3.4 Results
3.4.3 Dynamics
In this subsection we calculate the time-dependent reduced density matrixp(t)for the ini-tial preparation of the two-state system in the spin-up state: p(0)uu = 1, p(0)dd=p(0)ud =p(0)du = 0. From the result forΣ(z)fortc → ∞and from Eq. (3.32) we determine the Laplace transformp(z). According to Eq. (3.31)p(t)is then given by
p(t) = 1 2π
Z ∞+ic
−∞+ic
dz e−iztp(z)
= 1 2πect
Z ∞
−∞
dz e−iztp(z+ic) for c∈R+. (3.152) This Fourier integral is calculated numerically using the method described in AppendixE.3 and settingc= 0.1∆.
The solution for the time evolution of the reduced density matrix at vanishing temper-ature is shown in Figs.3.15-3.20for different couplings and for the unbiased and biased case. Here, we again chose ¯t → ∞, in fact these results are insensitive to the choice of t¯even for a coupling strength ofα = 0.2. Figs.3.15 and 3.16 show, that, for α = 0.1, = 0andD= 100∆we achieve a very good agreement with chromostochastic quantum dynamics (CSQD). Only the real parts of the nondiagonal elements, which correspond tohσxi, exhibit a deviation of approximately5%. However, in contrast to the results for hσziCSQD cannot give an error forhσxi[74]. The diagonal elements oscillate in time as well as the imaginary parts of the off-diagonal elements, in fact the latter do not contain any new information due to Eq. (3.151), so that we focus on the real part ofp(t) in the following. For the symmetric case the real parts of the off-diagonal elements show a pure decaying behaviour (Fig. 3.15). In Fig. 3.17 the correct scaling behaviour is checked forD = 100∆respectivelyD = 1000∆by rescaling the results using∆r = 0.599∆ re-spectively∆r = 0.464∆. One obtains coincidence of the diagonal elements, whereas the real parts of the off-diagonal elements have an extra factor of ∆r/∆, which is consistent with Ref. [1]. A comparison with the noninteracting blip approximation (NIBA) both for the unbiased and the biased case is drawn in Figs. 3.18 and 3.19. Note that a formal derivation of the NIBA results is presented in Appendix A. In Figs. 3.18and 3.19, p(t) is calculated from Eqs. (A.14) - (A.19), where we again used the technique presented in Appendix E.3. One recognizes that for = 0 the NIBA gives quite accurate results for the diagonal elements (see also AppendixA), but fails for the nondiagonal elements. The
0 10 20 30 40
∆t
0 0.5 1
p
RTRG CSQD puu=1−pdd
Re(pud)=Re(pdu)
Figure 3.15: Real part of the time-dependent reduced density matrix p(t) for α = 0.1, = 0,D= 100∆andT = 0. Solid lines: RTRG. Dashed lines: CSQD.
0 10 20 30 40
∆t
−0.4
−0.2 0 0.2 0.4
p
RTRG CSQD Im(pud)=−Im(pdu)
Figure 3.16: Imaginary part of the time-dependent reduced density matrixp(t)for α= 0.1,= 0,D= 100∆andT = 0. Solid lines: RTRG. Dashed lines: CSQD.
0 5 10 15
∆rt
0 0.5 1
p
puu=1−pdd
Re(pud)=Re(pdu)
D=100∆ D=1000∆
Figure 3.17: Rescaled real part of the time-dependent reduced density matrix p(t) for α = 0.1,= 0, T = 0andD = 100∆,1000∆. The nondiagonal elements have an extra factor∆r/∆ = 0.599∆,0.464∆.
0 10 20 30 40
∆t
0 0.5 1
p
RTRG NIBA
Re(pud)=Re(pdu) pdd
puu
Figure 3.18: Real part of the time-dependent reduced density matrix p(t) for α = 0.1, = 0,D= 100∆andT = 0. Solid lines: RTRG. Dashed lines: NIBA.
0 10 20 30 40
∆t
0
p 0.5
RTRG NIBA
puu
Re(pud)=Re(pdu) pdd
Figure 3.19: Real part of the time-dependent reduced density matrix p(t) for α = 0.1, = 0.5∆,D= 100∆andT = 0. Solid lines: RTRG. Dashed lines: NIBA.
NIBA even violates the bound
hσx(t)i2 ≤1, (3.153)
which follows fromhσx2(t)i − hσx(t)i2 ≥ 0. For finite bias (see Fig. 3.19) the diagonal elements also have a decaying part, as well as the nondiagonal elements also show oscil-lations. We see, that, for= 0.5∆the NIBA gives poor results both for the diagonal and the nondiagonal elements. In Fig. 3.20 our results for the unbiased and the biased case are shown forα= 0.2.
