (e.g. $ B = 0.5 $, 1.0), the system tends to exhibit a collective-mode behavior when the mean spin coupling $ \bar J $ is weak, and a central-peak behavior when $ \bar J $ are strong. However, when the asymmetry between $ {J_{j - 1}} $ and $ {J_j} $ is obvious, the system tends to exhibit a double- or multi-peak behavior. For the case of $ B \gt J $ (e.g. $ B = 1.5 $, 2.0), when $ \bar J $ is weak or the asymmetry between $ {J_{j - 1}} $ and $ {J_j} $ is not obvious, the system tends to exhibit a collective-mode behavior. When $ \bar J $ is strong, it tends to show a central-peak behavior. However, when the asymmetry between $ {J_{j - 1}} $ and $ {J_j} $ is evident, the bispectral feature (two spectral peaks appear at $ {\omega _1} \ne 0 $ and $ {\omega _2} \ne 0 $) dominates the dynamics. Under the regulating effect of link-impurities, the crossover between different dynamic behaviors can be easily realized, and it is easier to stimulate new dynamic modes, such as the double-peak behavior, the collective-mode-like behavior or bispectral feature one. The results in this work indicate that using link-impurity to manipulate the dynamics of quantum spin systems may be a new try. -
Keywords:
- Ising model /
- link-impurity /
- spin correlation function /
- spectral density
[1] Young A P 1997 Phys. Rev. B 56 11691
Google Scholar
[2] Florencio J, Sá Barreto F C 1999 Phys. Rev. B 60 9555
Google Scholar
[3] Liu Z Q, Kong X M, Chen X S 2006 Phys. Rev. B 73 224412
Google Scholar
[4] Yuan X J, Kong X M, Xu Z B, Liu Z Q 2010 Physica A 389 242
Google Scholar
[5] Chen S X, Shen Y Y, Kong X M 2010 Phys. Rev. B 82 174404
Google Scholar
[6] Nunes M E S, Florencio J 2003 Phys. Rev. B 68 014406
Google Scholar
[7] Nunes M E S, Plascak J A, Florencio J 2004 Physica A 332 1
Google Scholar
[8] Xu Z B, Kong X M, Liu Z Q 2008 Phys. Rev. B 77 184414
Google Scholar
[9] Li Y F, Kong X M 2013 Chin. Phys. B 22 037502
Google Scholar
[10] Laflorencie N, Rieger H, Sandvik A W, Henelius P 2004 Phys. Rev. B 70 054430
Google Scholar
[11] 李银芳, 申银阳, 孔祥木 2012 物理学报 61 107501
Google Scholar
Li Y F, Shen Y Y, Kong X M 2012 Acta Phys. Sin. 61 107501
Google Scholar
[12] Silva da Conceição C M S, Maia R N P 2017 Phys. Rev. E 96 032121
Google Scholar
[13] von Ohr S, Manssen M, Hartmann A K 2017 Phys. Rev. E 96 013315
Google Scholar
[14] Theodorakis P E, Georgiou I, Fytas N G 2013 Phys. Rev. E 87 032119
Google Scholar
[15] Crokidakis N, Nobre F D 2008 J. Phys. : Condens. Matter 20 145211
Google Scholar
[16] Liu Z Q, Jiang S R, Kong X M 2014 Chin. Phys. B 23 087505
Google Scholar
[17] Balcerzak T, Szałowski K, Jaščur M 2020 J. Magn. Magn. Mater. 507 166825
Google Scholar
[18] Silva R L, Guimarães P R C, Pereira A R 2005 Solid State Commun. 134 313
Google Scholar
[19] Sousa J M, Leite R V, Landim R R, Costa Filho R N 2014 Physica B 438 78
Google Scholar
[20] Huang X, Yang Z 2015 Solid State Commun. 204 28
Google Scholar
[21] Çağlar T, Nihat Berker A 2015 Phys. Rev. E 92 062131
Google Scholar
[22] Mazzitello K I, Candia J, Albano E V, 2015 Phys. Rev. E 91 042118
Google Scholar
[23] Hadjiagapiou I A, Velonakis I N 2018 Physica A 505 965
Google Scholar
