海森堡模型中概率及相应熵的计算分析

Calculation and analysis of probability and corresponding entropy of Heisenberg model

介绍了海森堡模型的不同位型[N,n](N为海森堡链总格点数,n为格点中自旋向下的电子数)中的体现本征值获取难易程度的本征值获取概率及其相应信息熵(香农所定义的)和体现模型体系关联程度的自旋向下电子发现概率、每一粒子的von Neumann及体系的平均von Neumann熵,可为量子计算与信息传递提供启示性信息。研究结果表明:1)事件发生概率大于(小于)50%时,信息熵随概率增加而减小(增加)。2)不同位型[N,n],当n(N)同,N(n)增加时:本征值获取概率减小,其相应的信息熵正确反映本征值获取的难易程度;模型参数一定时,格点中自旋向下电子发现概率与每一粒子的von Neumann熵及体系的平均vonNeumann熵都分别减小(增加)。3)位型[N,n]相同时,每一粒子的von Neumann熵及体系的平均vonNeumann熵随参数变化时出现拐点,显示体系发生量子相变的信息。4)同位型[N,n]且同参数时处于海森堡链对称位置粒子的von Neumann熵相同。

In the situation of[N,n]（N is the total number of sites of Heisenberg chain,n is the number of electrons at site spin down） of Heisenberg model,the acquisition probability of eigenvalues and corresponding information entropy definited by Shannon are showed firstly,then the finding probability of site electron spin down,the von Neumann of each particle and average von Neumann entropy of the model system（They can embody the relevancy extent of model system） are discussed.The studying results are the following. 1） When the probability of event occurrence is more（less） than 50%,the information entropy decreases （increases） along with the increase of the probability.2） When n（N） is same and N（n） increases,the acqu- isition probability of the eigenvalues of[N,n]decreases.The corresponding information entropy correctly reflects the obtaining difficulty of the eigenvalue.When n（N） is same,N（n） increases and the parameter is fixed,the finding probability of site electron spin down,the von Neumann entropy of each particle and average von Neumann entropy of the model system decrease（increase） respectively.3） When[N,n]is same, the von Neumann entropy of each particle and average von Neumann entropy of the model system change along with the change of the parameter,the points of inflection arise.These can embody the information of the quantum phase transition of the system.4） When[N,n]and the parameter are same,the von Neumann entropy of the particles of symmetric position of Heisenberg chain is same.