AI 中文总结
研究有限密度费米子系统符号问题，提出条件欧几里得 - 哈密顿（CEH）约化，结合两种描述，围绕参考问题和活跃部分计算，通过三个基准检验构造，支持从标量符号重加权到活跃空间近似转换，但未解决哈伯德符号问题及确定缩放比例。
AI 中文摘要
欧几里得蒙特卡罗方法在路径积分权重为实且非负时有效，但有限密度费米子系统在费米子部分被追踪后常产生变号或复标量权重。哈密顿公式避免了复权重采样问题，但面临希尔伯特空间快速增长。本文研究了一种结合这两种描述的条件欧几里得 - 哈密顿（CEH）约化。计算围绕一个蒙特卡罗可处理的参考问题和一个剩余的活跃部分展开。CEH 保持活跃部分为算符值，利用参考计算确定投影关联或转移矩阵，这些矩阵定义了一个有限有效哈密顿量。在有限秩时，结果是一个有效模型，其准确性须通过基扩展、度量条件、随机矩阵元误差和数扇区诊断来测试。在三个有限基准中检验了该构造。一个双通道振荡器测试条件基压缩；一个正测度随机计算测试关联矩阵收集和广义特征值问题提取；一个四格点哈伯德环结合了有限预算行列式符号应力测试和非目标活跃空间中的有限密度延续。基准支持从标量符号重加权到结构化有限模型中受监测的活跃空间近似的提议转换，但未提供哈伯德符号问题的端到端随机 CEH 处理或确定有利的缩放比例。实用性需要低秩近似性和有效提取所需的投影矩阵数据。
英文摘要
Euclidean Monte Carlo methods are effective when the path-integral weight is real and nonnegative, but finite-density fermion systems often produce sign-changing or complex scalar weights after the fermionic sector is traced out. Hamiltonian formulations avoid this complex-weight sampling problem but face rapid Hilbert-space growth. This paper studies a conditional Euclidean-Hamiltonian (CEH) reduction that combines these two descriptions. The calculation is organized around a Monte Carlo-tractable reference problem and a residual active sector. Instead of tracing the active sector into a determinant or scalar weight, CEH keeps it operator-valued and uses the reference calculation to determine projected correlation or transfer matrices. These matrices define a finite effective Hamiltonian, with the remaining finite-density dependence introduced after projection. At finite rank, the result is an effective model whose accuracy must be tested through basis enlargement, metric conditioning, stochastic matrix-element errors, and number-sector diagnostics. The construction is examined in three finite benchmarks. A two-channel oscillator tests conditional basis compression; a positive-measure stochastic calculation tests correlation-matrix harvesting and GEVP extraction, including a comparison with a PDMS-style estimator; and a four-site Hubbard ring combines a finite-budget determinant-sign stress test with finite-density continuation in non-target active spaces. The benchmarks support the proposed trade from scalar sign reweighting to a monitored active-space approximation in structured finite models, but they do not provide an end-to-end stochastic CEH treatment of the Hubbard sign problem or establish favorable scaling. Usefulness requires both low-rank approximability and efficient extraction of the required projected matrix data.
Comments20 pages, 3 figures, 4 tables