[24] Hadjiagapiou I A, Velonakis I N 2021 Physica A 578 126112
Google Scholar
[25] 袁晓娟 2023 物理学报 72 087501
Google Scholar
Yuan X J 2023 Acta Phys. Sin. 72 087501
Google Scholar
[26] Boechat B, Cordeiro C, Florencio J, Sá Barreto F C, de Alcantara Bonfim O F 2000 Phys. Rev. B 61 14327
Google Scholar
[27] De Souza W L, de Mello Silva É, Martins P H L 2020 Phys. Rev. E 101 042104
Google Scholar
[28] Nunes M E S, de Mello Silva É, Martins P H L, Plascak J A, Florencio J 2018 Phys. Rev. E 98 042124
Google Scholar
[29] Guimarães P R C, Plascak J A, De Alcantara Bonfim O F, Florencio J 2015 Phys. Rev. E 92 042115
Google Scholar
[30] Hu F M, Ma T, Lin H Q, Gubernatis J E 2011 Phys. Rev. B 84 075414
Google Scholar
[31] Liu Q, Liu C X, Xu C, Qi X L, Zhang S C 2009 Phys. Rev. Lett. 102 156603
Google Scholar
[32] Cirillo A, Mancini M, Giuliano D, Sodano P 2011 Nuclear Phys. B 852 235
Google Scholar
[33] Sindona A, Goold J, Lo Gullo N, Lorenzo S, Plastina F 2013 Phys. Rev. Lett. 111 165303
Google Scholar
[34] Li J, Wang Y P 2009 Europhys. Lett. 88 17009
Google Scholar
[35] Apollaro T J G, Francica G, Giuliano D, Falcone G, Palma G M, Plastina F 2017 Phys. Rev. B 96 155145
Google Scholar
[36] Giuliano D, Campagnano G, Tagliacozzo A 2016 Eur. Phys. J. B 89 251
Google Scholar
[37] Rommer S, Eggert S 2000 Phys. Rev. B 62 4370
Google Scholar
[38] Yuan X J, Zhao J F, Wang H, Bu H X, Yuan H M, Zhao B Y, Kong X M 2021 Physica A 583 126279
Google Scholar
[39] Eggert S, Affleck I 1992 Phys. Rev. B 46 10866
Google Scholar
[40] Schuster C, Eckern U 2002 Ann. Phys. 514 901
Google Scholar
[41] Huang X, Yang Z 2015 J. Magn. Magn. Mater. 381 372
Google Scholar
[42] Viswanath V S, Müller G 1994 The Recursion Method—Application to Many-body Dynamics (Berlin: Springe-Verlag
[43] Lee M H 1982 Phys. Rev. Lett. 49 1072
Google Scholar
[44] Lee M H 1982 Phys. Rev. B 26 2547
Google Scholar
[45] Lee M H 2000 Phys. Rev. E 62 1769
Google Scholar
[46] Yuan X J, Wang C Y, Kong X M, Zhao J F, Wang H, Bu H X 2023 J. Magn. Magn. Mater. 572 170632
Google Scholar
[47] Nunes M E S, Plascak J A 2024 Phys. Rev. E 109 014134
Google Scholar
[48] Florencio J, de Alcantara Bonfim O F 2020 Front. Phys. 8 557277
Google Scholar
[49] Florencio J, Lee M H 1987 Phys. Rev. B 35 1835
Google Scholar
-
图 1 对称型链接杂质在不同杂质耦合强度下的连分式系数$ {\varDelta _\nu } $, 其中横向磁场$ B = J \equiv 1 $, 杂质耦合强度取值为$ {J_{j - 1}} = {J_j} = 0.4 $, 0.6, 0.8, 1.0, 1.2, 1.4和1.6
Figure 1. Recurrants $ {\varDelta _\nu } $ for the symmetric type of link-impurity. The transverse magnetic field $ B = J \equiv 1 $, and the impurity coupling strength $ {J_{j - 1}} = {J_j} = 0.4 $, 0.6, 0.8, 1.0, 1.2, 1.4 and 1.6.
图 2 对称型链接杂质在不同杂质耦合强度下的自旋关联函数$C\left( t \right)$(a)及谱密度$\varPhi \left( \omega \right)$(b), 其中横向磁场$ B = J \equiv 1 $, 杂质耦合强度$ {J_{j - 1}} = {J_j} = 0.4 $, 0.6, 0.8, 1.0和1.2
Figure 2. Spin autocorrelation function $C\left( t \right)$ (a) and spectral density $\varPhi \left( \omega \right)$ (b) for the symmetric type of link-impurity under different impurity coupling strengths. The transverse magnetic field $ B = J \equiv 1 $, and the impurity coupling strength $ {J_{j - 1}} = {J_j} = 0.4 $, 0.6, 0.8, 1.0 and 1.2.
