Deformed Wigner crystal in a one-dimensional quantum dot
Deformed Wigner crystal in a one-dimensional quantum dot
Yasha Gindikin and Vladimir A. Sablikov
Kotel’nikov Institute of Radio Engineering and Electronics,
Russian Academy of Sciences, Fryazino, Moscow District, 141190, Russia
�e spatial Fourier spectrum of the electron density distribution in a �nite 1D system and the distribution
function of electrons over single-particle states are studied in detail to show that there are two universal
features in their behavior, which characterize the electron ordering and the deformation of Wigner crystal by
boundaries. �e distribution function has a δ -like singularity at the Fermi momentum kF . �e Fourier spectrum
of the density has a step-like form at the wavevector 2kF , with the harmonics being absent or vanishing above
this threshold. �ese features are found by calculations using exact diagonalization method. �ey are shown
to be caused by Wigner ordering of electrons, a�ected by the boundaries. However the common Lu�inger
liquid model with open boundaries fails to capture these features, because it overestimates the deformation of
the Wigner crystal. An improvement of the Lu�inger liquid model is proposed which allows one to describe
the above features correctly. It is based on the corrected form of the density operator conserving the particle
number.
I. INTRODUCTION
One-dimensional (1D) quantum dots a�ract a great deal
of a�ention as appealing model objects to study the e�ects
of electron-electron (e-e) interaction, which is principally im-
portant for 1D electron systems.1,2 �e interest to the many-
electron state in bounded 1D systems is presently increased
due to the recent progress in magnetotunnelling spectroscopy
studies of such structures.3–5 As a consequence of e-e inter-
action, electrons form the strongly correlated state, which is
referred to as Lu�inger liquid. Main distinctive features of a
Lu�inger liquid are the absence of fermionic quasiparticles,
which manifests itself in the absence of the Fermi step in the
momentum distribution function, the power-law behavior of
spectral functions near the Fermi energy with interaction-
dependent exponents,6,7 and Wigner-like correlations of elec-
trons.8
�ese properties were established for ideal, i.e. boundless
1D systems. However, whether they hold for realistic meso-
scopic systems of �nite length is not obvious. Indeed, the
presence of boundaries may strongly a�ect the electron state
and excitations since 1D electron correlation functions are
known to decay as a power of distance, hence there is no
characteristic length. Many observations performed on meso-
scopic structures containing a �nite 1D system do not con�rm
the theoretical predictions made for ideal systems. Just recall
the interaction-driven conductance renormalization9 in an
in�nite 1D system that does not actually occur because of the
contacts,10–12 the spin polarization13,14 that should not exist ac-
cording to the Ma�is-Lieb theorem, the ‘0.7’ anomaly,15 many
facts of the nonuniversality of conductance quantization,16–18
which existing theories fail to explain, and so on.
It is commonly believed that electrons in a bounded 1D sys-
tem form the Lu�inger liquid. �e boundary e�ect consists in
the change of electron correlations near the ends which are
described by additional boundary exponents. �is conclusion
was made in several theoretical works using the bosonization
approach.19–21 Unfortunately, this approach is based on a num-
ber of model assumptions, which are not well justi�ed. �ese
are (i) the extension of the linear electron spectrum to in�nite
negative energy, which results in a violation of the conserva-
tion laws,22,23 (ii) the linearization of the electron spectrum,
which can lead to a striking departure from the properties of
a real system with quadratic dispersion relation,24 and (iii)
neglecting some parts of the 2kF components of the electron
densities in the e-e interaction Hamiltonian.
�e goal of our paper is to investigate the electron state in
1D quantum dots beyondmodel assumptions. For this purpose
the exact diagonalization method is employed to calculate nu-
merically the electron density distribution in a �nite 1D wire
and the distribution function of electrons over single-particle
states. We have found that there are two unexpected features.
First, the distribution function has a δ -like singularity at the
Fermi energy at the background of a smooth dependence on
the energy. �e second feature relates to the Fourier spec-
trum of the spatial distribution of the electron density. �e
Fourier spectrum has a step-like form at the wavevector 2kF .
Above this threshold, the harmonics are absent or vanishing.
