利用交换子代数对量子多体伤疤的穷尽表征
Exhaustive Characterization of Quantum Many-Body Scars using Commutant Algebras
AI总结：
本文提出以交换子代数统一刻画量子多体伤疤，利用双交换子定理穷尽构造含伤疤哈密顿量代数，并阐明其扰动稳定性、形式体系等价性及对ETH违反的含义。
摘要：
我们以交换子代数的语言研究量子多体伤疤（QMBS），交换子代数被定义为局域哈密顿量族的对称代数。该框架解释了在具有精确QMBS的模型中观察到的动力学断连子空间的起源，即大的“热”子空间和小的“非热”子空间，它们归因于交换子中存在非传统的非局域守恒量；因此，这将传统对称性与弱遍历性破缺现象的研究统一到一个单一框架中。此外，这种语言使我们能够利用冯·诺依曼双交换子定理（DCT），形式化地写下具有一组所需QMBS的所有哈密顿量的穷尽代数，这证明QMBS在大类局域扰动下仍然存在。我们用若干QMBS的标准例子来说明这一点，包括spin-1/2铁磁体、AKLT、spin-1 XY π-双磁振子以及电子η配对态塔；在每一种情形中，我们都明确写下了具有这些QMBS的哈密顿量完整代数的一组生成元。理解QMBS哈密顿量中的这种隐藏结构，还使我们能够恢复以往对此类哈密顿量进行“暴力”数值搜索所得的结果。此外，这种语言清楚地证明了文献中提出的若干QMBS统一形式体系的等价性，并且阐明了两类表面上不同的QMBS哈密顿量之间的联系——即所谓Shiraishi-Mori构造所涵盖的哈密顿量，以及位于该构造之外的哈密顿量。最后，我们表明该框架促成了QMBS的精确定义，该定义自动意味着它们违反传统的本征态热化假说（ETH），并且我们讨论了其对动力学的意义。
英文摘要：
We study Quantum Many-Body Scars (QMBS) in the language of commutant algebras, which are defined as symmetry algebras of families of local Hamiltonians. This framework explains the origin of dynamically disconnected subspaces seen in models with exact QMBS, i.e., the large "thermal" subspace and the small "non-thermal" subspace, which are attributed to the existence of unconventional non-local conserved quantities in the commutant; hence this unifies the study of conventional symmetries and weak ergodicity breaking phenomena into a single framework. Furthermore, this language enables us to use the von Neumann Double Commutant Theorem (DCT) to formally write down the exhaustive algebra of all Hamiltonians with a desired set of QMBS, which demonstrates that QMBS survive under large classes of local perturbations. We illustrate this using several standard examples of QMBS, including the spin-1/2 ferromagnetic, AKLT, spin-1 XY $π$-bimagnon, and the electronic $η$-pairing towers of states; and in each of these cases we explicitly write down a set of generators for the full algebra of Hamiltonians with these QMBS.Understanding this hidden structure in QMBS Hamiltonians also allows us to recover results of previous "brute-force" numerical searches for such Hamiltonians. In addition, this language clearly demonstrates the equivalence of several unified formalisms for QMBS proposed in the literature, and also illustrates the connection between two apparently distinct classes of QMBS Hamiltonians -- those that are captured by the so-called Shiraishi-Mori construction, and those that lie beyond. Finally, we show that this framework motivates a precise definition for QMBS that automatically implies that they violate the conventional Eigenstate Thermalization Hypothesis (ETH), and we discuss its implications to dynamics.
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