量子格子模型的投影簇可加变换
Projective cluster-additive transformation for quantum lattice models
AI总结：
本文构建了一种基于投影的簇可加变换，可将广泛的格子哈密顿量块对角化，并推广了多种已知极小变换，在仅需较小基的条件下实现了微扰与非微扰链接簇展开的高效计算。
摘要：
我们构建了一种基于投影的簇可加变换，能够将广泛的格子哈密顿量 $\mathcal{H}=\mathcal{H}_0 +V$ 块对角化。其簇可加性是为 $\mathcal{H}_0$ 的简并激发子空间建立微扰或非微扰链接簇展开的关键要素。我们的变换推广了极小变换，该变换也以 Takahashi 变换、Schrieffer-Wolff 变换、des Cloiseaux 有效哈密顿量、正则 van Vleck 有效哈密顿量或双块正交化方法等名称为人所知。对于 $\mathcal{H}$ 的给定子空间（该子空间绝热连接到本征值为 $e_0^n$ 的 $\mathcal{H}_0$ 本征空间），其有效簇可加哈密顿量和变换仅依赖于与满足 $e_0^m\leq e_0^n$ 的 $e_0^m$ 相连接的 $\mathcal{H}$ 本征空间。相比之下，诸如多块正交化方法或微扰连续酉变换等其他簇可加变换则需要更大的基。这一点可被利用来在微扰和非微扰情况下高效地实现该变换。作为基准测试，我们在方格上横向场 Ising 模型的低场有序相中，针对单自旋翻转和双自旋翻转束缚态，执行了微扰和非微扰链接簇展开。
英文摘要：
We construct a projection-based cluster-additive transformation that block-diagonalizes wide classes of lattice Hamiltonians $\mathcal{H}=\mathcal{H}_0 +V$. Its cluster additivity is an essential ingredient to set up perturbative or non-perturbative linked-cluster expansions for degenerate excitation subspaces of $\mathcal{H}_0$. Our transformation generalizes the minimal transformation known amongst others under the names Takahashi's transformation, Schrieffer-Wolff transformation, des Cloiseaux effective Hamiltonian, canonical van Vleck effective Hamiltonian or two-block orthogonalization method. The effective cluster-additive Hamiltonian and the transformation for a given subspace of $\mathcal{H}$, that is adiabatically connected to the eigenspace of $\mathcal{H}_0$ with eigenvalue $e_0^n$, solely depends on the eigenspaces of $\mathcal{H}$ connected to $e_0^m$ with $e_0^m\leq e_0^n$. In contrast, other cluster-additive transformations like the multi-block orthognalization method or perturbative continuous unitary transformations need a larger basis. This can be exploited to implement the transformation efficiently both perturbatively and non-perturbatively. As a benchmark, we perform perturbative and non-perturbative linked-cluster expansions in the low-field ordered phase of the transverse-field Ising model on the square lattice for single spin-flips and two spin-flip bound-states.
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