Numerical challenges in Quantum Monte Carlo simulations in condensed matter physics

我们将解决凝聚态系统量子蒙特卡罗(QMC)模拟中的两个悬而未决的问题。第一个挑战是达到允许与实验直接比较的晶格大小。这些大型晶格模拟将使我们能够研究各种基于石墨烯的问题，包括拥有平带并受到强大的长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，这个问题经常阻碍对许多实验相关系统的研究。对于这两个挑战，都将采取统一的方法。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，来使用发展起来的形式来处理晶格量子色动力学中的超大型系统。在某些情况下，与标准辅助场QMC相比，它在系统大小上具有更好的伸缩性。正如我们在下面的一些示例中演示的那样，当在图形处理单元(GPU)上有效地实施该算法时，可以达到大小为102x102的晶格。例如，我们已经能够在非微扰QMC计算中第一次观察到费米速度的对数发散。使用连续辅助场的方法还允许我们以一种新颖和令人兴奋的方式来解决符号问题。柯西定理适用于连续场，它允许我们将泛函积分的积分域转移到复空间。证明了在某些情况下，我们可以构造一个符号大大减少的最优轮廓线问题。它由一组Lefschetz顶针组成，每个顶针对应于一组点，这些点在最陡峭的下降方案中一直作用到鞍座。由于被积函数的复相沿每个顶针是恒定的这一事实，符号问题沿着这个最优轮廓通常要弱得多。因此，向这个轮廓线转移到复数空间，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对复杂空间中曲线流形上的场配置进行采样。另一个好处是，我们可以洞察准确的鞍点，而不需要任何空间或欧几里得时间一致性的假设。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.