Grassmannian and Flag sigma models on interval: phase structure and L-dependence

We discuss the two-dimensional Grassmannian $SU(N)\mathrm{/}S(U(N-2)\times U(2))$ and the flag $SU(N)\mathrm{/}S(U(N-2)\times U(1)\times U(1))$ sigma models on a finite interval and construct analytical solutions of gap equations in the large N limit. We show that the flag model admits a homogeneous solution for `mixed’ Dirichlet-Neumann (DN) boundary conditions only for sufficiently large length $L$ and undergoes a phase transition from the phase of partly broken gauge symmetry ($U(1)$) to the symmetric phase ($U(1)\times U(1)$) for large $L$. On the other hand, the Grassmannian model has a detached phase with one massive and one massless non-zero condensates that completely break $U(2)$ gauge symmetry. This phase lives on a region of $L$ bounded from above and has to use the Robin boundary conditions. We also examine the L-dependence of the total energy and detect the linear growth inherent to confining string in all phases.