A spin chain with non-Hermitian $\mathscr{PT}-$symmetric boundary couplings: exact solution, dissipative Kondo effect, and phase transitions on the edge

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.