bibkey: araujoorlandoreffert2021quantumcrystals
authors: Thiago Araujo; Domenico Orlando; Susanne Reffert
year: 2021
title: "Quantum crystals, Kagome lattice, and plane partitions fermion-boson duality"
doi: 10.1103/PhysRevD.103.026020
url: https://arxiv.org/abs/2005.09103v2
claim: "The quantum crystal melting Hamiltonian H = −J(Γ + Π) + V(√q O₊ + O₋/√q) acts on plane-partition states; a fermion-boson correspondence (46) defines bosonic states through symmetric-group characters, and Section VI conjectures that Γ, Π and O± are linear combinations of the affine Yangian Y[gl̂(1)] generators, with the bosonic states forming its MacMahon representation, so that O± lie in the Cartan part B."
strata_touched:
  - D5/S3/StatisticalMechanics/QuantumCrystalYangianRefutation
license: citation-only
triage: anchor

Araujo, Orlando and Reffert, quantum crystals and plane partitions

T. Araujo, D. Orlando and S. Reffert, Quantum crystals, Kagome lattice, and plane partitions fermion-boson duality, Phys. Rev. D 103, 026020 (2021), DOI 10.1103/PhysRevD.103.026020; arXiv:2005.09103 [hep-th] (v1 18 May 2020, v2 27 January 2021).

Verified locator

DOI: 10.1103/PhysRevD.103.026020

Source: https://arxiv.org/abs/2005.09103v2; the text was read from the PDF.

• page 2: the Hamiltonian, Eqs. (2a)–(2b), and the meaning of Γ, Π and O±;
• page 8: the fermion-boson correspondence, Eqs. (42)–(46), the conjecture of Section VI and Eqs. (49)–(50);
• page 9: the consequence O± ∈ B and Eq. (51);
• page 10: the character table (A13);
• pages 11–12: Appendix C, the basis transformation for at most three boxes.

Source statements

Eq. (2b): “H = −J (Γ + Π) + V (√q O+(Π, Γ) + (1/√q) O−(Π, Γ)) … the operator Γ (Π) adds (removes) boxes in all possible ways. Moreover, the potential terms O+(Π, Γ) and O−(Π, Γ) count the number of positive and negative flippable plaquettes, respectively.” A negative flip removes a box from the plane partition.

Eq. (46): “|Λ(N K)⟩⟩ = Σ_Λ χ_Λ[C(K)] |Λ⟩”, where χ_Λ is the product over layers of characters of symmetric groups (Eq. (42c)), and “the correspondence between the plane partition basis preserves the number of boxes grading”.

Conjecture (Section VI): “The growth operators Γ, Π and O±(Γ, Π) are linear combinations of the generators of the algebra Y[gl̂(1)], and the states |N K⟩⟩ fulfill the Y[gl̂(1)] MacMahon representation.”

Eqs. (49)–(50): “e_j|Λ⟩⟩ = Σ_{□∈Q+} E_j(Λ ↦ Λ + □)|Λ + □⟩⟩, f_j|Λ⟩⟩ = Σ_{□∈Q−} F_j(Λ ↦ Λ − □)|Λ − □⟩⟩. … the operators ϕ_j act diagonally as ϕ_j|Λ⟩⟩ = ϕ_{j,Λ}|Λ⟩⟩.”

Page 9: “if the conjectured correspondence between the generators of Y[gl̂(1)] and with the growth operators Γ and Π is indeed correct, we immediately conclude that O±(Π, Γ) ∈ B since they can be written as linear combinations of ϕ_j.”

Character table (A13), S₃: χ₍₁,₁,₁₎, χ₍₂,₁₎, χ₍₃₎ take the values 1, 2, 1 on the identity class (3, 0, 0, …). Appendix C, Eq. (C2b), writes the corresponding bosonic state as the single-layer combination of the column, twice the hook, and the row.

Reading used by the repository

The state space has the plane partitions as basis, graded by the number of boxes, and O₋ acts on it diagonally by the number of removable boxes. The bosonic state of the identity class of S₃ is v = |(3)⟩ + 2|(2, 1)⟩ + |(1, 1, 1)⟩, by Eq. (46) with (A13) and as written in Appendix C. The repository refutes the consequence of the conjecture at v: a linear combination of operators satisfying (49)–(50) at v cannot agree with O₋ on v.