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arXiv:0907.4608v1 [cond-mat.quant-gas] 27 Jul 2009
Single-particle density matrix for a time-dependent strongly interacting
one-dimensional Bose gas
R. Pezer∗
Faculty of Metallurgy, University of Zagreb, 44103 Sisak, Croatia
T. Gasenzer
Institut f¨ ur Theoretische Physik, Universit¨ at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
H. Buljan
Department of Physics, University of Zagreb, PP 332, Zagreb, Croatia
(Dated: July 27, 2009)
We derive a 1/c-expansion for the single-particle density matrix of a strongly interacting time-
dependent one-dimensional Bose gas, described by the Lieb-Liniger model (c denotes the strength
of the interaction). The formalism is derived by expanding Gaudin’s Fermi-Bose mapping operator
up to 1/c-terms. We derive an efficient numerical algorithm for calculating the density matrix for
time-dependent states in the strong coupling limit, which evolve from a family of initial conditions in
the absence of an external potential. We have applied the formalism to study contraction dynamics
of a localized wave packet upon which a parabolic phase is imprinted initially.
PACS numbers: 05.30.-d,03.75.Kk
I.INTRODUCTION
One of the most attractive many-body quantum sys-
tems nowadays has been introduced by Lieb and Lin-
iger in their landmark paper more than forty years ago
[1]. The system is composed of N identical δ-interacting
bosons in one spatial dimension and is referred to as a
Lieb-Liniger (LL) gas. They have presented an explicit
form of the many-body wave function for a homogeneous
gas with periodic boundary conditions [1], including
equations describing the ground state and the excitation
spectrum. In the strongly interacting ”impenetrable-
core” regime [2], such a one dimensional (1D) system
is referred to as the Tonks-Girardeau (TG) gas; exact so-
lutions in this limit are obtained by Girardeau’s Fermi-
Bose mapping [2]. Following Ref. [1], Yang and Yang
[3] have eliminated a possible existence of phase tran-
sitions in the LL system by proving analyticity of the
partition function. After many recent experimental suc-
cesses [4, 5, 6, 7, 8, 9] in realization of effectively one
dimensional (1D) interacting gases, from the weak up to
the strongly interacting TG regime [5], the LL model has
attracted considerable attention of the physics commu-
nity. There is a clear reason for this; nontrivial quantum
many-body systems are notoriously oblique to a quan-
titative analysis, and therefore possibility of an exact
treatment in particular cases, together with experimental
realization, is of great value. Moreover, exact solutions
can be useful as a benchmark for approximate treatments
aiming to describe a broader range of physical systems.
Even though exact LL many-body wave functions can
∗Electronic address: rpezer@phy.hr
be constructed in some cases (e.g., stationary [1, 10, 11,
12, 13, 14, 15, 16] or time-dependent wave functions
[17, 18, 19, 20]), the calculation of observables (correla-
tion functions) from such solutions usually poses a major
difficulty in practice [21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32]. Various methods have been employed to over-
come this difficulty including, for example, the quantum
inverse scattering method (e.g., see [21, 29]) and quan-
tum Monte Carlo integration [25]. A recent discussion of
several exact methods for the calculation of correlation
functions of a nonequilibrium 1D Bose gas can be found
in Ref. [32]. In the TG limit, the momentum distribution
can be analytically studied for a ring geometry, and also
for harmonic confinement (e.g., see [33, 34]). Numerical
methods for the calculation of the reduced single-particle
density matrix (RSPDM) can be performed efficiently
for various TG states (ground state, excited and time-
dependent states, see Ref. [35] for hard-core bosons on
the lattice, and Ref. [36] for the continuous TG model
[2]).
Ultracold atoms in 1D atomic wave guides enter the
strongly interacting regime at low temperatures, in tight
transverse confinement, and with strong effective interac-
tions [37, 38, 39]. The correlations functions of a LL gas
can in this limit be calculated by using 1/c expansions
(e.g., see [22, 24, 27]) from the TG (c → ∞) regime.