�ese properties are universal in the sense that they do not de-
pend on the e-e interaction strength, interaction radius, wire
length, mean electron density. We argue that these features
are caused by the Wigner ordering of electrons. We compare
the obtained results with the calculations carried out within
the frame of the common Lu�inger liquid approach to �nd
that it fails to capture the above results. �e Lu�inger liquid
theory also gives a singularity of the distribution function at
the Fermi energy, but its form is incorrect. �e Fourier spec-
trum of the electron density distribution turns out to have
an incorrect singularity at 2kF and not to vanish above this
value. We have clari�ed that this discrepancy appears because
the Lu�inger liquid theory does not describe correctly the
deformation of the Wigner crystal by the boundaries. �e
reason of this shortcoming lies in the violation of the parti-
cle number conservation within the bosonization approach
used. We �nd a way to remedy the Lu�inger liquid theory by
introducing an improved expression for the electron density
operator. �e proposed approach allows one to describe cor-
rectly the above features of the distribution function and the
Fourier harmonics of electron density.
�e formulation of problem and exact diagonalization re-
sults are presented in section II. �e results of bosonization
with zero boundary conditions are given in section III. Sec-
tion IV contains a simple model of Wigner ordering, as well
as the interpretation of the results. Section V includes the
review of bosonization approach for the case of zero bound-
ary conditions, the construction of the density operator and
the calculation of the observables with the particle number
conservation taken into account. Technical details pertaining
to the exact diagonalization can be found in Appendix A.
II. EXACT DIAGONALIZATION STUDY OF 1D
CORRELATED STATE
A. Statement of problem
Consider N spinless electrons in a 1D quantum box with
zero boundary conditions for the many-electron wavefunc-
tion,
Ψ|x=0 = Ψ|x=L = 0 . (1)
�e Hamiltonian is
H = −
V (xi − x j ) +
U (xi ) , (2)
whereV (x) is the e-e interaction potential, andU (x) is the po-
tential of positively charged background, which is considered
as jelly.
Exact diagonalization method25 reduces to �nding the
Hamiltonian matrix in the appropriate basis Φp and solving
the eigenvalue problem by the standard methods of computa-
tional linear algebra.26 As a result we obtain themany-electron
wave function Ψ(x1, ..,xN ), expanded in this basis,
Ψ(x1, ..,xN ) =
apΦp(x1, ..,xN ) , (3)
and the spectrum. It is convenient to choose the basis func-
tions in the form of the Slater determinants
Φp(x1, ..,xN ) =
�������
ψα1 (x1) · · · ψαN (x1)
ψα1 (xN ) · · · ψαN (xN )
������� , (4)
built from the eigenfunctions of the non-interacting single-
particle Hamiltonian,
ψq(x) =
. (5)
�e quantum number q ∈ N, labelling single-particle states, is
analogous to momentum in translationally invariant systems,
and will be called so for brevity. �e many-particle state, la-
belled by the vector p = (α1, ..,αN ), is obtained by occupying
the single-particle states ψαi , i = 1 . . .N . We adhere to the
ordering convention that αi < αk for i < k .
�e Hamiltonian matrix elements are given in Appendix A.
�e interaction potential is chosen in the form V (x1 − x2) =
Ve2/(2ϵd) exp(−|x1 − x2 |/d), with d being the interaction ra-
dius, ϵ the permi�ivity of the medium. �is allows us to �nd
the matrix elements analytically to cut the calculation time.
Using such form of the potential is not essentially a limitation,
because in our calculations we are able to vary the parameters
V and d in a broad range to explore both the short-range and
long-range (on the system length scale) interaction cases.
Now let us express the observables we are going to �nd via
the coe�cients of expansion (3). �e momentum distribution
function is de�ned as
n(q) = 〈Ψ| c+qcq |Ψ〉 , (6)
where cq is electron destruction operator. Since
��Φp〉 is the
eigenvector of c+qcq ,
n(q) =
|ap |2θ
p , (7)
where the function θqp equals one if the many-particle state Φp
has the single-particle stateψq occupied, and zero otherwise.
�e average value of the particle density operator
δ (x − xi ) (8)
equals
ρ(x) = 〈Ψ| ρ |Ψ〉 =
Ψ∗(x ,x2, ..xN )Ψ(x ,x2, ..xN )dx2...dxN .
Using (3), one gets
ρ(x) =
p1,p2
a∗p1ap2γp1p2 (x) , (10)
where for p1 = p2 = (α1, ..,αN )
γp1p1 (x) =
|ψαi (x)|
2 , (11)
γp1p2 (x) = (−1)
k1+k2ψ ∗αk1
(x)ψαk2 (x) (12)
for the case when p1 and p2 have only two di�erent occupied
statesψαi in positions k1 and k2, respectively; γ = 0 otherwise.
