路径积分蒙特卡洛符号问题并非总是NP难的：谐振费米子可在二次时间内求解

中文摘要 AI 辅助

在谐振阱中费米子的路径积分蒙特卡洛（PIMC）模拟中，无论是否包含成对谐振相互作用，对于任意离散数量的虚时间切片（或珠子）以及任意选择的短时传播子，配分函数都可以从传播子的收缩行列式形式解析地获得。本工作表明，由此产生的递推关系可以用λ-环语言重新表述，从而在二维情况下得到以排列统计量表示的有限珠子配分函数的闭式表达式。该闭式配分函数可以通过一种特殊算法在O(n^2)时间内求值，为重现原始（不可行的）费米子PIMC模拟中n=10^4或更多费米子的能量提供了一种精确且数值稳定的方案。这一结果为完全绕过符号问题的数值不稳定性提供了一个具体框架，并作为反例驳斥了所有真正的费米子PIMC符号问题都是NP-hard的普遍观点。

英文摘要

In the Path Integral Monte Carlo (PIMC) simulation of fermions in a harmonic trap, with and without pairwise harmonic interactions, the partition functions for any discrete number of imaginary time slices (or beads) and for any choice of the short-time propagator can be analytically obtained from the contracted determinant form of the propagator. This work shows that the resulting recursion relation can be reformulated in the $λ$-ring language, yielding a closed-form finite-bead partition function in two dimensions in terms of permutation statistics. This closed-form partition function can be evaluated by a special algorithm in $O(n^2)$ time, providing an exact and numerically stable scheme for reproducing the energies of the original (undoable) fermion PIMC simulation for $n=10^4$ or more fermions. This result provides a concrete framework in which the numerical instability of the sign problem is completely bypassed, and serves as a counterexample to the prevailing view that all truly fermionic PIMC sign problems are NP-hard.

发表机构

• Hendrix Industries（亨德里克斯工业）
• Department of Physics and Astronomy, Texas A&M University（德州农工大学物理与天文学系）

机构由 AI 辅助整理，请以论文原文为准。