在此篇論文,我們主要研究⻑程有序在具有⻑程跳躍的一維的玻色-赫巴德模型 (Bose-Hubbard model)。我們使用量子蒙地卡羅(Quantum Monte Carlo) 來模擬系統,測量其關聯 (correlation) 和繞數 (winding number) 並用來計算凝聚係數 (condensate fraction) 和超流體密度 (superfluid density);此外,我們也利用密度矩陣重整化群(Density Matrix Renormalization Group) 來模擬系統,計算出零溫底下基態的糾纏熵 (entanglement entropy) 並用以推算出系統的中心電荷 (central charge)。

We study long-range orders in one-dimensional Bose-Hubbard model with power-law long-range hoppings. We measure the correlation and the winding numbers of hardcore bosons in one-dimension ring using quantum Monte Carlo simulations to calculate the condensate fraction and the superfluid density. On the other hand, we calculate the Entanglement entropy of open boundary zero-temperature ground states via density matrix renormalization group to estimate the central charge of the system.

Abstract i
摘要 ii
Contents iii
1 Introduction 1
2 Quantum Monte Carlo 2
2.1 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.4 Worm Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Density Matrix Renormalization Group 10
3.1 Tensor Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.1 Matrix Product States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.2 Graphic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.3 Matrix Product Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Results 16
4.1 Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1.1 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1.2 Condensate Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1.3 Superfluid Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.1 MPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.2 Entanglement Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Bibliography 42

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