3.2 Real-time renormalization-group formalism
3.2.2 RTRG method
Perturbative results for the spin-boson model (e.g. following from Redfield theory [61]) can be recovered by calculating the kernelΣ(z)up to first order in the couplingα. The aim of the present study, however, is to go beyond and calculate the kernel nonpertur-batively by a systematic RG procedure. The idea is to integrate out all contraction lines one after another, as we have already outlined in Section 2.3. This is accounted for by both a renormalized kernelΣ(z)and a renormalized propagator and vertices. We formally introduce the time scaletc, which the bath contractions depend on:
γp1,p2(t, tc) =γp1,p2(t)q(t, tc), t >0. (3.36) Starting withtc = 0 the RG flow is determined by the cutoff-functionq(t, tc). The stan-dard choice ofq(t, tc)is the sharp cutoff [13,14,24]
q(t, tc) = Θ(t−tc). (3.37) This choice has already been applied in Section2.3. With this form ofγp1,p2(t, tc)the RG flow can be described within the diagrammatic language. At a fixed value of tc all con-tractions with a time scalet < tc have already been integrated out, which again reminds us of the remarks in Section 2.3. Thus, in Fig. 3.4 (respectively Fig. 3.3) the shortest contraction line of each of the blocks (a)-(d) is integrated out first. In the diagrams (a)
and (b) this leads to a renormalization of L0, while in the blocks (c) and (d) a vertex is renormalized. The contractions connecting the upper and the lower propagator (as in (b)) may change the states on the upper and the lower propagator simultaneously. This gives rise to rates, so that the renormalized Liouvillian L0 cannot be represented by a commutator with a renormalized Hamiltonian H0 any more, i.e. L0 6= [H0,·]fortc > 0.
Thereby a non-Hamiltonian dynamics is generated. Correspondingly, due to diagrams as (d), a renormalized vertex generally acts on both the forward and backward propagator simultaneously. In the following we denote the rightmost (leftmost) vertex of the kernel Σ(z)byAp(z)(Bp(z)). They are renormalized in a different way than Gp and also be-come z dependent for tc > 0. It should be emphasized that, although the definition of the cutoff-function in Eqs. (3.36) and (3.37) has the advantage that it can be understood in this rather simple picture, this choice is not necessary. Formally, one may think of an arbitrary tc dependence of the function γp1,p2(t, tc), which only has to fulfill the initial condition
γp1,p2(t, tc = 0) =γp1,p2(t). (3.38) Furthermore, a convergent RG flow, where all diagrams are integrated out, can only be expected for
tclim→∞γp1,p2(t, tc) = 0. (3.39) Note that for the spin-boson model the above equation is fulfilled for the sharp cutoff defined in Eqs. (3.36) and (3.37), since
tlim→∞γp1,p2(t) = 0. (3.40) The renormalization for a general functionγp1,p2(t, tc) is based on the invariance of Σ(z). Before introducingtc the kernelΣis given by definition as some functionalF:
Σ(z) =F(L0, Gp, Ap, Bp, γp1,p2(t)) (3.41) with Ap = Bp = Gp. Introducing the time scale tc we postulate the invariance of the left-hand side of Eq. (3.41). This can be achieved by writing
Σ(z) = Σ(z, tc) +F(L0(tc), Gp(tc), Ap(z, tc), Bp(z, tc), γp1,p2(t, tc)). (3.42) With Eq. (3.38) the initial condition attc = 0is then given by Eq. (3.41):
Σ(z, tc = 0) = 0, L0(tc = 0) = L0,
Gp(tc = 0) = Ap(z, tc = 0) =Bp(z, tc = 0) =Gp. (3.43) Because of Eq. (3.39) andF =O(γp1,p2(t, tc))a solution is found fortc → ∞:
Σ(z) = lim
tc→∞Σ(z, tc). (3.44) In the following we derive the renormalization scheme determiningΣ(z, tc).
When incrementing the cutofftc →tc+dtc, the left-hand side of Eq. (3.42) stays the same, so that
0 = dΣ(z) + ∂F
∂L0dL0 +X
p
∂F
∂GpdGp+ ∂F
∂Ap(z)dAp(z) + ∂F
∂Bp(z)dBp(z)
+X
p1,p2
Z ∞ 0
dt ∂F
∂γp1,p2(t, tc)dγp1,p2(t, tc), (3.45) where we have omitted thetc dependence of the objectsΣ(z, tc),L0(tc),Gp(tc),Ap(z, tc) andBp(z, tc). According to Eqs. (3.29) and (3.31), a term contributing toF has the form
(−i)n Z ∞
0
dt1 . . . Z tn−2
0
dtn−1 Ap1(t1, z)· · ·Gpj(tj). . . Gpk(tk)· · ·Bpn(0, z), (3.46) where “ ” represents a contraction. The time-dependence ofGp(t),Ap(t, z)andBp(t, z) stems from the interaction picture respectively the Laplace transform, it is given by
Gp(t) = eiL0tGpe−iL0t, Ap(t, z) = eiztAp(z)e−iL0t, Bp(t, z) = eiL0tBp(z)e−izt.
