Implementation of the Lindbladian in Matrix Product States
A.J. de Bruyn (TU Delft - Applied Sciences)
Joseph M. Thijssen – Mentor (TU Delft - QN/Thijssen Group)
NV Budko – Mentor (TU Delft - Numerical Analysis)
JLA Dubbeldam – Graduation committee member (TU Delft - Mathematical Physics)
N. Chepiga – Graduation committee member (TU Delft - QN/Chepiga Lab)
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Abstract
In this thesis we implement the Lindblad equation in the matrix product state (MPS) formalism using an operator splitting method. We developed a second-order method based on a Trotter approximation and a third-order high-dimensional midpoint method and we proposed a new fourth-order method based on Duhamel’s principle and a nested RK4 method, all of which preserve positivity and Hermiticity of the density operator. We simulated spin transport through an XXZ-Hamiltonian Heisenberg chain, for which we found the magnetisation profile and measured a spin current of 0.04-0.05. The results obtained are consistent with the existing literature. The extensive error analysis shows that the time step ∆t is the main contributor to the error, if the bond dimension χ is set to at least 15. The third-order method is in general preferred to the second-order method, as only this method preserves trace. We also analysed the Hubbard model, including a spin orbit coupling, in order to propose a method for simulating the chiral induced spin selectivity (CISS) effect.