Kosterlitz-Thouless Phase Transition and Ground State Fidelity: a Novel Perspective from Matrix Product States

The Kosterlitz-Thouless transition is studied from the representation of the systems's ground state wave functions in terms of Matrix Product States for a quantum system on an infinite-size lattice in one spatial dimension. It is found that, in the critical regime for a one-dimensional quantum lattice system with continuous symmetry, the newly-developed infinite Matrix Product State algorithm automatically leads to infinite degenerate ground states, due to the finiteness of the truncation dimension. This results in \textit{pseudo} continuous symmetry spontaneous breakdown, which allows to introduce a pseudo-order parameter that must be scaled down to zero, in order to be consistent with the Mermin-Wegner theorem. We also show that the ground state fidelity per lattice site exhibits a \textit{catastrophe point}, thus resolving a controversy regarding whether or not the ground state fidelity is able to detect the Kosterlitz-Thouless transition.