Finite-temperature phase transition in a homogeneous one-dimensional gas of attractive bosons

In typical one-dimensional models the Mermin-Wagner theorem forbids long range order, thus preventing finite-temperature phase transitions. We find a finite-temperature phase transition for a homogeneous system of attractive bosons in one dimension. The low-temperature phase is characterized by a quantum bright soliton without long range order; the high-temperature phase is a free gas. Numerical calculations for finite particle numbers show a specific heat scaling as $N^2$, consistent with a vanishing transition region in the thermodynamic limit.