我們研究兩個題目，一個是ZN 晶格規範場論，這個工作使用密度矩陣重整
化群本重新計算了的[38]部分結果，主要復現了在梯子幾何晶格中的純ZN規範場
模型，當N ≤ 4時候，只存在禁閉相。在梯子幾何晶格中的純ZN 希格斯物質場
上，當N ≤ 4的情況，我們重現了無序相與希格斯相，而在N=5的時候，重現
了準長程距的BKT相變。在Z2希格斯場耦合規範場的情況複現了希格斯相與無
序相，但整個相空間都在禁閉相。這個複現的工作主要是想要未來研究非阿貝
爾晶格規範場論作訓練準備。
另一個工作是與冷原子實驗合作，調查了在可調實驗參數內，兩種玻色愛因
斯坦鋰銫原子凝聚的情況，呈現出分離相，並給出實驗與數值模擬的結果。

We have pursued two distinct research topics. The first job involves $\mathbb{Z}_{N}$ lattice gauge theory and reproduce part of the results of reference \cite{nyhegn2021z}. Specifically, the confinement phase for $N\leq5$ in the pure gauge $\mathbb{Z}_{N}$ ladder geometry are successfully reproduced by Density matrix renormalization group. Similarly, the pure matter $\mathbb{Z}_{N}$ ladder geometry, the disorder and order (Higgs) phases for $N\leq4$ are getted same result in the reference. Furthermore, the $N=5$ case, the quasi-long-range order characterized by the Berezinskii-Kosterlitz-Thouless (BKT) type phase transition verify by fidelity susceptibility. In the context of $\mathbb{Z}_{2}$ matter coupled with gauge field, we managed to reproduce the Higgs and disorder phases, with the entire phase space exhibiting confinement phase. This replication effort was primarily driven by the intention to explore Non-abelian lattice gauge theory in the future works.

In our second work, we collaborated with cold atom experiment to investigate the emergence of an immiscible phase in two species $Li^{7}$ and $Cs^{133}$ of Bose-Einstein condensates. It shows experimentally adjustable parameters and obtained immiscible phase that harmonized well with our numerical simulations.

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