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arXiv:1107.5632

On the initial stage of quasiparticle decay

Y. Pavlyukh, Ángel Rubio, Jamal Berakdar

Published Jul 28, 2011• 0 citations
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Abstract

The initial stages of the quasiparticle decay in a Fermi liquid are governed by a time-scale distinct from the scattering rates as derived from the Fermi golden rule approach. We show that the initial decay is nonexponential and that it is determined by the zeroth spectral moment of the electron self-energy. We analyzed numerically a number of approximations for the self-energy by comparing with exact configuration interaction calculations for small finite system with fragmented states. A numerically simple approach for computing the spreading of the quasiparticle states for large systems is devised.

DOI
10.48550/arxiv.1107.5632

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The initial stages of the quasiparticle decay in a Fermi liquid are governed by a time-scale distinct from the scattering rates as derived from the Fermi golden rule approach. We show that the initial decay is nonexponential and that it is determined by the zeroth spectral moment of the electron self-energy. We analyzed numerically a number of approximations for the self-energy by comparing with exact configuration interaction calculations for small finite system with fragmented states. A numerically simple approach for computing the spreading of the quasiparticle states for large systems is devised.
Generally, the addition or removal of a single particle in a many-body system does not correspond to an exact eigenstate of the system. Thus the resulting coherent excitation evolves in time. As discussed here, the evolution at short times upon the excitation with the energy $\ensuremath{\epsilon}$ exhibits a quadratic decay [with the rate constant ${\ensuremath{\sigma}}^{2}(\ensuremath{\epsilon})$]. Later on, after some time $\ensuremath{\tau}(\ensuremath{\epsilon})$, the exponential decay sets in. It is governed by another rate constant $\ensuremath{\gamma}(\ensuremath{\epsilon})$. This behavior is generic for many realistic finite and extended systems. For a finite system it is possible to assess this behavior full numerically using an exact solution of the many-body problem. We present a simple model for the electron spectral function that links together all three aforementioned parameters and give a prescription for how the energy uncertainty ${\ensuremath{\sigma}}^{2}(\ensuremath{\epsilon})$ can be computed within the many-body perturbation theory. Our numerical results demonstrate that the model approach accurately reproduces the exact spectral function in a large range of energies even in the case of fragmented many-body states. We show that the central quantity of this study ${\ensuremath{\sigma}}^{2}(\ensuremath{\epsilon})$ can easily be computed exactly or from approximate theories and, hence, can be used for their validation. We also point out how the set in time can be tested by means of attosecond spectroscopy.
We show that in low-dimensional disordered conductors, the quasiparticle decay and the relaxation of the phase are not exponential processes. In the quasi-one-dimensional case, both behave at small time as e(-(t/tau(in))3/2) where the inelastic time, tau(in), identical for both processes, is a power T-2/3 of the temperature. The nonexponential quasiparticle decay results from a modified derivation of the Fermi golden rule. This result implies the existence of an unusual distribution of relaxation times.
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.
We analyze the applicability of the Fermi-golden-rule description of quasiparticle relaxation in a closed diffusive quantum dot with electron-electron interaction. Assuming that single-particle levels are already resolved but the initial stage of quasiparticle disintegration can still be described by a simple exponential decay, we calculate the average inelastic energy relaxation rate of single-particle excitations and its mesoscopic fluctuations. The smallness of mesoscopic fluctuations can then be used as a criterion for the validity of the Fermi-golden-rule description. Technically, we implement the real-space Keldysh diagram technique, handling correlations in the quasi-discrete spectrum non-perturbatively by means of the non-linear supersymmetric sigma model. The unitary symmetry class is considered for simplicity. Our approach is complementary to the lattice-model analysis of Fock space: though we are not able to describe many-body localization, we derive the exact lowest-order expression for mesoscopic fluctuations of the relaxation rate, making no assumptions on the matrix elements of the interaction. It is shown that for the quasiparticle with the energy ε on top of the thermal state with the temperature T, fluctuations of its energy width become large and the Fermi-golden-rule description breaks down at max{ε,T}~δg, where δ is the mean level spacing in the quantum dot, and g is its dimensionless conductance.
Fermi's golden rule applies in the limit where an initial quantum state is weakly coupled to a continuum of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing; this question arises in a number of recently investigated many-body systems. For different symmetry classes, we analytically and/or numerically calculate the universal crossovers in the average decay of the initial state as the level spacing is varied, with the golden rule emerging in the limit of a continuum. Among the corrections to the exponential decay of the initial state given by Fermi's golden rule is the appearance of the spectral form factor in the longtime regime for small but nonzero level spacing.
We inspect the initial and the long-time evolution of excitations in Fermi liquids by analyzing the time structure of the electron spectral function. Focusing on the short-time limit we study the electron-boson model for the homogeneous electron gas and apply the first-order (in boson propagator) cumulant expansion of the electron Green's function. In addition to a quadratic decay in time upon triggering the excitation, we identify nonanalytic terms in the time expansion similar to those found in the Fermi edge singularity phenomenon. We also demonstrate that the exponential decay in time in the long-time limit is inconsistent with the GW\mathit{GW}GW approximation for the self-energy. The background for this is the Paley-Wiener theorem of complex analysis. To reconcile with the Fermi liquid behavior an inclusion of higher order diagrams (in the screened Coulomb interaction) is required.