unphysical solutions of the kadanoff-baym equations in linear response

7/25/2019 Unphysical Solutions of the Kadanoff-Baym Equations in Linear Response

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RAPID COMMUNICATIONS

PHYSICAL REVIEW B93, 041103(R) (2016)

On the unphysical solutions of the Kadanoff-Baym equations in linear response:

Correlation-induced homogeneous density-distribution and attractors

Adrian Stan*

Sorbonne Universit es, UPMC Universit e Paris VI, UMR8112, LERMA, F-75005 Paris, France;

LERMA, Observatoire de Paris, PSL Research University, CNRS, UMR8112, F-75014 Paris, France;

Laboratoire des Solides Irradies, Ecole Polytechnique, CNRS, CEA-DSM, F-91128 Palaiseau, France;

and European Theoretical Spectroscopy Facility (ETSF)

(Received 13 September 2015; revised manuscript received 12 December 2015; published 8 January 2016)

The Kadanoff-Baym equations, which allow for the calculation of time-dependent expectation values of all

one-particle observables, are found to yield unphysical electron density dynamics in the linear and nonlinear

response, for -derivable approximations, irrespective of interaction strength or type. In particular, we show

that when calculated from the Kadanoff-Baym equations using correlated self-energy approximations, the linear

response dynamics of isolated electron systems damps to an unphysical homogeneous density-distribution. The

damping is also present for Hartree or Hartree-Fock self-energies. These surprising results supplement previous

findings on the nonlinear response, and complement them by showing that the linear response is also plagued

by unphysical dynamics. Being universal, this additional feature indicates the possible presence of an attractor

that leads to amplitude death and a subsequent tendency to a homogeneous charge and density distribution.

This unveils a scenario in which the Kadanoff-Baym dynamics simply breaks down, drastically restricting the

parameter space for which the method can give physically meaningful insights. In addition to their relevance tothe field of ultrafast electron dynamics in isolated and open systems, these findings may also impact the results

obtained with the Bethe-Salpeter equation in linear response, due to the well-known equivalency between the

two methods. This suggests the need for a different approach to the dynamics of quantum systems.

DOI: 10.1103/PhysRevB.93.041103

In modern quantum many-body physics, the Kadanoff-

Baym equations (KBEs) [13] are a crucial component in the

treatment of strongly and weakly correlated systems far from

equilibrium [49]. They have the form

[it h(t)]G(t, t) = (t,t)+

[G](t,t)G(t ,t)dt, (1)

whereh(t) is the one-body part of the Hamiltonian, G(t,t) is

the nonequilibrium Green function, and the kernel (t, t) is

the self-energy approximation [1,10]. The integration is taken

over the so-called Keldysh contour [11]. The self-energy

is a functional of the Green function [G] and, when it

is obtained from a generating functional as [G] = [G]G

,

it satisfies sum rules and macroscopic conservation laws

[2,3,1214]. Well-known examples of such self-consistent

conserving approximations are Hartree-Fock (HF), Second

Born (2B) [1,15], GW[16], and T-matrix [1,17]. Unlike time-

dependent density functional theory, the use of KBEs is not

limited by the adiabatic approximation. This prompted their

use in calculating the properties of both isolated [4,8,1820]

and open[57,21]quantum systems, and as a complementary

method in the theoretical efforts to develop density functionals

[22]. Within the linear response, the time-propagation of

KBEs corresponds to solving the Bethe-Salpeter equation

with advanced kernels [8,23], constitutingfor long-time-

propagationsa benchmark for the latter. Hence, the KBEs

simultaneously became the method of choice and the state of

the art in ultrafast dynamics of correlated electron systems.

*Corresponding author: [email protected]

A dent in the hallmark of the KBEs was first made by

Friesen et al. [19], when they showed that in Hubbard systems,

the time-dependent density of conserving approximations

exhibits a so-called correlation damping in the nonlinear

regime. These findings were later reproduced by Uimonen

[24], and by Hermannset al.[9]. We can illustrate this feature

of KBEs by considering a system described by the generalHamiltonian

H(t) =ij,

hij(t)ci,ci,+

1

2

ijkl,

ci,c

j,ck,cl,, (2)

where i,j label a complete set of one-particle states, ,

are the spin indices (in this work we only consider spin-

unpolarized systems ), and c,c are creation and anni-

hilation operators, respectively. The one-body part of the

Hamiltonianhij(t) may have an arbitrary time dependence.

