First-principles diagrammatic Monte Carlo for electron-phonon interactions and polarons

Summing all Feynman diagrams with quantitative accuracy is a holy grail in theoretical physics. In condensed matter, the lattice vibration (phonon) field couples with the electrons, leading to the formation of entangled electron-phonon (e-ph) states called polarons. In the intermediate- and strong-coupling regimes common to many conventional and quantum materials, a many-body treatment of polarons requires adding up a large number of e-ph diagrams. Diagrammatic Monte Carlo (DMC) is an efficient method for diagram summation and has been employed to study polarons within simplified \e-ph models (Holstein, Frohlich, etc.). Here we show DMC calculations based on accurate first-principles e-ph interactions, enabling numerically exact results for ground-state and dynamical properties of polarons in real materials, including the polaron formation energy, effective mass, spectral weight, phonon cloud distribution, optical conductivity, and mobility. We demonstrate such DMC calculations in systems with polarons ranging from small (localized) to large (delocalized), including LiF, SrTiO3, and TiO2 rutile and anatase. This advance is enabled by our recently developed technique for compressing first-principles e-ph interaction matrices~\cite{SVD-paper}, together with a matrix-product formalism that mitigates the DMC sign problem from multiple electronic bands. Our work enables precise modeling of e-ph interactions and polarons in coupling regimes ranging from weak to strong, opening doors to studies of transport, linear response, and superconductivity in the strong e-ph coupling regime.

高精度求和所有费曼图（Feynman diagrams）是理论物理学领域的圣杯级目标。在凝聚态物理中，晶格振动场（声子，phonon）与电子发生耦合，进而形成纠缠态的电子-声子（electron-phonon, e-ph）束缚态，这类束缚态被称为极化子（polarons）。在众多传统材料与量子材料中普遍存在的中强耦合区域，对极化子开展多体理论处理时，需要对大量的电子-声子费曼图进行求和。图解蒙特卡洛（Diagrammatic Monte Carlo, DMC）是一种高效的费曼图求和方法，此前已被用于基于简化电子-声子模型（如Holstein模型、Fröhlich模型等）的极化子研究。本研究基于精确的第一性原理电子-声子相互作用开展图解蒙特卡洛计算，可获得真实材料中极化子基态与动力学性质的数值精确结果，涵盖极化子形成能、有效质量、谱权重、声子云分布、光导率以及迁移率等物理量。我们针对从小尺度（局域型）到大尺度（离域型）的各类极化子体系开展了上述图解蒙特卡洛计算，所涉及的材料包括氟化锂（LiF）、钛酸锶（SrTiO₃）以及金红石相（rutile）和锐钛矿相（anatase）二氧化钛（TiO₂）。这一研究进展得益于我们近期提出的第一性原理电子-声子相互作用矩阵压缩技术~cite{SVD-paper}，以及一套可缓解多电子能带带来的图解蒙特卡洛符号问题的矩阵乘积形式体系。本研究实现了从弱耦合到强耦合全区间内电子-声子相互作用与极化子的精准建模，为强电子-声子耦合区域内的输运性质、线性响应以及超导性等研究打开了全新的路径。