The quantum Monte Carlo method on asymptotic Lefschetz thimbles is a numerical algorithm devised specifically for alleviation of the sign problem appearing in the simulations of quantum many-body systems. In this method, the sign problem is alleviated by shifting the integration domain for the auxiliary fields, appearing, for example, in the conventional determinant quantum Monte Carlo method, from real space to an appropriate manifold in complex space. Here, we extend this method to quantum spin models with generic two-spin interactions, by using the Hubbard-Stratonovich transformation to decouple the exchange interactions and the Popov-Fedotov transformation to map the quantum spins to complex fermions. As a demonstration, we apply the method to the Kitaev model in a magnetic field whose ground state is predicted to deliver a topological quantum spin liquid with non-Abelian anyonic excitations. To illustrate how the sign problem is alleviated in this method, we visualize the asymptotic Lefschetz thimbles in complex space, together with the saddle points and the zeros of the fermion determinant. We benchmark our method in the low-temperature region in a magnetic field and show that the sign of the action is recovered considerably and unbiased numerical results are obtained with sufficient precision.

中文翻译：

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量子自旋系统渐近 Lefschetz 顶针的量子蒙特卡罗方法：在磁场中应用 Kitaev 模型

渐近 Lefschetz 顶针上的量子蒙特卡罗方法是一种数值算法，专门用于缓解量子多体系统模拟中出现的符号问题。在这种方法中，符号问题通过将辅助场的积分域移动来缓解，例如，出现在传统行列式量子蒙特卡罗方法中，从实空间到复空间中的适当流形。在这里，我们将这种方法扩展到具有通用双自旋相互作用的量子自旋模型，通过使用 Hubbard-Stratonovich 变换来解耦交换相互作用，并使用 Popov-Fedotov 变换将量子自旋映射到复杂的费米子。作为示范，我们将该方法应用于磁场中的 Kitaev 模型，该模型的基态预计会提供具有非阿贝尔任意子激发的拓扑量子自旋液体。为了说明这种方法如何缓解符号问题，我们将复杂空间中的渐近 Lefschetz 顶针与鞍点和费米子行列式的零点一起可视化。我们在磁场中的低温区域对我们的方法进行了基准测试，并表明动作的符号得到了显着恢复，并且以足够的精度获得了无偏的数值结果。