Hilbert Space Fragmentation and Commutant Algebras
Hilbert Space Fragmentation and Commutant Algebras
复制标题
DOI：
10.1103/physrevx.12.011050
复制
发表时间：
2021-08
影响因子：
12.5
通讯作者：
Sanjay Moudgalya;O. Motrunich
中科院分区：
文献类型：
--
作者：
Sanjay Moudgalya;O. Motrunich
我们使用通勤代数的语言研究了希尔伯特空间碎片的现象，这是所有操作员的代数的语言，这些代数的代数与汉密尔顿的每个术语或电路的每个门都通勤。在这种形式主义中，我们提供了希尔伯特空间碎片化的精确定义，即通勤代数的维度随系统大小而成倍增长。因此，碎片可以与具有传统对称性的系统（例如$ u（1）$或$ su（2）$区分开，其中通勤代数的尺寸随系统大小而多样地增长。此外，通勤代数语言还有助于区分“古典”和“量子”希尔伯特空间破碎化，前者在产品状态下指的是分裂。我们在表现出经典碎片化的几种系统中明确构建了换向代数，包括$ t-j_z $模型和Spin-1偶极旋转模型，我们说明了与先前研究的“统计上局部的运动积分”（SLIOMS）的连接（SLIOMS）。我们还重新审视了Temberley-Lieb旋转链，包括文献中广泛研究的Spin-1 Biquadratic链，并表明它们表现出量子Hilbert空间碎片。最后，我们研究了各种情况下，全部换向代数对Mazur界限的贡献。在零散的系统中，我们使用表达式来分析与先前数值结果一致的本地运算符的自相关功能，从而在分析上获得新的或改进的Mazur界限。此外，我们能够严格地显示在Spin-1偶极子连接模型中现场自旋算子的定位。
We study the phenomenon of Hilbert space fragmentation in isolated Hamiltonian and Floquet quantum systems using the language of commutant algebras, the algebra of all operators that commute with each term of the Hamiltonian or each gate of the circuit. We provide a precise definition of Hilbert space fragmentation in this formalism as the case where the dimension of the commutant algebra grows exponentially with the system size. Fragmentation can hence be distinguished from systems with conventional symmetries such as $U(1)$ or $SU(2)$, where the dimension of the commutant algebra grows polynomially with the system size. Further, the commutant algebra language also helps distinguish between"classical"and"quantum"Hilbert space fragmentation, where the former refers to fragmentation in the product state basis. We explicitly construct the commutant algebra in several systems exhibiting classical fragmentation, including the $t-J_z$ model and the spin-1 dipole-conserving model, and we illustrate the connection to previously-studied"Statistically Localized Integrals of Motion"(SLIOMs). We also revisit the Temperley-Lieb spin chains, including the spin-1 biquadratic chain widely studied in the literature, and show that they exhibit quantum Hilbert space fragmentation. Finally, we study the contribution of the full commutant algebra to the Mazur bounds in various cases. In fragmented systems, we use expressions for the commutant to analytically obtain new or improved Mazur bounds for autocorrelation functions of local operators that agree with previous numerical results. In addition, we are able to rigorously show the localization of the on-site spin operator in the spin-1 dipole-conserving model.