These calculations in the strongly interacting limit ex-
ploit the fact that a bosonic LL gas is dual to a fermionic
system [40], such that weakly interacting fermions cor-
respond to strongly interacting bosons and vice versa
[40, 41]. A strongly interacting 1D Bose gas was stud-
ied in Ref. [42] by using perturbation theory for the dual
fermionic system. In Ref. [27], the dynamic structure
factor was calculated for zero and finite temperatures.
Here we calculate the 1/c correction for the RSPDM of
a Lieb-Liniger gas, which adds upon a recently obtained
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formula for the RSPDM of a TG gas [36]. The method
is derived by using the 1/c term of the so-called Fermi-
Bose (FB) mapping operator introduced by Gaudin [17].
The FB operator method provides us with exact time-
dependent solutions of a LL model [17, 19] in the ab-
sence of external potentials and other boundary condi-
tions; it was recently used to study free expansion of a
LL gas [20]. We derive an efficient numerical algorithm
for calculating the RSPDM for time-dependent states in
the strong coupling limit, which evolve from a family of
initial conditions in the absence of an external potential.
We employ it to study the evolution of a many-body wave
packet with a parabolic phase imprinted at t = 0, which
corresponds to focusing with a lens in optics, a technique
experimentally feasible in 1D atomic gases [9].
This work complements the studies of nonequilibrium
dynamics of interacting Bose gases which have been ad-
dressed by use of the LL model away from [18, 19, 20, 32]
and in the TG limit [35, 36, 43, 44, 45, 46, 47, 48, 49].
The phenomenological relevance of these studies is un-
derlined by the fact that nonequilibrium dynamics is ac-
cessible experimentally [7, 8].
The paper is organized as follows: Section II gives a
detailed account of the formalism, where we outline the
procedure for the calculation of the RSPDM. In section
III we present the example of a Bose gas initially in the
ground state within a harmonic trap, effectively in the
TG regime; at t = 0 a parabolic phase is applied on the
initial state and the harmonic potential is turned off. The
subsequent dynamics leads to focusing of the cloud and
local increase of the 1/c correction. Details of the deriva-
tion and relevant mathematical identities are collected in
the Appendices.
II.
STRONGLY-INTERACTING LIEB-LINIGER GAS
FORMALISM FOR THE TIME-DEPENDENT
The Lieb-Liniger model describes a system of bosons
interacting via pointlike interactions; the many-body
Schr¨ odinger equation for the LL gas of N such bosons
reads [1]
i∂ψLL
∂t
= −
N
?
i=1
∂2ψLL
∂x2
i
+
?
1≤i<j≤N
2cδ(xi−xj)ψLL. (1)
Here, ψLL(x1,...,xN,t) is the time-dependent bosonic
wave function, c is the strength of the interaction. For
now, we assume the absence of any external potential and
boundary conditions.Under these circumstances, the
time-dependent LL model (1) can be solved by employing
the Fermi-Bose transformation [17, 19]. This method can
be applied, e.g., to exactly study free expansion from an
initially localized state [20]. If ψF(x1,...,xN,t) is an an-
tisymmetric (fermionic) wave function, which obeys the
Schr¨ odinger equation for a noninteracting Fermi gas,
i∂ψF
∂t
= −
N
?
i=1
∂2ψF
∂x2
i
, (2)
then the wave function
ψLL(x1,...,xN,t) =
?
N(c)
?
1≤r<l≤N
??
?
sgn(xl− xr)+
1
c
?
∂
∂xl
−
∂
∂xr
ψF(x1,...,xN,t) (3)
obeys Eq. (1) as pointed out in Ref. [17] (see also
Refs. [19, 20]). Here, N(c) is a normalization constant,
and the differential operator in the square brackets de-
notes the Fermi-Bose mapping operator [19]. The deriva-
tives do not act on any of the sign functions (the sign
functions can be avoided by working in only one sector of
the configuration space, e.g., for x1< x2... < xN, e.g.,
see Refs. [19, 20]). We assume that ψF(x1,...,xN,t)
is normalized to unity. Equation (3) can be reorganized
in a finite power series with terms of order 1/cm, where
m = 0,1,...,N(N−1)/2. By using the Fermi-Bose map-
ping operator in this form, one obtains a systematic ex-
pansion of the exact many-body wave function ψLL in
the inverse interaction strength 1/c.
