Permutation symmetry in large N Matrix Quantum Mechanics and Partition Algebras

Abstract

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.

64 pages + Appendices, 2 figures

References in corpus (18)

• Probing many-body dynamics on a 51-atom quantum simulator
• Quantum Many-Body Scars and Weak Breaking of Ergodicity
• Diagonal multi-matrix correlators and BPS operators in N=4 SYM
• Branes, Anti-Branes and Brauer Algebras in Gauge-Gravity duality
• Giant Gravitons - with Strings Attached (II)
• Enhanced symmetries of gauge theory and resolving the spectrum of local operators
• Group theoretic approach to many-body scar states in fermionic lattice models
• Spin Matrix Theory: A quantum mechanical model of the AdS/CFT correspondence
• Surprisingly Simple Spectra
• Integrability vs. Information Loss: A Simple Example
• Permutations and the Loop
• BPS coherent states and localization
• BPS states, conserved charges and centres of symmetric group algebras
• Spin Matrix Theory in near $\frac{1}{8}$-BPS corners of $\mathcal{N}=4$ super-Yang-Mills
• Flavour singlets in gauge theory as Permutations
• Topological String Correlators from Matrix Models
• Permutation invariant Gaussian 2-matrix models
• Hidden symmetries and Large N factorisation for permutation invariant matrix observables

Cited by in corpus (4)

• Majorana Scars as Group Singlets
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• Matrix and tensor witnesses of hidden symmetry algebras
• $\mathcal{N}=4$ SYM, (super)-polynomial rings and emergent quantum mechanical symmetries