Transfer matrix approach to quantum systems subject to certain Lindblad evolution

Solving for the time evolution of a many particle system whose dynamics is governed by Lindblad equation is hard. We extend the use of the transfer matrix approach to a class of Lindblad equations that admit a closed hierarchy of two point correlators. An example that we treat is the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>X</a:mi><a:mi>X</a:mi></a:mrow></a:math> spin chain, i.e., free fermions, subject to the local onsite dephasing, but can be extended to other Hermitian dissipators, e.g., nonlocal dephasing. We find a simple expression of the Green's function in the Laplace domain. The method can be used to get analytical results in the thermodynamic limit, for instance, to get the evolution of the magnetization density and to explicitly see the crossover between ballistic and diffusive behavior, or to show that the correlations between operators at distance <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>l</b:mi></b:math> decay with time as <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mn>1</c:mn><c:mo>/</c:mo><c:msup><c:mi>t</c:mi><c:mrow><c:mo>⌈</c:mo><c:mi>l</c:mi><c:mo>/</c:mo><c:mn>2</c:mn><c:mo>⌉</c:mo><c:mo>+</c:mo><c:mn>1</c:mn><c:mo>/</c:mo><c:mn>2</c:mn></c:mrow></c:msup></c:mrow></c:math>. It also provides a fast numerical method to determine the evolution of the density with a complexity scaling with the system size more favorably than in previous methods, easily allowing one to study systems with <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>∼</d:mo><d:msup><d:mn>10</d:mn><d:mn>6</d:mn></d:msup></d:mrow></d:math> spins.