本篇論文主要在探討大尺度的強關聯電子系統，更準確地說，我們使用
t − t′ − J 模型來模擬高溫超導銅氧化物中銅氧平面的微觀行為。在這個工
作中，我們使用Gutzwiller Approximation(GWA) 以及重整化平均場論方法
(Renormalized mean field theory) 來近似地計算t − J 漢米爾頓量。
一般而言，mean-field 在計算漢米爾頓的本徵波函數時使用的是精確數
值對角化，本篇論文採用了Kernal Polynomial Method(KPM) 來近似地計算
譜權重(spectral weigth)，進而得到密度矩陣(density matrix)。其次，本項工
作採用了CPU/GPU 平行計算的方法來加速。使用RMFT+KPM 方法並且加
上GPU 的加速，進而在短時間內計算出很大尺度的強關聯系統系統。
銅氧化物超導(cuprate) 中一直以來存在一個謎團，理論計算的純d-波超
導渦漩電子態應存在一個零偏壓電導峰（zero bias conductance peak,ZBCP)，
然而，大部分的Bi-系銅氧化物卻沒有相關的實驗報導，直到了2021 年
Renner 等人[1] 在過摻雜(overdoped) 的Bi-系銅氧化物樣品，並且極低的磁
場下找到了零篇壓電導峰。這引起了人們重新對銅氧化物渦漩態的關注。
此外，在2016 年的nat. comm. [3] 與2019 年的science 文章[2] 中，報
導了四個晶格週期與八個晶格週期的電荷密度波( charge density wave)，這
表明了配對密度波(pair density wave) 的存在，然而目前卻尚未有微觀的理
論模型來解釋這一結果。
在這篇論文中，我們會使用重整化平均場論方法來得到渦漩態，並且計
算它的電子能譜。利用前述KPM 方法，可以計算磁場低至此約3 特斯拉，
過者說96 × 96 的晶格大小系統，並且做了完整的磁場與電洞摻雜探討。另
外，我們也會展示雙向的電荷密度波與渦旋共存態，並且計算它的形狀因子(form factor) 並與實驗結果比較。

This thesis aims to solve the problem on the large scale of strongly correlated
electron systems. We simulate the physics for the copper-oxide plain of cuprate
by the t-t′-J model, which is approximated via renormalized mean field theory
(RMFT).
In general, exact diagonalization is used in calculating the eigenfunction of a
given mean-field Hamiltonian. In this thesis, Kernel Polynomial has been implemented
to estimate the spectral weight. Then we could get the density matrix and
the order parameters. On the other hand, parallel computation via CPU/GPU has
been applied. By using RMFT+KPM via acceleration of GPU, the system’s size
could be pushed to a huge scale (about 104 ∼ 105 electrons ).
A long-standing puzzle in the cuprate vortex problem is that the absence of zeros
bias conductance peak (ZBCP) in experimental observation until recent report
Renner et al. [1]. Another central issue is the relationship between PDW and PG.
Recent spectroscopic imaging scanning tunneling microscopy (SI-STM) results [2]
indicated the existence of CDW peaks QP and 2QP under magnetic field. In this
work, two critical findings will be addressed:
(1) The continuum-local density of state(c-LDOS), that convolute lattice Green’s
function with the Wannier function, is shown to be the sub-gap structure (SGS).
Also, our result indicates that the dopant- and field- dependence should not be ignored.
(2) A candidate checkerboard (CB) state is shown to simulate the STM experiment
[2, 3]. The result suggests that the dominating s′ form factor is related to the existence of antiferromagnetic order near the halo.

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