Ballistic to diffusive crossover in a weakly interacting Fermi gas

In the absence of disorder and interactions, fermions move coherently and their associated charge and energy exhibit ballistic spreading, even at finite energy density. In the presence of weak interactions and a finite energy density, fermion-fermion scattering leads to a crossover between early-time ballistic and late-time diffusive transport. The relevant crossover timescales and the transport coefficients are both functions of interaction strength, but the question of determining the precise functional dependence is likely impossible to answer exactly. In this work we develop a numerical method (fDAOE) which is powerful enough to provide an approximate answer to this question, and which is consistent with perturbative arguments in the limit of very weak interactions. Our algorithm, which adapts the existing dissipation-assisted operator evolution (DAOE) to fermions, is applicable to systems of interacting fermions at high temperatures. The algorithm approximates the exact dynamics by systematically discarding information from high <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>n</a:mi></a:math>-point functions, and is tailored to capture noninteracting dynamics exactly. Applying our method to a microscopic model of interacting fermions, we numerically determine crossover timescales and diffusion constants for a wide range of interaction strengths. In the limit of weak interaction strength (<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi mathvariant="normal">Δ</b:mi></b:math>), we demonstrate that the crossover from ballistic to diffusive transport happens at a time <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msub><d:mi>t</d:mi><d:mi>D</d:mi></d:msub><d:mo>∼</d:mo><d:mn>1</d:mn><d:mo>/</d:mo><d:msup><d:mi mathvariant="normal">Δ</d:mi><d:mn>2</d:mn></d:msup></d:mrow></d:math> and that the diffusion constant similarly scales as <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>D</f:mi><f:mo>∼</f:mo><f:mn>1</f:mn><f:mo>/</f:mo><f:msup><f:mi mathvariant="normal">Δ</f:mi><f:mn>2</f:mn></f:msup></f:mrow></f:math>. We confirm that these scalings are consistent with a perturbative Fermi's golden rule calculation, and we provide a heuristic operator-spreading picture for the crossover between ballistic and diffusive transport.