Download Real-time resolution of the causality paradox of time

PHYSICAL REVIEW A 77, 062511 共2008兲
Real-time resolution of the causality paradox of time-dependent density-functional theory
Giovanni Vignale
Department of Physics, University of Missouri-Columbia, Columbia, Missouri 65211, USA
共Received 29 January 2008; published 13 June 2008兲
I show that the so-called causality paradox of time-dependent density-functional theory arises from an
incorrect formulation of the variational principle for the time evolution of the density. The correct formulation
not only resolves the paradox in real time, but also leads to an expression for the causal exchange-correlation
kernel in terms of Berry curvature. Furthermore, I show that all the results that were previously derived from
symmetries of the action functional remain valid in the present formulation. Finally, I develop a model
functional theory which explicitly demonstrates the workings of the formulation.
DOI: 10.1103/PhysRevA.77.062511
PACS number共s兲: 31.15.E⫺, 71.15.Mb, 71.45.Gm
I. INTRODUCTION
Time-dependent density-functional theory 共TDDFT兲
关1–3兴 is becoming a standard tool for the computation of
time-dependent phenomena in condensed matter physics and
quantum chemistry. Naturally the growing number of applications has generated a new interest in the foundations of the
theory 共see, for example, the recent critique by Schirmer and
Dreuw 关4兴, and the rebuttal by Maitra, Burke, and van Leeuwen 关5兴兲. In this paper I address the so-called “causality
paradox,” a problem that has troubled TDDFT for many
years 关6兴, and has been the object of many discussions and
technically sophisticated resolutions 关1,7–10兴. I do not disagree with those resolutions, but I wish to propose another
one, which I find technically simpler, more direct, and closer
to the spirit of the original formulation of TDDFT.
As an introduction to the problem, let us recall that the
formal basis of TDDFT is the Runge-Gross 共RG兲 theorem
关2兴, which establishes a biunivocal correspondence between
the time-dependent particle density n共r , t兲 of a many-body
system and the potential v共r , t兲 that gives rise to that density
starting from an assigned quantum state 兩␺0典 at the initial
time t = 0. According to the RG theorem the potential that
gives rise to n共r , t兲 starting from 兩␺0典 is determined by n共r , t兲
and 兩␺0典 up to a time-dependent constant. Similarly, the timedependent quantum state 兩␺共t兲典 is determined by n共r , t兲 and
兩␺0典 up to a time-dependent phase factor. In this sense, both
v共r , t兲 and 兩␺共t兲典 are functionals of n共r , t兲 and 兩␺0典 over a
time interval 0 ⱕ t ⱕ T. They should be denoted by
v关n , 兩␺0典 ; r , t兴 and 兩␺关n , 兩␺0典兴典, respectively. From now on,
however, the dependence on the initial state will not be explicitly noted and we will simply write v关n ; r , t兴 and 兩␺关n兴典.
Admittedly, there is no proof that every reasonable density
can be produced by some local potential: but it is generally
assumed that the densities for which the theorem holds are
dense enough in the space of densities to provide an arbitrarily good approximation to the physical densities one
might encounter in real life. In this paper I will assume tout
court that all the time-dependent densities can be produced
by some local potential: i.e., all time-dependent densities are
v-representable.
Another theorem, proved by van Leeuwen in 1999, 关11兴
extends and strengthens the RG theorem. According to van
Leeuwen’s theorem, the density n共r , t兲 which evolves in an
1050-2947/2008/77共6兲/062511共9兲
interacting system under the action of an external potential
v共r , t兲 starting from an initial state 兩␺0典, can be reproduced in
a noninteracting system evolving under the action of an appropriate and uniquely determined potential vs共r , t兲, starting
from any initial state 兩␺s0典 that has the same density and
divergence of the current density as 兩␺0典. This theorem provides the basis for the extremely useful Kohn-Sham method
of calculating the density. The effective potential vs共r , t兲—a
functional of n共r , t兲, 兩␺0典, and 兩␺s0典—is known as the KohnSham potential. The difference v共r , t兲 − vs共r , t兲 − vH共r , t兲,
where vH共r , t兲 is the Hartree potential, is known as the
exchange-correlation
共xc兲
potential,
denoted
by
vxc关n ; r , t兴—also a functional of n共r , t兲, 兩␺0典, and 兩␺s0典.
In general the potentials v共r , t兲, vs共r , t兲, and vxc共r , t兲 depend on the density n共r⬘ , t⬘兲 at different positions and earlier
times t⬘ ⬍ t, but cannot be affected by changes in the density
at later times t⬘ ⬎ t. This obvious causality requirement implies that the functional derivatives of these potentials with
respect to n共r , t⬘兲 and, in particular, the exchange-correlation
kernel f xc共r , t ; r⬘ , t⬘兲 ⬅ ␦vxc关n ; r , t兴 / ␦n共r⬘ , t⬘兲 vanish for t
⬍ t ⬘.
An interesting question is whether the potentials v共r , t兲,
vs共r , t兲, and vxc共r , t兲 can be generated from functional derivatives of an action functional A关n , 兩␺0典兴 共denoted from now on
simply as A关n兴兲 with respect to the density, in close analogy
with static DFT, where the potentials are functional derivatives of energy functionals with respect to the density. The
existence of such a representation was suggested by RG in
their original paper 关2兴, and was subsequently used by this
author 关12,13兴 to derive several theorems in TDDFT. In the
mid-1990s, however, it became clear that the representation
was problematic to say the least 关6兴. If the potential could be
written as a functional derivative of an action functional
v关n;r,t兴 ⬅
␦A关n兴
,
␦n共r,t兲
共1兲
then we should also have
␦v关n;r,t兴
␦2A关n兴
=
.
␦n共r⬘,t⬘兲 ␦n共r,t兲␦n共r⬘,t⬘兲
共2兲
But this equation is patently false, because the left-hand side
is different from zero only for t ⬎ t⬘ 共by the causality require-
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PHYSICAL REVIEW A 77, 062511 共2008兲
GIOVANNI VIGNALE
ment兲, while the right-hand side is symmetric under interchange of t and t⬘.
