含阻挫的亚铁磁性Heisenberg模型的数值研究
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摘要
自从在一些有阻挫的海森堡(Heisenberg)自旋模型中发现了可能与高温超导机制有关的短程共振价键态(short-range resonating-valence-bond phase)后,含阻挫(frustration)的自旋系统引起了广泛而持续的关注。本论文研究的是一种含阻挫的S =1/2准一维反铁磁自旋链但却具有亚铁磁性的海森堡模型,即在S =1/2一维海森堡反铁磁链的上方每间隔一个格点周期性地掺入一个自旋为1/2的侧自旋,并且考虑侧自旋与链上次近邻格点间的交换作用(即阻挫αJ)。
严格对角化方法(ED)对于研究有限尺寸量子多体系统的物理特性,是非常有效而且很受欢迎的数值方法。而密度矩阵重整化群方法(DMRG)是处理一维和准一维量子自旋系统的强有力的工具,它克服了量子蒙特卡罗方法(QMC)中的负符号以及ED方法中只能处理有限格点的问题。本论文将利用ED和DMRG两种数值方法,计算周期掺杂侧自旋并引入阻挫后的准一维反铁磁自旋系统的基态能、自旋能隙、自旋关联函数和自旋态密度等物理量,研究阻挫( 0≤α≤1)对体系基态自旋图像以及磁序特性的影响。
首先利用简单的数学解析方法,得到体系随α变化的经典相图:亚铁磁序( 0≤α<1/3)、倾斜序( 1 /3<α<1)和直线序(α>1)。随后利用ED方法计算有限尺寸系统的基态总自旋S_G~T并外推到热力学极限情况,从而预测出体系的量子相图:亚铁磁相( 0≤α<0.401)、倾斜相( 0 .401<α<0.412)和无序相( 0 .412<α≤1)。明显看出在量子情形下由于量子涨落的影响,与经典对应的倾斜相区域变得非常狭窄,并且出现无经典对应的量子无序相。同时量子情形中亚铁磁相区域比经典情况下范围大,这又证实一个较为普遍的物理图像:量子涨落更倾向于自旋系统成奈尔序的自旋排列。
利用DMRG方法得到的系统基态能图和自旋关联图都证实了ED方法预测的量子相图,研究结果也进一步表明阻挫对系统基态的影响主要表现为:①在磁有序相区域(亚铁磁相+倾斜相),逐渐增强的阻挫只能减弱体系的磁有序性质,但还不能破坏其基态的磁性长程序;②在量子无序区域( 0 .412<α≤1),阻挫使体系基态波函数平移对称性发生自发破缺(平移对称性倍周期化),同时使体系在无序区伴随着自旋能隙的出现,而且TD (tetramer-dimer)自旋态密度< P >的权重取值以及实际基态与严格TD态下的单个晶胞基态能GSEPU (Ground-State Energy Per Unit Cell)比较都发现:体系在无序区的基态自旋状态可能和α=1出现的比dimer态更大的TD团簇态有关。
Since the evidence of the existence of a short-range resonating-valence-bond phase approach to the high-Tc superconductivity was found in a sufficiently frustrated Heisenberg model, a great deal of interest has been concentrated on the frustrated spin system. In this thesis, using ED (exact-diagonalization) and DMRG (density matrix renormalization group) methods,we study the effect of magnetic frustrations from next-nearest-neighbor bonds in a quasi-one-dimensional (QOD) antiferromagnetic Heisenberg (AFH) chain, which is ferrimagnetism for weak frustrations. This QOD model is constructed by periodically adding S =1/2 spins (side spin) beside every other spin site in a S= 1/ 2 AFH chain.
As the change of the frustration parameterα( = J 2 / J1), this model in its classical case exhibits different phases such as the ferrimagnetic phase (F, 0≤α<1/3), the canted phase (C, 1 /3<α<1) and the collinear phase (N,α>1). The calculation of S GT by ED method shows that the system in the quantum case exhibits the F phase ( 0≤α<0.401), the C phase ( 0 .401<α<0.412) and the disordered phase ( 0 .412<α≤1). Obviously, as a result of quantum fluctuations, the quantum C phase occurs only in a very small region. The quantum disordered phase has no counterpart in the classical case. Our calculations show that the quantum fluctuations favor the magnetic long-range order. A stronger frustration is needed to destroy this magnetic order in quantum case than in the classical case.
The DMRG data for the ground state energy per unit cell (GSEPU) and the different long-range spin-spin correlations further support the phase diagram suggested by ED method. Using these DMRG data, we also discuss the effect of frustrations from next-nearest-neighbor interactions in our model. We can conclude:①for the magnetic phase ( the quantum F phase + the quantum C phase), frustrations smoothly reduce the correlations between spins on the linear chain and so the magnetic order becomes weak. But moderate frustrations are able to destroy the magnetic order and to stabilize the quantum disordered phase;②for the disordered phase (0 .412<α≤1), the period of its ground state wave function doubles due to the spontaneous symmetry breaking and the spin gap is opened. As the frustrations increasing, the weight of TD (tetramer-dimer) spin-state density increases from about 0.77 to 1.0, so we can say in some sense that the overlap of the ground state and the TD state is large. Meanwhile the differences of energies between this two kinds of states become smaller and smaller. This result means that the TD state which is the ground state of this model atα=1, is dominant for the ground state in the disordered region.
