[论文解读] Solving the time-independent Schr\"odinger equation for chains of coupled excitons and phonons using tensor trains

Patrick Gelß, Rupert Klein+2 more|arXiv (Cornell University)|Sep 30, 2021

Quantum and electron transport phenomena参考文献 57被引用 11

一句话总结

本文提出一种基于张量-列车（TT）格式的方法，用于求解具有局域及最近邻相互作用的一维激子-声子系统的定态薛定谔方程。通过利用低秩张量分解，并结合带有Wielandt降秩的增强交替线性算法，该方法有效缓解了维度灾难问题，实现了对基态和激发态的精确计算，并成功再现了在不同耦合强度下Davydov的自陷现象。

ABSTRACT

We demonstrate how to apply the tensor-train format to solve the time-independent Schr\"{o}dinger equation for quasi one-dimensional excitonic chain systems with and without periodic boundary conditions. The coupled excitons and phonons are modeled by Frenkel-Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. We reduce the memory consumption as well as the computational costs significantly by employing efficient decompositions to construct low rank tensor-train representations, thus mitigating the curse of dimensionality. In order to compute also higher quantum states, we introduce an approach which directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems. Besides systems with coupled excitons and phonons, we also investigate uncoupled problems for which (semi-)analytical results exist. There, we find that in case of homogeneous systems the tensor-train ranks of state vectors only marginally depend on the chain length which results in a linear growth of the storage consumption. However, the CPU time increases slightly faster with the chain length than the storage consumption because the alternating linear scheme adopted in our work requires more iterations to achieve convergence for longer chains and a given rank. Finally, we demonstrate that the tensor-train approach to the quantum treatment of coupled excitons and phonons makes it possible to directly tackle the phenomenon of mutual self-trapping. We are able to confirm the main results of the Davydov theory, i.e., the dependence of the wavepacket width and the corresponding stabilization energy on the exciton-phonon coupling strength, though only for a certain range of that parameter. In future work, our approach will allow calculations also beyond the restrictions of the Frenkel-Holstein type Hamiltonians.

研究动机与目标

• 为解决准一维激子-声子系统求解定态薛定谔方程时面临的维度灾难问题。
• 开发一种数值稳定且高效的算法，用于计算基态以外的更高激发态。
• 实现基于Fröhlich-Holstein型哈密顿量的激子-声子系统中相互自陷现象的直接模拟。
• 在均匀系统中与半解析结果对比验证该方法，并确认波包局域化等已知物理现象。

提出的方法

• 采用张量-列车（TT）格式将量子态矢量和哈密顿矩阵表示为低秩张量网络，从而降低存储和计算成本。
• 使用截断奇异值分解（SVD）对TT核心进行正交化和秩截断，以保持低秩结构。
• 将交替线性算法（ALS）适配于特征值问题，并引入Wielandt降秩技术，实现多个激发态的顺序计算。
• 通过超核心分解构建哈密顿量的紧凑TT表示，实现高效的矩阵-向量运算。
• 应用专为TT设计的归一化与核心更新算法，确保迭代求解过程中的数值稳定性。
• 通过引入具有秩转置结构的特殊TT核心，实现周期性边界条件的循环系统扩展。

实验结果

研究问题

• RQ1 张量-列车格式能否以极低的存储和计算成本，高效表示并求解具有耦合激子-声子链的定态薛定谔方程？
• RQ2 在均匀系统中，量子态矢量的张量-列车秩如何随链长变化？
• RQ3 该方法在多大程度上能再现Davydov理论对激子-声子耦合下自陷行为的预测？
• RQ4 该方法能否可靠地计算更高激发态？收敛行为如何随系统尺寸和张量秩变化？

主要发现

• 在均匀系统中，态矢量的TT秩随链长增加仅略有上升，导致存储消耗呈线性增长。
• 由于在固定秩下长链需要更多ALS迭代才能收敛，计算时间的增长速度略快于存储消耗。
• 该方法成功再现了Davydov理论对波包宽度和稳定化能量随激子-声子耦合强度变化的预测。
• 该方法确认了在一定耦合参数范围内存在自陷态，验证了相互自陷的物理机制。
• 张量-列车框架使强关联体系中激发态的直接计算成为可能，克服了标准量子蒙特卡洛方法和基于网格方法的局限性。
• 该方法具有鲁棒性和可扩展性，未来可拓展至Fröhlich-Holstein模型和Davydov理论参数范围之外的应用。