Title: Distinct Types of Parent Hamiltonians for Quantum States: Insights from the $W$ State as a Quantum Many-Body Scar Show Chinese title

Title: 量子态的不同类型的母哈密顿量：从$W$态作为量子多体疤痕的见解

Abstract: The construction of parent Hamiltonians that possess a given state as their ground state is a well-studied problem. In this work, we generalize this notion by considering simple quantum states and examining the local Hamiltonians that have these states as exact eigenstates. These states often correspond to Quantum Many-Body Scars (QMBS) of their respective parent Hamiltonians. Motivated by earlier works on Hamiltonians with QMBS, in this work we formalize the differences between three distinct types of parent Hamiltonians, which differ in their decompositions into strictly local terms with the same eigenstates. We illustrate this classification using the $W$ state as the primary example, for which we rigorously derive the complete set of local parent Hamiltonians, which also allows us to establish general results such as the existence of asymptotic QMBS, and distinct dynamical signatures associated with the different parent Hamiltonian types. Finally, we derive more general results on the parent Hamiltonian types that allow us to obtain some immediate results for simple quantum states such as product states, where only a single type exists, and for short-range-entangled states, for which we identify constraints on the admissible types. Altogether, our work opens the door to classifying the rich structures and dynamical properties of parent Hamiltonians that arise from the interplay between locality and QMBS. Show Chinese abstract

Abstract: 构造具有给定状态作为其基态的母哈密顿量是一个被广泛研究的问题。 在本工作中，我们通过考虑简单的量子态并检查这些态作为精确本征态的局部哈密顿量来推广这一概念。 这些态通常对应于它们各自母哈密顿量的量子多体疤痕（QMBS）。 受早期关于具有QMBS的哈密顿量工作的启发，在本工作中我们形式化了三种不同类型的母哈密顿量之间的差异，这些类型在将其分解为具有相同本征态的严格局部项时有所不同。 我们使用$W$态作为主要例子来说明这种分类，对于该态，我们严格推导出所有局部母哈密顿量的完整集合，这也使我们能够建立一些一般性结果，例如渐近QMBS的存在性，以及与不同母哈密顿量类型相关的不同动力学特征。 最后，我们得出了关于母哈密顿量类型的更一般的结果，使我们能够获得一些简单量子态的即时结果，例如乘积态，其中只存在一种类型，以及短程纠缠态，对于这些态，我们确定了可接受类型的约束。 总的来说，我们的工作为分类由局域性和QMBS相互作用产生的母哈密顿量的丰富结构和动力学特性打开了大门。