Bethe ansatz solution to integrable bosonic cube networks

We study two extended Bose-Hubbard-type Hamiltonians representing bosonic networks restricted to the graph of a cube. For both Hamiltonians, we demonstrate that Bethe ansatz methods of solution can be employed after applying a canonical transformation of operators. We provide the resulting Bethe ansatz equations, and corresponding formulae for states and energies of both Hamiltonians.

💡 Research Summary

The paper investigates two extended Bose‑Hubbard‑type Hamiltonians defined on the eight‑site cube graph (Q₃). The first Hamiltonian, H₁, contains a global quadratic interaction term, a linear combination of number operators on opposite vertices (1,8,2,7), and a standard nearest‑neighbour hopping term with amplitude J. The second Hamiltonian, H₂, is similar but couples a different set of opposite vertices (1,4,6,7). Both models include parameters U₀, U₁, and U governing interaction strengths, and they preserve the total particle number operator N.

A central achievement of the work is the identification of two canonical (linear) transformations that render each Hamiltonian amenable to Bethe‑ansatz techniques. Transformation I mixes the original bosonic operators in four pairs (a₁,a₂), (a₃,a₄), (a₅,a₆), (a₇,a₈) into new operators b₁…b₈ via a 45° rotation. After this change of basis, H₁ decomposes into two disjoint square sub‑graphs, each involving four b‑operators. Transformation II mixes the operators in a different pairing (a₁,a₆), (a₅,a₂), (a₃,a₈), (a₇,a₄) to produce operators e b₁…e b₈. In the new basis H₂ becomes a single square plus two independent dimers (two‑site bonds). Both transformations preserve the original conserved quantity N₁−N₈+N₂−N₇ (or its analogue) and keep the total particle number N invariant.

In the transformed representation each Hamiltonian can be expressed entirely in terms of generators of one or more copies of the su(2) Lie algebra. For H₁, two commuting su(2) algebras are introduced: e₁ = b₁†b₄ + b₁†b₅ + b₄†b₈ + b₅†b₈, f₁ = b₄†b₁ + b₅†b₁ + b₈†b₄ + b₈†b₅, h₁ = 2(N_{b₁}−N_{b₈}), and similarly e₂, f₂, h₂ acting on the second square. The Hamiltonian then reads H₁ = U₀ N² + U₁ N (h₁+h₂) + U (h₁+h₂)² − J (e₁+e₂+f₁+f₂). The Casimir operators C₁ = ½ h₁² + e₁ f₁ + f₁ e₁ and C₂ = ½ h₂² + e₂ f₂ + f₂ e₂ are conserved. Together with H₁, N, and the two linear combinations h₁, h₂, this yields eight mutually commuting integrals of motion, establishing complete integrability.

For H₂ three commuting su(2) algebras are constructed (one for the square, two for the dimers). Denoting their generators by e₁,f₁,h₁ (square) and e₂,f₂,h₂, e₃,f₃,h₃ (dimers), the Hamiltonian becomes H₂ = U₀ N² + U₁ N (h₁+h₂−h₃) + 4U (h₁+h₂−h₃)² − J (e₁+e₂+f₁+f₂). Again the corresponding Casimirs C₁, C₂, C₃ together with N and the Hamiltonian provide eight independent commuting conserved quantities.

Having identified the su(2) structure, the authors develop a Bethe‑ansatz solution. They first construct lowest‑weight states for each su(2) sector. For the square face (1,4,5,8) they introduce an auxiliary su(2) algebra (e_α, f_α, h_α) and define |ψ_{ℓ m}⟩ = Σ_{k=0}^{⌊m/2⌋} C_{ℓ,m,k} (e_α)^k (b₄†−b₅†)^{m−2k} (b₈†)^{ℓ−m} |0⟩, which satisfies f₁|ψ⟩ = f_α|ψ⟩ = 0. Using the actions of e₁, f₁, h₁ on these states they generate a recursive basis |N,k,ℓ,m⟩, where N+ℓ = total particle number n and k runs from 0 to 2(ℓ−m). The algebraic actions are simple: e₁|N,k,ℓ,m⟩ = (2(ℓ−m)−k) |N,k+1,ℓ,m⟩, f₁|N,k,ℓ,m⟩ = k |N,k−1,ℓ,m⟩, h₁|N,k,ℓ,m⟩ = 2(k−ℓ+m) |N,k,ℓ,m⟩. These relations lead directly to a set of Bethe equations for the spectral parameters associated with each excitation. Solving these equations yields explicit expressions for the eigenstates and eigenvalues of H₁. An analogous construction is carried out for H₂, employing the three su(2) copies and their respective lowest‑weight vectors.

The paper thus provides:

1. Explicit canonical transformations that map the original cube‑graph Bose‑Hubbard models onto decoupled su(2) structures.
2. A full set of eight commuting integrals of motion for each model, proving complete integrability.
3. Detailed Bethe‑ansatz formulations, including the construction of lowest‑weight states, recursive basis, and the resulting Bethe equations.
4. Closed‑form expressions for the energy spectra in terms of the Bethe roots.

These results extend the class of exactly solvable many‑body bosonic lattice models to non‑complete, symmetric bipartite graphs such as the cube. The methodology—canonical transformation followed by identification of hidden su(2) symmetry and Bethe‑ansatz solution—offers a template for tackling more complex networks (e.g., higher‑dimensional hypercubes or other symmetric graphs) where integrability may be hidden behind a suitable change of basis. The work also connects to recent interests in engineered quantum devices, where controllable interactions on small networks are essential for quantum simulation and quantum control applications.