  We construct an exactly solvable $\mathscr{PT}-$symmetric non-Hermitian model
where a spin$-\frac{1}{2}$ isotropic quantum Heisenberg spin chain is coupled
to two spin$-\frac{1}{2}$ Kondo impurities at its boundaries with coupling
strengths that are complex conjugates of each other. Solving the model by means
of a combination of the Bethe Ansatz and density matrix renormalization group
(DMRG) techniques, we show that the model exhibits three distinct boundary
phases: a $\mathscr{PT}$ symmetric phase with a dissipative Kondo effect, a
phase with bound modes and spontaneously broken $\mathscr{PT}$ symmetry, and a
phase with an effectively unscreened spin (i.e. a free local moment). In the
Kondo and the unscreened phases, the $\mathscr{PT}-$symmetry is unbroken, and
hence all states have real energies, whereas in the bound mode phases, in
addition to the states with real energies, there exist states with complex
energy eigenvalues that appear in complex conjugate pairs, signaling
spontaneous breaking of the $\mathscr{PT}-$symmetry. The exact solution is used
to provide an accessible benchmark for DMRG with a non-Hermitian matrix product
operator representation that demonstrates an accuracy comparable to its
Hermitian limit thus showing the power of DMRG to handle non-Hermitian many
body calculations.