The expansion of the wave function (3) in the inverse
coupling strength 1/c is particularly useful in the strong
coupling limit, as it allows an approximate calculation
of one-body observables contained within the reduced
single-particle density matrix (RSPDM). In the Tonks-
Girardeau limit, where c = ∞, a formula for efficient cal-
culation of the RSPDM has recently been derived [36].
Here we generalize that result to include the 1/c term
in the expansion. By keeping only the 1/c terms in the
expansion, Eq. (3) reduces to
ψLL(x1,...,xN,t) ≃
?
?∂ψF
∂xl
N(c)
?
1≤i<j≤N
−∂ψF
∂xr
sgn(xj−xi)
?
ψF+
1
c
?
1≤r<l≤N
sgn(xl− xr)
??
(x1,...,xN,t)
(4)
The first term is simply the TG gas wave function [2],
while the second term gives the 1/c correction to the TG
wave function when the coupling constant c is finite. This
expression is the starting point for all results that will be
derived in this paper. The RSPDM is defined as
ρLL(x,y,t) =N
?
dx2···dxNψLL(x,x2,...,xN,t)∗
× ψLL(y,x2,...,xN,t). (5)
On inserting Eq. (4) into Eq. (5)) we obtain a formal
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expression for the O(1/c) correction of the RSPDM:
ρLL(x,y,t) = N(c)ˆI(X)ψ∗
+
cN(c)
1≤r<l≤N
F(x,X,t)ψF(y,X,t)
1
?
ˆI(X)
?
sgn(xl− xr)
?∂ψF
∂xl
+ sgn(xl− xr)ψ∗
?∂ψF
∂xl
−∂ψF
∂xr
?∗
(x,X,t)
ψF(y,X,t)
F(x,X,t)
−∂ψF
∂xr
?
(y,X,t)
?
+ O(1/c2).
(6)
Here, X = (x2,...,xN) and the integral operatorˆI(X)
is defined as:
ˆI(X) = N
N
?
n=2
?∞
−∞
dxnsgn(x − xn)sgn(y − xn). (7)
The first term on the right hand side of Eq. (6) is the
TG gas RSPDM [36]. It can be proven (as is done in Ap-
pendix A) that only the partial derivatives with respect
to the first coordinate x1in Eq. (6) give a nonvanishing
contribution. After eliminating the vanishing terms from
Eq. (6) we are left with
ρLL(x,y,t) = N(c)ρTG(x,y,t) +1
N
?
l=2
cN(c)
?∗
×
ˆI(X)
?
sgn(x − xl)
?∂ψF
∂x1
(x,X,t)
ψF(y,X,t)+
sgn(y − xl)ψ∗
F(x,X,t)
?∂ψF
∂x1
?
(y,X,t)
?
+ O(1/c2)
= ρTG(x,y,t) +1
c[η(x,y,t) + η(y,x,t)∗] + O(1/c2).
(8)
In Eq.
η(x,y,t):
(8) we have implicitly defined the quantity
η(x,y,t) =
N
?
l=2
ˆI(X)sgn(x − xl)
?∂ψF
∂x1
?∗
(x,X,t)
?∗
(x,X,t)
ψF(y,X,t)
= (N − 1)ˆI(X)sgn(x − x2)
?∂ψF
∂x1
ψF(y,X,t).
(9)
By using the calculation presented in Appendix A it fol-
lows that
?
dx[η(x,x,t) + η(x,x,t)∗] = 0,
that is, the 1/c correction to the single-particle density
ρLL(x,x,t) increases the density in some regions of space,
but also lowers it in others such that the integral over
the terms of order 1/c is zero. By using this result we
see that in the leading 1/c approximation we can take
N(c) = 1; this fact has already been utilized in the last
line of Eq. (8). It is straightforward to verify that the
integralsˆI(X)sgn(x−xl)
pendent of l (2 ≤ l ≤ N), which yields the second identity
in Eq. (9). The last line of Eq. (8) verifies that to or-
der 1/c the RSPDM possesses as required the symmetry
ρLL(x,y,t) = ρLL(y,x,t)∗.