�e cosine Fourier-transform of the density is
ρ(q) =
ρ(x) cos
dx , q ∈ N , (13)
and sine Fourier-transform is zero, since according to
Eqs. (11), (12), ρ(x) contains only cosine harmonics.
B. Results
Below we present the results obtained for N = 12 electrons.
�e system length is L = 333aB , the interaction radius d =
n ( q )
0 4 8 1 2 1 6 2 0 2 40
0 . 2
0 . 4
0 . 6
0 . 8
FIG. 1. Momentum distribution function, calculated for the system
of N = 12 electrons. �e value of q = 12 corresponds to kF .
33aB , with aB being e�ective Bohr’s radius, V = 3.6. �e
corresponding value of the RPA parameter rs = (2naB )−1 is
13.9, the estimate of the Lu�inger liquid interaction parameter
according to д = (1 +Vq=0/π~vF )−0.5 gives 0.3.
�e distribution function n(q) over the single-particle states
ψq is shown in Fig. 1. Far from the Fermi surface, the form
of the n(q) curve is smoothed by the interaction, which is
familiar from the Lu�inger model. However, right at the
Fermi surface there appears an unexpected δ -type singularity,
with the value of the distribution function being close to 1 at
this point.
�e presence of the singularity was checked by changing
the number N of electrons from 3 to 20, varying the length
parameters and the interaction strength by the two orders of
magnitude. �e result proved to be perfectly stable against
the change in the system parameters (L,d,V ,N ). Hence, the
δ -singularity in the momentum distribution n(q) at q = kF is a
universal property of �nite 1D systems. Its origin is explained
in section IV.
Electron density ρ(x) in the ground state is an oscillatory
function, the amplitude of which decays o� the boundaries. To
analyze the ordering in the electron state, consider the Fourier-
transform of electron density ρ(q), which is shown in Fig. 2.
For comparison, the results for non-interacting electrons are
also provided.
For free electrons, ρ(q) has a step-like structure, with
ρ(q) = −0.5 for 0 < q ≤ 2kF , and ρ(q) = 0 for q > 2kF .
We emphasize that such threshold behavior holds even for a
strongly interacting system, where ρ(q) remains very close to
zero for q > 2kF . Electron-electron interaction modi�es the
values of density harmonics only for 0 < q ≤ 2kF .
�e harmonics ρ(q) for 0 < q < 2kF are suppressed by the
interaction. �e q = 2kF harmonic of the density is enhanced
by interaction, reaching the values comparable to one half of
the background density. �is re�ects the well-known fact that
e-e interaction leads to the strong electron correlations on the
scale of average inter-particle distance, or in other words, to
the Wigner-type ordering in the system.
� 1 . 5
� 0 . 5
0 0 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8
I n t e r a c t i o n o nI n t e r a c t i o n o f f
FIG. 2. �e Fourier-transform of electron density. �e value of q = 24
corresponds to 2kF .
�e presented results are exact, since they are based on a
precise many-particle wave function. In the next section we
compare them to the results of the Lu�inger liquid theory.
III. BOUNDED LUTTINGER LIQUID THEORY
One of the most advanced analytical theories of a strongly
correlated electron state in 1D quantum dots is a Lu�inger
liquid theory, based on the bosonization with zero (open)
boundary conditions.19–21 In this theory the spatial distribu-
tion of electron density and the distribution of electrons over
single-particle states are expressed through the Green func-
tion of chiral fermions G+(x ,y) = 〈ψ++ (y)ψ+(x)〉 via
n(q) =
dxdyG+(x ,y)eiq(y−x ) , (14)
and ρ(x) = −e−2ikF xG+(−x ,x) + c .c .�e Green function is
G+(x ,y) =
(eβ − 1)
(д+1)2
4д +1
eβ − ei
L (x−y)
(cosh β − cos 2πxL )(cosh β − cos
] д−1−д
cosh β − cos 2π (x−y)L
] (д+1)2
cosh β − cos 2π (x+y)L
] 1−д2
with the dimensionless cuto� parameter β ≈ N −1.
�e momentum distribution function is presented in Fig. 3.
A comparison with exact diagonalization shows that this re-
sult is qualitatively incorrect. �e momentum distribution
function, calculated within bosonization, also has a singular-
ity at q = kF , but instead of a single δ -peak at q = kF , n(q)
deviates from a smoothed step in a �nite band around kF ,
where the form of the curve is close to the derivative of the
δ -peak.27
n ( q )
0 . 1
0 . 2
0 . 3
0 . 4
B o xR i n g
FIG. 3. Momentum distribution in the Lu�inger model with zero
(�lled circles) and periodic (empty circles) boundary conditions, the
interaction parameter д = 0.3.
