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 ${\cal Y}[\widehat{\mathfrak{gl}}(1)]$.
Highlights
Random partitions appear in many contexts in mathematics and physics
In this work we study quantum crystal melting in three space dimensions
We focus on the three-dimensional version of the quantum crystal melting, which can be rephrased in terms of random plane partitions, and it has been partly addressed in [9]
Summary
Random partitions appear in many contexts in mathematics and physics. This omnipresence is partly explained by the fact that they are part of the core of number theory, the queen of mathematics [1]. Random partitions can be successfully realized in terms of fermionic operators living on a chain where particles and holes are labeled by half-integers along the real line [3,7,8,9]. In this formalism, the empty partition is equivalent to a configuration where all negative holes (or better yet, holes on negative positions) are occupied and all positive holes are vacant. We present a number of equivalent formulations of the 3D quantum crystal melting problem, such as in terms of particle-hole hopping, a fermionic description on a Kagome lattice, and a tensor product representation. In Appendix C, we detail basis transformations for various states and in Appendix D, we give amplitudes for transitions between various states
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