This startling observation became quickly known as the
causality paradox and prompted several sophisticated resolutions 关7–10兴. The best known is the van Leeuwen’s construction of a “Keldysh action” in pseudotime 关8,9兴. More recently Mukamel 关10兴 has shown how to construct causal
response functions from symmetrical functional derivatives
corresponding to “Liouville space pathways.” The gist of
these resolutions is that causality is not violated, but one
must use a more abstract mathematical apparatus 共Keldysh
formalism or the Liouville superoperator method兲 in order to
connect functional derivatives of the action to causal response functions.
In this paper I reexamine the “paradox” from a more elementary point of view. I show that the variational principle
for the time evolution of the wave function, when properly
implemented as a variational principle for the density, yields
an expression for the potential as the sum of two terms: 共1兲
the functional derivative of the RG action and 共2兲 a correction term, which cannot be expressed as a functional derivative, but is still simple enough to be included in all the formal proofs. So the gist of the present resolution is that we
learn to write the potential as functional derivative of an
action plus a boundary term. Among other benefits, this approach explains why theorems that were originally proved
under the incorrect assumption 共1兲 关12,13兴, turned out to be
true after all. Furthermore, it leads to interesting expressions
for the inverse of the density-density response function and
the xc kernel in terms of “Berry curvature.”
This paper is organized as follows. In the next section I
discuss in detail the failure of the stationary action principle
for the density and show how the correct causal expressions
for the xc potential and the xc kernel are derived from a
modified variational principle. In Sec. III, I explain why in
many cases one can still pretend that the xc potential is the
functional derivative of the xc action and get correct results.
Appendix A clears up a technical point about the equal-time
singularities of causal response functions. Finally, Appendix
B presents a pedagogical “time-dependent position densityfunctional theory,” which is conceptually equivalent to the
full-fledged TDDFT but can be solved exactly, illustrating
the workings of the new formulation.
ĤV = Ĥ0 +
冕
is the sum of the internal Hamiltonian Ĥ0 共kinetic
+ potential兲 and the interaction with an external timedependent potential field V共r , t兲.
The proof is straightforward. The variation of AV induced
by a variation ␦兩␺典 ⬅ 兩␦␺典 is
冕
冕
T
␦AV关兩␺典兴 =
具␦␺共t兲兩i⳵t − ĤV兩␺共t兲典dt
0
T
+
具␺共t兲兩i⳵t − ĤV兩␦␺共t兲典dt
and the second term on the right-hand side can be integrated
by parts to yield
␦AV关兩␺典兴 =
冕
T
具␦␺共t兲兩i⳵t − ĤV兩␺共t兲典dt +
0
冕
具共i⳵t − ĤV兲
共6兲
The last term on the right-hand side vanishes by virtue of the
boundary conditions on 兩␦␺共t兲典, and the vanishing of the first
two terms is equivalent to the time-dependent Schrödinger
equation 共i⳵t − ĤV兲兩␺共t兲典 = 0.
Since 兩␺共t兲典 is, by virtue of the RG theorem, a functional
of n and 兩␺0典, Runge and Gross suggested that a stationary
action principle for the density could be formulated in terms
of the functional
AV关n兴 =
冕
T
具␺关n兴兩i⳵t − Ĥ兩␺关n兴典dt
0
= A0关n兴 −
冕
T
V共r,t兲n共r,t兲drdt,
共7兲
0
where the “internal action”
A0关n兴 ⬅
冕
T
具␺关n兴兩i⳵t − Ĥ0兩␺关n兴典dt
共8兲
0
is a universal functional of the density and the initial state.
Then, setting ␦AV = 0 for arbitrary variations of the density
we easily find
␦A0关n兴
,
␦n共r,t兲
共9兲
when n共r , t兲 is the density corresponding to V共r , t兲. This implies that the external potential, viewed as a functional of the
density, is the functional derivative of the internal action with
respect to the density
共3兲
v关n;r,t兴 ⬅
0
共ប = 1兲 be stationary 共␦AV = 0兲 with respect to arbitrary variations of the wave function which vanish at the ends of the
time interval 0 ⱕ t ⱕ T, i.e., 兩␦␺共0兲典 = 兩␦␺共T兲典 = 0. Here
T
t=T
.
⫻␺共t兲兩␦␺共t兲典兩dt + i具␺共t兲兩␦␺共t兲典兩t=0
T
具␺共t兲兩i⳵t − ĤV兩␺共t兲典dt
冕
0
V共r,t兲 =
AV关兩␺典兴 =
共5兲
0
II. VARIATIONAL PRINCIPLE FOR THE DENSITY
The starting point is the time-dependent quantum variational principle 关14兴 according to which the time-dependent
Schrödinger equation is equivalent to the requirement that
the action
共4兲
V共r,t兲n̂共r兲dr
␦A0关n兴
,
␦n共r,t兲
共10兲
and the time evolution of the density is determined by requiring v关n ; r , t兴 = V共r , t兲, where V共r , t兲 is the actual external potential. The only problem with Eq. 共10兲, which would otherwise be very useful, is that it plainly contradicts causality, as
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REAL-TIME RESOLUTION OF THE CAUSALITY PARADOX ...
discussed in the Introduction. What went wrong?
The problem arises from the fact that the Frenkel variational principle ␦AV = 0 is valid only for variations of 兩␺典 that
vanish at the end points of the time interval under consideration, i.e., at t = 0 and t = T. But a variation of the density at
any time t ⬍ T inevitably causes a change in the quantum
state at time T. Therefore we can only set 兩␦␺共0兲典 = 0, but
have no right to set 兩␦␺共T兲典 = 0. Taking this into account, and
going back to Eq. 共6兲 we see that the correct formulation of
the variational principle for the density is not ␦AV = 0 but
␦AV关n兴 = i具␺T关n兴兩␦␺T关n兴典.