� 2 0
� 1 0
2 k F
FIG. 4. �e Fourier-transform of electron density in the Lu�inger
model with zero boundary conditions, the interaction parameter
д = 0.3.
Electron density, calculated according to (15), equals
ρ(x) =
1 − 2
2 sinhд(β/2)
cos(2kFx − 2f (x))
[cosh β − cos(2πx/L)]д/2
with f (x) being
f (x) =
arctan
sin(2πx/L)
eβ − cos(2πx/L)
. (17)
Its Fourier-transform is presented in Fig. (4). �e qualitative
error of this result is that ρ(q) does not vanish at q > 2kF ,
but, on the contrary, grows rapidly as q → 2kF + 0. In other
words, a whole branch appears in the region q > 2kF , which
is absent in the exact solution.
�us bosonization breaks down for the momenta close to
kF and multiples of it. �is scale corresponds to the mean
inter-particle distance. Hence, the bosonization with zero
boundary conditions incorrectly treats the short-ranged elec-
tron correlations, responsible for the formation of the ordered,
Wigner-like state in 1D quantum dots. In the next section
we demonstrate that the bosonization results describe the
deformed Wigner crystal.
IV. MODEL OF WIGNER ORDERING
Consider a simple model that takes into account theWigner
ordering from the very beginning. In this model, the many-
electronwavefunctionΦ(y1, ..,yN ) is represented by the Slater
determinant
Φ(y1, ..,yN )=
�������
ϕ(y1−x1) · · · ϕ(y1−xN )
ϕ(yN −x1) · · · ϕ(yN −xN )
������� , (18)
built on the single-particle wavefunctions ϕ(x) =
π−1/4a−1/2 exp(−x2/2a2), localized at positions xk , k = 1 . . .N .
�e wavefunction ‘width’ a is assumed to be smaller than the
distance between the particles L/N , so the wavefunctions do
not overlap and form an orthonormal set.
�e Fourier-transform of the density is
ρ(q) = e−
π 2q2a2
. (19)
�e momentum distribution function n(q) can be expressed
through the electron Green function, similarly to Eq. (14),
n(q) =
dxdyG(x ,y)ψ ∗q (x)ψq(y) . (20)
�e Green function G(x ,y) = 〈ψ+(y)ψ (x)〉 is related to the
one-particle density matrix
ρ(x ,y) =
Φ∗(y, z2, .., zN )Φ(x , z2, .., zN )dz2..dzN , (21)
via G(x ,y) = N ρ(x ,y).28 Substitute this into Eq. (20) to get
n(q) = N
����∫ dy Φ∗(y, z)ψq(y)����2 , (22)
where z = (z2, .., zN ). Using Eq. (18), we �nally obtain
n(q) =
����∫ dy ϕ∗(y − xk )ψq(y)����2
2q2a2
First consider the case of a Wigner crystal, i.e. when xk =
(k − 12 )
N . �e Fourier-transform of the density ρ(q) is non-
zero only for multiples of 2kF ,
ρ(n · 2kF ) = (−1)nNe−a
2k2Fn
. (24)
�e momentum distribution function n(q) equals
n(q) =
(−1)mδq,mN
a2π 2
. (25)
n ( q )
0 . 2
0 . 4
0 . 6
0 . 8
2 k Fk F
FIG. 5. Momentum distribution function in the model of localized
electrons.
n ( q )
0 5 1 0 1 50
0 . 2
0 . 4
0 . 6
0 . 8
FIG. 6. Momentum distribution function, calculated by exact diag-
onalization for the system of N = 5 electrons. �e value of q = 5
corresponds to kF .
Fig. 5 shows that n(q) is fully consistent with the results of
exact diagonalization, including the δ -singularity at the Fermi
surface. Moreover, the additional δ -singularities at multiples
of kF , predicted by the model, do exist in the exact solution
for stronger interaction (д ≈ 0.1), see Fig. 6.
