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

Abstract

Abstract We discuss the two-dimensional Grassmannian SU(N)/S(U(N − 2) × U(2)) and the flag SU(N )/S(U(N − 2) × U(1) × 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) × 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.