Numerical challenges in Quantum Monte Carlo simulations in condensed matter physics

我们将讨论凝聚态系统的量子蒙特卡罗（QMC）模拟中的两个突出问题。第一个挑战是达到可以与实验直接比较的晶格尺寸。这些大晶格模拟将使我们能够研究各种基于石墨烯的问题，包括承载平带和受强远程库仑排斥的超晶格结构，以及流体动力学流动的研究。第二个挑战解决了符号问题，这通常阻碍了许多实验相关系统的研究。将采取统一的方法应对这两项挑战。我们将利用具有连续辅助场的量子多体系统配分函数的特殊表示。这种方法对我们有利，因为我们可以通过采用费米子行列式的随机表示，使用晶格量子色动力学中开发的形式来处理超大系统。在某些情况下，与标准辅助字段QMC相比，它会导致系统大小的优越缩放。正如我们在下面的一些示例中所演示的那样，当在图形处理单元（gpu）上有效实现时，该算法可以达到102x102大小的格。例如，我们已经能够在非微扰QMC计算中首次观察到费米速度的对数散度。使用连续辅助字段的方法也使我们能够以一种新颖而令人兴奋的方式解决符号问题。柯西定理对于连续域是有效的，它允许我们将泛函积分的积分域转移到复空间中。结果表明，在某些情况下，我们可以用实质约简问题构造最优轮廓。它由Lefschetz顶针的集合组成，每个顶针对应于最陡下降方案中向下作用到马鞍的点集。由于被积函数的复相位在每个顶针上都是恒定的，因此符号问题在这个最优轮廓上通常要弱得多。因此，向这个轮廓转移到复空间，可以给我们一个有效的算法来抑制符号问题。我们打算研究各种算法方法来有效地采样复空间中弯曲流形上的场构型。一个额外的好处是，我们可以洞察精确的鞍点，而不需要任何假设，比如空间或欧几里得时间的均匀性。这使我们能够系统地建立更精确的准经典近似。

We will address two outstanding problems in quantum Monte Carlo (QMC) simulations of condensed matter systems. The first challenge is to reach lattices sizes that allow direct comparison with experiments. These large lattice simulations will allow us to study various graphene-based problems including, superlattice structures hosting flat bands and subject to strong long-range Coulomb repulsion, and the study of hydrodynamic flow. The second challenge addresses the sign problem, which often prevents the study of many experimentally relevant systems. A unified approach will be taken for both challenges. We will exploit the special representation of the partition function of the quantum many-body system with continuous auxiliary fields. This approach works to our advantage, since we can use the formalism developed to deal with extremely large systems in lattice quantum chromodynamics, by adopting a stochastic representation of the fermionic determinant. In some cases, it leads to a superior scaling in system size as compared to the standard auxiliary field QMC. As we demonstrate on some examples below, this algorithm, when implemented efficiently on Graphic Processing Units (GPUs) can reach lattices as large as 102x102. For instance, we already have been able to observe the logarithmic divergence of the Fermi velocity for the first time in non-perturbative QMC calculations. The approach using continuous auxiliary fields also allows us to address the sign problem in a novel and exciting way. Cauchy's theorem, which is valid for continuous fields, allows us to shift the integration domain of the functional integral into complex space. It was shown that in some cases we can construct an optimal contour with substantially reduced sign problem. It consists of an ensemble of Lefschetz thimbles, each corresponding to the set of points that role down to a saddle within a steepest descent scheme. Due to the fact that the complex phase of the integrand is constant along each thimble, the sign problem is often substantially weaker along this optimal contour. Thus, the shift into complex space towards this contour, can give us an efficient algorithm to suppress the sign problem. We intend to study various algorithmic approaches to efficiently sample field configurations over curved manifolds in complex space. An additional benefit is that we gain insight on exact saddle points, without any assumptions like uniformity in space or Euclidean time. This allows us to systematically build more accurate quasi-classical approximations.