Extension of quantum symmetric simple exclusion process model for entanglement spreading in stochastic diffusive quantum systems

We investigate out-of-equilibrium entanglement dynamics in a generalization of the so-called <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>Q</a:mi> <a:mi>S</a:mi> <a:mi>S</a:mi> <a:mi>E</a:mi> <a:mi>P</a:mi> </a:mrow> </a:math> model, which is a free-fermion chain with stochastic in space and time hopping amplitudes. In our setup, the noisy amplitudes are spatially modulated satisfying a <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>ν</b:mi> </b:math> -site translation invariance but retaining their randomness in time. For each noise realization, the dynamics preserves Gaussianity, which allows to obtain noise-averaged entanglement-related quantities. The statistics of the steady-state correlators satisfy nontrivial relationships that are of topological nature. They reflect the Haar invariance under multiplication with structured momentum-dependent random <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>S</c:mi> <c:mi>U</c:mi> <c:mo>(</c:mo> <c:mi>ν</c:mi> <c:mo>)</c:mo> </c:mrow> </c:math> matrices. We discuss in detail the case with <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>ν</d:mi> <d:mo>=</d:mo> <d:mn>1</d:mn> </d:mrow> </d:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>ν</e:mi> <e:mo>=</e:mo> <e:mn>2</e:mn> </e:mrow> </e:math> . For <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>ν</f:mi> <f:mo>=</f:mo> <f:mn>1</f:mn> </f:mrow> </f:math> , i.e., spatially homogeneous noise, we show that the entanglement dynamics is describable by a stochastic generalization of the quasiparticle picture. Precisely, entanglement is propagated by pairs of quasiparticles. The entanglement content of the pairs is the same as for the deterministic chain. However, the trajectories of the quasiparticles are random walks, giving rise to diffusive entanglement growth.