多体系统基态的变分下界与密度矩阵的平方参数化
A variational lower bound on the ground state of a many-body system and the squaring parametrization of density matrices
摘要：
量子系统$E_{\rm gs} \leqslant \langle Ψ|H| Ψ\rangle$的基态能量$E_{\rm gs}$的变分上界是众所周知的（这里$H$是系统的哈密顿量，$Ψ$是任意波函数）。鲜为人知的是变的{ \it下限}上的基态。我们考虑一个这样的界限，这是有效的一个多体平行四边形不变的晶格系统。这样的晶格可以被划分为直到平移都相同的簇。这种系统的哈密顿量可以写为$H=\sum_{i=1}^M H_i$，其中项$H_i$在第$i$簇上被支持。边界读作$E_{\rm gs}\geqslant M \inf\limits_{ρ_{cl} \in {\mathbb S_{cl}^G}} {\rm tr}_{cl}ρ_{cl} \, H_{cl} $，其中${\mathbb S_{cl}^G}$是单个簇的一些明智选择的简化密度矩阵的集合。后一种变分原理的实现可能会受到集合$\mathbb M$参数化的困难的阻碍，这是变分过程的必要先决条件。这个困难的根本原因是非线性正约束$ρ>0$，这是由密度矩阵满足。密度矩阵的平方参数化，$ρ=τ^2/{\rm tr}\,τ^2$，其中$τ$是一个任意的（不一定是正的）厄米算子，占积极的自动。我们讨论了如何平方参数化可以用来找到变分下界的基态的双折射不变的多体系统。作为一个例子，我们考虑一维海森堡反铁磁体。
英文摘要：
A variational upper bound on the ground state energy $E_{\rm gs}$ of a quantum system, $E_{\rm gs} \leqslant \langle Ψ|H| Ψ\rangle$, is well-known (here $H$ is the Hamiltonian of the system and $Ψ$ is an arbitrary wave function). Much less known are variational {\it lower} bounds on the ground state. We consider one such bound which is valid for a many-body translation-invariant lattice system. Such a lattice can be divided into clusters which are identical up to translations. The Hamiltonian of such a system can be written as $H=\sum_{i=1}^M H_i$, where a term $H_i$ is supported on the $i$'th cluster. The bound reads $E_{\rm gs}\geqslant M \inf\limits_{ρ_{cl} \in {\mathbb S_{cl}^G}} {\rm tr}_{cl}ρ_{cl} \, H_{cl} $, where ${\mathbb S_{cl}^G}$ is some wisely chosen set of reduced density matrices of a single cluster. The implementation of this latter variational principle can be hampered by the difficulty of parameterizing the set $\mathbb M$, which is a necessary prerequisite for a variational procedure. The root cause of this difficulty is the nonlinear positivity constraint $ρ>0$ which is to be satisfied by a density matrix. The squaring parametrization of the density matrix, $ρ=τ^2/{\rm tr}\,τ^2$, where $τ$ is an arbitrary (not necessarily positive) Hermitian operator, accounts for positivity automatically. We discuss how the squaring parametrization can be utilized to find variational lower bounds on ground states of translation-invariant many-body systems. As an example, we consider a one-dimensional Heisenberg antiferromagnet.
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