考慮較長作用力下的二維硬核玻色子哈伯模型中的奇異量子相態

Abstract

在這篇論文中，我們使用隨機續列展開(Stochastic Series expansion, SSE)量子蒙地卡羅的方法來探討在二維方形晶格中硬核玻色子哈伯模型(Hardcore Boson Hubbard Model)。在同時考慮最鄰近(nearest-neighbor, NN)以及次鄰近(next nearest-neighbor, NNN)的交互作用時，有一些奇異的量子相態被我們觀察到，像是超固體態(Supersolid)。同時我們也使用古茨維勒平均場理論(Gutzwiller mean-field method)的方法來幫助我們探討我們的問題。 我們觀察到當把次鄰近的躍遷項加入考慮時，我們能夠觀察到一個原本被認為是不穩定的棋盤式超固體(checkerboard supersolid)；其次，當同時考慮足夠大的最鄰近與次鄰近赤力時，我們可以觀察到四分之一填滿的固體以及同型式的超固體(quarter-filled solid and quarter-filled supersolid)。 在我們所研讀的範圍裡，當在半填滿狀態時可以觀察到棋盤式以及條紋式的固體在四分之一填滿狀態時能觀察到四分之一填滿型式的固體。而相對應的超固體只能在系統的填滿密度遠離這些填滿狀態時才能觀察到；這強烈的建議我們在方形晶格中所觀察到的這些超固體都和他們的晶格缺陷有很大的關係。

Supersolid state of matter, which exhibits both the superfluid and the solid orders, was first proposed half a century ago. Since the theoretical predictions in the early 70s, there has been enormous amount of experimental attempts to search for its existence. Experimental evidence of the supersolid phase, however, is not present until the observation of superflow in bulk He^4 in 2004. This intriguing result renews the interests of this topic, and intensive research has been carried out to find the existence of the supersolid phase and understand possible mechanisms. However, the situation is still quite controversial. On the other hand, numerical studies on simple lattice models provide a useful theoretical prediction for this problem. With the help of Quantum Monte Carlo (QMC) simulation, physical behaviors of the boson Hubbard model, which captures the competition between the kinetic and the potential energies, can be analyzed exactly. Over the past decade, several numerical studies on the square lattice have been proposed. In the hardcore limit, the checkerboard supersolid is not a stable supersolid phase with only the nearest neighbor(NN)repulsion. In contrast, it is possible to observe a stable striped supersolid with only large next-nearest neighbor(NNN) repulsion. Such 'lattice supersolid' in the lattice models is not only an idealized prediction; nowadays, it is possible to trap ultracold atoms in optical potential.It is possible to make connection between lattice models and experiments on the optical lattice. This provides other possibilities to observe supersolid state. In this thesis, we study the phase diagram of the hardcore boson-Hubbard model on the 2D square lattice with both the nearest and the next-nearest neighbor hopping and repulsion. We study the model using bosonic Gutzwiller mean-field theory and the stochastic series expansion (SSE) QMC simulation with directed-loop algorithm. We observe a stable checkerboard supersolid when NNN hopping is introduced to the model. Slowly turning on the NN hopping, we see that checkerboard supersolid melts into a superfluid. We also find the quarter-filled solid and the quarter-filled supersolid emerge when the NN and the NNN repulsions are both large near quarter-filling. The quarter-filled and checkerboard supersolids are observed when the system is doped away from solid states at these fillings. In the intermediate region between the t'-V_1 model and t-V_2 model, we find that there is always a superfluid phase between the two different solid structures. It suggests that the superfluid is the preferred state in the strong competition region.