Exact solutions of dissipative quantum spin chains using Majorana fermions

The Lindblad equation is the well-known quantum master equation which describes the evolution of open quantum systems. While Lindblad equations have been used in the past mostly to describe few-particle systems in e.g., quantum optics, recent years have witnessed a growing interest in many-body systems in the Lindblad setting. However, very few exact results are available for many-body systems. The main difficulty is that we often need to deal with effective interactions arising from dissipation [1] even when the Hamiltonian itself is reducible to that of a free-particle system. This prevents us from understanding the full dynamics of the system.

In this talk, we construct a new exactly solvable dissipative spin model which maps to an effective non-Hermitian many-body system. By Kitaev's technique [2], this model can be seen as free Majorana fermions in a static Z₂ gauge field, allowing us to obtain the full dynamics of this open system sector by sector. With this method, we obtain the Liouvillian gap (the inverse of relaxation time) numerically and exactly. We also obtain a closed formula for the auto-correlation of the edge spin by relating it to the enumeration of lattice paths.

[1] M. Znidaric, Phys. Rev. E 89, 042140 (2014)
[2] A. Kitaev, Ann. Phys. 321, 2 (2006)