Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia.
J Chem Phys. 2023 Jan 28;158(4):044108. doi: 10.1063/5.0133597.
Ultracold-atom simulations of the Hubbard model provide insights into the character of charge and spin correlations in and out of equilibrium. The corresponding numerical simulations, on the other hand, remain a significant challenge. We build on recent progress in the quantum Monte Carlo (QMC) simulation of electrons in continuous space and apply similar ideas to the square-lattice Hubbard model. We devise and benchmark two discrete-time QMC methods, namely the fermionic-propagator QMC (FPQMC) and the alternating-basis QMC (ABQMC). In FPQMC, the time evolution is represented by snapshots in real space, whereas the snapshots in ABQMC alternate between real and reciprocal space. The methods may be applied to study equilibrium properties within the grand-canonical or canonical ensemble, external field quenches, and even the evolution of pure states. Various real-space/reciprocal-space correlation functions are also within their reach. Both methods deal with matrices of size equal to the number of particles (thus independent of the number of orbitals or time slices), which allows for cheap updates. We benchmark the methods in relevant setups. In equilibrium, the FPQMC method is found to have an excellent average sign and, in some cases, yields correct results even with poor imaginary-time discretization. ABQMC has a significantly worse average sign, but also produces good results. Out of equilibrium, FPQMC suffers from a strong dynamical sign problem. On the contrary, in ABQMC, the sign problem is not time-dependent. Using ABQMC, we compute survival probabilities for several experimentally relevant pure states.
利用冷原子模拟 Hubbard 模型可以深入了解非平衡和平衡条件下的电荷和自旋关联特性。然而，相应的数值模拟仍然是一个重大挑战。我们基于连续空间中电子量子蒙特卡罗（QMC）模拟的最新进展，并将类似的思想应用于正方形晶格 Hubbard 模型。我们设计并基准测试了两种离散时间 QMC 方法，即费米子传播子 QMC（FPQMC）和交替基 QMC（ABQMC）。在 FPQMC 中，时间演化由实空间中的快照表示，而 ABQMC 中的快照在实空间和倒空间之间交替。这些方法可用于研究巨正则或正则系综中的平衡性质、外场淬火，甚至纯态的演化。各种实空间/倒空间相关函数也在其范围内。这两种方法都处理大小等于粒子数的矩阵（因此与轨道数或时间片无关），这允许进行廉价的更新。我们在相关设置中对方法进行基准测试。在平衡条件下，FPQMC 方法的平均符号非常好，在某些情况下，即使在较差的虚时间离散化下也能得到正确的结果。ABQMC 的平均符号明显较差，但也能得到较好的结果。在非平衡条件下，FPQMC 受到强烈的动力学符号问题的困扰。相反，在 ABQMC 中，符号问题与时间无关。使用 ABQMC，我们计算了几个与实验相关的纯态的存活概率。