Up to this point we did not make any assumptions
on the structure of ψF (except that it is antisymmet-
ric and normalized), that is, the derivation was general,
valid even if the wave functions should describe the gas
in an external potential etc. Quite generally, ψFcan also
be considered as a function of c, and one could expand
it in powers of 1/c. ¿From this point on, ψF will be
represented as a Slater determinant formed from single
particle wave functions,
?
∂ψF
∂x1
?∗
(x,X)ψF(y,X) are inde-
ψF =
1
√N!
?
P
(−)PφP1(x1,t)···φPN(xN,t);(10)
such wave functions can be used to study dynamics on an
infinite line, which arises from initial conditions given by
Eqs. (4) and (10). In Eq. (10), φj(x,t) (j = 1,...,N) are
orthonormal single-particle wave functions which obey
the Schr¨ odinger equation i∂φj/∂t = −∂2φj/∂x2. P de-
notes a permutation P(1,...,N) = (P1,...,PN) of the
particle number indices, and (−)Pis its signature. In
this form ψF enables us to rewrite Eq. (8) such that it
involves 1D integrals only, resulting in certain algebraic
cofactors suitable for numerical calculation.
A.Algorithm for RSPDM calculation
For the sake of clarity of the presentation, we first de-
scribe the algorithm for calculation of the RSPDM, and
only afterwards provide its derivation. Without loss of
generality we consider the case x < y. The first step is
to calculate the integrals
Ik,l(x,y,t) = δkl− 2
?y
x
dx′φ∗
k(x′,t)φl(x′,t)(11)
and
Ik,l(y,t) = −δkl+ 2
?y
−∞
dx′φ∗
k(x′,t)φl(x′,t);(12)
for k,l = 1,...,N.
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These integrals are arranged in the following matrices:
P(x,y,t) =
I1,1(x,y,t) I1,2(x,y,t) ... I1,N(x,y,t)
I2,1(x,y,t) I2,2(x,y,t) ... I2,N(x,y,t)
...
IN,1(x,y,t) IN,2(x,y,t) ... IN,N(x,y,t)
...
...
...
, (13)
and
P(l)(x,y,t) =
I1,1(x,y,t) I1,2(x,y,t) ... I1,l(y,t) ... I1,N(x,y,t)
I2,1(x,y,t) I2,2(x,y,t) ... I2,l(y,t) ... I2,N(x,y,t)
...
IN,1(x,y,t) IN,2(x,y,t) ... IN,l(y,t) ... IN,N(x,y,t)
...
...
...
.(14)
Let us define the column vector
Ψ(x,t) =
φ1(x,t)
...
φN(x,t)
,(15)
and its first spatial derivative
Ψ′(x,t) =
φ′
1(x,t)
...
φ′
N(x,t)
. (16)
The TG reduced single-particle density matrix is given
by [36]
ρTG(x,y,t) = det[P(x,y,t)]
Ψ†(x,t)?P(x,y,t)−1?TΨ(y,t). (17)
The quantity η(x,y,t) can also be written in a convenient matrix form (suitable for efficient numerical implemen-
tation):
η(x,y,t) =
N
?
l=1
?
det
?
P(l)(x,y,t)
?
Ψ′†(x,t)
?
P(l)(x,y,t)−1?T
Ψ(y,t)
?
− det[P(x,y,t)]Ψ′†(x,t)?P(x,y,t)−1?TΨ(y,t) (18)
If any of the matrices P(l)happen to be singular, we can
resort to a direct calculation via algebraic cofactors (see
the proof of the algorithm in Appendix B). This hap-
pens rarely and only for some particular high-symmetry
points. Equations (13)–(18) provide the grounds for an
efficient numerical method for calculating the RSPDM,
which is a generalization of the previously introduced
method for the TG gas [36].
