Quantum crystals, Kagome lattice and plane partitions fermion-boson duality

TL;DR

This paper explores quantum crystal melting in three dimensions, establishing a fermion-boson duality for plane partitions and proposing a connection to the affine Yangian algebra.

Contribution

It introduces a fermion-boson duality for plane partitions and conjectures a representation of quantum growth operators via the affine Yangian ${\

Findings

Fermion-boson duality for plane partitions established.

Quantum Hamiltonian growth operators conjectured to relate to affine Yangian.

Recasting crystal melting as a dimer occupancy problem in Kagome lattice.

Abstract

In this work, we study quantum crystal melting in three space dimensions. Using an equivalent description in terms of dimers in a hexagonal lattice, we recast the crystal melting Hamiltonian as an occupancy problem in a Kagome lattice. The Hilbert space is spanned by states labeled by plane partitions and writing them as a product of interlaced integer partitions, we define a fermion-boson duality for plane partitions. Finally, based upon the latter result we conjecture that the growth operators for the quantum Hamiltonian can be represented in terms of the affine Yangian ${\cal Y}[\widehat{\mathfrak{gl}}(1)]$.