The two-body part accounts for interactions between the elec-

trons where, in the case of a Coulomb-like interaction[25,26],

vijkl = vijiljk withvij= vii = Ufor i = jand vij= U

2(ij)for i = j and, for the Hubbard model, vij= vii = U for

i = j andv ij= 0.0 for i = j. In this Rapid Communication

we will consider both Coulomb and Hubbard systemswith

on-site energiesh ii = 0 and hoppings hij= 1.0 between the

neighboring sitesi and j in linear and nonlinear responses.

All parameters are given in terms of hoppings.

In the inset of Fig. 1 we reproduce the time-dependent

density, of a Hubbard dimer, within the 2B approximation, as

shown in Fig.3(b)of Ref. [19]. It is obvious that the density at

Site 1, where the strong, steplike perturbation was applied

[w(t) = w0(t) = E = 5.0], quickly damps to an unphysi-

cal value. (Note here the upwards trend.) The mechanism

underlying this situation was labeled correlation-induced

2469-9950/2016/93(4)/041103(6) 041103-1 2016 American Physical Society
http://dx.doi.org/10.1103/PhysRevB.93.041103http://dx.doi.org/10.1103/PhysRevB.93.041103


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ADRIAN STAN PHYSICAL REVIEW B93, 041103(R) (2016)

0 25 50 75 100 125 150time (a.u.t.)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Site

density

Site 1 (Hubbard dimer; U = 4.0, E=5.0)

Site 1 (Coulomb dimer; U = 4.0, E=5.0)

0 1 2 3 4 5time (a.u.t.)

0

0.2

0.4

0.6

0.8

Site

density

FIG. 1. Long-time density dynamics at the perturbed site of

Hubbard (solid line) and Coulomb (dotted line) dimers. Inset: The

short-time density dynamics of the Hubbard dimer at the perturbed

Site 1 (legend and axes in the notation of Ref. [19]).

damping and was attributed to the induction of an infinite

number of poles by the use of partial-summation schemes and

the subsequent generation of diagrams to all orders. These

diagrams are thought to introduce an unsustainable number of

particles and holes in the finite, isolated system, and act as an

artificial reservoir to which the system couples, dampening its

dynamics. In contrast, in an infinite system, these unphysical

excitations would perfectly cancel[19,20]. As in this case the

damping is faster with increasing the perturbation strength,

the findings have restricted the applicability of the KBEs in

finite, isolated systems to a narrow range of external fields,

i.e., linear response. It was argued that as the dynamics in

linear response is described by the Bethe-Salpeter equationwith a kernel /G that has a discrete spectrum, the resulting

response will be undamped. Here, we show that the damping

is not a simple matter of coupling to an artificial bath, but

rather a pathological behavior of the KBEs that manifests in

linear response as well, and even for uncorrelated self-energy

approximations.

Let us note that the results in the inset of Fig. 1, aswellas in

all calculations in Refs. [19,20], areon a very shorttime-scales,

i.e., 510 atomic units of time (a.u.t.). This short time-scale

is insufficient for determining the value of the artificial steady

state to which the time-dependent density damps. Consider

the same Hubbard system as in the inset of Fig. 1 and,

in addition, a Coulomb dimer at half-filling, with the same

U= 4.0, and a constant perturbation E = 5.0 at Site 1. As

seen in Fig. 1 (main plot), after the initial fast damping, the

value of the density at the perturbed site gradually increases up

to 0.5, and stabilizes. Albeit slower, this also holds true for the

Coulomb dimer. Not only the time-dependent density suffers

an amplitude death, but it also shows a tendency towards

a completely unphysical homogeneous density-distribution

(HD-D). This homogeneity is achieved for both Coulomb

and Hubbard interactions, despite the large constant shift of

Site 1 caused by the field applied all throughout the time-

propagation. Clearly unphysical, this additional feature points

out a second unphysical trait of the KBEs in the nonlinear

regime: For strongly correlated, strongly perturbed systems,

0 25 50 75 100 125 1500.475

0.48

0.485

0.49

0.495

0.5

0.505

Site

density

Coulomb

0 25 50 75 100 125 1500.47

0.48

0.49

0.5

0.51

0.52

0.53

Hubbard

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0.48

0.485

0.490.495

0.5

0.505

Site

densi

ty

0 25 50 75 100 125 1500.475

0.48

0.485

0.490.495

0.5

0.505

U = 1.0

U = 2.0

U = 3.0

0 25 50 75 100 125 150

time (a.u.t.)