�is model proves that the origin of the singularities of n(q)
is the ordering of electrons in bounded 1D systems. In in�nite
1D systems, the ordering is destroyed by �uctuations, and
short-range electron correlations manifest themselves only
in dynamic response to external perturbation.22 If the system
is �nite, the boundaries pin the charge density waves, giving
rise to Friedel oscillations, the amplitude of which is enhanced
by e-e interaction. �is results in the increase of the weight
of the state with q = kF , which is re�ected in the momentum
distribution function, as well as in the high value of 2kF -
harmonic of density. �e conclusion about the important role
of short-range electron correlations in bounded 1D systems
is con�rmed by the calculation of momentum distribution of
electrons in a ring, i.e. in a system without boundaries. In this
case, the singularity of the momentum distribution disappears,
as can be seen from Fig. (3).
n ( q )
0 . 2
0 . 4
0 . 6
0 . 8
FIG. 7. Momentum distribution of the deformed Wigner crystal.
� 2 0
� 1 0
2 k F
FIG. 8. �e Fourier-transform of electron density for the deformed
Wigner crystal.
Now let us squeeze electrons down to the center of the
system by introducing a displacement δxk ∝ −(k − N2 )
�e momentum distribution function and the density Fourier-
spectrum, calculated according to (19), (23) for this case, are
presented in Figs. 7, 8. �e curves are clearly identical to
those from the previous section. �is con�rms our suggestion
that bosonization describes the electron state that is deformed
in comparison with the exact solution. �e origin of such
deformation is discussed in the next section.
V. BOSONIZATIONWITH PARTICLE NUMBER
CONSERVATION
A. Formalism
Let us brie�y review the theory of the bounded Lu�inger
liquids, following19–21. Decompose the electron �eld operator
ψ (x) =
cqψq(x) (26)
into the �eldsψr (x) of the so-called r -fermions (r = ±),
ψ (x) = eikF xψ+(x) + e−ikF xψ−(x) , (27)
where
ψr (x) = −
irkx . (28)
�e �elds are not independent, since
ψ+(x) = −ψ−(−x) , (29)
so we will deal withψ+(x) alone. �e la�er has the property
ψ+(L) = ψ+(−L) , (30)
i.e. it satis�es periodic boundary conditions on the interval
[−L,L]. Hence, it can be bosonized in a conventional way.
�e main assumptions of bosonization are the lineariza-
tion of r -fermion spectrum near the Fermi surface and its
extrapolation to in�nity. �ey allow one to solve the model
exactly.
Consider the r -fermion density operator,
ρ(q) =
: c+k+qck :=
c+k+qck − δq,0
c+k ck
for q , 0. Zero harmonic ρ(q = 0) is the number of particles
operator ∆N . �e density operator obeys the commutation
relation
[ρ(q), ρ(−q′)] = −δq,q′
, (32)
which allows one to introduce bosons
b+q = i
ρq , q > 0 . (33)
�e r -fermion �eld operator has the following boson repre-
sentation,
ψ+(x) = −
U ei(ϕ(x )+
L ∆N ) , (34)
where the bosonic phase ϕ equals
ϕ(x) =
e−iqx−α |q |
ρq , (35)
U stands for the ladder operator, α denotes the ultraviolet
cuto�, which by the order of magnitude equals k−1F .
�e r -fermion spatial density is related to bosonic phase
ρ+(x) =: ψ++ (x)ψ+(x) :=
, (36)
and ρ−(x) = ρ+(−x).
To obtain the density operator of real electrons, one has to
expressψ (x) in terms ofψr (x). Using Eq. (27), one gets
ρ(x) = ρlw(x) + ρCDW(x) , (37)
where the �rst term is the long-wave component of the den-
sity,
ρlw(x) = ρ+(x) + ρ−(x) , (38)
and the second term is the charge density wave (CDW) com-
ponent
ρCDW(x) = e−2ikF xψ++ (x)ψ−(x) + h.c . (39)
which describes short-range electron correlations. In terms
of bosonic phase, the electron density operator equals
ρ(x) =
cos(2kFx − 2φ(x) − 2f (x)), (40)
where φ(x) = 12 (ϕ(−x) − ϕ(x)), and function f (x) is an addi-
tional phase due to zero boundary conditions,
f (x) =
[ϕ(x),ϕ(−x)] =
arctan
sin(2πx/L)
eβ − cos(2πx/L)
, (41)
with dimensionless cuto� β = πα/L ≈ N −1.
�e kinetic energy
H0 = vF
k : c+k ck : (42)
is bosonized using Kronig’s identity to give
H0 = vF
q : b+qbq : +
(∆N )2 . (43)
�e interaction part of the Hamiltonian equals
dxdy ρ(x)V (x − y)ρ(y) , (44)
where ρ(x) is given by Eq. (40). In the model considered,19–21
the interaction operator is simpli�ed by retaining only the
direct
dxdy (ρ+(x)ρ+(y) + ρ−(x)ρ−(y))V (x − y) (45)
and cross terms
dxdy (ρ+(x)ρ−(y) + ρ−(x)ρ+(y))V (x − y) . (46)
�e cross term contains the non-local contribution
ρ+(x)ρ−(y) = ρ+(x)ρ+(−y), which makes the diagonalization
of the Hamiltonian impossible, unless the interaction poten-
tial is assumed short-ranged, that is of the form Vδ (x − y).
