Bethe Ansatz solution for a model of global-range interacting bosons on the square lattice

TL;DR

The paper extends Bethe Ansatz solvability to a two-dimensional square-lattice boson model with global-range interactions by mapping the lattice to a spectrally weighted set of disjoint two-vertex graphs through the adjacency matrix $B$. The Bethe Ansatz solution is obtained after a unitary diagonalization of $B$, yielding energy formulas $E=UN^2+4U\sum_{j,n} \frac{N_j\varepsilon_j^2}{v_n-\varepsilon_j^2}$ and Bethe equations that depend on $\varepsilon_j^2$; explicit results for open, cylindrical, and toroidal boundaries are given, along with parity-sensitive imbalance fluctuations. The work also analyzes an integrable boundary defect, showing that integrability can persist while the Bethe Ansatz ceases to be expressible in closed form, illustrating a distinction between integrability and exact solvability. Overall, the study demonstrates a route to exact solutions for higher-dimensional bosonic lattices with long-range interactions and clarifies how the eigenstructure of the adjacency matrix controls the solvability landscape, with potential relevance to cavity-mediated interactions in cold-atom systems.

Abstract

Quantum systems on a one-dimensional lattice are ubiquitous in the study of models exactly-solved by Bethe Ansatz techniques. Here it is shown that including global-range interaction opens scope for Bethe Ansatz solutions that are not constrained to one-dimensional quantum systems. A bosonic model on a square lattice is defined, and the exact Bethe Ansatz solution is provided for open, cylindrical, and toroidal boundary conditions. Generalising the result for an integrable defect leads to a Bethe Ansatz solution that is not expressible in an exact, closed-form manner.

Figures (4)

• Figure 1: Schematic representation of the hopping terms in the Hamiltonian (\ref{['ham']}) for open boundary conditions. (a) Example using the $4\times 4$ square lattice, with sites coloured red and blue to highlight the bipartite structure. The 4 corner vertices have 2 edges, the remaining 8 boundary vertices have 3 edges, while the 4 internal vertices have 4 edges. (b) Equivalent bipartite graph representation that allows the adjacency submatrix $B$ to be determined, as given by Eq. (\ref{['bmatrixobc']}). Observe that in this representation there are 4 vertices that have 2 edges (1st left, 1st right, 7th left, 7th right), 4 vertices that have 4 edges (4th left, 4th right, 6th left, 6th right), while the remaining 8 vertices have 3 edges. (c) Transformed bipartite graph representation resulting from the diagonalisation of $B$. This results in a union of disconnected, two-vertex bipartite graphs, with the eigenvalues of $B$ weighting the edges. This leads to the Bethe Ansatz solution (\ref{['nrg']},\ref{['bae']}).
• Figure 2: Schematic representation of the hopping terms in the Hamiltonian (\ref{['ham']}) for cylindrical boundary conditions. (a) Example using the $4\times 4$ square lattice, with sites coloured red and blue to highlight the bipartite structure. Compared to the previous Fig. \ref{['fig1']}, there are four additional edges connecting the top and bottom of the lattice. (b) The equivalent bipartite graph representation that leads to Eq. (\ref{['bmatrixcbc']}) also has four additional edges compared to the corresponding graph of Fig. \ref{['fig1']}. (c) Transformed bipartite graph representation resulting from the diagonalisation of $B$. The different colours used, compared to the previous figure, are only to emphasise that a different basis transformation is required.
• Figure 3: Schematic representation of the hopping terms in the Hamiltonian (\ref{['ham']}) for toroidal boundary conditions. (a) Example using the $4\times 4$ square lattice, with sites coloured red and blue to highlight the bipartite structure. (b) Equivalent bipartite graph representation that allows the adjacency submatrix $B$ to be determined, as given by Eq. (\ref{['bmatrixtbc']}). All vertices have four edges, so each row and column of (\ref{['bmatrixtbc']}) has four entries, reflecting the horizontal and vertical translational invariance of the lattice. (c) Transformed bipartite graph representation resulting from the diagonalisation of $B$. As before, the different colours used are only to emphasise that the basis transformation is different from those required for the other boundary conditions.
• Figure 4: Schematic representation of the square lattice with a boundary defect. (a) An example using the $8\times 8$ square lattice, with open boundary conditions. (b) Illustration of a boundary defect through removal of an edge of the graph from the top boundary.