自旋投影矩阵乘积态（SP-MPS）：强关联系统的通用工具
Spin-projected matrix product states (SP-MPS): a versatile tool for strongly correlated systems
摘要：
我们提出了一个新的波函数分析，结合自旋投影的强度与矩阵乘积态（MPS）和矩阵乘积算子（MPO）的语言中使用的密度矩阵重整化群（DMRG）。具体地，自旋投影矩阵乘积态（SP-MPS）被构造为$|Ψ^{(N,S,M)}_{SP-MPS}\rangle=\mathcal{P}_S|Ψ_{MPS}^{(N,M)}\rangle$，其中$\mathcal{P}_S$是总自旋$S$的自旋投影器，而$|Ψ_{MPS}^{(N,M)}\rangle$是具有给定粒子数$N$和自旋投影$M$的MPS波函数。这种新方法具有几个吸引人的特点：（1）与显式地将非阿贝尔SU（2）对称性引入MPS相比，它提供了一种更简单的实现自旋自适应（即创建$\hat{S}^2$的本征函数）的途径。特别地，由于SP-MPS中的基本状态仅使用阿贝尔对称性，因此不需要如通常在自旋自适应DMRG中采用的用于非单态的单态嵌入方案来实现单个一致的变分优化状态。(2)由于使用$|Ψ_{MPS}^{(N,M)}\rangle$作为其基础状态，SP-MPS可以紧密地连接到对称性破缺的平均场态。这允许直接产生大量的破缺对称性猜测，需要探索磁系统中复杂的电子景观。此外，这种联系可以在开壳层系统的量子嵌入理论的未来发展中加以利用。(3)哈密顿量和自旋投影器$\mathcal{P}_S$的MPO表示的总和自然导致用于计算期望值和优化SP-MPS的并行算法。(4)优化SP-MPS属于自旋投影理论的投影后变分（VAP）类。
英文摘要：
We present a new wavefunction ansatz that combines the strengths of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as $|Ψ^{(N,S,M)}_{SP-MPS}\rangle=\mathcal{P}_S|Ψ_{MPS}^{(N,M)}\rangle$, where $\mathcal{P}_S$ is the spin projector for total spin $S$ and $|Ψ_{MPS}^{(N,M)}\rangle$ is an MPS wavefunction with a given particle number $N$ and spin projection $M$. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin-adaptation (i.e. to create eigenfunctions of $\hat{S}^2$) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for non-singlet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) Due to the use of $|Ψ_{MPS}^{(N,M)}\rangle$ as its underlying state, the SP-MPS can be closely connected to broken symmetry mean-field states. This allows to straightforwardly generate the large number of broken symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector $\mathcal{P}_S$ naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin projected theories.
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