0.47

0.48

0.49

0.5

0.51

0.52

0.53

Site

density

0 25 50 75 100 125 150

time (a.u.t.)

0.47

0.48

0.49

0.5

0.51

0.52

0.53

(a) (b)

(c) (d)

(e) (f)

FIG. 2. Time-propagation up to 150 atomic units of time (a.u.t.):

(a) Exact resultfor a Coulomb dimerU= 2 with a constant field E =

0.05. (b) Exact result for a Hubbard dimer U= 2 and a kick ofE =

0.05 at t= 0. (c), (d) Coulomb and Hubbard dimers, respectively,

with a constant field ofE = 0.05 within the Second Born self-energy

approximation (U= 1 dotted line; U= 2 dashed line; U= 3 solid

line). (e), (f) Coulomb and Hubbard dimers, respectively, with a kick

ofE = 0.05 applied at t= 0, within the Second Born self-energy

approximation.

the time-dependent density converges to a homogeneous

density-distribution, irrespective of the interaction type. This

result is extremely surprising and constitutes the first important

findingof the present work. It is hard to attribute this additional

feature to the coupling of an artificial reservoir generated by

partial-summation schemes, as argued in Ref. [19].

We further wish to determine the maximum strength of

the external perturbation applied to an interacting, correlatedsystem, such that a conserving self-energy approximation

could still be deemed good with respect to exact. For this we

consider the same dimers at half-filling, as in Fig. 1, with the

difference that we drastically reduce the external perturbations

by twoorders of magnitude, i.e.,E = 5.0 E = 0.05,falling

back onto the linear response regime. In this regime, the many-

body approximations have been shown to perform well, and

without any visible damping, on short time-propagations [19].

In Fig. 2 we investigate thetime-dependent densityof Coulomb

and Hubbard dimers, with increasing the interaction U. The

twotypes of perturbation, constant and kick, are applied to Site

1 only. First, in Fig. 2(a),we show the exact time-dependent

density for a Coulomb dimer with a constant field E = 0.05

applied during the time-propagation. Figure2(b)features the

exact time-dependent density of a Hubbard dimer with a kick

E = 0.05 applied at t= 0. As expected, irrespective of the

interaction or the applied perturbation, an isolated quantum

system presents no damping of the density oscillations.

In what concerns the many-body approximations, in

Figs. 2(c) and 2(d) we plot, for increasing U, the time-

dependent densityobtained from the time-propagation of the

KBEs within the 2B self-energy approximationof a system

subject to a constant field E = 0.05 applied to Site 1. ForU= 0 the damping is absent (not shown), but as Uincreases,

for both interactions, a now familiar damping of the density

oscillations becomes apparent. This damping is accompanied

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0.48

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0.495

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0.52

Site

densi

ty

Coulomb

0 50 100 150 200

time (a.u.t.)

0.49

0.492

0.495

0.497

0.5

0.502

0.505

0.507

0.51

Site 1 (E =0.00, E =0.05)

Site 2 (E =0.00, E =0.00)

Site 1 (E =0.05, E =0.00)

Site 2 (E =0.00, E =0.00)

Hubbard

(a) (b)

FIG. 3. Time-dependent densitiesat Sites 1 and 2 of Coulomb and

Hubbard dimers, with U= 2 and constant field E tp applied to Site

1 during time-propagation (dashed-dotted lines). The same densities

starting from a ground state with a constant shift Egs = 0.05 applied

to Site 1, and no further time-dependent perturbation (solid lines).

