Theoretical and Numerical Study of Many-Body Interaction and Quantum Phase Transition in Low-Dimensional Magnets

用新的积分方程对XXX模型的高温展开利用有限聚类方法对一维Heisenberg模型的比热和磁化率等热力学量进行了24阶的高温展开。另一方面，Bethe ansatz法的高温膨胀被认为是困难的。但我们成功地利用新的热力学Bethe ansatz方程扩展到100阶。采用Pade近似，并与量子转移矩阵法的数值结果进行了比较。2) Bethe ansatz方程与Hubbard模型高温展开的比较热力学Bethe ansatz方程有无数个未知函数。对非线性积分方程进行了截断和牛顿法求解。一般来说，热力学量是由大势的微分表示的。然后我们必须解线性积分方程来得到热力学量。但是这个方程也可以用牛顿法来解。通过一次线性运算，我们可以得到能量、熵、磁化和粒子密度等一阶热力学量。3)一维各向同性XY模型的空区形成概率采用数值和解析方法研究了一维XY模型的空区形成概率，空区形成概率是多个点相关函数之一

1) High temperature expansion of XXX model by new integral equationHigh temperature expansion of thermodynamic quantities such as specific heat and magnetic susceptibility for one-dimensional Heisenberg model has been done up to 24-th order using finite cluster method. On the other hand high temperature expansion by Bethe ansatz method was believed to be difficult. But we succeeded to expand to 100th order using new thermodynamic Bethe ansatz equation. We apply Pade approximation and compared with the numerical results of quantum transfer matrix method.2) Comparison of Bethe ansatz equation with high-temperature expansion for Hubbard modelThermodynamic Bethe ansatz equation has infinite number of unknown functions. We made cut off and used Newton's method to solve non-linear integral equation. Generally speaking thermodynamic quantities are expressed by the differentiation of grand potential. Then we must solve linear integral equation to get thermodynamic quantities. But this equation can be done in the same method in Newton's method. Only by one time linear operation we can get first order thermodynamic quantities such as energy, entropy, magnetization and particle density.3) Emptiness formation probability of one-dimensional isotropic XY modelWe investigated the emptiness formation probability which is one of many points corellation functions for one-dimensional XY model using numerical and analytical methods