Breakdown of Linear Spin-Wave Theory in a Non-Hermitian Quantum Spin Chain

We present the spin-wave theory of the excitation spectrum and quench dynamics of the non-Hermitian transverse-field Ising model. The complex excitation spectrum is obtained for a generic hypercubic lattice using the linear approximation of the Holstein-Primakoff transformation together with the complex bosonic Bogolyubov transformation. In the one-dimensional case, our result compares very well with the exact quasiparticle dispersion relation obtained via a fermionic representation of the problem, at least in the regime of large dissipation and transverse field. When applied to the quench dynamics we show however that the linear spin-wave approximation breaks down and the bosonic theory is plagued by a divergence at finite times. We understand the origin of this instability using a single mode approximation. While limited to short times, we show that this approach allows us to characterize the dynamics arising from the quench of the dissipative term and the structure of the Lieb-Robinson light-cone of the propagation quantum correlations. Furthermore, for the one-dimensional case, the linear spin-wave dynamics shows good agreement with the exact fermionic solution, both for the local magnetization and the spin-spin correlations.