by a tendency towards a HD-D, whichbecomes more andmore

apparent with increasing U and is particularly striking after

amplitude death. We remind the reader that we are now in the

linear response regime, while observing the same unphysical

behavior as in nonlinear one (see Fig. 1). On the time-scale

used here, the unphysical HD-D obtained for Coulomb (U=

3) and Hubbard (U 2), shown in Figs. 2(c) and 2(d), is

identically equal to 0.5. A slight tendency towards this latter

value can be observed also for weaker interactions. With

increasingU, both the damping and the tendency mentioned

above become apparent on an ever shorter time-scale. Thislinear response behavior is reminiscent of the case of strong

external perturbationsshown in Fig.1and in line with the

observations in Ref. [27]. In Figs.2(e)and2(f),we show the

time-dependent density following a kick ofE = 0.05 applied

at t= 0. We note the same time-scale to amplitude death as

in the case of a constant perturbation. Overall, no matter the

perturbation, the density-dynamics of Hubbard dimers exhibits

a damping with an amplitude half-life approximately three

times shorter than that of Coulomb dimers. In other words, in

linear and nonlinear response regimes, for a given interaction,

the time to amplitude death is inversely proportional with the

strength of the electron-electron interaction.

In the surprising amplitude death found in Fig. 2, the

damping of the oscillations occurs around a mean value that

changes in time. Even during the damping process, this mean

value shifts towards an unphysical HD-D. It is interesting

to determine what this mean value is. We consider again a

Coulomb and a Hubbard dimer at half-filling, and we focus

on the case ofU= 2. The results are shown in Fig. 3. In

the first setup we start by calculating the ground state of

the dimer (Coulomb or Hubbard) and then we propagate

in time with a constant field (Etp = 0.05) applied to Site

1 [see also Figs. 2(c) and 2(d)]. The second setup consists

of calculating the ground state of the dimer with an applied

constant field (Egs = 0.05) to Site 1, and then propagate this

ground state in time without applying any further perturbation.

0 5 10 15 20 25 30 35 40 45

(a)

(b)

500.488

0.49

0.492

0.494

0.496

0.498

0.5

Site

density

U = 4.0, t = t = 1.0

U = 4.0, t = 1.0, t = 2.0

0 5 10 15 20 25 30 35 40 45 50

time (a.u.t.)

0.488

0.49

0.492

0.494

0.496

0.498

0.5

Site

density

E = 0. 01

E = 0. 05

E = 0.1

FIG. 4. (a) Time-dependent 2B densities at Site 1

of a symmetric and asymmetric four-site Hubbard dimer, with

a constant Etp = 0.05 field applied during time-propagation. (b)

Time-dependent GWdensities at Site 1 of a four-site Hubbard dimer,

with constant external fields of different strengths applied to Site 1.

The time-dependent densities of the first setup oscillate around

the values of the time-dependent densities of the second setup

until the amplitude death occurs, while following the same

trend to HD-D (see Fig. 3). For U= 2, on the time-scale

shown here, this tendency is much more visible for the

Hubbard dimeras we noted before, the amplitude half-life

is approximately three times shorter for Hubbardbut it can

also be observed for the Coulomb dimer, on this time-scale,

at a close inspection. The results in Fig. 3 bring in a new

element: For long-time-propagations, the KBEs are incapable

of propagating an isolated system, with a frequency-dependent

self-energy, without converging to an unphysical steady state,irrespective of the interaction used andthe presence or absence

of a time-dependent field. The opposite of this constitutes a

common misconception and a widespread assumption in the

field.

To investigate if the symmetry of the Hamiltonian plays

a role in the unphysical features underlined above, we can

alter the symmetry of the system while increasing the number

of sites (see Fig. 4). The first system we consider is a

completely symmetric, four-site Hubbard chain with t= 1.0,U= 4.0. A constant fieldEtp = 0.05 is applied to Site 1 during

time-propagation. In the second system the symmetry of the

Hubbard chain withU= 4.0 is altered by making the hopping

between sites 3 4,t2 = 2.0t1, where t1 = 1.0. The field is

applied at the same site. The results are shown in Fig. 4(a).

Although the dynamics of the system is changed due to the

increased kinetic energy of the electrons between sites 3 4,

the damping still follows the same pattern and the HD-D is

reached at a slightly later time. These results also show that

the unphysical feature is not restricted to two-site dimers.

