arXiv:cond-mat/0404361 · Physical Review Letters

Nonexponential Quasiparticle Decay and Phase Relaxation in Low-Dimensional Conductors

Gilles Montambaux, Éric Akkermans

Published Jun 29, 2005• 16 citations

Abstract

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.

DOI: 10.1103/physrevlett.95.016403

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arXiv:cond-mat/0404361 · Physical Review Letters

Nonexponential Quasiparticle Decay and Phase Relaxation in Low-Dimensional Conductors

Gilles Montambaux, Éric Akkermans

Published Jun 29, 2005• 16 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.95.016403

Related papers

Non-exponential relaxations in disordered conductors

Gilles Montambaux, Éric Akkermans

We show that, in low dimensional conductors, the quasiparticle decay and the relaxation of the phase are not exponential processes. In quasi-one dimension, they scale as $e^{- (t/τ_N)^{3/2}}$ where the characteristic time $τ_{in}$ , identical for both processes, is a power $T^{2/3}$ of the temperature. This result implies a distribution of relaxation times.We show that, in low dimensional conductors, the quasiparticle decay and the relaxation of the phase are not exponential processes. In quasi-one dimension, they scale as $e^{- (t/τ_N)^{3/2}}$ where the characteristic time $τ_{in}$, identical for both processes, is a power $T^{2/3}$ of the temperature. This result implies a distribution of relaxation times.

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On the initial stage of quasiparticle decay

Y. Pavlyukh, Ángel Rubio, Jamal Berakdar

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.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.

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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.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.

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

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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 ( $Δ$ ), 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.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.

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Benjamin Reichert, Aleksandra Petković, Zoran Ristivojević

In a one-dimensional weakly interacting Bose-Fermi mixture, one branch of elementary excitations is well described by the Bogoliubov spectrum. Here we use the microscopic theory to study the decay of such quasiparticle excitations. The main scattering process which leads to their decay is the backscattering of a Bogoliubov quasiparticle off the Fermi sea, where a particle-hole pair is excited. For a low-momentum quasiparticle (phonon) of momentum $q$ , we find that the decay rate scales as ${q}^{3}$ provided $q$ is smaller than the Fermi momentum ${k}_{F}$ , while in the opposite case the decay behaves as ${q}^{2}$ . If the ratio of the masses of fermions and bosons is equal to the ratio of the boson-fermion and the boson-boson interaction strengths, the decay rate changes dramatically. It scales as ${q}^{7}$ for $q<{k}_{F}$ , while we find ${q}^{6}$ scaling at $q>{k}_{F}$ . For a high-momentum Bogoliubov quasiparticle, we find a constant decay rate for $q<{k}_{F}$ , while it scales as $1/q$ for $q>{k}_{F}$ . We also find an analytic expression for the decay rate in the crossover region between low and high momenta. The decay rate is a continuous, but nonanalytic function of the momentum at $q={k}_{F}$ . In the special case when the parameters of our system correspond to the integrable model, we observe that the decay rate vanishes.In a one-dimensional weakly interacting Bose-Fermi mixture, one branch of elementary excitations is well described by the Bogoliubov spectrum. Here we use the microscopic theory to study the decay of such quasiparticle excitations. The main scattering process which leads to their decay is the backscattering of a Bogoliubov quasiparticle off the Fermi sea, where a particle-hole pair is excited. For a low-momentum quasiparticle (phonon) of momentum $q$, we find that the decay rate scales as ${q}^{3}$ provided $q$ is smaller than the Fermi momentum ${k}_{F}$, while in the opposite case the decay behaves as ${q}^{2}$. If the ratio of the masses of fermions and bosons is equal to the ratio of the boson-fermion and the boson-boson interaction strengths, the decay rate changes dramatically. It scales as ${q}^{7}$ for $q<{k}_{F}$, while we find ${q}^{6}$ scaling at $q>{k}_{F}$. For a high-momentum Bogoliubov quasiparticle, we find a constant decay rate for $q<{k}_{F}$, while it scales as $1/q$ for $q>{k}_{F}$. We also find an analytic expression for the decay rate in the crossover region between low and high momenta. The decay rate is a continuous, but nonanalytic function of the momentum at $q={k}_{F}$. In the special case when the parameters of our system correspond to the integrable model, we observe that the decay rate vanishes.

arXiv:1611.01143Physical review. B./Physical review. BJan 27, 20177 citations

We show that, in low dimensional conductors, the quasiparticle decay and the relaxation of the phase are not exponential processes. In quasi-one dimension, they scale as

e^{- (t/τ_N)^{3/2}}

where the characteristic time

τ_{in}

, identical for both processes, is a power

T^{2/3}

of the temperature. This result implies a distribution of relaxation times.

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.

q

, we find that the decay rate scales as

{q}^{3}

provided

q

is smaller than the Fermi momentum

{k}_{F}

, while in the opposite case the decay behaves as

{q}^{2}

. If the ratio of the masses of fermions and bosons is equal to the ratio of the boson-fermion and the boson-boson interaction strengths, the decay rate changes dramatically. It scales as

{q}^{7}

for

q<{k}_{F}

, while we find

{q}^{6}

scaling at

q>{k}_{F}

. For a high-momentum Bogoliubov quasiparticle, we find a constant decay rate for

q<{k}_{F}

, while it scales as

1/q

for

q>{k}_{F}

. We also find an analytic expression for the decay rate in the crossover region between low and high momenta. The decay rate is a continuous, but nonanalytic function of the momentum at

q={k}_{F}

. In the special case when the parameters of our system correspond to the integrable model, we observe that the decay rate vanishes.