一維物理系統的數值研究

Numerical study of one-dimensional physics

薛志龍

指導教授 ： 陳柏中

國立清華大學/理學院/物理學系/博士(2013年)

摘要

本文對於兩個有趣的一維物理系統進行了數值研究。我們首先嘗試了解在一維光晶格中自旋為2玻色子的絕緣態相圖。利用密度重整化群(DMRG)計算，我們指出了三個可能的基態︰鐵磁態(ferromagnetic phase)、二聚化態(dimerized phase)及三聚化態(trimerized phase)。藉由檢驗激發態能隙與系統長度的關係，我們推定三聚化態為一臨界相(critical phase)。此外，我們定義了合宜描述聚化態的序參量。利用文獻中所提供的s波散射長度，我們亦推估了冷原子實驗中常用自旋為2原子的可能基態。最後，我們提出了一個在實驗上可簡易驗證數值計算結果的方法。

我們接著對於一維自旋1/2 XXZ模型的量子糾纏熵(entanglement entropy)尺度行為(scaling behaviors)進行了系統性地研究。利用三種不同的數值方法︰密度重整化群(DMRG)、無限尺度密度重整化群(iDMRG)及精確對角化法(exact diagonalization)，我們計算出量子糾纏熵及系統基態能量。經由適當的數據處理我們可以得出與共形場論(conformal field theory)相關的尺度常數中心荷(central charge)。我們的計算發現中心荷的估算會受到鄰近鐵磁相的影響。

最後，利用估計中心荷與尺度維(scaling dimension)，我們推斷一維光晶格中自旋為2玻色子的三聚化態可以經由共形場論中 $ SU(3)_{1} $ WZW模型所描述。同時,我們亦發現了三聚化態中奇異的量子糾纏熵尺度行為。

關鍵字

密度重整化群 ； 量子糾纏熵 ； 精確對角化法 ； 共形場論

並列摘要

In this thesis, we perform numerical studies on two special one-dimensional physical systems. We first investigate the insulating phases of spin-2 bosons loaded in optical lattice by non-Abelian DMRG calculation. Three possible phases are identified in our calculation, namely, ferromagnetic, dimerized and trimerized. By finite-size extrapolations, the exotic trimerized phase is believed to be a critical phase. We here also define dimerized and trimerized order parameters to describe the exotic phases. By using the scattering lengths in literature, we point out the possible phases of related spin-2 elements in cold atom experiments. Finally we propose an easy-implemented method to verify the ternary phase diagram experimentally.

Secondly, We carry out a systematic study in entanglement entropy scaling of the XXZ spin-1/2 chain. We evaluate the scaling constant, central charge $ c $, from the scaling of both entanglement entropy and finite-size ground state energy by three different numerical methods: finite-size DMRG, infinite-size DMRG (iDMRG) and exact diagonalization (ED). Our calculation shows that all the estimations of $ c $ in the critical XY phase are influenced by the neighboring ferromagnetic phase and result in crossover behavior.

In the end, we certify the universality class of the critical trimerized phase as a $ SU(3)_{1} $ WZW model by evaluating central charge $ c $ and scaling dimension $ x $. In addition, unusual scaling behavior of entanglement and R'enyi entropy is observed.

參考文獻

[76] Elisa Ercolessi, Stefano Evangelisti, Fabio Franchini, and Francesco Ravanini, Phys. Rev. B 85, 115428 (2012).

[72] Elisa Ercolessi, Stefano Evangelisti, Fabio Franchini, and Francesco Ravanini, Phys. Rev. B 83, 012402 (2011).

[70] D. Zheng, G. M. Zhang, T. Xiang, and D. Lee, Phys. Rev. B 83, 014409 (2011).

[63] Jonas A. Kjall, Michael P. Zaletel, Roger S. K. Mong, Jens H. Bardarson, Frank Pollmann, Phys. Rev. B 87, 235106 (2013).

[47] Fields, Strings and Critical Phenomena, (Les Houches, Session XLIX, 1988) ed. by E. Brezin and J. Zinn Justin, 1989.