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arXiv:2508.00254

Breakdown of Fermi's Golden Rule in 1d systems at non-zero temperature

Young, Thomas, Jerome Lloyd, Curt W. von Keyserlingk

Published Aug 1, 2025• 0 citations
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Abstract

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 ($Δ$), a naive application of Fermi's Golden Rule (FGR) predicts an $\mathcal{O}(Δ^{2})$ quasiparticle decay rate. However, for 1d fermions on the lattice at $T>0$, 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 $τ^{-1} \sim Δ^{2} \log Δ^{-2}$. 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.

DOI
10.48550/arxiv.2508.00254

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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.
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.
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 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
We investigate the decoherence of the electron wave packet in purely ballistic one-dimensional systems described through the Luttinger liquid (LL). At a finite temperature TTT and long times ttt, we show that the electron Green's function for a fixed wave vector close to one Fermi point decays as $\mathrm{exp}(\ensuremath{-}t∕{\ensuremath{\tau}}_{F})$ ---as opposed to the power-law behavior occurring at short times---and the emerging electron lifetime obeys ${\ensuremath{\tau}}_{F}^{\ensuremath{-}1}\ensuremath{\propto}T$ for spinful as well as spinless electrons. For strong interactions, $(T{\ensuremath{\tau}}_{F})\ensuremath{\ll}1$, reflecting that the electron is not a good Landau quasiparticle in LL's. We justify that fractionalization is the main source of electron decoherence for spinful as well as spinless electrons clarifying the peculiar electron mass renormalization close to the Fermi points. For spinless electrons and weak interactions, our intuition can be enriched through a diagrammatic approach or Fermi golden rule and through a Johnson-Nyquist noise picture. We stress that the electron lifetime (and the fractional quasiparticles) can be revealed from Aharonov-Bohm experiments or momentum resolved tunneling. We aim to compare the results with those of spin-incoherent and chiral LL's.
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The decay rate of quasiparticles in quantum dots is studied through the real time calculation of the single-particle Green function in the self-consistent approximation. The method avoids exact diagonalization, transforming the problem into a system of coupled non-linear integral equations which may be solved iteratively. That allows us to study systems larger than previously treated in the literature. Our results for the inverse participation ratio of many-body states show that the threshold energy for the quasiparticle disintegration is ${E}^{*}\ensuremath{\sim}\sqrt{g}\ensuremath{\Delta}.$ The delocalization transition is soft rather than sharp. Three different regimes as function of the effective interaction strength may be clearly identified at high energies.