冓 冏 冔
␦A0关n兴
␦␺T关n兴
− i ␺T关n兴
,
␦n共r,t兲
␦n共r,t兲
共12兲
冓 冏 冔 冓 冏 冔
␺T关n兴
␦␺T关n兴
␦␺T关n兴
=−
␺ 关n兴
␦n共r,t兲
␦n共r,t兲 T
where
v̂关n;t兴 ⬅
冕
v关n;r⬘,t兴n̂共r⬘兲dr⬘ ,
具␺关n;t⬘兴兩v̂关n;t⬘兴兩␺关n;t⬘兴典dt⬘ ,
⌬A0关n兴 = ⌬T具␺T关n兴兩v̂关n;T兴兩␺T关n兴典.
共14兲
共15兲
n̂共r兲 is the density operator, and v关n ; r , t兴 is the potential that
yields n共r , t兲. Now in view of Eq. 共14兲 the internal action
over the time-domain 关0 , T兴 can be written as
共16兲
共17兲
Taking the functional derivative with respect to n共r , t兲, with t
within the interval 关0 , T兴, we see that
⌬
␦v̂关n;T兴
␦A0关n兴
= ⌬T具␺T关n兴兩
兩␺ 关n兴典.
␦n共r,t兲
␦n共r,t兲 T
共18兲
The reason why we could take the functional derivative
inside the expectation value on the right-hand side
of
this
equation
is
that
具␺T关n兴兩v̂关n ; T兴兩␺T关n兴典
= 兰v关n ; r⬘ , T兴n共r⬘ , T兲dr⬘ depends on n共r , t兲 only through the
potential functional v关n ; r⬘ , T兴: the density n共r⬘ , T兲 is, by
definition, unaffected by a variation of the density at the
earlier time t 关21兴.
Consider now the variation of the boundary term of Eq.
共12兲 again due to the change of the upper limit of the time
interval from T to T + ⌬T. To first order in ⌬T we have
兩␺T+⌬T关n兴典 − 兩␺T关n兴典 = − i⌬TĤ关n;t兴兩␺T关n兴典,
共19兲
where Ĥ关n , t兴 = Ĥ0 + v̂关n ; T兴 is the full time-dependent Hamiltonian regarded as a functional of the density. Substituting
this in the variation of the boundary term we get
冓 冏 冔
− i⌬ ␺T关n兴
␦␺T关n兴
␦
= ⌬T具␺T关n兴兩Ĥ关n;T兴
␦n共r,t兲
␦n共r,t兲
共13兲
is a purely imaginary quantity.
At first sight, however, Eq. 共12兲 is still problematic because it appears to depend on the arbitrary upper limit of the
time interval 共T兲 and therefore also on the density at times
t⬘ ⬎ t. However, this is only appearance. The point is that
both the functional derivative of A0关n兴 and the boundary
term, considered separately, have a noncausal dependence on
the density, but the dependence on n共t⬘兲 with t⬘ ⬎ t cancels
out exactly when the two terms are combined.
Let us show this in detail. Consider, for example, an increment of the upper limit of the time interval from T to T
+ ⌬T. The density n共r , t兲 must be smoothly continued to the
larger time interval 关0 , T + ⌬T兴, and the quantum state 兩␺关n兴典
satisfies in this time interval the Schrödinger equation
共i⳵t − Ĥ0 − v̂关n;t兴兲兩␺关n;t兴典 = 0,
T
and its variation, due to the extension of the upper limit from
T to T + ⌬T is, to first order in ⌬T given by
␦␺ 关n兴
T
典 is a compact representation for the functional
where 兩 ␦n共r,t兲
derivative of 兩␺T关n兴典 with respect to density. This is the main
result of this paper, since it shows that the external potential
共and hence also the Kohn-Sham potential and the xc potential兲 is not merely a functional derivative of the Runge-Gross
action A0关n兴. Notice that the additional “boundary term” is
real, in spite of the i, because the quantum state 兩␺T关n兴典 is
normalized to 1 independent of density, implying that
冕
0
共11兲
Here 兩␺T关n兴典 ⬅ 兩␺关n ; T兴典 is the quantum state at time T regarded as a functional of the density 共and of course of the
initial state兲. So we see that the action functional is not stationary, but its variation must be equal to another functional
of the density, which is given on the right-hand side of Eq.
共11兲.
Taking the functional derivative of Eq. 共11兲 with respect
to n共r , t兲 and making use of Eqs. 共7兲 and 共10兲 we get
v关n;r,t兴 =
A0关n兴 =
−
␦
␦n共r,t兲
Ĥ关n;T兴兩␺T关n兴典. 共20兲
This seemingly complicated expression contains the commutator between the Hamiltonian Ĥ关n ; T兴 and the functional
derivative with respect to n共r , t兲. This commutator is simply
␦v̂关n;T兴
− ␦␦Ĥ关n;T兴
n共r,t兲 , which is evidently equal to − ␦n共r,t兲 . Thus the variation of the boundary term is given by
冓 冏 冏冔
− i⌬ ␺T关n兴
␦␺T关n兴
␦n共r,t兲
= − ⌬T具␺T关n兴兩
␦v̂关n;T兴
兩␺ 关n兴典.
␦n共r,t兲 T
共21兲
Combining Eqs. 共18兲 and 共21兲 we see that net variation of the
potential v关n ; r , t兴 is exactly zero. This means that we have
the freedom to change at will the upper limit T of the time
interval: the value of v关n ; r , t兴 will not change, as long as T
remains larger than or equal to t. But this means that we can
always choose T = t, and this proves that the potential at time
t does not depend on what happens to the density at times
later than t. QED.