For one of these plots of p(t) we calculated 512 points of the Laplace transform p(z + ic) for z ∈ [−10∆,10∆]. For z < −10∆ respectively z > 10∆ we extrap-olated p(z + ic) algebraically (see also Appendix E.3). For such an integration prob-lem a parallelization of the computer program may strongly improve the performance.
We applied the “pvm” package to parallelize our program [75]. Using a cluster of four PIII (500MHz)-machines, one of these plots ofp(t)is generated within about five hours only.
From the above results for p(t)we also determine the oscillation frequencyΩ ∼ ∆r
of the diagonal elements. We compare it with the analytical result Ω0, which is valid in the limit of smallαonly. It reads [1]
Ω0 =
2 + (Γ(1−2α) cos(πα))1/1−α∆2r1/2
. (3.154)
Furthermore, for the symmetric case, conformal field theory (CFT) applied to the anisotropic
0 10 20 30 40
∆t
0 0.5 1
p
ε=0 ε=0.5∆ pdd
puu
Re(pud)=Re(pdu)
Figure 3.20: Real part of the time-dependent reduced density matrix p(t) for α = 0.2, D= 100∆andT = 0. Solid lines: = 0. Dashed lines: = 0.5∆.
Kondo model, leads to the solution [29,1]
ΩCF T = sin (πα/2(1−α)) cos (πα/2(1−α)) Γ (α/2(1−α))
√πΓ (1/2(1−α))
× Γ 12 +α
Γ (1−α) Γ(1−2α) cos(πα)
√π
!1/2(1−α)
∆r, (3.155) which is valid for0< α <0.5. Figs.3.21and3.22showΩas a function of the coupling strengthαfor the unbiased and the biased case withD= 100∆andT = 0. As expected, in the limit of smallαour results are consistent withΩ0, while for largeαthere are devia-tions. In comparison to the CFT result for the symmetric case we find a good agreement, only for largeαthere are small deviations.
The long-time behaviour ofp(t)can be approximated by an exponential decay, which is parametrized by the dephasing timeτdephand the relaxation timeτrel:
p(t) =p0+p1eiΩte−t/τdeph+p2e−t/τrel. (3.156) The results forα 1read [1]
τ0deph = 2 (Γ(1−2α) cos(πα))1/(α−1)Ω0
πα∆2r , (3.157)
τ0rel = (Γ(1−2α) cos(πα))1/(α−1)Ω0
πα∆2r . (3.158)
0 0.1 0.2
α
0.2 0.4 0.6 0.8 1
Ω/∆
RTRG α<<1 CFT
Figure 3.21: Oscillation frequencyΩas a function of the coupling constant αfor = 0, D= 100∆andT = 0. Solid line: RTRG. Dashed line:Ω0 (α1). Dotted line: ΩCF T.
0 0.05 0.1 0.15 0.2
α
0.6 0.7 0.8 0.9 1 1.1 1.2
Ω/∆
RTRG α<<1
Figure 3.22: Oscillation frequencyΩas a function of the coupling constantαfor = 0.5∆,D= 100∆andT = 0. Solid line: RTRG. Dashed line: Ω0 (α1).
0 0.1 0.2
α
0 20 40 60 80
∆τ
RTRG α<<1 CFT
τrel τdeph
Figure 3.23: Decay constantsτdeph andτrel as a function of the coupling constantαfor = 0, D = 100∆andT = 0. Solid line: RTRG. Dashed line: τ0deph andτ0rel (α 1).
Dotted line: τCF Tdeph.
0 0.05 0.1 0.15 0.2
α
0 20 40 60 80
∆τ
RTRG α=<<1
τrel
τdeph
Figure 3.24: Decay constantsτdeph andτrel as a function of the coupling constantαfor = 0.5∆,D = 100∆andT = 0. Solid line: RTRG. Dashed line:τ0dephandτ0rel(α1).
For the symmetric case, a solution for the diagonal elements has been found using the CFT [29]. But as the diagonal elements exhibit a purely oscillating behaviour for = 0, CFT yields only a result forτdeph:
τCF Tdeph = sin−2(πα/2(1−α))
√πΓ (1/2(1−α)) Γ (α/2(1−α))
× Γ 12 +α
Γ (1−α) Γ(1−2α) cos(πα)
√π
!1/2(α−1)
∆−1r , (3.159) which again is valid for 0 < α < 0.5. We calculate the dephasing and relaxation times by studying the poles ofp(z). The α dependence ofτdeph andτrel for the unbiased and the biased case is shown in Figs.3.23and3.24. Again our results coincide withτ0dephand τ0rel only in the limit of small α, and the result ofτdeph for = 0 is in good agreement with CFT.