It is convenient to calculate the diagonal correction
to the RSPDM. In this case, the matrices appearing in
Eq. (18) are very simple and it is straightforward to ob-
tain
c
?ρLL(x,x,t)
N(c)
− ρTG(x,x,t)
?Tr(ρ′)Tr(I) − Tr(ρ′· I)?
?
≃
(x,t)
(19)
where on the right hand side ρ is N ×N matrix given by
ρk,l(x,t) = φ∗
k(x,t)φl(x,t)
and primes denote the first spatial derivative. We can
also omit the normalisation constant from Eq. (19) since
N(c) = 1 within the 1/c approximation considered here.
III. CONTRACTION DYNAMICS OF
STRONGLY INTERACTING 1D BOSE GAS
In this section we apply the presented formalism to
investigate time-evolution of a many-body wave packet
which contracts in space after a parabolic phase was im-
printed on it. More specifically, we consider N strongly
interacting bosons which are initially in the ground state
in the presence of a harmonic potential. We assume that
this initial state is well described by the Tonks-Girardeau
ground state wave function, a symmetrized Slater deter-
minant. As described in more detail below, at time t = 0,
a parabolic phase is suddenly imprinted onto this state,
and the harmonic potential is turned off. Imprinting the
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parabolic phase is experimentally feasible [9] and is equiv-
alent to the action of a focusing lens on the travelling-
wave state of a light beam in optics. In the context of
atomic gases this can be achieved by applying, over a
short period of time, a tight longitudinal harmonic po-
tential to the one-dimensional gas. This results in pro-
viding different initial momenta to different parts of the
wave packet, with the momenta being proportional to the
distance from the center of the wave packet and directed
towards the center.
In order to connect the model written in Eq. (1) to
physical units we first write the 1D Schr¨ odinger equation
for this system:
i?∂ψLL
∂τ
=
N
?
i=1
?
−?2
2m
∂2
∂ξ2
i
+mω2
HO
2
ξ2
i
?
ψLL
+
?
1≤i<j≤N
2gδ(ξi− ξj)ψLL,
where g is the 1D scattering length [37], ξ is the spa-
tial coordinate, and τ denotes time. Here we denote the
axial trapping frequency by ωHO. By multiplying the
above expression with 2/?ωHOwe obtain a dimensionless
equation suitable for numerical calculation and a physical
interpretation of the relevant scales:
i∂ψLL
∂t
=
N
?
i=1
?
−∂2
∂x2
i
+ x2
i
?
ψLL
+
?
1≤i<j≤N
2
2g
aHOEHO
δ(xi− xj)ψLL. (20)
where we have denoted zero point energy of the har-
monic trap with EHO =
2/ωHO, while space is in units aHO =
other words xi = ξi/aHO, and t = ωHOτ/2.
set c = 2g/aHOEHO we easily verify that it is exactly
the interaction strength of the LL model c, see Eq. (1).
For concreteness, let us assume that the harmonic trap
has frequency ωHO= 60 Hz, which is loaded with87Rb
atoms, N = 12. These values yield aHO = 1.38µm. As
we have already stated, we assume that the effective in-
teraction strength c is sufficiently large such that the sys-
tem is initially in the Tonks-Girardeau regime, and that
the 1/c correction is negligible at t = 0. Under these
conditions the ground state is given by
?ωHO
2
. The unit of time is
??/mωHO; in
If we
ψg.s.=
?
1≤i<j≤N
sgn(xj− xi)
N
det
j,m=1[φj(xm)]/
√
N!,
(21)
where φj(x) is the jth eigenstate of the SP harmonic
oscillator −φ′′
parabolic phase to the system. Technically, each of the
SP wave functions in the determinant (21) is multiplied
by
j+ x2φj = Ejφj. At t = 0 we imprint a
P(x) = e−i(x/b)2,
where we have chosen b =
function which starts to evolve in the absence of a trap-
ping potential is
√2. Thus, the initial wave
ψLL(x1,...,xN,0) ≈
?
