自旋为1/2的双层平方晶格阻挫模型的基态相变

卿煜林; 彭小莉; 文林; 胡爱元

doi:10.7498/aps.71.20211584

摘要

基于线性自旋波理论, 研究了二维各向异性双层平方晶格阻挫模型的基态性质. 探讨了各向异性和自旋间交换作用对基态相图及系统可能发生相变的影响. 结果显示: 对于各向同性, 当层间耦合相互作用$ {J_{\text{c}}} \geqslant $$ 0.21 $或$ {J_{\text{c}}} \leqslant - 0.175 $时, Nèel态(NS)和Collinear态(CS)能共存; 对于各向异性, NS和CS也可以共存. 对弱的各向异性, NS更稳定; 对于强的各向异性, 系统将发生由NS向CS转换的一阶相变.

关键词:

Abstract

In this paper, we investigate the ground state phase transition of the spin-1/2 frustrated model on a stacked square lattice by the linear spin wave method. The effects of the anisotropy and the interplay of neighboring couplings on phase diagram and phase transformation between the Nèel state and collinear state are explored. Our results show that both the Nèel state and collinear state can exist due to the interplay of interlayer coupling $ {J_{\text{c}}} \geqslant 0.21 $ or $ {J_{\text{c}}} \leqslant - 0.175 $ for an isotropic system. For an anisotropic system, both the Nèel state and collinear state may also exist. In this case, for the weak anisotropy, the Nèel state is more stable. For the strong anisotropy, the system will undergo a first-order phase transition from the Nèel state to the collinear state.

Keywords:

antiferromagnetic /
magnetic anisotropy /
general theory and models of magnetic ordering /
exchange and superexchange interactions

作者及机构信息

重庆师范大学物理与电子工程学院, 重庆　401331

通信作者: 胡爱元, huaiyuan@cqnu.edu.cn

基金项目: 国家自然科学基金(批准号: 11875010, 12175027)和重庆市自然科学基金(批准号: cstc2021jcyj-msxmX0168, cstc2019jcyj-msxmX0217, cstc2019jcyj-msxmX0251)资助的课题

Authors and contacts

College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing 401331, China

Corresponding author: Hu Ai-Yuan, huaiyuan@cqnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China ( Grant Nos. 11875010, 12175027) and the Chongqing Natural Science Foundation, China (Grant Nos. cstc2021jcyj-msxmX0168, cstc2019jcyj-msxmX0217, cstc2019jcyj-msxmX0251).

