Inverse Participation Ratios in the XX spin chain

TL;DR

This paper investigates the inverse participation ratios (IPRs) in the XX Heisenberg spin chain, deriving exact expressions and asymptotics using symmetric function theory, and relating IPRs to physical models and combinatorial structures.

Contribution

It introduces a new discrete Hall inner product representation for IPRs of excited states and explores their asymptotic behavior and connections to physics and combinatorics.

Findings

IPRs can be indexed by partitions.

Asymptotically, IPRs of a partition equal those of its conjugate.

Connections established between IPRs, Dyson ensembles, and permutohedra.

Abstract

We continue the study of the Inverse Participation Ratios (IPRs) of the XXZ Heisenberg spin chain initiated by Misguich, Pasquier and Luck (2016) by focusing on the case of the XX Heisenberg Spin Chain. For the ground state, Misguich et al. note that calculating the IPR is equivalent to Dyson's constant term ex-conjecture. We express the IPRs of excited states as an apparently new "discrete" Hall inner product. We analyze this inner product using the theory of symmetric functions (Jack polynomials, Schur polynomials, the standard Hall inner product and $\omega_{q,t}$) to determine some exact expressions and asymptotics for IPRs. We show that IPRs can be indexed by partitions, and asymptotically the IPR of a partition is equal to that of the conjugate partition. We relate the IPRs to two other models from physics, namely, the circular symplectic ensemble of Dyson and the Dyson-Gaudin two-dimensional Coulomb lattice gas. Finally, we provide a description of the IPRs in terms of a signed count of diagonals of permutohedra.