1≤i<j≤N
sgn(xj− xi)
N
det
j,m=1[φj(xm)P(xm)]/
√
N!. (22)
After this initial excitation the wave packet starts to con-
tract. Figure 1 displays the evolution of the SP x-space
density and the momentum distribution (SP k-space den-
sity) for c = 50; the momentum distribution is defined as
[33]
nLL(k,t) = (2π)−1
?∞
−∞
dx
?∞
−∞
dx′e−ik(x−x′)ρLL(x,x′,t).
It should be emphasized that the results are scalable, that
is, the same results can be obtained for smaller values of
c provided that the initial state is broader (this corre-
sponds to the fact that for the LL gas in equilibrium
the regime is determined by the interaction strength c
divided by the linear density [1]). As the gas becomes
more dense during the contraction, the 1/c correction
to the TG density becomes larger and clearly visible in
the x-space density. Note that the gas compresses more
strongly for the finite c in comparison to the TG results.
After the wave packet reaches the maximally dense con-
figuration, it starts to freely expand and the 1/c correc-
tion starts to become less visible.
note that while the changes in the x-space density are
nonnegligible in contraction, the 1/c correction in the k-
space density is practically negligible at all times. ¿From
our simulations we find quite generally that the 1/c cor-
rection to the k-space density is much less sensitive to
changes in the coupling strength than the x-space density.
Note that the total x-space density in the presented form
of the 1/c approximation can become negative in some
parts of the space, if the approximation is used beyond
its range of validity (e.g., if c is not sufficiently large).
This gives a clear indication that higher order terms, of
order O(1/c2), become relevant, at least in those parts
of the space where the density is negative.
We emphasize again that the results presented here,
for an evolution in one dimension without further bound-
ary conditions, are correct up to 1/c for all times of the
evolution, i.e., there is no accumulation of the error in
time due to the approximation and we can extract the
asymptotic behaviour of the LL gas.
It is interesting to
IV.CONCLUSION
We have studied the 1/c expansion in the strong cou-
pling limit of the Lieb-Liniger model, exploiting the for-
malism based on a mapping between free Fermi and inter-
acting Bose gas. More specifically we have developed an
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ρLL
x
2.7 ms
1.8 ms
1.3 ms
0.8 ms
0 ms
−6
−4
−20246
0
0.1
0.2
0.3
0.4
nLL
k
2.7 ms
1.8 ms
1.3 ms
0.8 ms
0 ms
−3
−2
−1
0123
0
1
2
3
4
5
FIG. 1:
gas with N=12 bosons and c = 50 at different times of the propagation. At the time of 1.3 ms the cloud attains the maximal
compression. The corresponding shapes for the TG gas are also shown as black dot-dashed lines. The difference between
red solid and black dot-dashed lines corresponds to the 1/c correction. The curves are shifted on the ordinate axis for better
visibility. See text for details.
(color online) The x-space density (left, red solid line), and momentum distribution (right, red solid line) of the LL
efficient algorithm to calculate the reduced one body den-
sity matrix of the strongly interacting LL gas, which is
based on the method previously developed for the Tonks-
Girardeau gas [36], and the Fermi-Bose transformation
[17, 19, 20]. The method is suitable to describe dynamics
on an infinite line, in the absence of external potentials,
and for initial states which can be described by Equations
(4) and (10). An attractive feature is stable accuracy for
all times since the error is not being accumulated dur-
ing the evolution. For the x-space density we provide
a very compact formula Eq. (19) that can be employed
for quite large numbers of particles enabling fast calcu-
lation of the nonequilibrium x-space density dynamics.
In addition, the efficiency of the method is demonstrated
in an example discussed in Sec. III; we have examined
nonequilibrium dynamics where a parabolic phase is im-
printed onto a localized wave packet after which the wave
packet contracts for a while. Our simulation studies the
setup which was experimentally realized in Ref. [9] for a
weakly interacting Bose gas at finite temperature.