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参考文献

[1]	Chubukov A V, Jolicoeur T 1991 Phys. Rev. B 44 12050 Google Scholar
[2]	Bishop R F, Farnell D J J, Parkinson J B 1998 Phys. Rev. B 58 6394 Google Scholar
[3]	Metavitsiadis A, Sellmann D, Eggert S 2014 Phys. Rev. B 89 241104 Google Scholar
[4]	Haghshenas R, Sheng D N 2018 Phys. Rev. B 97 174408 Google Scholar
[5]	Sadrzadeh M, Haghshenas R, Langari A 2019 Phys. Rev. B 99 144414 Google Scholar
[6]	Choo K, Neupert T, Carleo G 2019 Phys. Rev. B 100 125124 Google Scholar
[7]	Doretto R L 2020 Phys. Rev. B 102 014415 Google Scholar
[8]	Roscher D, Gneist N, Scherer M M, Trebst S, Diehl S 2019 Phys. Rev. B 100 125130 Google Scholar
[9]	Bishop R F, Li P H Y, Darradi R, Schulenburg J, Richter J 2008 Phys. Rev. B 78 054412 Google Scholar
[10]	Wang L, Sandvik A W 2018 Phys. Rev. Lett. 121 107202 Google Scholar
[11]	Haghshenas R, Lan W W, Gong S S, Sheng D N 2018 Phys. Rev. B 97 184436 Google Scholar
[12]	Sadrzadeh M, Haghshenas R, Jahromi S S, Langari A 2016 Phys. Rev. B 94 214419 Google Scholar
[13]	Ferrari F, Becca F 2020 Phys. Rev. B 102 014417 Google Scholar
[14]	Carretta P, Papinutto N, Azzoni C B, Mozzati M C, Pavarini E, Gonthier S, Millet P 2002 Phys. Rev. B 66 094420 Google Scholar
[15]	Carretta P, Melzi R, Papinutto N, Millet P 2002 Phys. Rev. Lett. 88 047601 Google Scholar
[16]	Nath R, Tsirlin A A, Rosner H, Geibel C 2008 Phys. Rev. B 78 064422 Google Scholar
[17]	Zhao J, Yao D X, Li S, Hong T, Chen Y, Chang S, Ratcliff W, Lynn J W, Mook H A, Chen G F, Luo J L, Wang N L, Carlson E W, Hu J, Dai P 2008 Phys. Rev. Lett. 101 167203 Google Scholar
[18]	Ewings R A, Perring T G, Bewley R I, Guidi T, Pitcher M J, Parker D R, Clarke S J, Boothroyd A T 2008 Phys. Rev. B 78 220501 Google Scholar
[19]	Johnston D C, McQueeney R J, Lake B, Honecker A, Zhitomirsky M E, Nath R, Furukawa Y, Antropov V P, Singh Y 2011 Phys. Rev. B 84 094445 Google Scholar
[20]	Holt M, Sushkov O P, Stanek D, Uhrig G S 2011 Phys. Rev. B 83 144528 Google Scholar
[21]	Fan Z, Jie Q L 2014 Phys. Rev. B 89 054418 Google Scholar
[22]	Schmalfuß D, Darradi R, Richter J, Schulenburg J, Ihle D 2006 Phys. Rev. Lett. 97 157201 Google Scholar
[23]	Nunes W A, Ricardo de Sousa J, Roberto Viana J, Richter J 2010 J. Phys.: Condens. Matter. 22 146004 Google Scholar
[24]	Majumdar K 2011 J. Phys.: Condens. Matter 23 046001 Google Scholar
[25]	Thalmeier P, Zhitomirsky M E, Schmidt B, Shannon N 2008 Phys. Rev. B 77 104441 Google Scholar
[26]	Rojas O, Hamer C J, Oitmaa J 2011 J. Phys.: Condens. Matter. 23 416001 Google Scholar
[27]	Isaev L, Ortiz G 2012 Phys. Rev. B 86 100402 Google Scholar
[28]	Bishop R F, Li P H Y, Götze O, Richter J 2019 Phys. Rev. B 100 024401 Google Scholar
[29]	Holstein T, Primakoff H 1940 Phys. Rev. 58 1098 Google Scholar
[30]	Dyson F J 1956 Phys. Rev. 102 1217 Google Scholar
[31]	Oitmaa J, Zheng W H 1996 Phys. Rev. B 54 3022 Google Scholar
[32]	Majumdar K 2010 Phys. Rev. B 82 144407 Google Scholar
[33]	Kalza A, Honecker A, Fuchs S, Pruschke T 2008 Eur. Phys. J. B 65 533 Google Scholar
[34]	Jin S, Sen A, Sandvik A W 2012 Phys. Rev. Lett. 108 045702 Google Scholar

施引文献

图 1 NS和CS的磁构型　(a), (b) 层间耦合为反铁磁相互作用; (c), (d)层间耦合为铁磁相互作用. 实心和空心圆圈分别描述的是自旋取向相上和向下

Fig. 1. Spin configurations of the NS and CS: (a), (b) Interlayer coupling as antiferromagnetic interactions; (c), (d) interlayer coupling as ferromagnetic interactions. The solid and empty circles represent the up-spins and down-spins, respectively.

下载: 全尺寸图片幻灯片

图 2 当$ \eta = 1 $时, 不同$ {J_{\text{c}}} $下的子晶格磁化强度$ m $与$ {J_2} $之间的关系　(a) $ {J_{\text{c}}} = 0, 0.21, 1 $; (b) $ {J_{\text{c}}} = 0, - 0.175, - 1 $; (c) $ {J_{\text{c}}} = 0 $; (d) $ {J_{\text{c}}} = 1 $. 图(c)和(d)中, 短划线、空心方格、实心圆、实心三角形和破折号分别描述的是线性自旋波理论(本文的结果, LSW)、系列展开法到9阶(SE I)^[31]、系列展开法到12阶(SE II)^[26]、耦合团簇方法(CCM)^[31]和二阶自旋波理论(SOSW)^[32]的结果

Fig. 2. Sublattice magnetization $ m $ as a function of $ {J_2} $ for different $ {J_{\text{c}}} $ values at $ \eta = 1 $: (a) $ {J_{\text{c}}} = 0, 0.21, 1 $; (b) $ {J_{\text{c}}} = $$ 0, - 0.175, - 1 $; (c) $ {J_{\text{c}}} = 0 $; (d) $ {J_{\text{c}}} = 1 $. In panel (c) and (d), the shorted-dashed lines, open squares, filled circles, filled triangles and dashed-doted lines represent the results of the linear spin wave (this paper, LSW), the series expansion up to the 9^th order (SE I)^[31], the series expansion up to the 12^th order (SE II)^[26], the couple-cluster method (CCM)^[31] and second-order spin wave (SOSW)^[32], respectively.