Now that we have the correct expression for the potential
as a functional of the density it is easy to construct the other
two potentials of interest, namely, the Kohn-Sham potential
and the xc potential. For the Kohn-Sham potential we simply
have
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GIOVANNI VIGNALE
vs关n;r,t兴 =
冓 冏 冏冔
␦A0s关n兴
␦␺sT关n兴
,
− i ␺sT关n兴
␦n共r,t兲
␦n共r,t兲
共22兲
the first and the second derivative of the ␦ function with
respect to its own argument. Thus the complete expression
for ␦v / ␦n has the following form:
冓
where A0s is the internal action for a noninteracting system
and 兩␺sT关n兴典 is the noninteracting version of 兩␺T关n兴典 共starting
from an initial state 兩␺s0典兲. For the exchange-correlation potential we get
冓 冏 冏冔
冓 冏 冏冔
␦Axc关n兴
␦␺T关n兴
− i ␺T关n兴
vxc关n;r,t兴 =
␦n共r,t兲
␦n共r,t兲
+ i ␺sT关n兴
␦␺sT关n兴
,
␦n共r,t兲
共23兲
where Axc关n兴 is the usual xc action defined in the RungeGross paper as the difference A0关n兴 − A0s关n兴 − AH关n兴, where
the last term is the Hartree action. Notice that vxc depends
not only on the density, but also on the two initial states 兩␺0典
and 兩␺s0典.
The above formulas allow us to obtain an elegant expression for the functional derivative of the potential with respect
to the density. Consider, for example, the functional derivative of v共n ; r , t兲. From Eq. 共12兲 we see that this is given by
冓 冏
冏 冏冔
␦v关n;r,t兴
␦2A0关n兴
␦2␺T关n兴
=
− i ␺T关n兴
␦n共r⬘,t⬘兲 ␦n共r,t兲␦n共r⬘,t⬘兲
␦n共r,t兲␦n共r⬘,t⬘兲
冓
␦␺T关n兴 ␦␺T关n兴
.
−i
␦n共r⬘,t⬘兲 ␦n共r,t兲
冏冔
冓 冏 冏冔册
冓 冏 冏冔
␦v关n;r,t兴
=−i
␦n共r⬘,t⬘兲
−
冋冓
冏
␦␺T关n兴 ␦␺T关n兴
␦n共r,t兲 ␦n共r⬘,t⬘兲
= 2Im
␦␺T关n兴 ␦␺T关n兴
␦n共r⬘,t⬘兲 ␦n共r,t兲
共27兲
where Ŝ⬁关nt ; r , r⬘兴 is the differential operator C0 + C1 dtd
2
+ C2 dtd 2 . The coefficient of the leading term C2 is independent
of interactions, and the coefficient of the linear term C1 vanishes in the linear response limit. An explicit demonstration
of the equal-time singularities is provided in Eq. 共B18兲 of
Appendix B.
Equal-time singularities also enter the expression of the
exchange-correlation kernel. Taking into account the fact that
C2 is independent of interactions we find
f xc关n;r,t,r⬘,t⬘兴 ⬅
␦vxc关n;r,t兴
␦n共r⬘,t⬘兲
再冓
= 2␪共t − t⬘兲 Im
冓
− Im
共24兲
␦␺T关n兴 ␦␺T关n兴
␦n共r⬘,t⬘兲 ␦n共r,t兲
冏冔
+ Ŝ⬁关nt ;r,r⬘兴␦共t − t⬘兲,
冏冔
The first two terms on the right-hand side are symmetric
under interchange of r , t and r⬘ , t⬘, while the last term, which
has the structure of a Berry curvature, is antisymmetric under
the same interchange. Let us then subtract from Eq. 共24兲 the
same equation with r , t and r⬘ , t⬘ interchanged. When t ⬎ t⬘
⬘,t⬘兴
then ␦v␦关n;r
vanishes because of causality, so on the leftn共r,t兲
关n;r,t兴
hand side only ␦␦vn共r
is left. And on the right-hand side the
⬘,t⬘兲
symmetric terms cancel out, leaving only the Berry curvature
term. The result is
冏
␦v关n;r,t兴
␦␺T关n兴 ␦␺T关n兴
= 2␪共t − t⬘兲Im
␦n共r⬘,t⬘兲
␦n共r⬘,t⬘兲 ␦n共r,t兲
冏
␦␺T关n兴 ␦␺T关n兴
␦n共r⬘,t⬘兲 ␦n共r,t兲
冏
␦␺sT关n兴 ␦␺sT关n兴
␦n共r⬘,t⬘兲 ␦n共r,t兲
冔冎
冏冔
+ ⌬C1关nt ;r,r⬘兴␦˙ 共t − t⬘兲
+ f xc,⬁关nt ;r,r⬘兴␦共t − t⬘兲,
where ⌬C1 is the difference between the coefficients C1 in
the interacting and noninteracting systems and f xc,⬁ denotes
the difference between the coefficients C0 in the interacting
and noninteracting system. In the linear response regime, i.e.,
when the time-dependent potential is a weak perturbation to
the ground-state, ⌬C1 vanishes and f xc,⬁ reduces to the wellknown infinite-frequency xc kernel of linear response theory
关15,16兴. This behavior is demonstrated in Appendix B for our
exactly solved model—see Eq. 共B21兲.
Finally, I note that the adiabatic approximation to the xc
kernel is given by
ad
共r,t,r⬘,t⬘兲 ⯝ f xc,0共r,r⬘,t兲␦共t − t⬘兲,
f xc
共t ⬎ t⬘兲. 共25兲
Notice that the right-hand side of this formula is independent
of T 共as long as T is larger that t and t⬘兲 since, as we have
shown, v关n ; r , t兴 satisfies the causality requirements.
关n;r,t兴
The above argument determines ␦␦vn共r
for t ⬎ t⬘, but
⬘,t⬘兲
leaves open the possibility of singular contributions at t = t⬘.
