Permutation symmetry in large N Matrix Quantum Mechanics and Partition Algebras

TL;DR

This paper explores how permutation symmetry influences the structure and dynamics of large N matrix quantum systems, providing exact solutions, algebraic descriptions, and connections to many-body scars and AdS/CFT correlators.

Contribution

It introduces a comprehensive framework for permutation invariant quantum matrix models using partition algebras, including explicit solutions and basis constructions.

Findings

Solved the general permutation invariant quantum matrix harmonic oscillator.

Connected permutation invariant sectors to partition algebra diagrams.

Identified symmetry mechanisms for quantum many-body scars and selection rules for correlators.

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.