MPS流形上包含Riemann度量的最简非齐次矩阵积态的显式性质
Explicit properties of the simplest inhomogeneous Matrix-Product-State including the Riemann metric of the MPS manifold
摘要：
我们考虑了一个N个量子自旋的开链的最简单的非均匀矩阵积态，其中每个位置只涉及两个角，每个键只涉及两个角，具有以下直接的物理意义。与位点$k$相关联的两个角度是两个布洛赫角，其参数化单独自旋$k$的约化密度矩阵$ρ_k$的两个正交归一化本征向量。与键$(k,k+1)$相关的两个角度参数化了键$(k,k+1)$上施密特分解的纠缠特性。给出了两个连续位置的约化密度矩阵$ρ_{k,k+1}$的显式结果，这是需要评估两体哈密顿量的能量，并为约化密度矩阵$ρ_{k,k+r}$的两个站点在距离$r$，这是需要评估的自旋-自旋相关距离$r$。由这些$(4N-2)$角参数化的MPS流形的全局结构然后由其显式黎曼度量表征。最后，讨论了对任意无环树结构和周期边界条件链的推广。
英文摘要：
We consider the simplest inhomogeneous Matrix-Product-State for an open chain of N quantum spins that involves only two angles per site and two angles per bond with the following direct physical meanings. The two angles associated to the site $k$ are the two Bloch angles that parametrize the two orthonormal eigenvectors of the reduced density matrix $ρ_k$ of the spin $k$ alone. The two angles associated to the bond $(k,k+1)$ parametrize the entanglement properties of the Schmidt decomposition across the bond $(k,k+1)$. Explicit results are given for the reduced density matrix $ρ_{k,k+1}$ of two consecutive sites that is needed to evaluate the energy of two-body Hamiltonians, and for the reduced density matrix $ρ_{k,k+r}$ of two sites at distance $r$ that is needed to evaluate the spin-spin correlations at distance $r$. The global structure of the MPS manifold as parametrized by these $(4N-2)$ angles is then characterized by its explicit Riemann metric. Finally, the generalizations to any tree-like structure without loops and to the chain with periodic boundary conditions are discussed.
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