图 3 对称型链接杂质在不同杂质耦合强度下的谱密度$\varPhi \left( \omega \right)$, 图(a)—(d)中横向磁场的取值分别为$ B = 0.4 $, 0.7, 1.5和2.0, 主体格点自旋耦合$ J \equiv 1 $
Figure 3. Spectral densities $\varPhi \left( \omega \right)$ for symmetric type of link-impurity under different impurity coupling strength. Without loss of generality, the parameter $ J \equiv 1 $, and the transverse magnetic field $ B = 0.4 $, 0.7, 1.5 and 2.0 in panel (a)–(d).
图 4 固定横场$ B = J \equiv 1 $, 非对称型链接杂质在不同杂质耦合强度下的谱密度$\varPhi \left( \omega \right)$, 其中固定$ {J_{j - 1}} = J' $, 图(a)—(d)中$ {J_{j - 1}} $分别取值为0.2, 0.5, 1.0和1.4; $ {J_j} = J'' $的取值从0.2变化到1.8
Figure 4. Spectral densities for non-symmetric type of link-impurity under different impurity coupling strength. The transverse magnetic field $ B = J \equiv 1 $, and the impurity coupling strength $ J_{j-1}=J'=0.2 $, 0.5, 1.0 and 1.4 are set in panels (a)–(d), respectively. The other impurity coupling strength $ {J_j} ({J_j} = J'')$ changes from 0.2 to 1.8.
图 5 固定横场$ B = 0.5 = J/2 $, 非对称型链接杂质在不同杂质耦合强度下的谱密度, 其中固定$ {J_{j - 1}} = J' $, 图(a)—(f)中$ {J_{j - 1}} $分别取值为0.2, 0.5, 0.8, 1.2, 1.6和2.0, $ {J_j} = J'' $的取值从0.2变化到1.8
Figure 5. Spectral densities for non-symmetric type of link-impurity under different impurity coupling strength. The transverse magnetic field $ B = 0.5 = J/2 $, and the impurity coupling strength $ {J_{j - 1}} =J'= 0.2 $, 0.5, 0.8, 1.2, 1.6 and 2.0 are set in panels (a)–(f), respectively. The other impurity coupling strength $ {J_j} ({J_j} = J'')$ changes from 0.2 to 1.8.
图 6 固定横场$ B = 1.5 = 1.5 J $, 给出非对称型链接杂质在不同杂质耦合强度下的谱密度, 固定$ {J_{j - 1}} = J' $, 图(a)—(d)中的$ {J_{j - 1}} $分别取值为0.4, 0.8, 1.2和1.6; $ {J_j} = J'' $的取值从0.2变化到1.8
Figure 6. Spectral densities for non-symmetric type of link-impurity under different impurity coupling strength. The transverse magnetic field $ B = 1.5 = 1.5 J $, and the impurity coupling strength $ J_{j-1}=J'=0.4 $, 0.8, 1.2和1.6 are set in panels (a)–(d), respectively. The other impurity coupling strength $ {J_j} ({J_j} = J'')$ changes from 0.2 to 1.8.
-
[1] Young A P 1997 Phys. Rev. B 56 11691
Google Scholar
[2] Florencio J, Sá Barreto F C 1999 Phys. Rev. B 60 9555
Google Scholar
[3] Liu Z Q, Kong X M, Chen X S 2006 Phys. Rev. B 73 224412
Google Scholar
[4] Yuan X J, Kong X M, Xu Z B, Liu Z Q 2010 Physica A 389 242
Google Scholar
[5] Chen S X, Shen Y Y, Kong X M 2010 Phys. Rev. B 82 174404
Google Scholar
[6] Nunes M E S, Florencio J 2003 Phys. Rev. B 68 014406
Google Scholar
[7] Nunes M E S, Plascak J A, Florencio J 2004 Physica A 332 1
Google Scholar
[8] Xu Z B, Kong X M, Liu Z Q 2008 Phys. Rev. B 77 184414
Google Scholar
[9] Li Y F, Kong X M 2013 Chin. Phys. B 22 037502
Google Scholar
[10] Laflorencie N, Rieger H, Sandvik A W, Henelius P 2004 Phys. Rev. B 70 054430
Google Scholar
[11] 李银芳, 申银阳, 孔祥木 2012 物理学报 61 107501
Google Scholar
Li Y F, Shen Y Y, Kong X M 2012 Acta Phys. Sin. 61 107501
Google Scholar
[12] Silva da Conceição C M S, Maia R N P 2017 Phys. Rev. E 96 032121
Google Scholar
[13] von Ohr S, Manssen M, Hartmann A K 2017 Phys. Rev. E 96 013315
Google Scholar
[14] Theodorakis P E, Georgiou I, Fytas N G 2013 Phys. Rev. E 87 032119