Under this assumption, the direct and cross terms transform
dx ρ2+(x) , (47)
dx ρ+(x)ρ+(−x) . (48)
Combining the terms, and using bosonic representation (36)
of the density operator, we arrive at the following model
Hamiltonian:
H = H0 +Vd +Vc = (vF +
b+qbq +
q + bqbq
(vF +
)(∆N )2.
�e Hamiltonian is diagonalized by Bogoliubov transforma-
b̃q = e
iSbqe
−iS , (50)
where
q − bqbq
. (51)
�e Hamiltonian is diagonal if e2ξ = (1 + V /πvF )−1/2. �e
interaction parameter д ≡ e2ξ belongs to (0, 1) for repulsive
interaction, and equals unity for free electrons. �e diagonal
form of the Hamiltonian is
qv(q) b̃+q b̃q +
vN (∆N )2 , (52)
v(q) =
vF +V /2π
cosh 2ξ
, vN = vF +
. (53)
�e bosonic representation (34) of the �eld operator, the den-
sity operator (40) and quadratic Hamiltonian (52) are su�cient
to obtain the Green function (15) with all the ensuing prob-
lems for the momentum distribution (14) and density (16).
�e root of these problems lies in the density operator (40),
commonly used in the bosonization approach.
B. Density operator
�e bosonization approach as presented above violates the
particle number conservation. To see this, just notice that
the integral of the density �uctuation (the second term in
Eq. (16)) over the system length is not zero. �is is a well-
known problem, which exists not only in the case of zero
boundary conditions, but also in a standard bosonization on
the ring.22 �e problem arises at the level of the electron
density operator (40), the CDW component of which does
not conserve the number of particles in an isolated system.
�e physical reason of the violation of the particle number
conservation is that the CDW component of (40) includes the
response of the in�nite positron sea, which is not completely
eliminated when using the approximate relation (27).
In the case of zero boundary conditions, the situation is
reacher because now there appear problems even with the
long-wave component of the density. Indeed, Eq. (16) of the
common Lu�inger liquid theory does not give a correct tran-
sition to the case of non-interacting electrons in the box. �e
density of free electrons
ρfree(x) =
sin(2kF + πL )x
2 sin πxL
contains an additional term 1/2L, missing in (16). �is term,
being integrated over the length of the system, gives an extra
charge of e/2.
�us the microscopic theory leads to the density operator
that violates the particle number conservation. We will obtain
the correct operator, following the harmonic-�uid approach
by Haldane.29
Introduce the phase θ (x) that increases by π each time x
passes the location xk of a k-th particle. �e particle density
operator (8) becomes
ρ(x) = ∂xθ
δ (θ (x) − kπ ) =
e2imθ (x ) . (55)
According to Eq. (40),
θ (x) = kFx − φ(x) − f (x) . (56)
�us the density operator takes the form
ρ(x) =
) cos(2kFx − 2φ(x) − 2f (x)),
where we retain onlym = 0,±1 harmonics, and halve the am-
plitude of the CDW component to obtain the correct transition
to the non-interacting case.22
�e density operator of Eq. (57) has the form of a full dif-
ferential, which guarantees that the integral of the density
�uctuation over the length of the system is zero, i.e. the num-
ber of particles is conserved. �e long-wave part of the oper-
ator contains an additional term −∂x f /π that gives the 1/2L
component, missing in (16), since in the bulk of the system
f (x) ≈ π4 −
C. Hamiltonian and observables
�e interaction part (44) of the Hamiltonian, having been
calculated with the density operator of Eq. (57), gets additional
terms of the form30
dx ∂xφ∂x f = iV
(b̃+2q − b̃2q). (58)
By shi�ing bosons
d2q = b̃2q + i
we obtain a diagonalized full Hamiltonian H + H1,
H + H1 =
qv(q)d+qdq . (60)
�e bosonic phase (35) is transformed into ϕ(x) = ϕ0(x) +
ϕ1(x), where ϕ0(x) is linear in new bosons,
ϕ0(x)=
(ceiqx−se−iqx )dq + (ce−iqx−seiqx )d+q
with c and s being, respectively, cosh ξ and sinh ξ . �e func-
tion ϕ1(x) is the new phase, speci�c to the case of zero bound-
ary conditions, ϕ1(x) = Af (x), A = Vд2/2πvF .
