Tensor Network Studies on Antiferromagnetic Ising Chain in External Fields

在本論文中，我們針對處於外加橫場與縱場中的反鐵磁易辛鏈進行一系列的研究。在均勻系統中，我們利用以矩陣乘積態和矩陣乘積算符詮釋的密度矩陣重整化群找到系統的基態，並且結合我們所發現之高次方序參量矩陣乘積算符來計算出基態的Binder比值。計算結果顯示系統的反鐵磁-順磁相變在縱場非零時仍然存在。此外，我們也使用樹狀張量網絡語言下的強無序重整化群計算出無序系統中的基態，並從其計算出的Binder比值結果中發現臨界點在縱場出現時即被破壞。我們認為這樣的現象是來自於縱場在無序系統中生成的鐵磁局域對整體反鐵磁量子相變的破壞。此外，我們也計算了能隙和端點關聯函數對無序組態的分布，並由此觀察無窮隨機不動點在臨界點的行為以及Griffiths相在無序相裡的表現。無序系統的動力學指數也可以由這些分布的有限尺寸標度來得到。In this thesis, we study the quantum phase transitions of 1D antiferromagnetic (AFM) Ising model in the presence of transverse and longitudinal fields. We first consider the clean model which is homogeneous over the whole chain. The ground state is obtained by implementing the density matrix renormalization group (DMRG) in the fashion of matrix product states (MPS) and matrix product operators (MPO). With the discovery of MPO for powers of order paramter, we compute the Binder cumulant for ground state and use it to determine the critical point. The results show that the separation of AFM and paramagnetic (PM) phases extends to the nonzero longitudinal field regime. Next, we add disorder to the system by randomly choosing spin couplings and transverse fields in certain ranges. After deriving the ground state with tree tensor network strong disorder renormalization group (TSDRG), we find that the behavior of Binder cumulant shows the destruction of phase transition in nonzero longitudinal field regime. This phenomena is explained by the emergence of ferromagnetic (FM) rare regions. We also consider the distribution of energy gaps and end-end correlation functions over disorder realizations, which shows infinite randomness behavior at the critical point and Griffiths phase behavior in the disordered phase. The dynamical exponent can be determined from the finite-size scaling