Objetivos fundamentales de la psicomotricidad en el ámbito de la

The mentioned fourth order corrections to the flow equations calculation describe the char- acteristic dynamics of a third time regime on a scalet ∼ρ−_F3U−4. They originate from two sources: firstly from a more accurate implementation of the diagonalizing transformation. These are corrections to the quasiparticle picture. Since Fermi liquid theory is exact only strictly at the Fermi energy, they are expected in any microscopic foundation of Fermi liquid theory. In this work they may, moreover, indicate a possible higher order difference between two quasiparticle pictures, namely of the Landau quasiparticles on the one hand and the energy-diagonal degrees of freedom of the flow equation approach on the other. Secondly, the energy diagonal interaction Hamiltonian generates an additional time evolution which is based on the influence of energy diagonal (i.e. elastic) scattering processes. Therefore fourth order secular terms can be expected. The full calculation of these corrections is beyond the scope of this work. Here only an effective treatment can be given which leaves many questions open for a further analysis.

11.2.1 Self-consistent treatment of the momentum distribution

However, the existence of an additional dynamics can be already seen from a simple moti- vation: One can read (10.7) as an implicit equation for the momentum distribution. Then substitutingni =NiNEQ in the phase space factor Jk(E;n) leads to

N_kNEQ(t) =nk−4U2 Z 3D −3D dE sin 2_((E− k)t/2) (E−k)2 Jk(E;n=NNEQ) (→ 10.7)

This self-consistent treatment induces an additional dynamics of the momentum distribution. Since the initial buildup of correlations has redistributed momentum mode occupations such thatN_iNEQ 6=ni, the evaluation of the phase space factor for zero temperature as it was done

in section (9.2.2) is not exact any more. Hence, the denominator in (10.7) is not exactly cancelled at the Fermi energy any more. These corrections, however, become relevant only for higher orders of the perturbative expansion which correspond to longer time scales. Since then the short-time regime has already ended the time dependent kernel in (10.7) can be treated analogously to the derivation of Fermi’s golden rule in section (5.3.1), i.e. in a long time limit for which limt→∞ sin

2_((E−

k)t/2)

(E−k)2 =

π

2tδ(E−k). There it motivates a differential

equation for the momentum distribution which is driven by a nonvanishing scattering integral for elastic two-particle scattering processes.

dN_kNEQ(t) dt ≈hlimt1i N_kNEQ(t)−nk t =−2πtU 2_J k(E =k;n=NNEQ) (11.1)

This suggests a quantum Boltzmann description of the residual dynamics. However, one should not na¨ıvely solve (11.1) since it represents only one among many fourth order correction terms. It is only presented to indicate the direction of the following argument.

11.2Long-time behavior – Thermalization 113

11.2.2 Quantum Boltzmann equation

As it has been explained in the introduction a quantum Boltzmann description aims at the residual dynamics of quasiparticles. This fits well to the scenario of the quenched Fermi liquid: Since a quasiparticle picture has been established by a second order calculation, corrections in fourth order can be treated as such an residual interaction.

Quasiparticle picture for later times

However, one has to ensure that the quasiparticle picture is pertained also for (much) later times. This is a delicate prerequisite of any quantum Boltzmann approach but can be justified in a small neighborhood around the Fermi energy. There scattering processes between quasi- particles are suppressed by phase space arguments. Therefore the lifetime of quasiparticles is extended on a time scaleτk = (k−F)−2. Sufficiently close to the Fermi energy this scale

becomes large. If quasiparticle stability is required, say, on a time related to fourth order perturbation theory inU, this can be safely assumed for an environment of |k−F|< ρU2

around the Fermi energy. Making statements only within this energy window, no decay processes of quasiparticles need to be considered.