Albeit 2B is a correlated approximation from the same

class of correlated, conserving approximations as GW andT-matrix, it only contains four diagrams, while the others

result from an infinite summation of bubble or ladder diagrams

[13,28]. In Fig.4(b)we show that the present findings are not

limited to 2B, but alsoextend to the conserving approximations

resulting from infinite summations of a type of diagram, e.g.,

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GW. The system considered is a four-site Hubbard dimer

witht= 1.0,U= 4.0 to which fields of different strengths

but within linear responsewere applied to Site 1. For this

case, in Fig. 4(b), we show that the density at Site 1 damps

and tends to HD-D. It can be seen, from a comparison of

the case Etp = 0.05 with Fig. 4(a), that in GW the lifetime

of the excitations is shorter, i.e., HD-D is reached slightly

faster. We note here that the GW, as well as the T-matrixapproximation, were also found to damp the electron dynamics

of Hubbard systems in the nonlinear response regime [20],

although the propagation time was too short for a HD-D to be

observed (see also Fig. 1 andthe discussion in the text). Hence,

the unphysical behavior remains unchanged across correlated,

conserving approximations and regimes, and it is present in

systems of different sizes.

We point out here that the numerical proofs in this Rapid

Communication are not a result of numerical inaccuracies. In

the Supplemental Material we provide the proper numerical

checks for particle-number conservation and time-reversal

invariance[29].

The convergence to HD-D, across both the nonlinear(Fig. 1) and the linear (Fig. 2 and 3) response regimes,

points to a pathological trait of the KBEs, reminiscent of

an unphysical (global) attractor, independent of interaction

strength, external perturbation, or self-energy approximation.

Although unexpected, the high nonlinearity of theseequations,

i.e., complex, integrodifferential equations with nonlinear

kernels that depend on two times, makes the emergence of such

behavior less surprising. Whenever conserving approxima-

tions are employed, the complexity of the self-energy kernels

of the KBEs can be broken down into two pieces,

[G(t, t)] = HF[G(t,t)]+c[G(t,t

)], (3)

whereHF[G(t,t

)] is the Hartree-Fock part and c[G(t,t

)]is the correlation part [15,30]. Diagrammatically, this simply

means that any correlated conserving approximation, e.g., 2B,GW,T-matrix, contains the HF approximation. Furthermore,HF[G(t,t

)] can be broken down into two diagrams: Hartree

(H) and exchange. In this way, by systematically considering

an ever smaller selection of diagrams, we can gauge the role of

the kernel in thedamping, andwe choose to directlyinvestigate

the behavior of KBEs for the uncorrelated approximations

H and HF. We compute the time-dependent Hartree-Fock

(TDHF) densities for Coulomb and Hubbard dimers, and

propagate to800 a.u.t. As inthe case of 2B(see Fig. 5), wealso

calculate the time-dependent densities produced by a setup of

a ground state calculated with an applied Egs = 0.01 field to

Site 1, followed by a time-propagation without any further

perturbation (see also Fig.3). For both Coulomb and Hubbard

dimers, the time-dependent densities of the system with a

constant time-dependent field ofEtp = 0.01 oscillate around,

and ultimately damp to, the values produced by the setup with

only Egs. This is similar to what is observed in the case of

2B. However, in contrast with correlated approximations, no

tendency towards HD-D is present. Another difference is that

the amplitude half-life is comparable for both interactions.

Simplifying thekernel even more leads to the same damping

pattern. In the insets of Fig. 5 we show the time-dependent

Hartree (TDH) densities for the same systems, but withU= 2.5. The density dynamics is damped also in this case.

0 200 400 600 8000.497

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0.501

0.502

0 200 400 600 800

time (a.u.t.)

0.497

0.498

0.499

0.5

0.501

0.502

0.503

Site

density

Site 1 (E =0.00, E =0.01)

Site 2 (E =0.00, E =0.00)

Site 1 (E =0.01, E =0.00)

Site 2 (E =0.00, E =0.00)

Coulomb

0 200 400 600 800

time (a.u.t.)

0.498

0.499

0.5

0.501

0.502

Hubbard

(a) (b)

FIG. 5. (a) Time-dependent Hartree-Fock densities of a Coulomb

dimer (U= 5.0) starting from the ground state, and from a ground

state with an applied field. Inset: Time-dependent Hartree densities

of a Coulomb dimer (U= 2.5). (b) Time-dependent Hartree-Fock

densities of a Hubbard dimer (U= 5.0) starting from a ground stateand from a ground state with an applied field. Inset: Time-dependent

Hartree densities of a Hubbard dimer (U= 2.5).

