量子多体系统蒙特卡罗模拟中离散逆拉普拉斯变换的病态性分析

中文摘要 AI 辅助

将虚时量子蒙特卡罗数据解析延拓到实频谱，需要对严重不适定的双边拉普拉斯变换求逆，这在量子多体动力学性质计算中自然出现。本文区分了连续逆双边拉普拉斯问题的内在不适定性与有限维离散化的条件数。对于等距采样和重建网格，我们通过具有指数分布节点的对角缩放单项式范德蒙德矩阵来表达离散问题。施加物理细致平衡对称性将离散化转化为对角缩放切比雪夫-范德蒙德系统。利用这些结构，我们以物理和离散化参数推导出条件数的显式下界和上界。对于无约束离散化，我们的界揭示了条件数随重建维度呈超指数增长，这不能仅通过增加虚时样本数来消除。细致平衡显著改善了条件数，尤其是在实际相关的预渐近区域，尽管渐近超指数依赖性仍然存在。在两种情况下，我们的界识别出一个低维区域，在该区域中超指数贡献被抑制，稳定重建仍然可行。这些结果为解析延拓中低维谱表示和细致平衡的有效性提供了数学解释。虽然它们没有消除根本的不适定性，但显著减轻了其有限维离散化的病态性，延迟了灾难性超指数增长的开始，从而扩大了数值可访问的重建范围。

英文摘要

Analytic continuation of imaginary-time quantum Monte Carlo data to real-frequency spectra requires the inversion of a severely ill-posed two-sided Laplace transform and arises naturally in quantum many-body calculations of dynamic properties. In this work, we distinguish the intrinsic ill-posedness of the continuous inverse two-sided Laplace problem from the conditioning of its finite-dimensional discretization. For equidistant sampling and reconstruction grids, we express the discrete problem via a diagonally scaled monomial Vandermonde matrix with exponentially distributed nodes. Imposing the physical detailed-balance symmetry transforms the discretization into a diagonally scaled Chebyshev-Vandermonde system. Exploiting these structures, we derive explicit lower and upper bounds on the condition numbers in terms of the physical and discretization parameters. For the unconstrained discretization, our bounds reveal super-exponential growth of the condition number with the reconstruction dimension, which cannot be removed by increasing the number of imaginary-time samples alone. Detailed-balance substantially improves the conditioning, especially in the practically relevant pre-asymptotic regime, although the asymptotic super-exponential dependence remains. In both cases, our bounds identify a low-dimensional regime in which the super-exponential contribution is suppressed and stable reconstruction remains feasible. These results provide a mathematical explanation for the effectiveness of low-dimensional spectral representations and detailed-balance in analytic continuation. While they do not remove the fundamental ill-posedness, they substantially mitigate the ill-conditioning of its finite-dimensional discretization, delay the onset of its catastrophic super-exponential growth and thereby enlarge the range of numerically accessible reconstructions.

发表机构

• Center for Advanced Systems Understanding, Helmholtz-Zentrum Dresden-Rossendorf（Helmholtz-Zentrum Dresden-Rossendorf 高级系统理解中心）
• Institute of Radiation Physics, Helmholtz-Zentrum Dresden-Rossendorf（Helmholtz-Zentrum Dresden-Rossendorf 辐射物理研究所）
• Mathematical Institute, University of Wrocław（弗罗茨瓦夫大学数学研究所）

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