In Appendix A I show that indeed the functional derivative
␦v关n ; r , t兴 / ␦n共r⬘ , t⬘兲 contains an equal-time singularity of the
form
C0␦共t − t⬘兲 + C1␦˙ 共t − t⬘兲 + C2␦¨ 共t − t⬘兲,
共26兲
where C0, C1, and C2 are functionals of the density at time t
共⬅nt兲 and functions of r and r⬘. ␦˙ and ␦¨ denote, respectively,
共28兲
共29兲
where
f xc,0共r,r⬘,t兲 = f xc,⬁关nt ;r,r⬘兴 +
冕
t
f xc关n;r,t,r⬘,t⬘兴dt⬘
0
共30兲
is the integral of the exchange-correlation kernel over all
times t⬘ earlier than t. The implicit assumption here is that
the retardation range of the xc kernel is much shorter than
the time scale of variation of the density, so that the xc kernel
can effectively be approximated as a ␦ function on that time
scale. The first term on the right-hand side of this expression
is the contribution of the “true” ␦-function terms of Eq. 共28兲.
The time integral in the second term is restricted to times
strictly less than t.
062511-4
PHYSICAL REVIEW A 77, 062511 共2008兲
REAL-TIME RESOLUTION OF THE CAUSALITY PARADOX ...
␦Axc[n]
III. WHY DID vxc[n ; r , t] = ␦n(r,t) WORK?
The incorrect representation of the xc potential as a functional derivative of the xc action played a significant role in
the early development of TDDFT, particularly in the proof of
theorems that depend on symmetries of the action functional.
Consider, for example, the “zero-force theorem” 关12,13兴, according to which the net force exerted by the xc potential on
the system is zero. This theorem was originally derived from
the apparent invariance of the xc action under a homogeneous time-dependent translation of the density
Axc关n⬘兴 = Axc关n兴,
共31兲
where n⬘共r , t兲 = n关r + x共t兲 , t兴, and x共t兲 is an arbitrary timedependent displacement that vanishes at t = 0. The invariance
of the action under this transformation implies
冕
␦Axc关n兴 ជ
ⵜ n共r兲dr = 0
␦n共r,t兲 r
共32兲
and an integration by parts leads to
冕
ជ ␦Axc关n兴 dr = 0.
n共r兲ⵜ
r
␦n共r,t兲
共33兲
This would be the zero-force theorem if we could identify
␦Axc关n兴
␦n共r,t兲 with vxc关n ; r , t兴 which, of course, is incorrect.
Fortunately, the resolution of the puzzle is now at hand.
The point is that we are making two errors, which are luckily
compensating each other, leaving us with the correct result.
The first error is in Eq. 共31兲: it is not true that the xc action
remains invariant under the transformation n → n⬘. The invariance of Axc was “derived” in Ref. 关12兴 by showing that
the change of the internal action under this transformation
depends only the density, not on the wave function, and
therefore cancels out in the difference A0 − A0s. However, we
failed to include the boundary term i具␺T关n兴 兩 ␦␺T关n兴典, which
does depend on the wave function and therefore does not
cancel out, causing the xc action to vary, to first order in
n⬘ − n, by
␦Axc = i具␺T关n兴兩␦␺T关n兴典 − i具␺sT关n兴兩␦␺sT关n兴典,
provided we calculate that functional derivative incorrectly,
i.e., ignoring the boundary contribution. This is exactly what
we did 共unwittingly兲 in our earlier papers.
IV. CONCLUSION
I believe that the foregoing analysis provides a straightforward and pedagogically transparent resolution of the causality paradox in TDDFT. Compared to the resolutions proposed in Refs. 关8,10兴 the present approach is obviously much
closer to the spirit of the original RG paper. Furthermore, our
approach allows us to understand why in many cases we can
get correct results from an incorrect representation of the xc
potential.
In closing I wish to emphasize that what we have derived
here is a variational principle for the time-dependent density.
A variational principle is not as strong as a minimum principle, yet it is strong enough to formulate a dynamical theory.
While the absolute numerical value of the RG action has no
physical meaning 共because a multiplication of the wave
function by an arbitrary phase factor changes its value by an
arbitrary constant兲, it must be borne in mind that the action
determines the dynamics through its variations, and those
variations are independent of the arbitrary additive constant
共a similar situation occurs in classical mechanics, since the
Lagrangian is defined up to an arbitrary total derivative with
respect to time兲.
ACKNOWLEDGMENTS
I am very grateful to Ilya Tokatly for a critical reading of
the manuscript and for suggesting the analysis of equal-time
singularities in Appendix A, and to Carsten Ullrich for pressing a discussion of the adiabatic limit, and to Robert van
Leeuwen for his insightful comments. This work has been
supported by the DOE under Grant No. DE-FG0205ER46203.
共34兲
where 兩␦␺T关n兴典 ⬅ 兩␺T关n⬘兴典 − 兩␺T关n兴典 and 兩␦␺sT关n兴典 ⬅ 兩␺sT关n⬘兴典
− 兩␺sT关n兴典. Therefore, Eq. 共31兲 must be amended as follows:
Axc关n⬘兴 = Axc关n兴 + i具␺T关n兴兩␦␺T关n兴典 − i具␺sT关n兴兩␦␺sT,关n兴典
APPENDIX A: EQUAL TIME SINGULARITIES
IN ␦v[n ; r , t] Õ ␦n(r⬘ , t⬘)
Following Tokatly 关17,18兴 we write the exact local conservation laws for particle number and momentum
共35兲
and Eq. 共32兲 is replaced by
冕再
冎 冓 冏 冏冔
冓 冏 冏冔
共A1兲
m⳵t ji + ⳵ j共mnuiu j + Pij兲 + n⳵iv = 0,
共A2兲
and
␦Axc关n兴
␦␺T关n兴
− i ␺T关n兴
␦n共r,t兲
␦n共r,t兲
␦␺sT关n兴 ជ
+ i ␺sT关n兴
ⵜrn共r兲dr = 0.
␦n共r,t兲
⳵ tn + ⵜ · j = 0
共36兲
Integrating by parts, and using the correct formula for vxc,
Eq. 共23兲 we do indeed recover the zero force theorem
ជ v 关n ; r , t兴dr = 0.
