Composite order in SU(N) theories coupled to an Abelian gauge field

We consider $\mathrm{SU}(N)$-symmetric Ginzburg-Landau models coupled to a noncompact 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 $\mathrm{SU}(2)$ case, the neutral state cannot be mapped onto an $\mathrm{O}(M)$ model. We term this state the $\mathrm{SU}{(N)}_{\mathrm{n}}$ phase.