As a result, the �eld operator (34) acquires the factor of
exp(iAf (x)), and the Green function (15) transforms as
Gnew(x ,y) = G+(x ,y)eiA(f (x )−f (y)) . (62)
�e average value of the density equals
〈ρ(x)〉 =
− (1 −A)
sinhд(β/2)
21−д/2π
sin(2kFx − 2(1 −A)f (x))
[cosh β − cos 2πxL ]д/2
Note that the additional phase (1 −A)f (x), which appears
in Eq. (63), changes the period of density oscillations. Since
∂x f < 0 everywhere in the system except narrow regions near
the ends, the Wigner crystal is squeezed by the boundaries.
�e deformation is determined by the coe�cient (1−A). In the
common Lu�inger liquid theory A = 0. Our approach with
corrected expression (57) for the density operator gives A > 0.
Hence, restoring total neutrality results in the reduction of
the Wigner crystal compression.
�e distribution function n(q) and the Fourier spectrum of
the density calculated with the use of the above expression for
A coincide with the numerical calculations quite well. How-
ever this formula is justi�ed for weak interaction (1 − д � 1).
We propose a generalized expression for any д taking into
account that A is known for three limiting cases:
i) For д = 1, A = 0 to provide the correct transition to the
case of non-interacting electrons.
ii) For weak interaction, A should be proportional to
V /2πvF , in agreement with our model.
iii) For strong interaction д → 0, A → 1 to provide the
transition to the limiting case of the Wigner crystal with
strictly periodic density.
�e simplest choice of A = 1 − д satis�es these require-
ments and proves to be highly successful, as is demonstrated
below. Figs. 9, 10 show the momentum distribution function
and the density Fourier-transform, calculated according to
Eqs. (62), (63). �ey are seen to agree nicely with the exact
results of section II.
Fig. 11 shows the spatial distribution of the density, calcu-
lated according to Eqs. (16), (54), (63). It is seen that according
to the standard bosonization the electron density maxima al-
most coincide with those of free electrons, even for strong
n ( q )
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
FIG. 9. Momentum distribution function, the violation of the particle
number conservation is �xed.
0 2 k F
FIG. 10. �e Fourier-transform of electron density, the violation of
the particle number conservation is �xed.
interaction. In contrast, according to Eq. (63) the electron loca-
tions are shi�ed towards the periodic positions as Wigner or-
dering prescribes at strong interaction. �us the bosonization
approach that does not respect the particle number conserva-
tion leads to the picture of the state with electrons squeezed
to the center of the system.
VI. CONCLUDING REMARKS
In this work, the ground state of interacting electrons in a
1D quantum dot was investigated using the exact diagonal-
ization. An unexpected δ -like singularity was discovered in
the momentum distribution function at the Fermi energy. A
threshold behavior was found for the spatial Fourier-spectrum
of electron density, with the step at 2kF . �ese e�ects are sta-
ble against the change in the system length, interaction radius,
the number of electrons, and the interaction strength. �us we
suggest that they are inherent in �nite 1D electronic systems.
We proposed a simple model which proved that these e�ects
originate from the formation of the Wigner molecule in a 1D
quantum dot. Comparison of exact results with the Lu�inger
0 0 . 5 10
FIG. 11. �e spatial distribution of the density, calculated according
to Eqs. (16) (dashed line), Eq. (54) (thin solid line), Eq. (63) (solid line).
model with zero boundary conditions shows that the la�er
does not correctly describe both the momentum distribution
near the Fermi energy and the density Fourier-spectrum. �e
problem is that bosonization overestimates the deformation
of the Wigner crystal caused by the boundaries by introduc-
ing the excessive positive charge into the 1D system, which
a�racts electrons to the center of the system. �is is a result
of using the density operator that violates the number of par-
ticles conservation. We derived the density operator devoid of
the mentioned shortcoming, corrected the Hamiltonian, and
calculated the observables to �nd a nice agreement with the
exact results.
We emphasize that our results are speci�c to �nite 1D sys-
tems with boundaries. �e above features of the momentum
distribution function and density Fourier-spectrum are both
kept with increasing the system length. As far as the spec-
trum remains discrete, the δ -singularity of n(q) refers to a
single point q = kF . Formally, the features survive even at
L → ∞. However, this is not equivalent to the thermody-
namic limit, which also requires that dephasing processes be
taken into account. In the correct thermodynamic limit the
features disappear together with the boundary e�ects.
