Quantum Spin Chains and Symmetric Functions

TL;DR

This paper explores how quantum spin chains encode rich combinatorial and algebraic structures, revealing connections to symmetric functions, integrability, and potential applications in quantum computing.

Contribution

It introduces a fermionic representation linking quantum spin chains to symmetric functions and demonstrates how operators extract combinatorial data from the Hilbert space.

Findings

Quantum spin chains encode skew Kostka numbers and Littlewood-Richardson coefficients.

Operators diagonalized by the Bethe basis extract combinatorial data.

Quantum integrable systems relate to symmetric functions and quantum computing.

Abstract

We consider the question of what quantum spin chains naturally encode in their Hilbert space. It turns out that quantum spin chains are rather rich systems, naturally encoding solutions to various problems in combinatorics, group theory, and algebraic geometry. In the case of the XX Heisenberg spin chain these are given by skew Kostka numbers, skew characters of the symmetric group, and Littlewood-Richardson coefficients. As we show, this is revealed by a fermionic representation of the theory of "quantized" symmetric functions formulated by Fomin and Greene, which provides a powerful framework for constructing operators extracting this data from the Hilbert space of quantum spin chains. Furthermore, these operators are diagonalized by the Bethe basis of the quantum spin chain. Underlying this is the fact that quantum spin chains are examples of "quantum integrable systems." This is somewhat analogous to bosons encoding permanents and fermions encoding determinants. This points towards considering quantum integrable systems, and the combinatorics associated with them, as potentially interesting targets for quantum computers.