- 學位論文
對稱張量網路對Kitaev自旋液體之研究
Symmetric Tensor Network Studies of Kitaev Spin Liquids
摘要
本論文使用對稱張量網路研究 Kitaev 自旋液體。藉由將虛時間演化產生的波函數投影到無渦流空間，我們獲得了各向同性自旋-$1$ Kitaev 蜂巢狀模型的可靠基態。通過計算張量網路中虛擬希爾伯特空間上定義的虛擬序參數，我們確認了其$\mathbb{Z}_2$ 量子自旋液體的性質。此外，我們發現不同於自旋-$1/2$ 情況下的馬約羅納費米子色散激發，各向同性自旋-$1$ Kitaev 模型具有電荷任意子的色散激發。我們使用轉移矩陣光譜和低能量激發態之間的對應關係識別單粒子和雙粒子激發態的下緣。另一方面，我們研究了自旋-$1/2$ 星形 Kitaev 模型的拓撲相變。我們發現類似於任意子凝聚態機制，$\mathbb{Z}_2$-單射投影糾纏對態的物理變形也可用於研究從亞伯群到非亞伯群自旋液體的相變。此外，我們展示在非亞伯群相中的基態總是具有無限大的關聯長度，這與手性投影糾纏對態永遠具有無間隙父哈密頓量的不可行定理吻合。
並列摘要
In this thesis, we study the Kitaev spin liquids using symmetric tensor network. By projecting the state generated from imaginary time evlolution to the vortex-free sector, we obtain the reliable ground states of the isotropic spin-$1$ Kitaev honeycomb model. The $\mathbb{Z}_2$ quantum spin liquids nature is then confirmed by computing the virtual order parameters defined on the virtual Hilbert space in the tensor network formalism. Besides, we find that contrary to the dispersive Majorana excitation in the spin-$1/2$ case, the isotropic spin-$1$ Kitaev model possesses dispersive charge anyon excitaion. We identify the lower edge of the single- and two-particle excitations using the correspondence between the transfer matrix spectrum and low-lying excitations. On the other hand, we study the topological phase transition of the spin-$1/2$ star lattice Kitaev model. We find that similar to the anyon condensation, physical deformation of $\mathbb{Z}_2$-injective projected entangled pair states (PEPS) can also be used to study the transtion from Abelian to non-Abelain spin liquid phases. Furthermore, we show that the in the non-Abelian regime, the ground states always have infinite correlation length, consistent with the no-go theorem that a chiral PEPS has a gapless parent Hamiltonian.