Numerical methods for detecting symmetries and commutant algebras
Numerical methods for detecting symmetries and commutant algebras
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DOI：
10.1103/physrevb.107.224312
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发表时间：
2023-02
影响因子：
3.7
通讯作者：
Sanjay Moudgalya;O. Motrunich
中科院分区：
文献类型：
--
作者：
Sanjay Moudgalya;O. Motrunich
对于由局部部分定义的哈密顿族，对称代数的最一般定义是交换代数，即与每个局部部分交换的算子的代数。将对称代数视为交换代数允许在相同的代数基础上处理常规对称和非常规对称（例如，那些负责弱遍历破缺现象的对称）。本文讨论了从哈密顿算子族出发，用数值方法构造该交换代数的两种方法。首先，我们利用该问题与给定局部算子集的同时块对角化问题的等价性，讨论了一种概率方法，该方法对阿贝尔对称和非阿贝尔对称或交换代数都具有1的概率。其次，我们将这个问题映射到确定某些哈密顿量的无挫折基态的问题上，并利用张量网络算法的思想在一维中有效地解决了这个问题。这些数值方法可用于检测哈密顿量族中的标准和非标准守恒量，其中包括规则对称，希尔伯特空间碎片和量子多体伤痕的例子，我们展示了许多这样的例子。此外，它们对于验证我们在以前的工作中提出的关于这些情况下交换代数结构的几个猜想是必要的。最后，我们还讨论了具有给定对称或交换代数的局部算子逆问题的类似方法，这些方法与文献中已有的方法相联系。这种构造的一种特殊情况简化为众所周知的“特征态到哈密顿”方法，用于构造具有给定状态作为特征态的厄米局域算子。
For families of Hamiltonians defined by parts that are local, the most general definition of a symmetry algebra is the commutant algebra, i.e., the algebra of operators that commute with each local part. Thinking about symmetry algebras as commutant algebras allows for the treatment of conventional symmetries and unconventional symmetries (e.g., those responsible for weak ergodicity breaking phenomena) on equal algebraic footing. In this work, we discuss two methods for numerically constructing this commutant algebra starting from a family of Hamiltonians. First, we use the equivalence of this problem to that of simultaneous block-diagonalization of a given set of local operators, and discuss a probabilistic method that has been found to work with probability 1 for both Abelian and non-Abelian symmetries or commutant algebras. Second, we map this problem onto the problem of determining frustration-free ground states of certain Hamiltonians, and we use ideas from tensor network algorithms to efficiently solve this problem in one dimension. These numerical methods are useful in detecting standard and non-standard conserved quantities in families of Hamiltonians, which includes examples of regular symmetries, Hilbert space fragmentation, and quantum many-body scars, and we show many such examples. In addition, they are necessary for verifying several conjectures on the structure of the commutant algebras in these cases, which we have put forward in earlier works. Finally, we also discuss similar methods for the inverse problem of determining local operators with a given symmetry or commutant algebra, which connects to existing methods in the literature. A special case of this construction reduces to well-known ``Eigenstate to Hamiltonian"methods for constructing Hermitian local operators that have a given state as an eigenstate.