這篇論文大略包含兩個部分。在第一部分我們討論光晶格中的玻色子的量子相,主要是使用量子蒙地卡羅的數值方法。冷原子在光晶格中的實驗提供了一個模擬凝態材料系統的可能性,其系統是乾淨且受良好控制的,我們主要關注兩類玻色子的系統,其中會出現有奇特的相。例如配對超流以及配對超固體。在第二部分我們以量子蒙地卡羅為基礎提出了一些數值方法來計算強關聯系統中的糾纏性質。利用複製品技巧,我們藉由計算雷尼熵來重建糾纏譜。另外我們研究了部分轉置密度矩陣的蹟數,是個與混合態糾纏的測量—負值—相關的量。下面我們簡短的總結每一個章節。
在第一章我們非常簡短的簡介冷原子系統以及它與量子蒙地卡羅計算之間的關係。
在第二章我們回顧路徑積分表相的蒙地卡羅方法。我們專注在方向性蟲演算法,是一個對玻色子以及自旋系統都很有力的方法。這個論文裡的其他方法—包括兩類玻色子、糾纏譜、以及部分轉置的計算—都是基於這章所介紹的演算法衍伸的。
在第三章我們討論兩類玻色子在方晶格中。我們展示出異類間的吸引力以及同類間的排斥力能使系統形成配對超固體,其中對角線的固體序並存於非對角線的配對超流序。我們表徵了超固體到其他相的量子相變以及溫度相變。我們發現從配對超固體到雙超流的相變是個一階相變且沒有任何中間相。另外,配對超固體的融化過程分為兩階段。首先配對超流經過一個 KT 相變被溫度摧毀,然後是固體序經由 Ising相變的融化。
在第四章我們提供了一個新的方法,利用量子蒙地卡羅重新建立量子多體系統的部分糾纏譜。此方法原則上能用在二維或更高維度。此方法建立在複製品技巧上,來計算粒子數解析的約化密度矩陣的 n 次方的蹟數。利用這些資訊我們使用多項式解根重新建立了前 n 個糾纏譜的譜帶。我們演示我們的方法在一維的延展玻色哈伯模型,能解析出Haldane 絕緣體的糾纏譜中的亞簡併。整體來說,這個方法能讓我們重建出最大的幾個本徵值。
在第五章我們設計了一個蒙地卡羅方法來計算有限溫度下部分轉置約化密度矩陣的某次方的蹟數。這個量能用來建立一些與負值有關的尺度不變量。負值是一個能真正測量一個線段中的兩個區段的糾纏的量。具體來說,我們討論了在一維硬核玻色子模型下的幾個尺度不變的結合量。對於兩個連接的區段,我們展示了異常的有限尺度修正,顯示了周長與填滿相關的震盪的奇偶效應。在大尺度下我們得到與保角場論的計算完全符合的結果。另一方面,對於兩個分開的區段有限尺度修正較為嚴重,而保角場論的結果只能在取極限下被得到。同時我們的證據也顯示了修正的指數項與互訊息的修正是一樣的。另外我們研究了一維玻色子模型的超流相。在此系統中有限尺度修正較小,稍大系統的量子蒙地卡羅的結果已經能良好的符合保角場論的計算。

This thesis is divided into two parts. In the first part we study quantum phases in bosonic systems in an optical lattice by using numerical unbiased Quantum Monte Carlo (QMC) method. Ultracold atoms experiments in optical lattices provide a possibility to simulate the materials of condensed matter in a clean and well-controled way. We focus on two components bosonic systems where exotic phases such as pair-superfluid and pair-supersolid appear. In the second part we provide numerical methods based on QMC simulations to study the entanglement properties of strong correlated systems. By employing the replica trick, we reconstruct the entanglement spectrum from the Renyi entropies. Furthermore, we study the trace of the power of the partial transposed reduced density matrix, which is related to the entanglement measurement negativity for mixed states. In the following we briefly summarize each chapter.
In Chapter 1 we very briefly introduce the cold atom systems and the connection with QMC simulations.
In Chapter 2 we review the QMC method based on the path integral (world line) representation. We focus on the directed worm algorithm which is an efficient algorithm with global updtae for bosonic and spin systems. All the following methods in this thesis – the method of two component bosonic systems, the reconstruction of entanglement spectrum and the measuring of partial transposed quantities, are based on directed worm algorithm introduced in this chapter.
In Chapter 3 we discuss two component bosons in a square lattice. We show that the interspecies attraction and nearest-neighbor intraspecies repulsion result in the pair-supersolid phase, where a diagonal solid order coexists
with an off-diagonal pair-superfluid order. The quantum and thermal transitions out of the pair-supersolid phase are characterized. It is found that there is a direct first-order transition from the pair-supersolid phase to the double-superfluid phase without an intermediate region. Furthermore, the melting of the pair-supersolid occurs in two steps. Upon heating, first the pair-superfluid is destroyed via a Kosterlitz-Thouless transition, then the solid order melts via an Ising transition.
In Chapter 4 we represent a new method to reconstruct a subset of the entanglement spectrum of quantum many body systems by QMC, where the method can in principle be applied to two or higher dimension. The approach builds on the replica trick to evaluate particle number resolved traces
of the first n of powers of a reduced density matrix. From this information we reconstruct first n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector.
In Chapter 5 we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two
intervals embedded in a chain. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity
effects that oscillate with a filling dependent period. For large chains we find perfect agreement with conformal field theory (CFT) calculations. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered
only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. The finite-size effects are smaller and QMC data are already in impressive
agreement with CFT at moderate large sizes.

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