The interplay of charge-4e and nematic order: a study based on generalized XY model

Previous studies suggested that charge-4e superconductivity can emerge from a nematic superconductor by the proliferation of nematic vortices. In this work, we study a generalized XY model whose ground state breaks U(1) and rotation symmetry simultaneously as with the nematic superconductor. Upon increasing temperature, we show that quasi-long range order of U(1) charge $e^{i heta}$ is first destroyed with $e^{i2 heta}$ (charge-4e order parameter) remaining quasi-long range ordered. Interestingly, at the time when $e^{i heta}$ is disordered, the nematic long-range order phase changes into a critical phase with nematic power-law correlation. With the temperature further raised, there can be a pure $e^{i2 heta}$ quasi-long range ordered phase before the fully disordered phase. The phase diagram is different from the previous prediction of a nematic superconductor with an additional nematic power-law phase here. We believe that domain wall proliferation and the interplay of charge-4e and nematic order are responsible for this nematic power-law phase. Nevertheless, vortex proliferation is still responsible for the formation of the pure charge-4e phase and the direct transition to a fully disordered phase.