It is worth mentioning here that for Hubbard systems, the

densities within TDHF with 2Ushould be exactly the same as

those resulting from TDH withU (this provides an additional

consistency numerical check). This is indeed the case, as can

be seen from Fig. 5(b) and the corresponding inset. Hence,

when used in the KBEs, and the propagation times are large,

the H and HF self-energy approximations lead to an amplitude

death of the time-dependent densities, without the tendency to

a HD-D, in this type of system and regime[31]. On a shorter

time-scale, the unphysical damping is much harder to observein this case. As the damping also occurs in H and HF, we can

hypothesize about the existence of an attractor in KBEs.

A system is said to have an attractor if, for a wide range of

initial conditions, its dynamics tends towards the same set of

numerical values [32]. A dynamical systemthat hasreached (or

is in the vicinity of) a stable fixed point will tend back towards

this point if a further perturbation is applied.We investigate this

aspect for a correlated system that has reached an unphysical

HD-D. In Fig. 6 we show the time-dependent density of a

Coulomb dimer withU= 3.0, following a kick ofE = 0.01att= 0. When the density ofboth sites is close to0.5, we applied

a constant field ofE = 0.05 to Site 1. The resulting dynamics

is quickly damped (in about 30 a.u.t.). This emphasizes thestability of the unphysical solution at 0.5, i.e., the HD-D, and

is consistent with the dynamical behavior of a system found in

the vicinity of a fix point.In this Rapid Communication we showed that the KBEs

yield, for -derivable self-energies, an unphysical behaviorof the electron density-dynamics in both linear and nonlinearresponses. While in the nonlinear response the damping toan unphysical steady state has been known for some time[19,20], we supplemented their findings by pointing out thatthe steady state is a peculiar homogeneous density-distributionand by showing that the previous findings in the nonlinearregime also apply to systems with long-range Coulombinteractions. Furthermore, we showed that the same unphysical

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0 25 50 75 100 125 150

time (a.u.t.)

0.496

0.498

0.5

0.502

0.504

Sited

ensity

Site 1

Site 2

FIG. 6. Time-dependentdensities of a Coulomb dimer (U= 3.0).

Att= 0, a deltalike perturbation ofE = 0.01 was applied to Site 1.

At t= 90, a constant field ofE = 0.05 was switched on at the

same site.

behavior is found within 2B and GW, in linear response,for sufficiently long propagation times, irrespective of theinteraction (Hubbard or Coulomb), and in both weakly andstrongly correlated systems. Also, theT-matrix approximationand conserving subsets of diagrams show the same unphysical

traits[27]. We shelve for later a more detailed analysis. Fora simpler kernel, e.g.,H[G] or HF[G], the Kadanoff-Baymdynamics still damps, but lacks the tendency towards a HD-Dshown by correlated approximations in these types of systems.

Taken altogether, the findings in Refs. [19,20] and hereinshow that the unphysical behavior is universal, i.e., across allregimes, and points to the existence of attractors in KBEs.

We investigated this hypothesis by showing that, once theunphysical steady state is reached, perturbing it will lead toan even faster damping. This emphasizes the stability of theunphysical solution.

Because in the linear response regime the time-propagationof the KBEs corresponds to solving the Bethe-Salpeterequation with advanced kernels [8,23], the numerical evidencehere suggests that a self-consistent Bethe-Salpeter approachmay suffer from similar pathologies. Whether or not this is thecase remains an openquestion. However, these findings furtherlimit the applicability of the KBEs to very weakly interactingsystems in linear response and to systems coupled to externalleads, e.g., for which the lifetime of the excitations is shorter

than the artificial damping[20]. This work adds to the recentfindings of other unphysical solutions in many-body perturba-tion theory[3337] and suggests the need for a fundamentallydifferent approach to quantum many-body systems.

The author acknowledges funding by the Academy ofFinland under Grant No. 140327/2010, andwould like to thankAna Rotili and Lucia Reining for useful discussions.

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unphysical solutions of the kadanoff-baym equations in linear response

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