非监督学习高维多体波函数矢量轨迹所在低维子空间

Low-dimensional trajectory subspace of high-dimensional many-body wavefunctions by unsupervised learning

Frame等提出采用特征矢量延拓方法求解关联量子模型的高维多体波函数:当模型哈密顿矩阵包含光滑变化的参数时,其特征矢量随参数变化的轨迹集中在1个低维的子空间中,因此可以将哈密顿量投影到该子空间的一组基矢量来简化求解(Frame D,He R Z,Ipsen I,Lee D,Lee D,Rrapaj E 2018 Phys.Rev.Lett.121032501).但是他们没有明确给出轨迹子空间的维度以及其与模型大小之间的联系.本文系统研究了大小不同的反铁磁Heisenberg链模型,其交换相互作用随参数光滑变化.首先通过主成分分析方法分别确定了包含4个自旋的模型和包含6个自旋的模型的基态多体波函数矢量轨迹子空间,并分别绘制了子空间中的轨迹.然后分析了包含8,…,14个自旋的模型基态矢量轨迹的主成分分量,并指出:当采用特征矢量延拓方法求解反铁磁Heisenberg链模型基态时,所需基矢数目随模型所包含自旋个数的增加而增加.本文研究可用于指导采用特征矢量延拓方法求解包含更多自旋的反铁磁Heisenberg链模型哈密顿量.

Frame et al.(Frame D,He R Z,Ipsen I,Lee D,Lee D,Rrapaj E 2018 Phys.Rev.Lett.121032501)proposed to use eigenvector continuation to solve high-dimensional many-body wavefunctions of relevant quantum models.When a model’s Hamiltonian matrix includes smoothly varying parameters,the corresponding eigenvector trajectory spans only a low-dimensional subspace.Therefore,it is possible to simplify the calculations by projecting the Hamiltonian onto a set of basis vectors of this subspace.However,the dimension of the trajectory subspace and its relationship with the size of the model are still unclear.In this paper,we study the antiferromagnetic Heisenberg chain models of different sizes systematically;their exchange interactions change with parameters smoothly.We first use principal component analysis to determine the subspaces of ground state many-body wavefunction vector trajectories of a 4-spin model and a 6-spin model,and plot the trajectories in the subspaces,respectively;we then analyze the principal components of ground state vector trajectories of models including 8,…,14 spins,and reveal that when using eigenvector continuation to solve the ground state of an antiferromagnetic Heisenberg chain model,the number of basis vectors required increases with the number of spins in the model increasing.Our study can guide the application of eigenvector continuation in solving the Hamiltonian of a Heisenberg chain model containing more spins.