兰n共r兲ⵜ
r xc
The lesson is quite general: we are allowed to pretend that
the xc potential is the functional derivative of the action,
where j is the current density, ji is its ith Cartesian component, ui = ji / n is the velocity field, ⳵ j denotes the derivative
with respect to r j 共with implied summation over repeated
indices兲, and finally Pij is the stress tensor. These equations
are valid both for interacting and noninteracting systems and
together define the time-dependent potential v as a functional
of the density, provided the velocity field and the stress tensor are regarded as functionals of the density. Interaction
effects enter implicitly through the form of these functionals.
062511-5
PHYSICAL REVIEW A 77, 062511 共2008兲
GIOVANNI VIGNALE
Taking the divergence of the second equation and making
use of the first, we recast the system in the more explicit
form
⳵i共n⳵iv兲 = m⳵2t n + m⳵i共ui⳵tn兲 − m⳵i共j · ⵜui兲 − ⳵i⳵ j Pij .
共A3兲
This equation can be formally solved, yielding
v = Ĝ关m⳵2t n + m⳵i共ui⳵tn兲 − m⳵i共j · ⵜui兲 − ⳵i⳵ j Pij兴, 共A4兲
where Ĝ is the inverse of the operator ⳵in⳵i.
For the limited purpose of identifying the equal-time singularities in ␦v / ␦n we can ignore any retardation in the functional dependence of Pij and ui on the density. Then the
right-hand side of Eq. 共A4兲 depends on the density and its
first two derivatives ⳵tn and ⳵2t n at time t. No higher derivatives are involved. Then taking the functional derivative with
respect to n共r⬘ , t⬘兲 we get a singularity proportional to ␦¨ 共t
− t⬘兲 from ⳵2t n共r , t兲, a singularity proportional to ␦˙ 共t − t⬘兲 from
⳵tn共r , t兲, and, of course, a singularity proportional to ␦共t
− t⬘兲 from the terms that do not contain time derivatives of
the density.
We can furthermore say that the coefficient of ␦¨ 共t − t⬘兲 is
completely free of interaction effects, since the interactions
enter only in the functional Pij and ui 关22兴. And we observe
that the ␦˙ singularity vanishes in the linear response regime
共small perturbations around the ground state兲 because the
current vanishes in the ground state.
Very little can be said in general about the explicit form of
the ␦-function singularity. In the linear response regime, the
dependence of Pij on density has been extensively studied,
but only in local or semilocal approximations 关19兴. A fully
nonlinear, but still local approximation to Pij, known as nonlinear elastic local deformation approximation, has been formulated by Tokatly 关17兴 and studied by Ullrich and Tokatly
关20兴 in a model calculation. An accessible review of this
theory can be found in Chapter 8 of Ref. 关1兴.
APPENDIX B: TIME-DEPENDENT POSITION
FUNCTIONAL THEORY
time-dependent force, which acts only on particle 1. The two
particles are subjected to a parabolic potential well, and interact with each other with a harmonic force with “elastic
constant” k ⬎ 0. The idea is that x̂1 plays the role of the
density operator, its expectation value x1共t兲 is the timedependent density, and −F共t兲 is the external potential. As in
TDDFT, one can show that the time-dependent position x1共t兲
and the initial state of the system at t = 0 uniquely determine
the force f共t兲 that produces it; but in this case the functional
f关x1 ; t兴 can be explicitly constructed.
For definiteness, we start from an initial state described by
the wave function
2
␺0共x1,x2兲 = Ce−X e−
冑1+2kx2/4
,
共B2兲
共1+2k兲1/8
where X ⬅ 共x1 + x2兲 / 2, x ⬅ x1 − x2, and C = ␲1/2 is the normalization constant. This is the ground-state of the Hamiltonian for F = 0. The time evolution of this state under the
full time-dependent Hamiltonian is
2
␺共x1,x2,t兲 = Cei␾共t兲e−关X − Xc共t兲兴 e2i关X−Xc共t兲兴Ẋc共t兲
⫻ e−
冑1+2k关x − xc共t兲兴2/4 i关关x−xc共t兲兴ẋc共t兲/2兴
e
,
共B3兲
where Xc共t兲 and xc共t兲 are the solutions of the classical equations of motion
Ẍc共t兲 + Xc共t兲 = F共t兲/2,
ẍc共t兲 + 共1 + 2k兲xc共t兲 = F共t兲
共B4兲
with initial conditions Xc共0兲 = Ẋc共0兲 = 0 and xc共0兲 = ẋc共0兲 = 0.
The phase factor ␾共t兲 is the classical action 共including the
zero-point energy兲
␾共t兲 = −
1 + 冑1 + 2k
t+
2
where
L = Ẋ2c − X2c + FXc +
冕
t
L共t⬘兲dt⬘ ,
共B5兲
0
冉 冊
ẋ2c
1 + 2k 2 F
xc + xc
−
4
4
2
共B6兲
is the classical Lagrangian.
The solution of the equations of motion is
The evolution of electronic systems subjected to timedependent potentials is in general too complicated to allow
us to construct the functionals v关n ; r , t兴, 兩␺T关n兴典, etc., even in
the simplest nontrivial case of a two-electron system. However, a simpler “position-functional theory” can be easily formulated, which is conceptually equivalent to the full-fledged
theory and allows us to demonstrate explicitly all the main
points of the theory.
Our model is based on a two-particle system in onedimension, with a time-dependent Hamiltonian of the form
1
k
ĤF共t兲 = 关p̂21 + p̂22 + x̂21 + x̂22兴 + 共x̂1 − x̂2兲2 − F共t兲x̂1 ,
2
2
共B1兲
Xc共t兲 =
xc共t兲 =
冕
t
0
1
2
冕
t
sin共t − t⬘兲F共t⬘兲dt⬘ ,
共B7兲
0
sin关冑1 + 2k共t − t⬘兲兴
冑1 + 2k
F共t⬘兲dt⬘ .