Google Scholar
[15] Crokidakis N, Nobre F D 2008 J. Phys. : Condens. Matter 20 145211
Google Scholar
[16] Liu Z Q, Jiang S R, Kong X M 2014 Chin. Phys. B 23 087505
Google Scholar
[17] Balcerzak T, Szałowski K, Jaščur M 2020 J. Magn. Magn. Mater. 507 166825
Google Scholar
[18] Silva R L, Guimarães P R C, Pereira A R 2005 Solid State Commun. 134 313
Google Scholar
[19] Sousa J M, Leite R V, Landim R R, Costa Filho R N 2014 Physica B 438 78
Google Scholar
[20] Huang X, Yang Z 2015 Solid State Commun. 204 28
Google Scholar
[21] Çağlar T, Nihat Berker A 2015 Phys. Rev. E 92 062131
Google Scholar
[22] Mazzitello K I, Candia J, Albano E V, 2015 Phys. Rev. E 91 042118
Google Scholar
[23] Hadjiagapiou I A, Velonakis I N 2018 Physica A 505 965
Google Scholar
[24] Hadjiagapiou I A, Velonakis I N 2021 Physica A 578 126112
Google Scholar
[25] 袁晓娟 2023 物理学报 72 087501
Google Scholar
Yuan X J 2023 Acta Phys. Sin. 72 087501
Google Scholar
[26] Boechat B, Cordeiro C, Florencio J, Sá Barreto F C, de Alcantara Bonfim O F 2000 Phys. Rev. B 61 14327
Google Scholar
[27] De Souza W L, de Mello Silva É, Martins P H L 2020 Phys. Rev. E 101 042104
Google Scholar
[28] Nunes M E S, de Mello Silva É, Martins P H L, Plascak J A, Florencio J 2018 Phys. Rev. E 98 042124
Google Scholar
[29] Guimarães P R C, Plascak J A, De Alcantara Bonfim O F, Florencio J 2015 Phys. Rev. E 92 042115
Google Scholar
[30] Hu F M, Ma T, Lin H Q, Gubernatis J E 2011 Phys. Rev. B 84 075414
Google Scholar
[31] Liu Q, Liu C X, Xu C, Qi X L, Zhang S C 2009 Phys. Rev. Lett. 102 156603
Google Scholar
[32] Cirillo A, Mancini M, Giuliano D, Sodano P 2011 Nuclear Phys. B 852 235
Google Scholar
[33] Sindona A, Goold J, Lo Gullo N, Lorenzo S, Plastina F 2013 Phys. Rev. Lett. 111 165303
Google Scholar
[34] Li J, Wang Y P 2009 Europhys. Lett. 88 17009
Google Scholar
[35] Apollaro T J G, Francica G, Giuliano D, Falcone G, Palma G M, Plastina F 2017 Phys. Rev. B 96 155145
Google Scholar
[36] Giuliano D, Campagnano G, Tagliacozzo A 2016 Eur. Phys. J. B 89 251
Google Scholar
[37] Rommer S, Eggert S 2000 Phys. Rev. B 62 4370
Google Scholar
[38] Yuan X J, Zhao J F, Wang H, Bu H X, Yuan H M, Zhao B Y, Kong X M 2021 Physica A 583 126279
Google Scholar
[39] Eggert S, Affleck I 1992 Phys. Rev. B 46 10866
Google Scholar
[40] Schuster C, Eckern U 2002 Ann. Phys. 514 901
Google Scholar
[41] Huang X, Yang Z 2015 J. Magn. Magn. Mater. 381 372
Google Scholar
[42] Viswanath V S, Müller G 1994 The Recursion Method—Application to Many-body Dynamics (Berlin: Springe-Verlag
[43] Lee M H 1982 Phys. Rev. Lett. 49 1072
Google Scholar
[44] Lee M H 1982 Phys. Rev. B 26 2547
Google Scholar
[45] Lee M H 2000 Phys. Rev. E 62 1769
Google Scholar
[46] Yuan X J, Wang C Y, Kong X M, Zhao J F, Wang H, Bu H X 2023 J. Magn. Magn. Mater. 572 170632
Google Scholar
[47] Nunes M E S, Plascak J A 2024 Phys. Rev. E 109 014134
Google Scholar
[48] Florencio J, de Alcantara Bonfim O F 2020 Front. Phys. 8 557277
Google Scholar
[49] Florencio J, Lee M H 1987 Phys. Rev. B 35 1835
Google Scholar
DownLoad:
Catalog
Metrics
- Abstract views: 250
- PDF Downloads: 13
- Cited By: 0
