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We study the boundary-dissipated transverse field Ising model described by aLindblad Master equation and exactly solve its Liouvillian spectrum in thewhole parameter space. By mapping the Liouvillian into a Su-Schrieffer-Heegermodel with imaginary boundary potentials under a parity constraint, we solvethe rapidity spectrum analytically and thus construct the Liouvillian spectrumstrictly with a parity constraint condition. Our results demonstrate that theLiouvillian spectrum displays four different structures, which arecharacterized by different numbers of segments. By analyzing the properties ofrapidity spectrum, we can determine the phase boundaries between differentspectrum structures analytically and prove the Liouvillian gap fulfilling aduality relation in the weak and strong dissipation region. Furthermore, weunveil the existence of a dynamical duality, i.e., the long-time relaxationdynamics exhibits almost the same dynamical behavior in the weak and strongdissipation region as long as the duality relation holds true.

Exact solution of the boundary-dissipated transverse field Ising model: Structure of Liouvillian spectrum and dynamical duality