The solution of the quantum-mechanical problem is obtained
by substituting Eqs. 共B7兲 into Eqs. 共B3兲, 共B5兲, and 共B6兲. It is
immediately evident that xc共t兲 and Xc共t兲 are the expectation
values of the quantum-mechanical center-of-mass operator
X̂ = 共x̂1 + x̂2兲 / 2 and relative position x̂ = x̂1 − x̂2, respectively.
The expectation values of x̂1 and x̂2 are given by
where x̂1, x̂2 are the position operators of the two particles p̂1
and p̂2 the canonical momentum operators, and F共t兲 is a
062511-6
具␺共t兲兩x̂1兩␺共t兲典 ⬅ x1共t兲 = Xc共t兲 +
xc共t兲
,
2
共B8兲
PHYSICAL REVIEW A 77, 062511 共2008兲
REAL-TIME RESOLUTION OF THE CAUSALITY PARADOX ...
具␺共t兲兩x̂2兩␺共t兲典 ⬅ x2共t兲 = Xc共t兲 −
冓 冏 冏冔
xc共t兲
.
2
− i ␺T关x1兴
␦␺T关x1兴
␦x1共t兲
=
冕
T
0
␦ f关x1 ;t⬘兴
x 共t⬘兲dt⬘ 共B14兲
␦x1共t兲 1
Our task is now to express the external force and the wave
function as functionals of x1共t兲—the “density” of our model.
To do this, we observe that x2共t兲 is related to x1共t兲 by the
classical equation of motion
from which Eq. 共B13兲 follows at once.
Similarly, we can show that the “Berry curvature”
␦␺ 关x 兴 ␦␺ 关x 兴
2Im具 ␦xT共t⬘1兲 兩 ␦xT1共t兲1 典 is given by
ẍ2共t兲 + x2共t兲 = − k关x2共t兲 − x1共t兲兴
␦x2共T兲 ␦ẋ2共T兲 ␦x2共T兲 ␦ẋ2共T兲
−
,
␦x1共t⬘兲 ␦x1共t兲 ␦x1共t兲 ␦x1共t⬘兲
共B9兲
with initial condition x2共0兲 = ẋ2共0兲 = 0. The solution of this
equation, for given x1共t兲, is
x2共t兲 = k
冕
t
sin关冑1 + k共t − t⬘兲兴
冑1 + k
0
x1共t⬘兲dt⬘ .
f关x1 ;t兴 = ẍ1共t兲 + 共1 + k兲x1共t兲
冕
t
0
sin关冑1 + k共t − t⬘兲兴
冑1 + k
x1共t⬘兲dt⬘ .
共B11兲
Observe how the force is uniquely and causally determined
by x1共t兲. Knowing the force we can construct the phase ␾共t兲
关Eq. 共B5兲兴 as a functional of x1, by substituting F = f关x1兴 in
the Lagrangian 共B6兲. Finally, we construct the internal action
functional
冕
冕
T
A0关x1兴 =
具␺关x1 ;t兴兩i⳵t − Ĥ0兩␺关x1,t兴典dt
0
T
=−
f关x1 ;t⬘兴x1共t⬘兲dt⬘ ,
冓 冏 冏冔
␦␺T关x1兴 ␦␺T关x1兴
␦x1共t⬘兲 ␦x1共t兲
2Im
− f关x1,t兴 =
冓 冏 冏冔
␦A0关x1兴
␦␺T关x1兴
− i ␺T关x1兴
,
␦x1共t兲
␦x1共t兲
sin关冑1 + k共t − t⬘兲兴
冑1 + k
from which the arbitrary time T has disappeared. Armed with
this result it is an easy matter to verify that
−
冓 冏 冏冔
␦ f关x1 ;t兴
␦␺T关x1兴 ␦␺T关x1兴
,
= 2Im
␦x1共t⬘兲
␦x1共t⬘兲 ␦x1共t兲
共B17兲
for t ⬎ t⬘, in agreement with Eq. 共25兲. We can also verify the
presence of the singularity at t = t⬘ discussed in Sec. II after
Eq. 共27兲. Indeed, the functional derivative of the first two
terms in the expression of our force functional 共B11兲 gives
冏
␦ f关x1 ;t兴
␦x1共t⬘兲
冏
= ␦¨ 共t − t⬘兲 + 共1 + k兲␦共t − t⬘兲.
共B18兲
sing
Notice that there is no term proportional to ␦˙ in this simple
model.
Finally, we observe that the analog of the xc potential—an
xc force in this case—is Fxc关x1 ; t兴 = f s关x1 , t兴 − f关x1 , t兴, where
the noninteracting force functional f s关x1 , t兴 is obtained from
Eq. 共B11兲 simply by putting k = 0, so that
共B12兲
f s关x1,t兴 = ẍ1共t兲 + x1共t兲
共B19兲
and
Fxc关x1 ;t兴 = − kx1共t兲 + k2
冕
t
sin关冑1 + k共t − t⬘兲兴
冑1 + k
0
= − k兵x1共t兲 − x2关x1 ;t兴其.
共B13兲
where the state 兩␺T关x1兴典 is described by the wave function
共B3兲, evaluated at time T and expressed as a functional of
x1关t兴 关the negative sign on the left-hand side comes from the
fact that the force enters the Hamiltonian ĤF 共B1兲 with a sign
opposite to that of the potential 共4兲兴.
The calculation is greatly simplified by the following two
observations. 共i兲 All the terms that involve an expectation
value of X̂ − Xc or x̂ − xc are obviously zero and 共ii兲 because
xc共t兲 and Xc共t兲 are solutions of the classical equation of motion, the variation of the phase ␾共T兲 comes only from the
variation of the force 共regarded as a functional of x1兲, and
from the variation of x2共t兲 at the upper limit of integration. In
this way we easily arrive at
= k2
共B16兲
0
where f关x1 , t兴 is given by Eq. 共B11兲.