Acknowledgments
R.P. and H.B. acknowledge support from the Croat-
ian Ministry of Science (Grant No. 119-0000000-1015).
T.G. acknowledges support by the Deutsche Forschungs-
gemeinschaft. This work is also supported in part by
the Croatian-German scientific collaboration funded by
DAAD and MZOˇS, and the Croatian National Founda-
tion for Science.
APPENDIX A: FORMULAS
Integrals in Eq. (6) involving derivatives in integration
coordinates all vanish due to following expression for k ?=
1 and for any l ?= k:
?∂ψF
∂xk
ˆI(X)sgn(xk− xl)
?∗
(x,X)
ψF(y,X)
+ˆI(X)sgn(xk− xl)ψ∗
F(x,X)
?∂ψF
∂xk
?
(y,X)
=ˆI(X)sgn(xk− xl)
∂
∂xk
[ψ∗
F(x,X)ψF(y,X)]
This is obtained by collecting the corresponding terms
from the first and the second part of the 1/c correction
to the RSPDM in Eq. (6). Let us now look at the kth
integration only. Integrating by parts we have:
[sgn(xk− xl)ψ∗
F(x,X)ψF(y,X)]
???
∞
−∞−
?∞
−∞
dxk2δ(xk− xl)ψ∗
F(x,X)ψF(y,X)
The first term is zero due to boundary condition at infin-
ity and the second term is zero because of the antisym-
metry of ψF. By using the method presented above it is
Page 7
7
straightforward to see that
?∞
−∞
dx[η(x,x,t) + η(x,x,t)∗] = 0
where η is defined in Eq.
is that the normalization constant N(c) = 1 in the 1/c
approximation cosidered here.
(9).A direct consequence
APPENDIX B: DETERMINANT EXPANSION
In this appendix we derive Eq. (18). The derivation
is similar to that for the RSPDM of a TG gas [36]. To
simplify the notation, we will suppress the time variable
in all formulas. By inserting Eq. (10) for ψF in Eq. (9)
for η we obtain
η(x,y) =N(N − 1)
N!
?
ij
(−)i+j[φ
′
i(x)]∗φj(y)
?
P,Q
(−)P(−)QIp2,q2Ip3,q3...IpN,qN. (B1)
Here, P ∈ SN−1 is a permutation of the N − 1 indices
not including i: (p2,...,pN) = P(1...i − 1 i + 1...N).
Similarly, Q ∈ SN−1 permutes the N − 1 indices not
including j: (q2,...,qN) = Q(1...j − 1 j + 1...N).
We considerthe sum
Let (q′′
N) be the ordered series of integers
{1,...,N}\{l,j}, where l ?= j, and Q′∈ SN−2 a
permutation of these N − 2 numbers, (q′
over thepermutations.
3,...,q′′
3,...,q′
N) =
Q′(q′′
in Eq. (B1) can be written as
3,...,q′′
N). With these, the sum over permutations
1
(N − 2)!
?
P,Q
(−)P+QIp2,q2Ip3,q3···IpN,qN
=
1
(N − 2)!
N
?
l=1
(l?=j)
(−)l+1?
P,Q′
(−)P+Q′Ip2,lIp3,q′
3···IpN,q′
N
=
N
?
l=1
(l?=j)
?
P
(−)l+1(−)PIp2,lIp3,q′′
3···IpN,q′′
N
=
N
?
l=1
(l?=j)
detP(l)
i,j(x,y). (B2)
Here, P(l)
inating the ith row and the jth column from the matrix
P(l)defined in Eq.(13). This yields the following result
for η(x,y):
i,jis a N − 1 × N − 1 matrix obtained by elim-
η(x,y) =
?
ij
[φ
′
i(x)]∗φj(y)
?
l=1
(l?=j)
(−)i+jdetP(l)
i,j(x,y)
(B3)
The final result, Eq. (18), is obtained from (B3) with the
use of standard matrix algebra connecting minors, the
cofactor matrix, and the matrix inverse.
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