Vortices and composite order in $\mathrm{S}\mathrm{U}(N)$ theories coupled to Abelian gauge field

We consider $\mathrm{S}\mathrm{U}(N)$-symmetric Ginzburg-Landau models coupled to non-compact Abelian gauge field focusing on the case $N>2$ at finite temperature. We show that, at least for sufficiently large gauge-field coupling constants, these models have two phase transitions. The intermediate phase between the symmetric and low-temperature phases is a state with composite neutral order and no Meissner effect. In this neutral phase the system spontaneously breaks only the symmetry associated with phase differences and density differences between components. For $N>2$, in contrast to the $SU(2)$ case, the neutral state cannot be mapped onto an $\mathrm{O}(M)$ model. We term this state ${\mathbb{C}P}^{N-1}$-neutral phase. We also show that while $\mathrm{S}\mathrm{U}(N)$-symmetric Ginzburg-Landau models are not superconductors or superfluids in the usual sense, their state in external field at sufficiently low temperature is a vortex lattice.