  We introduce a variational algorithm to simulate quantum many-body states
based on a tree tensor network ansatz which releases the isometry constraint
usually imposed by the real-space renormalization coarse-graining: This
additional numerical freedom, combined with the loop-free topology of the tree
network, allows one to maximally exploit the internal gauge invariance of
tensor networks, ultimately leading to a computationally flexible and efficient
algorithm able to treat open and periodic boundary conditions on the same
footing. We benchmark the novel approach against the 1D Ising model in
transverse field with periodic boundary conditions and discuss the strategy to
cope with the broken translational invariance generated by the network
structure. We then perform investigations on a state-of-the-art problem, namely
the bilinear-biquadratic model in the transition between dimer and
ferromagnetic phases. Our results clearly display an exponentially diverging
correlation length and thus support the most recent guesses on the peculiarity
of the transition.
