Integrable model of bosons in a four-well ring with anisotropic tunneling

We introduce an integrable, four-well ring model for bosons where the tunneling couplings between nearest-neighbour wells are not restricted to be equal. We show how the model may be derived through the Quantum Inverse Scattering Method from a solution of the Yang–Baxter equation, and in turn solved by algebraic Bethe Ansatz means. The model admits multiple pseudovaccum states. Numerical evidence is provided to indicate that all pseudovacua are required to obtain a complete set of Bethe eigenstates. The model has the notable property that there is a class of eigenstates which admit a simple, closed-form energy expression.

💡 Research Summary

The authors present a novel integrable quantum many‑body model describing bosons confined in a four‑well ring geometry, where the tunnelling amplitudes between nearest‑neighbour wells are allowed to be anisotropic. The Hamiltonian reads

H = U (N₁+N₃−N₂−N₄)² + μ (N₁+N₃−N₂−N₄) + Σ⟨i,j⟩ t_{ij}(a_i† a_j + a_j† a_i),

with the bosonic creation/annihilation operators a_i†, a_i (i=1,…,4) and number operators N_i = a_i† a_i. The tunnelling couplings are not independent; they are parametrised as

t_{ij}=−κ α_i α_j, i,j∈{1,2,3,4},

which imposes only the single constraint t₁₂ t₃₄ = t₂₃ t₁₄ while still providing a large anisotropic regime. The interaction strength U, the external potential μ, and the overall scale κ are real parameters, as are the four α‑coefficients.

To demonstrate integrability the authors employ the Quantum Inverse Scattering Method (QISM). Starting from the standard SU(2) R‑matrix

R(u)=(\begin{pmatrix}1&0&0&0\0&b(u)&c(u)&0\0&c(u)&b(u)&0\0&0&0&1\end{pmatrix})

with b(u)=u/(u+η) and c(u)=η/(u+η), they construct a Lax operator

L_{i,j}(u)=u I + η N_{i,j} A_{i,j}A_{i,j}†/(α_i²+α_j²) − η I,

where A_{i,j}=α_i a_i + α_j a_j and N_{i,j}=N_i+N_j. The Lax operator satisfies the Yang‑Baxter equation, allowing the definition of a monodromy matrix

T(u)=L_{1,3}(u+ω) L_{2,4}(u−ω) = (\begin{pmatrix}A(u)&B(u)\ C(u)&D(u)\end{pmatrix}).

Its trace τ(u)=A(u)+D(u) is the transfer matrix. Because τ(u) commutes for different spectral parameters, it generates a family of commuting conserved quantities. By expanding τ(u) in powers of u one finds

τ(u)=c₀ + c₁ u + c₂ u²,

with c₁ proportional to the total particle number N, c₂ the identity, and c₀ containing the Hamiltonian. Matching coefficients yields the identifications

U = κ η⁴/4, μ = κ ω/η, t_{ij}=−κ α_i α_j.

Thus the Hamiltonian is a linear combination of τ(u) and known constants, establishing its integrability.

The algebraic Bethe Ansatz is then applied. Two commuting operators

Γ†{1,3}=α₃ a₁†−α₁ a₃†, Γ†{2,4}=α₄ a₂†−α₂ a₄†

are introduced; they commute with the A‑operators of the monodromy matrix and raise the quantum numbers N_{1,3} and N_{2,4} respectively. Consequently a whole family of “pseudovacua” can be built:

|φ_{k,l}⟩ = (Γ†{1,3})^k (Γ†{2,4})^l |0⟩, k+l ≤ N.

Each pseudovacuum is annihilated by B(u) and is an eigenstate of A(u) and D(u). Bethe states are generated by acting with the C‑operator:

|ψ_{k,l}⟩ = C(v₁)…C(v_{N−k−l}) |φ_{k,l}⟩,

where the rapidities {v_j} satisfy Bethe equations that depend on the numbers k and l. Two distinct regimes appear:

1. k+l = N (all particles are created by the Γ† operators). In this case the transfer‑matrix eigenvalue simplifies to

λ_{k,l}(u) = (u+ω+kη)(u−ω+lη) + const,

and the energy becomes

E = U (l−k)² + μ (l−k).

Remarkably, this expression does not involve the tunnelling parameters t_{ij}; it is a closed‑form result for a whole class of eigenstates.

2. k+l < N. Here the eigenvalue contains a product over the Bethe roots, and the rapidities obey the non‑linear equations

η² (v_i+ω+kη)(v_i−ω+lη) = ∏{j≠i}(v_i−v_j−η) ∏{j≠i}(v_i−v_j+η).

Solving these equations yields the remaining part of the spectrum.

Beyond H and the total particle number N, the model possesses two additional conserved operators

Q_{1,3}= (Γ†{1,3}Γ{1,3})/(α₁²+α₃²), Q_{2,4}= (Γ†{2,4}Γ{2,4})/(α₂²+α₄²),

which commute with each other, with H, and with N. Hence the system has four independent commuting integrals of motion, satisfying the requirement for complete integrability.

The authors validate the analytical results with exact diagonalisation for the sectors N=1 and N=2. For N=1 the Hilbert space dimension is 4; three distinct pseudovacua are required to reproduce all four eigenvalues, and the Bethe equations give the same energies as the direct diagonalisation. For N=2 the dimension is 10; six pseudovacua are needed, and each produces a set of Bethe roots that generate the full spectrum, including the simple closed‑form energies for the k+l=N sector. The numerical evidence confirms that all pseudovacua are essential for completeness.

In summary, the paper introduces an anisotropic four‑well Bose‑Hubbard model that remains exactly solvable via the Quantum Inverse Scattering Method. The presence of multiple pseudovacua, extra conserved quantities, and a class of eigenstates with a simple energy formula enriches the structure compared with the standard isotropic Bose‑Hubbard model. The work opens avenues for exploring more complex lattice geometries, time‑dependent protocols, or additional interactions (e.g., dipolar terms) while retaining integrability, and suggests potential applications in quantum simulation and atom‑interferometry where controllable anisotropic tunnelling is desirable.