Quantum phase transitions in three-leg spin tubes

We investigate the properties of a three-leg quantum spin tube using several techniques such as the density-matrix renormalization-group (DMRG) method, strong-coupling approaches and the nonlinear sigma model. For integer S, the model proves to exhibit a particularly rich phase diagram consisting of an ensemble of 2S phase transitions. They can be accurately identified by the behavior of a nonlocal string order parameter associated to the breaking of a hidden symmetry in the Hamiltonian. The nature of these transitions is further elucidated within the different approaches. We carry a detailed DMRG analysis in the specific cases S = 1. The numerical data confirm the existence of two Haldane phases with broken hidden symmetry separated by a trivial singlet state. The study of the gap and of the von Neumann entropy suggest a first-order phase transition but at the close proximity of a tricritical point separating a gapless and a first-order transition line in the phase diagram of the quantum spin tube.