二级相变临界理论研究的微妙之处
Subtlety of Studying the Critical Theory of a Second Order Phase Transition
摘要：
利用第一性原理Monte Carlo方法研究了蜂窝晶格上扩展的Bose-哈伯德模型从超固相到ρ = 1/2固相的量子相变.我们研究的动机是定量地了解理论输入的影响，特别是动力学临界指数z，在计算临界指数nu。因此，我们分别进行了四组模拟，其中beta = 2N^{1/2}，beta = 8 N ^{1/2}，beta = N/2和beta = N/4。这里β是温度的倒数，N是模拟中使用的晶格位置的数量。通过对第二空间方向上可观测到的超流密度rho_{s2}应用数据塌陷，我们证实了这种转变确实受超流-绝缘体普适性类的支配。然而，我们发现确定临界点的精确位置是微妙的。例如，对于使用β = 2N^{1/2}获得的数据，临界化学势（mu/V）_c出现在（mu/V）_c = 2.3239（3）处，而从使用β = N/2模拟的数据确定的（mu/V）_c被发现为（mu/V）_c = 2.3186（2）。此外，虽然可以利用在模拟中使用β = N/4确定的数据来获得rho_{s2} N的良好数据崩溃，但是对于从具有β = 8 N ^{1/2}的另一组模拟计算的相同可观察量，几乎不能达到合理的数据崩溃质量。令人惊讶的是，假设在蒙特卡罗计算中首先确定该相变的z为2，则对于（mu/V）_c ~ 2.3184和nu ~ 0.7，使用β = 8 N ^{1/2}获得的数据可以实现rho_{s2} N的高质量数据崩溃。我们的研究结果意味着，人们可能需要重新考虑一些模型的相图，如果临界点的准确位置是至关重要的，在获得一个结论。
英文摘要：
We study the quantum phase transition from a super solid phase to a solid phase of rho = 1/2 for the extended Bose-Hubbard model on the honeycomb lattice using first principles Monte Carlo calculations. The motivation of our study is to quantitatively understand the impact of theoretical input, in particular the dynamical critical exponent z, in calculating the critical exponent nu. Hence we have carried out four sets of simulations with beta = 2N^{1/2}, beta = 8N^{1/2}, beta = N/2, and beta = N/4, respectively. Here beta is the inverse temperature and N is the numbers of lattice sites used in the simulations. By applying data collapse to the observable superfluid density rho_{s2} in the second spatial direction, we confirm that the transition is indeed governed by the superfluid-insulator universality class. However we find it is subtle to determine the precise location of the critical point. For example, while the critical chemical potential (mu/V)_c occurs at (mu/V)_c = 2.3239(3) for the data obtained using beta = 2N^{1/2}, the (mu/V)_c determined from the data simulated with beta = N/2 is found to be (mu/V)_c = 2.3186(2). Further, while a good data collapse for rho_{s2}N can be obtained with the data determined using beta = N/4 in the simulations, a reasonable quality of data collapse for the same observable calculated from another set of simulations with beta = 8N^{1/2} can hardly be reached. Surprisingly, assuming z for this phase transition is determined to be 2 first in a Monte Carlo calculation, then a high quality data collapse for rho_{s2}N can be achieved for (mu/V)_c ~ 2.3184 and nu ~ 0.7 using the data obtained with beta = 8N^{1/2}. Our results imply that one might need to reconsider the established phase diagrams of some models if the accurate location of the critical point is crucial in obtaining a conclusion.
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