We are now in a position to demonstrate explicitly the
connection between the force and the functional derivative of
the action. Namely, we can prove that
共B15兲
so making use of Eq. 共B10兲 we obtain
共B10兲
From this we can express both Xc共t兲 and xc共t兲 as functionals
of x1共t兲, and hence the whole time-dependent wave function
␺共x1 , x2 , t兲 as a functional of x1共t兲. Furthermore, the force f,
which produces the evolution x1共t兲 is given by f共t兲 = ẍ1共t兲
+ x1共t兲 + k关x1共t兲 − x2共t兲兴. Upon substituting the functional dependence of x2共t兲 on x1共t⬘兲 in the expression for f共t兲 we
obtain the force functional
− k2
1
x1共t⬘兲
共B20兲
Of course, Fxc is nothing but the force exerted by the second
particle on the first, expressed as a functional of the basic
variable x1共t兲. It is worth noting that the singular term ẍ1共t兲
has canceled out in Fxc. The singular part of the functional
derivative of Fxc is simply a ␦ function
冏
␦Fxc关x1 ;t兴
␦x1共t⬘兲
冏
= − k␦共t − t⬘兲,
共B21兲
sing
in agreement with the discussion following Eq. 共28兲.
Let us now demonstrate explicitly how, given the knowledge of the exact functional Fxc关x1 ; t兴 关Eq. 共B20兲兴, one can
calculate the evolution of x1共t兲 within a “Kohn-Sham
scheme.” First of all, we introduce the “Kohn-Sham Hamiltonian”
062511-7
PHYSICAL REVIEW A 77, 062511 共2008兲
GIOVANNI VIGNALE
具␺s共t兲兩x̂1兩␺s共t兲典 = x1c共t兲.
1
Ĥs共t兲 = 关p̂21 + p̂22 + x̂21 + x̂22兴 − F共t兲x̂1 − Fxc关x1 ;t兴x̂1共t兲,
2
共B22兲
which describes two noninteracting particles with an effective force F + Fxc acting only on particle “1.” Then we solve
the time-dependent Schrödinger equation 关i⳵t − Ĥs共t兲兴兩␺s共t兲典
= 0, starting with the noninteracting ground state
␺s0共x1,x2兲 =
1
冑␲
2
2
e−x1/2e−x2/2 ,
共B23兲
which clearly has the same expectation values of x̂1 and p̂1 as
its interacting counterpart 共B2兲. The solution is the timedependent “Kohn-Sham wave function”
␺s共x1,x2,t兲 =
ei␾1共t兲
冑␲
e
−关x1 − x1c共t兲兴2/2 i关x1−x1c共t兲兴ẋ1c共t兲 −x22/2
e
e
The equation of motion 共B25兲 for x1c共t兲 can be rewritten
explicitly as an integrodifferential equation
ẍ1c共t兲 + 共1 + k兲x1c共t兲 = F共t兲 + k2
共s2 + 1 + k兲x1c共s兲 = F共s兲 +
␾1共t兲 = −
t
+
2
冕
=
共B26兲
x1c共t⬘兲,
k2
x1c共s兲
s2 + 1 + k
共B30兲
s2 + 1 + k
F共s兲
共s2 + 1兲共s2 + 1 + 2k兲
冋
册
1
1
F共s兲
.
+ 2
s + 1 s + 1 + 2k 2
2
共B31兲
Going back to the time domain we finally obtain
1
2
冕
t
0
sin共t − t⬘兲F共t⬘兲 +
1
2
冕
t
0
sin关冑1 + 2k共t − t⬘兲兴
冑1 + 2k
F共t⬘兲,
共B32兲
t
L1共t⬘兲dt⬘
冑1 + k
and finally
x1c共t兲 =
with initial conditions x1c共0兲 = ẋ1c共0兲 = 0, and
sin关冑1 + k共t − t⬘兲兴
with initial conditions x1c共0兲 = ẋ1c共0兲 = 0. This equation can
be solved by Laplace transformation. Denoting by x1c共s兲 the
Laplace transform of x1c共t兲 we get
,
共B25兲
t
共B29兲
共B24兲
ẍ1c共t兲 + x1共t兲 = F共t兲 + Fxc关x1 ;t兴
冕
0
x1c共s兲 =
where x1c共t兲 is the solution of the equation of motion
共B28兲
Finally, we make use of the Kohn-Sham wave function to
calculate the expectation value of x̂1:
which of course agrees with the exact solution x1共t兲 = Xc共t兲
+ xc共t兲 / 2 obtained from Eq. 共B7兲.
In TDDFT we do not have the luxury of knowing the
exact xc force functional. But this example shows that, if we
knew it, we could use it to predict the exact evolution of the
density. So there is every reason to believe that a good approximation to the exact xc potential, obtained by whatever
means, would enable us to make good predictions for the
time evolution of the density.
关1兴 Time-Dependent Density Functional Theory, edited by M. A.
L. Marques, C. A. Ullrich, F. Nogueira, A. Rubio, K. Burke,
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1996兲, p. 199.
P. Kramer and M. Saraceno, Geometry of the Time-Dependent
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␦n共r,T兲
More formally, ␦n共r⬘,t兲 = ␦共r − r⬘兲␦共T − t兲, which vanishes if T
0
with
L1共t兲 =
2
x2
ẋ1c
− 1c + 兵F共t兲 + Fxc关x1c ;t兴其x1c共t兲.
2
2
共B27兲
关14兴
关15兴
关16兴
关17兴
关18兴
关19兴
关20兴
关21兴
062511-8
PHYSICAL REVIEW A 77, 062511 共2008兲
REAL-TIME RESOLUTION OF THE CAUSALITY PARADOX ...
⬎ t. It should be noted that the nature of our variational principle for the density is such that we are allowed to impose the
condition ␦n共r , t兲 = 0 at t = 0 and t = T, just as in the formulation
of the variational principle for the wave function we can assume that the variations of the wave function vanish at t = 0
and t = T. This makes the derivation of Eq. 共18兲 even simpler.
关22兴 It is worth noting that in a time-dependent current densityfunctional theory, ui = ji / n would not be a functional but a basic variable, so interaction effects would enter only through
Pij.
062511-9