The last term in Eq. (3.45) leads to terms where one contraction line γp1,p2(t, tc) is re-placed by dγp1,p2(t, tc). This will be indicated by a cross “ × ”. We write Cj for Gj ≡ Gpj(tj) respectively Aj ≡ Apj(tj, z) or Bj ≡ Bpj(tj, z) and define a
“cross contraction” by
×
Cj· · ·Ck=−dγpj,pk
dtc (tj −tk, tc)dtcCj· · ·Ck. (3.47) We have included a minus sign in Eq. (3.47) in order to later identify a cross con-traction with the renormalization contributionsdΣ(z), dL0, dGp, dAp(z), dBp(z), so that Eq. (3.45) is fulfilled.
Let us now consider terms of the form Z
t1>t2>t3>t4
dt2dt3 C1
×
G2G3 C4. (3.48)
Such terms contribute to a renormalization ofL0, which corresponds to the second term in Eq. (3.45). The propagation at a given time pointt∗witht1 > t∗ > t4for an infinitesimal time intervaldtcan be expanded
exp(−iL0dt) = 1−iL0dt . (3.49) Thus, the second term in Eq. (3.45) gives rise to a contributiondL0(t∗) =eiL0t∗dL0e−iL0t∗ to the renormalized Liouvillian at any time pointt∗. Regarding the above term (3.48) one
has to choose this time point t∗, which the cross contraction corresponds to. We fix that time ordering variable t∗ at the smaller time point of the cross contraction. To do that formally the term (3.48) is written as
Z The first term on the right-hand side of Eq. (3.50) is identified withdL0(t3). Including the factors(−i)and the sums overp2, p3 we obtain
Shifting the integration boundaries leads to
−idL0(0) = (−i)2 The second term on the right-hand side of Eq. (3.50) is a correction term due to the choice of the time ordering. Its occurence can be understood, if one multipliesdL0(t3)from the left by some vertexC1. Then time ordering requirest1 > t3. Thus,t1 is decoupled from the integration variablet2, and terms witht2 > t1 > t3 occur. Since such diagrams have not been present before calculatingdL0, we have to substract the corresponding correction term in Eq. (3.50). It is interpreted as a contribution to a renormalized vertexC1, where C1 =G1respectivelyC1 =A1. SinceBis the vertex with the smallest time argument in Σ(z),C1 =B1is not allowed. Thus, we account for this term using the third (respectively fourth) term in Eq. (3.45). In these terms the vertexGp (Ap) is replaced by dGp (dAp).
Denoting the correction term bydC1(c)we obtain
−idC1(c) =−(−i)3
The last term on the right-hand side of Eq. (3.50) is interpreted as a double vertex, since both t1 and t4 lie within the contraction interval [t2, t3]. Such objects are neglected. A further vertex renormalization stems from terms of the form
Z
t2>t1>t3
dt2dt3
×
C2G1G3 . (3.54)
If C2 = G2 this contributes to dG1, whereas C2 = A2 leads to a renormalization dA1. Correspondingly we obtain a contributiondB1 from
Z
t2>t1>t3
dt2dt3
×
G2G1B3 . (3.55)
The terms (3.54) and (3.55) again cause correction terms. However, they correspond to double and higher-order vertex objects, which are neglected. Thus, with shifted integra-tion boundaries the total renormalizaintegra-tion contribuintegra-tionsdGp,dAp anddBpread
−idGp(0) = (−i)
The renormalization scheme is completed by the terms connecting boundary vertex ob-jects. The terms
×
A1G2 and
×
G1B2
do not occur, since we consider only irreducible diagrams. The remaining term
×
A1B2
is accounted for by the renormalization dΣ(z) in Eq. (3.45). With the definition of the Laplace transform this yields
Using the definition in Eq. (3.47), the renormalization in Eq. (3.52) and Eqs. (3.56)
-(3.59) leads to the RG equations the right-hand sides of the above equations becomes trivial (e.g. the sharp cutoff given in Eqs. (3.36) and (3.37)). In principle, the integration variables only refer to the interaction picture, so that we deal with pure differential equations.