  In this paper, we consider the sharp interface limit of a matrix-valued
Allen-Cahn equation, which takes the form: $$\partial_t A=\Delta
A-\varepsilon^{-2}( A A^{\mathrm{T}}A-
A)~~\text{with}~~A:\Omega\subset\mathbb{R}^m\to\mathbb{R}^{n\times n}.$$ We
show that the sharp interface system is a two-phases flow system: the interface
evolves according to the motion by mean curvature; in the two bulk phase
regions, the solution obeys the heat flow of harmonic maps with values in
$O^+(n)$ and $O^-(n)$ (represent the sets of $n\times n$ orthogonal matrices
with determinant $+1$ and $-1$ respectively); on the interface, the phase
matrices in two sides satisfy a novel mixed boundary condition.
  The above result provides a solution to the Keller-Rubinstein-Sternberg's
(conjecture) problem in this $O(n)$ setting. Our proof relies on two key
ingredients. First, in order to construct the approximate solution by matched
asymptotic expansions as the standard approach does not seem to work, we
introduce the notion of quasi-minimal connecting orbits. They satisfy the usual
leading order equations up to some small higher order terms. In addition, the
linearized systems around these quasi-minimal orbits needs to be solvable up to
some good remainders. These flexibilities are needed for the possible
"degenerations" or higher dimensional kernels for the linearized operators due
to intriguing boundary conditions at the sharp interface. The second key point
is to establish a spectral uniform lower bound estimate for the linearized
operator around approximate solutions. To this end, we introduce additional
decompositions to reduce the problem into the coercive estimates of several
linearized operators for scalar functions and some singular product estimates
which is accomplished by exploring special cancellation structures between
eigenfunctions of these linearized operators.