Application of the quantum Boltzmann equation

Instead of calculating the other fourth order correction terms to the dynamics of the momen- tum distribution for physical fermions which would complement the self-consistency treatment in (11.1) I perform all further analysis in a quasiparticle representation. This equals approxi- mately working in the diagonal basis of the flow equations approach. According to the remark made above the quasiparticle picture can be used to describe the subsequent dynamics of low energy excitations around the Fermi energy even beyond the time regime of the quasi-steady state. This is done by studying the time evolution of the momentum distribution which is ini- tialized in the quasi-steady, transient state. Its quasiparticle momentum distribution can be made explicit by the approximationN_kNEQ,QP=N_kEQU,PFwhich serves as an initial condition. The effective kinetic equation which describes the residual interaction between Landau quasi- particles is the quantum Boltzmann equation [254]. For a translationally invariant system it can be written as

∂N_kQP(t)

∂t =−ρF U

2_J

k(E=k, NQP(t)) . (11.2)

The characteristic features of the dynamics induced by the quantum Boltzmann equation can be read off its right hand side which is commonly referred to as the collision integral [167].

Thermalization of the momentum distribution

Since insertingN_kQP:NEQintoJk(E =k, n) allows nonzero phase space for scattering processes

in the vicinity of the Fermi surface, linearizing the phase space factor in the collision integral shows that the initial quasiparticle distribution function starts to evolve on the time scale

t ∝ ρ−_F3U−4. This implies that the quasi-steady fermionic distribution function depicted in Fig. 10.2d) starts to decay on the same time scale. The further dynamics of the quasiparticle momentum distribution function follows, again, from the collision integral. Since Jk(E =

k, n) vanishes for Fermi-Dirac distributions (n = nFD) these are the stable fixed points of

is retained on a long time scale the prediction of thermalization of the momentum distribution can be concluded from the overall quantum dynamics of a Boltzmann equation on a lattice. For nonequilibrium initial conditions with energies far below the Fermi energy it describes a flow towards a thermal distribution which is its only attractive fixed point or, mathematically speaking, its unique solution [147].

However, the fourth order corrections which motivate a quantum Boltzmann treatment on a long time scale are present at all times. Hence they cause an obliteration of the Fermi surface discontinuity already at the onset of the dynamics. For short times after the quench, the momentum distribution still shows a steep descent; therefore its widening can be safely neglected. Yet the assumption of a persistent quasiparticle picture for all later times may be questioned. This is a general shortcoming of a quantum Boltzmann approach and is usually accepted.

Final temperature

Moreover, the collision integral conserves the kinetic energy such that the evolution towards a fixed point is constrained to an energy hypersurface in phase space. As the quantum Boltzmann dynamics continues until it reaches a stable fixed point, thermalization of the momentum distribution can be expected. This implies that the excitation energy, which has relaxed into an excess of kinetic energy already at an earlier stage, is redistributed among the momentum modes until a thermal distribution is achieved. The corresponding tempera- ture Tth of the thermal momentum distribution follows directly from fitting its Sommerfeld

expansion [8] to the excitation energy. Equally, one relates the total kinetic energy to the temperature via the specific heatCV of a Fermi liquid which depends linearly on temperature

(cf. table 2.2.6). This reads

ENEQ,KIN_{= 2E}EXC_{= 2ρ}

FαU2 =CVTth ⇒ Tth =

s

6ρFα

m∗kFk_B2

U (11.3)

Quantum Boltzmann equation and the flow equation approach

In the following I will point out that a quantum Boltzmann description of a dynamics caused by a residual interactions is a most natural extension of a flow equation implementation of unitary perturbation theory. The matching point of both approaches are energy diagonal, i.e. elastic scattering processes described by the interaction term in the Hamiltonian. On the one hand, they remain untouched by a flow equation transformation such that an energy diagonal interaction Hamiltonian generates a nontrivial time evolution. On the other hand, only elastic two-quasiparticle scattering processes contribute to the scattering integral of the quantum Boltzmann equation which is responsible for the further dynamics of the momentum distribution. Since the flow equation transformation implements approximately Landau’s quasiparticle mapping, connecting interacting physical fermions to diagonal degrees of freedom which are described by the same quantum numbers as a gas of noninteracting fermions, the nontrivial energy diagonal time evolution in the final representation of a flow equation treatment and the quasiparticle scattering contribution to the evolution of the momentum distribution can be approximately identified. Hence a quantum Boltzmann approach appears as the most adequate extension of a real-time analysis based on the flow equation technique. It stands in a historic line with prior treatments of residual interactions (c.f. 1.4.5).