ACKNOWLEDGMENTS
�is work was supported by Russian Foundation for Basic
Research (project No. 05-02-16854), Russian Academy of Sci-
ences (programs “�antum Nanostructures” and “Strongly
Correlated Electrons in Semiconductors, Metals, Supercon-
ductors, and Magnetic Materials”), RF Ministry of Education
and Science, and Russian Science Support Foundation.
Appendix A: Hamiltonian matrix elements
�e Hamiltonian matrix elements
Hp1p2 =
Φ∗p1HΦp2 dx1..dxN (A1)
are calculated by substituting Eqs. (2), (4) into Eq. (A1), expand-
ing the Slater determinants, and using the orthogonality prop-
erty of single-particle functions (5) during integration.31–34
�e matrix element of the kinetic energy equals
Tp1p2 =
~2π 2
k2i δp1p2 . (A2)
For the matrix elements of the pair e-e interaction, four
situations are possible, depending on p1 and p2.
1. If p1 = p2 = (α1, ..,αN ), i.e. diagonal matrix elements
are calculated, then
Vp1p1 =
i>j=1
Vαiα j ,αiα j , (A3)
where
Vαiα j ,αkαl =∫
ϕ∗αiα j (x1,x2)V (x1 − x2)ϕαkαl (x1,x2)dx1dx2,
ϕαiα j (x1,x2) =
����ψαi (x1) ψα j (x1)ψαi (x2) ψα j (x2)
���� . (A5)
2. If p1 and p2 are identical except two states, numbered
k1 and k2, respectively, i.e. if p1 = (α1, ..αk1 , ..αN ), p2 =
(α1, ..βk2 , ..αN ), then
Vp1p2 = (−1)
k1+k2
Vαk1αi ,βk2αi , (A6)
3. If p1 and p2 are identical except four states, numbered
ki , i = 1 . . . 4, i.e. if p1 = (α1, ..αk1 , ..αk2 , ..αN ), p2 =
(α1, ..βk3 , ..βk4 , ..αN ), then
Vp1p2 = (−1)
Vαk1αk2,βk3 βk4 . (A7)
4. If p1 and p2 contain more than four not coincident states
αi , then
Vp1p2 = 0 . (A8)
Matrix elements of the interaction between electrons and
positive background are as follows.
1. If p1 = p2 = (α1, ..,αN ), then
Up1p1 =
fαiαi , (A9)
where
fαiαk =
ψ ∗αi (x)U (x)ψαk (x)dx . (A10)
2. If p1 and p2 are identical except two states, numbered
k1 and k2, respectively, i.e. if p1 = (α1, ..αk1 , ..αN ), p2 =
(α1, ..βk2 , ..αN ), then
Up1p2 = fαk1 βk2 . (A11)
3. Otherwise,Up1p2 = 0.
�e matrix element of the pair e-e interaction (A4) equals
Vl j,km =
[f (l , j,k,m) − f (l , j,k,−m)−
f (l , j,−k,m) + f (l , j,−k,−m) − f (l ,−j,k,m)+
f (l ,−j,k,−m) + f (l ,−j,−k,m) − f (l ,−j,−k,−m)−
f (j, l ,k,m) + f (j, l ,k,−m) + f (j, l ,−k,m)−
f (j, l ,−k,−m) + f (j,−l ,k,m) − f (j,−l ,k,−m)−
f (j,−l ,−k,m) + f (j,−l ,−k,−m)],
(A12)
where f (l ,n,k,m) = д(l + k,n +m), and д(p,q) = 0 for odd
(p + q), while for even (p + q)
д(p,q) = (e−a(−1)p − 1)
1 + π
a2 pq
(1 + π 2a2 p
2)(1 + π 2a2 q
1 + π
δp,−q ,
(A13)
with a = L/d . Matrix element (A10) equals
fln = N
(д(l + n, 0) − д(l − n, 0)) . (A14)
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	Deformed Wigner crystal in a one-dimensional quantum dot
	Abstract
	I Introduction
	II Exact diagonalization study of 1D correlated state
	A Statement of problem
	B Results
	III Bounded Luttinger liquid theory
	IV Model of Wigner ordering
	V Bosonization with particle number conservation
	A Formalism
	B Density operator
	C Hamiltonian and observables
	VI Concluding remarks
	 Acknowledgments
	A Hamiltonian matrix elements
	 References