下载: 全尺寸图片幻灯片

图 3 当$ \eta = 1 $时, 系统在参数$ {J_2} $和$ {J_{\text{c}}} $空间的基态相图

Fig. 3. Ground state phase diagram in the ${J_2} \text{-} {J_{\text{c}}}$ plan for$ \eta = 1 $.

下载: 全尺寸图片幻灯片

图 4 不同参数时的磁化强度$ m $与$ {J_2} $的关系　(a) $ {J_{\text{c}}} = 0, \eta = 0.956 $; (b) ${J_{\rm c}} = 0.025, \eta = 0.9718$; (c)$ {J_{\text{c}}} = - 0.058, \eta = 0.987 $; (d) $ {J_{\text{c}}} = 0.3, \eta = 0.8 $; (e) $ {J_{\text{c}}} = 0.3, \eta = 0.8 $; (f) $ {J_{\text{c}}} = - 0.3, \eta = 0, 0.8, 0.9, 1 $

Fig. 4. Sublattice magnetization $ m $ as a function of $ {J_2} $ for different parameter values: (a) $ {J_{\text{c}}} = 0, \eta = 0.956 $; (b) ${J_{\rm c}} = 0.025, $$ \eta = 0.9718$; (c) $ {J_{\text{c}}} = - 0.058, \eta = 0.987 $; (d) $ {J_{\text{c}}} = 0.3, \eta = 0.8 $; (e) $ {J_{\text{c}}} = 0.3, \eta = 0.8 $; (f) $ {J_{\text{c}}} = - 0.3, \eta = 0, 0.8, 0.9, 1 $.

下载: 全尺寸图片幻灯片

图 5 在参数$ \eta $-$ {J_{\text{c}}} $空间中两个态共存所对应的区域. 平面分成了两个区域: 顺磁相和两个态共存的区域. 在共存区域$ {J_2} $的取值范围是$ J_2^1 \leqslant {J_2} \leqslant J_2^2 $. 相应的例子是图4(e)

Fig. 5. Area corresponding to the coexistence of the two states in the $\eta \text{-} {J_{\text{c}}}$ space. The plane is divided into the two areas: paramagnetic phase and the coexistence of the two states. In the coexistence area of the two states , the value range of $ {J_2} $ is $ J_2^1 \leqslant {J_2} \leqslant J_2^2 $. The corresponding example is Fig. 4(e).

下载: 全尺寸图片幻灯片

图 6 (a)当$ {J_2} = 0.5 $时, NS和CS的基态能$ {E_0} $与$ {J_{\text{c}}} $的关系; (b)当$ {J_{\text{c}}} = 0, 1 $时, NS和CS的基态能$ {E_0} $与$ {J_2} $的关系. 实心方块和实心圆是系列展开法(SE II)的结果^[26]

Fig. 6. (a) Ground state energy $ {E_0} $ of the Néel and collinear states as a function of $ {J_{\text{c}}} $ for $ {J_2} = 0.5 $; (b) ground state energy $ {E_0} $ of the Néel and collinear states as a function of $ {J_2} $ for $ {J_{\text{c}}} = 0, 1 $. The filled squares and filled circles are the results of the series expansion up to the 12^th order (SE II)^[26].

下载: 全尺寸图片幻灯片

图 7 当$ {J_{\text{c}}} = 0.03 $时, 不同$ \eta $值下两个态的基态能$ {E_0} $与$ {J_2} $的关系　(a) $ \eta = 0.97 $; (b) $ \eta = 0.7703 $; (c) $ \eta = 0.5 $; (d) $\eta = 0$

Fig. 7. Ground state energy $ {E_0} $ of the two states as a function of $ {J_2} $ for different $ \eta $ values when $ {J_{\text{c}}} = 0.03 $: (a) $\eta = $$ 0.97$; (b) $ \eta = 0.7703 $; (c) $ \eta = 0.5 $; (d) $ \eta = 0 $.

下载: 全尺寸图片幻灯片

图 8 当$ J_2^1 \leqslant {J_2} \leqslant J_2^2 $时, 两个态的基态能在参数${J_{\text{c}}} \text{-} \eta$空间中的比较

Fig. 8. Comparison of the ground state energy of the two states in the ${J_{\text{c}}} \text{-} \eta$ space when $ J_2^1 \leqslant {J_2} \leqslant J_2^2 $.

