迭代固定节点动力学的几何视角

中文摘要 AI 辅助

量子多体基态的符号结构是固定节点方法中缓解费米子符号问题的核心量。对于连续电子结构哈密顿量，正确的符号结构足以计算精确的基态性质；另一方面，晶格固定节点方法保留了对方程试探波函数振幅的额外依赖，这些振幅在保持其符号结构的同时以非最优方式被改进。在此，我们考虑一个迭代映射，该映射通过反复将固定节点试探态替换为其关联的晶格固定节点哈密顿量的基态来获得（无论实际算法是否以及如何实现此迭代）。我们证明了该映射的不动点是哈密顿量和约化哈密顿量的本征态，导致每个符号室内部至多有一个不动点。我们证明了基态不动点是稳定的，且基态符号室的边界是排斥的。这表明，在自洽迭代下，超出正确基态符号结构的振幅依赖性消失。在其他符号室中，能量下降可能被强加的符号结构所阻碍；这种张力导致迭代将选定振幅驱动至零，产生向室边界的“支撑坍缩”，并对应于约化哈密顿量。因此，激发态不动点可证明是不稳定的，并流向能量更低的边界不动点。我们证明了符号室边界具有方向稳定性；在一个方向上吸引的边界点在符号翻转下是排斥的。通过揭示迭代固定节点动力学的几何和稳定性结构，我们的结果阐明了费米子符号问题核心方法的基础，并激发了潜在的新算法策略。

英文摘要

The sign-structure of a quantum many-body ground state is a central quantity in fixed-node approaches to mitigating the Fermion sign problem. For continuum electronic-structure Hamiltonians, the correct sign-structure is sufficient to compute exact ground state properties; on the other hand, lattice fixed-node approaches retain an additional dependence on trial wave-function amplitudes which are improved upon non-optimally while preserving their sign structure. Here we consider an iterative map obtained by repeatedly replacing the fixed-node trial state with the ground state of its associated lattice fixed-node Hamiltonian (independent of whether and how a practical algorithm could implement this iteration). We show the fixed points of this map are eigenstates of the Hamiltonian and reduced Hamiltonian resulting in at most one fixed point in the interior of each sign chamber. We prove that the ground state fixed point is stable and that the boundary of the ground state sign chamber is repulsive. This shows the amplitude dependence beyond having the correct ground state sign-structure disappears under self-consistent iteration. In other sign chambers, energy descent can be obstructed by the imposed sign structure; this tension leads to the iteration driving select amplitudes to zero producing `support collapse' onto chamber boundaries and corresponding to reduced Hamiltonians. Excited-state fixed points are therefore provably unstable and flow toward lower-energy boundary fixed points. We show sign chamber boundaries have a directional stability; boundary points attractive in one direction are repulsive under a sign flip. By revealing the geometry and stability structure of iterative fixed-node dynamics, our results clarify the foundations of a central approach to the Fermion sign problem and motivate potential new algorithmic strategies.

发表机构

• University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）

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