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arXiv:2310.16043

The ballistic to diffusive crossover in a weakly-interacting Fermi gas

Jerome Lloyd, Tibor Rakovszky, Frank Pollmann, Curt W. von Keyserlingk

Published Oct 24, 2023• 0 citations
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Abstract

Charge and energy are expected to diffuse in interacting systems of fermions at finite temperatures, even in the absence of disorder, with the interactions inducing a crossover from the coherent and ballistic streaming of quasi-particles at early times, to incoherent diffusive behavior at late times. The relevant crossover timescales and the transport coefficients are both controlled by the strength of interactions. In this work we develop a numerical method to simulate such systems at high temperatures, applicable in a wide range of interaction strengths, by adapting Dissipation-assisted Operator Evolution (DAOE) to fermions. Our fermion DAOE, which approximates the exact dynamics by systematically discarding information from high $n$-point functions, is tailored to capture non-interacting dynamics exactly, thus providing a good starting point for the weakly interacting problem. Applying our method to a microscopic model of weakly interacting fermions, we numerically demonstrate that the crossover from ballistic to diffusive transport happens at a time $t_D\sim1/Δ^{2}$ and that the diffusion constant similarly scales as $D \sim 1/Δ^2$, where $Δ$ is the interaction strength. We substantiate this scaling with a Fermi's golden rule calculation in the operator spreading picture, interpreting $t_D$ as the fermion-fermion scattering time and lifetime of the single-particle Green's function.

DOI
10.48550/arxiv.2310.16043

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Charge and energy are expected to diffuse in interacting systems of fermions at finite temperatures, even in the absence of disorder, with the interactions inducing a crossover from the coherent and ballistic streaming of quasi-particles at early times, to incoherent diffusive behavior at late times. The relevant crossover timescales and the transport coefficients are both controlled by the strength of interactions. In this work we develop a numerical method to simulate such systems at high temperatures, applicable in a wide range of interaction strengths, by adapting Dissipation-assisted Operator Evolution (DAOE) to fermions. Our fermion DAOE, which approximates the exact dynamics by systematically discarding information from high nnn-point functions, is tailored to capture non-interacting dynamics exactly, thus providing a good starting point for the weakly interacting problem. Applying our method to a microscopic model of weakly interacting fermions, we numerically demonstrate that the crossover from ballistic to diffusive transport happens at a time tD∼1/Δ2t_D\sim1/Δ^{2}tD​∼1/Δ2 and that the diffusion constant similarly scales as D∼1/Δ2D \sim 1/Δ^2D∼1/Δ2, where ΔΔΔ is the interaction strength. We substantiate this scaling with a Fermi's golden rule calculation in the operator spreading picture, interpreting tDt_DtD​ as the fermion-fermion scattering time and lifetime of the single-particle Green's function.
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 n -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 ( Δ ), we demonstrate that the crossover from ballistic to diffusive transport happens at a time tD∼1/Δ2 and that the diffusion constant similarly scales as D∼1/Δ2 . 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. Published by the American Physical Society 2024
Interacting lattice Hamiltonians at high temperature generically give rise to energy transport governed by the classical diffusion equation; however, predicting the rate of diffusion requires numerical simulation of the microscopic quantum dynamics. For the purpose of predicting such transport properties, computational time evolution methods must be paired with schemes to control the growth of entanglement to tractably simulate for sufficiently long times. One such truncation scheme---dissipation-assisted operator evolution (DAOE)---controls entanglement by damping out components of operators with large Pauli weight. In this paper, we generalize DAOE to treat fermionic systems. Our method instead damps out components of operators with large fermionic weight. We investigate the performance of DAOE, the new fermionic DAOE (FDAOE), and another simulation method, density matrix truncation (DMT), in simulating energy transport in an interacting one-dimensional Majorana chain. The chain is found to have a diffusion coefficient scaling like interaction strength to the fourth power, contrary to naive expectations based on Fermi's Golden rule---but consistent with recent predictions based on the theory of weak integrability breaking. In the weak interaction regime where the fermionic nature of the system is most relevant, FDAOE is found to simulate the system more efficiently than DAOE.
In interacting quantum systems, the single-particle Green's function is expected to decay in time due to the interaction induced decoherence of quasiparticles. In the limit of weak interaction strengths (ΔΔΔ), a naive application of Fermi's Golden Rule (FGR) predicts an O(Δ2)\mathcal{O}(Δ^{2})O(Δ2) quasiparticle decay rate. However, for 1d fermions on the lattice at T>0T>0T>0, this calculation gives a divergent result and the scaling of the quasiparticle lifetime with interaction strength remains an open question. In this work we propose a solution to this question: combining numerical simulations using the recently introduced dissipation-assisted operator evolution (DAOE) method, with non-perturbative diagrammatic re-summations, we predict a logarithmic enhancement of the quasiparticle decay rate τ−1∼Δ2log⁡Δ−2τ^{-1} \sim Δ^{2} \log Δ^{-2}τ−1∼Δ2logΔ−2. We argue that this effect is present in a wide variety of well-known weakly interacting quantum fermionic and bosonic systems, and even in some classical systems, provided the non-interacting limit has quasiparticles with a generic dispersion.
Using artificial dissipation to tame entanglement growth, we chart the emergence of diffusion in a generic interacting lattice model for varying U(1) charge densities. We follow the crossover from ballistic to diffusive transport above a scale set by the scattering length, finding the intuitive result that the diffusion constant scales as D ∝ 1 / ρ at low densities ρ . Our numerical approach generalizes the Dissipation-Assisted Operator Evolution algorithm: in the spirit of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, we effectively approximate nonlocal operators by their ensemble averages, rather than discarding them entirely. This greatly reduces the operator entanglement entropy, while still giving accurate predictions for diffusion constants across all density scales. We further construct a minimal model for the transport crossover, yielding charge correlation functions which agree well with our numerical data. Our results clarify the dominant contributions to hydrodynamic correlation functions of conserved densities, and serve as a guide for generalizations to low-temperature transport.
We present an exact solution of the quantum kinetic equation of a weakly interacting Fermi gas in the crossover from the degenerate Fermi-liquid regime to the classical Boltzmann gas. We construct families of orthogonal polynomials tailored to each angular momentum channel, enabling a fast and systematically improvable decomposition of the phase-space distribution. This approach yields accurate, non-variational predictions for the shear viscosity, thermal diffusivity, and spin diffusivity to leading order in the scattering length. We demonstrate that the commonly used relaxation-time approximation fails dramatically at low temperature--by up to 25%. Our method provides a numerically efficient framework for benchmarking transport in strongly correlated regimes and for simulating the kinetics of quantum gases beyond hydrodynamics.
We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators. We represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision.