Elementos de la psicomotricidad

From the time-dependent fermion operator the nonequilibrium time-dependent momentum distribution function for the initial state|Ω0ican be constructed in analogy to the equilibrium

case (9.21). N_kNEQ(t) :=hΩ0| Nk(B = 0, t)|Ω0i=nk−4U2 Z 3D −3D dE sin 2_((E− k)t/2) (E−k)2 Jk(E;n) (10.7)

For convenience, the correlation-induced time-dependent correction to the momentum distri- bution is defined as ∆N_kNEQ(t) = N_kNEQ(t)−nk=−4U2 Z ∞ −∞ dE sin2 (k−E)t 2 (k−E)2 Jk(E;n) (10.8)

and the long-time limit of its time average is performed.

10.2.1 Comparison with a golden rule argument and long-time limit This result invites for a comparison with the ’derivation’ of Fermis golden rule in section (5.3.1): The modulus of the transition matrix element |hn|Hint|ii| seems to translate into

the constant U, the time-dependent energy kernel has a similar [sin(t∆E)/∆E]2 structure. However, a phase space factorJk(E;n) characteristic for fermionic many-particle systems ap-

pears. It describes the analogue to the selection rules which a constant matrix element cannot provide; these turn out to be extremely restrictive and dramatically modify the behavior at the Fermi surface.

SinceJkF(E;n)∼ρ

3_(E−

F)2the phase space factor compensates for the energy denominator

in (10.8). Then the energy kernel does no longer represent a regularization of the Dirac delta distribution. Instead, a plain sinusoidal time dependence remains. The long-time limit is no longer given by a delta function but, effectively, by the time average limt→∞hsin2(αt)it= 1/2.

For the correction to the momentum distribution around the Fermi surface, this means

∆N_kNEQ_≈k F := lim_t→∞h∆N NEQ k≈kF(t)it = −2U 2 Z 3D −3D dE Jk≈kF(E;n) (k≈kF −E) 2 (9.22) = 2 ∆N_kEQU_≈k F (10.9)

10.2.2 Plot of the time dependent momentum distribution function

To illustrate the initial dynamics of the momentum distribution it is helpful to plot equation (10.8) as a function of energy k for different points in time. Unfortunately, the internal

momentum summations present in the phase space factorJk(E;n) may cause severe restric-

tions to a numerical evaluation. Moreover, any explicit evaluation requires the specification of a particular lattice geometry. Since the aim of this work is not to draw attention to par- ticularities of certain implementations of the Hubbard model in unconventional lattices but to conclude on the generic behavior of a Fermi liquid by studying the Hubbard model the simplest lattice geometry may be chosen.

For computational convenience a hypersquare lattice in the limit of infinite dimensions is considered here. It is generally assumed as well as confirmeda posteriori that in this limit the generic features of a Fermi liquid are retained. However, a dramatic simplification results

10.2Nonequilibrium momentum distribution 101

Figure 10.2: (a)-(d): Time evolution ofNNEQ() plotted around the Fermi energy forρFU =

0.6. A fast reduction of the discontinuity and 1/t-oscillations can be observed. The arrow in (d) indicates the size of the quasiparticle residue in the quasi-steady regime. In (e) the universal curves for ∆Nk=Nk−nkare given for both equilibrium and for the nonequilibrium

quasi-steady state in the weak-coupling limit.

from the fact that in infinite dimensions momentum sums can be evaluated as energy integrals over a Gaussian density of states given by (3.6). For lower dimensions than infinity this can be understood in the spirit of a dynamical mean-field approximation.

For three time steps explicit results are depicted in Fig. 10.2.

10.2.3 Findings for the 2nd order nonequilibrium momentum distribution The relation (10.9) contains the main observations of this thesis. Its physical origins and implications will be discussed in detail in chapter (11). Here only statements are made to give an overview of the relevant features.

• Since equation (10.8) is a second order perturbative result, it depicts the time evolu- tion reliable on a time scale set by tP T ∼ 1/U2. This includes the initial buildup of

correlations. Plotting the time dependent momentum distribution for various times in Fig. 10.2 shows that for small values of U within this time frame both a buildup phase and a later (quasi-)steady regime can be observed. Since a nontrivial evolution of the momentum distribution seems to cease already for times smaller thantP T this motivates

• Comparing the modifications of the nonequilibrium momentum distribution function in this formal long-time limit with the corresponding equilibrium result exhibits a decisive

factor of two in (??. Since the calculation is valid only around the Fermi surface, the main conclusion is that the quasiparticle residue Z is characterized by a similar mismatch: In a second order perturbative calculation, its reduction due to correlation effects is doubled in nonequilibrium compared to the equilibrium result 1−ZNEQ = 2(1−ZEQU). Nonetheless, a nonvanishing quasiparticle residue indicates a picture of Landau quasiparticles. A thorough discussion of these consequences can be found in chapter (11).

• Equation (10.8) defines the momentum distribution function in the initial representa- tion, i.e. forphysical particles. However, it allows to conclude on the related momentum distribution in a quasiparticle picture.

• Obviously, the phase space factorJk≈kF(E;n) depends on the correlatornk =hΩ0|c †

kck|Ω0i.

This is a tricky point. On the one hand, this dependence originates largely from a con- sistent application of the normal ordering prescription. According to its definition in section (8.3.2) the correlator is frozen at its initial value. This seems to be plausible for the flow equation transformation at zero time. On the other hand, however, a physical interpretation of the phase space factor as the restriction of scattering processes due to the Pauli principle suggests to replace the fixed correlator by thetime evolved momen- tum distribution. This would describe the opening of phase space because of correlation effects. Then the relations (10.7) and (10.9) are considered as self-consistent equations for a determination of Nk and ∆Nk, respectively.