下载: 全尺寸图片幻灯片

[1]	Chubukov A V, Jolicoeur T 1991 Phys. Rev. B 44 12050 Google Scholar
[2]	Bishop R F, Farnell D J J, Parkinson J B 1998 Phys. Rev. B 58 6394 Google Scholar
[3]	Metavitsiadis A, Sellmann D, Eggert S 2014 Phys. Rev. B 89 241104 Google Scholar
[4]	Haghshenas R, Sheng D N 2018 Phys. Rev. B 97 174408 Google Scholar
[5]	Sadrzadeh M, Haghshenas R, Langari A 2019 Phys. Rev. B 99 144414 Google Scholar
[6]	Choo K, Neupert T, Carleo G 2019 Phys. Rev. B 100 125124 Google Scholar
[7]	Doretto R L 2020 Phys. Rev. B 102 014415 Google Scholar
[8]	Roscher D, Gneist N, Scherer M M, Trebst S, Diehl S 2019 Phys. Rev. B 100 125130 Google Scholar
[9]	Bishop R F, Li P H Y, Darradi R, Schulenburg J, Richter J 2008 Phys. Rev. B 78 054412 Google Scholar
[10]	Wang L, Sandvik A W 2018 Phys. Rev. Lett. 121 107202 Google Scholar
[11]	Haghshenas R, Lan W W, Gong S S, Sheng D N 2018 Phys. Rev. B 97 184436 Google Scholar
[12]	Sadrzadeh M, Haghshenas R, Jahromi S S, Langari A 2016 Phys. Rev. B 94 214419 Google Scholar
[13]	Ferrari F, Becca F 2020 Phys. Rev. B 102 014417 Google Scholar
[14]	Carretta P, Papinutto N, Azzoni C B, Mozzati M C, Pavarini E, Gonthier S, Millet P 2002 Phys. Rev. B 66 094420 Google Scholar
[15]	Carretta P, Melzi R, Papinutto N, Millet P 2002 Phys. Rev. Lett. 88 047601 Google Scholar
[16]	Nath R, Tsirlin A A, Rosner H, Geibel C 2008 Phys. Rev. B 78 064422 Google Scholar
[17]	Zhao J, Yao D X, Li S, Hong T, Chen Y, Chang S, Ratcliff W, Lynn J W, Mook H A, Chen G F, Luo J L, Wang N L, Carlson E W, Hu J, Dai P 2008 Phys. Rev. Lett. 101 167203 Google Scholar
[18]	Ewings R A, Perring T G, Bewley R I, Guidi T, Pitcher M J, Parker D R, Clarke S J, Boothroyd A T 2008 Phys. Rev. B 78 220501 Google Scholar
[19]	Johnston D C, McQueeney R J, Lake B, Honecker A, Zhitomirsky M E, Nath R, Furukawa Y, Antropov V P, Singh Y 2011 Phys. Rev. B 84 094445 Google Scholar
[20]	Holt M, Sushkov O P, Stanek D, Uhrig G S 2011 Phys. Rev. B 83 144528 Google Scholar
[21]	Fan Z, Jie Q L 2014 Phys. Rev. B 89 054418 Google Scholar
[22]	Schmalfuß D, Darradi R, Richter J, Schulenburg J, Ihle D 2006 Phys. Rev. Lett. 97 157201 Google Scholar
[23]	Nunes W A, Ricardo de Sousa J, Roberto Viana J, Richter J 2010 J. Phys.: Condens. Matter. 22 146004 Google Scholar
[24]	Majumdar K 2011 J. Phys.: Condens. Matter 23 046001 Google Scholar
[25]	Thalmeier P, Zhitomirsky M E, Schmidt B, Shannon N 2008 Phys. Rev. B 77 104441 Google Scholar
[26]	Rojas O, Hamer C J, Oitmaa J 2011 J. Phys.: Condens. Matter. 23 416001 Google Scholar
[27]	Isaev L, Ortiz G 2012 Phys. Rev. B 86 100402 Google Scholar
[28]	Bishop R F, Li P H Y, Götze O, Richter J 2019 Phys. Rev. B 100 024401 Google Scholar
[29]	Holstein T, Primakoff H 1940 Phys. Rev. 58 1098 Google Scholar
[30]	Dyson F J 1956 Phys. Rev. 102 1217 Google Scholar
[31]	Oitmaa J, Zheng W H 1996 Phys. Rev. B 54 3022 Google Scholar
[32]	Majumdar K 2010 Phys. Rev. B 82 144407 Google Scholar
[33]	Kalza A, Honecker A, Fuchs S, Pruschke T 2008 Eur. Phys. J. B 65 533 Google Scholar
[34]	Jin S, Sen A, Sandvik A W 2012 Phys. Rev. Lett. 108 045702 Google Scholar

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自旋为1/2的双层平方晶格阻挫模型的基态相变