用混合蒙特卡洛形式重新表述 Pfaffian 量子蒙特卡洛
Reformulating Pfaffian Quantum Monte Carlo with the Hybrid Monte Carlo formalism
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中文总结 AI 辅助
本文提出将混合蒙特卡洛与 Pfaffian 量子蒙特卡洛结合，实现高效采样，并验证其在 Kitaev 链中的准确性。
中文摘要 AI 辅助
Pfaffian 量子蒙特卡洛将辅助场方法扩展到反常费米子哈密顿量，从而可以访问配对系统和 Hubbard-Stratonovich 分解，而这些在行列式形式中是不可用的。我们表明，这种灵活性可以与混合蒙特卡洛（HMC）的全局更新相结合，产生一种缩放为 $\mathcal{O}(N_\tau L^3)$ 的算法，类似于没有赝费米子的行列式量子蒙特卡洛（DQMC）对应算法。核心观察是分子动力学力仅依赖于 Pfaffian 权重的幅度，该权重允许行列式表示，并导致简化的力表达式。这种构造还以几乎零额外成本产生不等时马约拉纳关联函数。对于没有方便解析指数导数的相互作用核，我们采用截断的力展开，其误差由精确的 Metropolis 接受-拒绝步骤消除。应用于相互作用的 Kitaev 链时，该方法与精确对角化一致，并解析出位于相反边界上的低能模式，以及它们在拓扑转变附近不断增加的叠加。这些结果确立了 HMC 作为 Pfaffian 辅助场模拟的高效采样框架。
英文摘要
Pfaffian quantum Monte Carlo extends auxiliary field methods to anomalous fermionic Hamiltonians, opening access to pairing systems and Hubbard-Stratonovich decompositions that are unavailable in determinant formulations. We show that this flexibility can be combined with the global updates of Hybrid Monte Carlo (HMC), leading to an algorithm scaling as $\mathcal{O}(N_τL^3)$, similar to its determinant quantum Monte Carlo (DQMC) counterpart without pseudofermions. The central observation is that the molecular dynamics force depends only on the magnitude of the Pfaffian weight, which admits a determinant representation and leads to a simplified force expression. This construction also yields unequal-time Majorana correlators at essentially no additional cost. For interaction kernels with no convenient analytic exponential derivative, we employ a truncated force expansion whose errors are removed by the exact Metropolis accept-reject step. Applied to the interacting Kitaev chain, the method agrees with exact diagonalization and resolves low-energy modes localized at opposite boundaries, together with their increasing overlap near the topological transition. These results establish HMC as an efficient sampling framework for Pfaffian auxiliary-field simulations.
