Fractal decompositions and tensor network representations of Bethe wavefunctions

We investigate the entanglement structure of a generic $M$-particle Bethe wavefunction (not necessarily an eigenstate of an integrable model) on a 1d lattice by dividing the lattice into $L$ parts and decomposing the wavefunction into a sum of products of $L$ local wavefunctions. Using the fact that a Bethe wavefunction accepts a \textit{fractal} multipartite decomposition -- it can always be written as a linear combination of $L^M$ products of $L$ local wavefunctions, where each local wavefunction is in turn also a Bethe wavefunction -- we then build \textit{exact, analytical} tensor network representations with finite bond dimension $\chi=2^M$, for a generic planar tree tensor network (TTN), which includes a matrix product states (MPS) and a regular binary TTN as prominent particular cases. For a regular binary tree, the network has depth $\log_{2}(N/M)$ and can be transformed into an adaptive quantum circuit of the same depth, composed of unitary gates acting on $2^M$-dimensional qudits and mid-circuit measurements, that deterministically prepares the Bethe wavefunction. Finally, we put forward a much larger class of \textit{generalized} Bethe wavefunctions, for which the above decompositions, tensor network and quantum circuit representations are also possible.