Spacetime picture for entanglement generation in noisy fermion chains

Studies of random unitary circuits have shown that the calculation of Rényi entropies of entanglement can be mapped to classical statistical mechanics problems in spacetime. In this paper, we develop an analogous spacetime picture of entanglement generation for random free or weakly interacting fermion systems without conservation laws. We first study a free-fermion model, namely a one-dimensional chain of Majorana modes with nearest-neighbor hoppings, random in both space and time. We analyze the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi><a:mi>th</a:mi></a:math> Rényi entropy of entanglement using a replica formalism, and we show that the effective model is equivalent to an <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>SO</b:mi><b:mo>(</b:mo><b:mn>2</b:mn><b:mi>N</b:mi><b:mo>)</b:mo></b:mrow></b:math> Heisenberg spin chain evolving in imaginary time. By applying a saddle-point approximation to the coherent states path integral for the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>N</c:mi><c:mo>=</c:mo><c:mn>2</c:mn></c:mrow></c:math> case, we arrive at a semiclassical picture for the dynamics of the entanglement purity, in terms of two classical fields in spacetime. The classical solutions involve a smooth domain wall that interpolates between two values, with the width of this smooth domain wall spreading diffusively in time. We then study how adding weak interactions to the free-fermion model modifies this spacetime picture. Interactions reduce the symmetry of the effective continuum description. As a result the width of the entanglement domain wall remains finite, rather than growing diffusively in time. This yields a crossover from diffusive to ballistic spreading of information.