大N矩阵量子力学和分划代数中的置换对称性
Permutation symmetry in large N Matrix Quantum Mechanics and Partition Algebras
摘要：
我们描述了一般大小的矩阵$N$的量子力学系统的状态空间和动力学的置换对称性的影响。我们求解了一般的11参数置换不变量子矩阵谐振子哈密顿量，计算了正则配分函数。对于一般的哈密顿算子，希尔伯特空间的置换不变扇区可以用划分代数图来描述，划分代数图形成了划分代数塔$P_k(N)$的基底。整数$k$在量子力学中被解释为矩阵振子多项式的次数。家庭的相互作用的哈密顿被对角化的表示理论的基础上，我们构造的置换不变的子空间$ N \ge 2k $。这些包括低能量状态是置换不变的哈密顿量，可以引起大的基态退化有关的分区代数的尺寸。文献中讨论的量子多体伤疤的一种基于量子力学的机制可以在这些具有置换对称性的矩阵系统中实现。的矩阵索引值的晶格位置的映射允许在修改后的玻色-哈伯德模型的上下文中的机制的实现。极端的adS/CFT研究类似于那些遵守选择规则的基础上Clebsch-Gordan多重性（克罗内克系数）的对称群。
英文摘要：
We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size $N$. We solve the general 11- parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras $P_k(N)$. The integer $k$ is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Families of interacting Hamiltonians are described which are diagonalised by a representation theoretic basis for the permutation invariant subspace which we construct for $ N \ge 2k $. These include Hamiltonians for which the low-energy states are permutation invariant and can give rise to large ground state degeneracies related to the dimensions of partition algebras. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realised in these matrix systems with permutation symmetry. A mapping of the matrix index values to lattice sites allows a realisation of the mechanism in the context of modified Bose-Hubbard models. Extremal correlators analogous to those studied in AdS/CFT are shown to obey selection rules based on Clebsch-Gordan multiplicities (Kronecker coefficients) of symmetric groups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