严格对角化方法(ED)对于研究有限尺寸量子多体系统的物理特性,是非常有效而且很受欢迎的数值方法。而密度矩阵重整化群方法(DMRG)是处理一维和准一维量子自旋系统的强有力的工具,它克服了量子蒙特卡罗方法(QMC)中的负符号以及ED方法中只能处理有限格点的问题。本论文将利用ED和DMRG两种数值方法,计算周期掺杂侧自旋并引入阻挫后的准一维反铁磁自旋系统的基态能、自旋能隙、自旋关联函数和自旋态密度等物理量,研究阻挫( 0≤α≤1)对体系基态自旋图像以及磁序特性的影响。
首先利用简单的数学解析方法,得到体系随α变化的经典相图:亚铁磁序( 0≤α<1/3)、倾斜序( 1 /3<α<1)和直线序(α>1)。随后利用ED方法计算有限尺寸系统的基态总自旋S_G~T并外推到热力学极限情况,从而预测出体系的量子相图:亚铁磁相( 0≤α<0.401)、倾斜相( 0 .401<α<0.412)和无序相( 0 .412<α≤1)。明显看出在量子情形下由于量子涨落的影响,与经典对应的倾斜相区域变得非常狭窄,并且出现无经典对应的量子无序相。同时量子情形中亚铁磁相区域比经典情况下范围大,这又证实一个较为普遍的物理图像:量子涨落更倾向于自旋系统成奈尔序的自旋排列。
利用DMRG方法得到的系统基态能图和自旋关联图都证实了ED方法预测的量子相图,研究结果也进一步表明阻挫对系统基态的影响主要表现为:①在磁有序相区域(亚铁磁相+倾斜相),逐渐增强的阻挫只能减弱体系的磁有序性质,但还不能破坏其基态的磁性长程序;②在量子无序区域( 0 .412<α≤1),阻挫使体系基态波函数平移对称性发生自发破缺(平移对称性倍周期化),同时使体系在无序区伴随着自旋能隙的出现,而且TD (tetramer-dimer)自旋态密度< P >的权重取值以及实际基态与严格TD态下的单个晶胞基态能GSEPU (Ground-State Energy Per Unit Cell)比较都发现:体系在无序区的基态自旋状态可能和α=1出现的比dimer态更大的TD团簇态有关。
Since the evidence of the existence of a short-range resonating-valence-bond phase approach to the high-Tc superconductivity was found in a sufficiently frustrated Heisenberg model, a great deal of interest has been concentrated on the frustrated spin system. In this thesis, using ED (exact-diagonalization) and DMRG (density matrix renormalization group) methods,we study the effect of magnetic frustrations from next-nearest-neighbor bonds in a quasi-one-dimensional (QOD) antiferromagnetic Heisenberg (AFH) chain, which is ferrimagnetism for weak frustrations. This QOD model is constructed by periodically adding S =1/2 spins (side spin) beside every other spin site in a S= 1/ 2 AFH chain.
As the change of the frustration parameterα( = J 2 / J1), this model in its classical case exhibits different phases such as the ferrimagnetic phase (F, 0≤α<1/3), the canted phase (C, 1 /3<α<1) and the collinear phase (N,α>1). The calculation of S GT by ED method shows that the system in the quantum case exhibits the F phase ( 0≤α<0.401), the C phase ( 0 .401<α<0.412) and the disordered phase ( 0 .412<α≤1). Obviously, as a result of quantum fluctuations, the quantum C phase occurs only in a very small region. The quantum disordered phase has no counterpart in the classical case. Our calculations show that the quantum fluctuations favor the magnetic long-range order. A stronger frustration is needed to destroy this magnetic order in quantum case than in the classical case.
The DMRG data for the ground state energy per unit cell (GSEPU) and the different long-range spin-spin correlations further support the phase diagram suggested by ED method. Using these DMRG data, we also discuss the effect of frustrations from next-nearest-neighbor interactions in our model. We can conclude:①for the magnetic phase ( the quantum F phase + the quantum C phase), frustrations smoothly reduce the correlations between spins on the linear chain and so the magnetic order becomes weak. But moderate frustrations are able to destroy the magnetic order and to stabilize the quantum disordered phase;②for the disordered phase (0 .412<α≤1), the period of its ground state wave function doubles due to the spontaneous symmetry breaking and the spin gap is opened. As the frustrations increasing, the weight of TD (tetramer-dimer) spin-state density increases from about 0.77 to 1.0, so we can say in some sense that the overlap of the ground state and the TD state is large. Meanwhile the differences of energies between this two kinds of states become smaller and smaller. This result means that the TD state which is the ground state of this model atα=1, is dominant for the ground state in the disordered region.
引文
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