Persistent current of Luttinger liquid in one-dimensional ring with weak link:
Continuous model studied by configuration interaction and quantum Monte Carlo
R. N´meth1,2 , M. Moˇko1,∗ R. Krˇm´r1, A. Gendiar1 , M. Indlekofer3 , and L. Mitas4
e
s
c a

arXiv:0902.2225v1 [cond-mat.mes-hall] 12 Feb 2009

1

Institute of Electrical Engineering, Slovak Academy of Sciences, 841 04 Bratislava, Slovakia
2
Institute for Bio and Nanosystems, CNI, Research Center J¨lich, 52425 J¨lich, Germany
u
u
3
Wiesbaden, University of Applied Sciences, ING/ITE, 65428 R¨sselsheim, Germany and
u
4
Department of Physics, North Carolina State University, Raleigh, NC 27695
(Dated: June 15, 2018)

We study the persistent current of correlated spinless electrons in a continuous one-dimensional
ring with a single weak link. We include correlations by solving the many-body Schrodinger equation
for several tens of electrons interacting via the short-ranged pair interaction V (x − x′ ). We solve
this many-body problem by advanced configuration-interaction (CI) and diffusion Monte Carlo
(DMC) methods which rely neither on the renormalisation group techniques nor on the Bosonisation
technique of the Luttinger-liquid model. Our CI and DMC results show, that the persistent current
(I) as a function of the ring length (L) exhibits for large L the power law typical of the Luttinger
liquid, I ∝ L−1−α , where the power α depends only on the electron-electron (e-e) interaction. For
strong e-e interaction the previous theories predicted for α the formula α = (1 + 2αRG )1/2 −1, where
αRG = [V (0) − V (2kF )]/2π vF is the renormalisation-group result for weakly interacting electrons,
with V (q) being the Fourier transform of V (x − x′ ). Our numerical data show that this theoretical
result holds in the continuous model only if the range of V (x − x′ ) is small (roughly d
1/2kF ,
2
more precisely 4d2 kF ≪ 1). For strong e-e interaction (αRG 0.3) our CI data show the power law
I ∝ L−1−α already for rings with only ten electrons, i.e., ten electrons are already enough to behave
like the Luttinger liquid. The DMC data for αRG
0.3 are damaged by the so-called fixed-phase
approximation. Finally, we also treat the e-e interaction in the Hartree-Fock approximation. We
find the exponentially decaying I(L) instead of the power law, however, the slope of log(I(L)) still
depends solely on the parameter αRG as long as the range of V (x − x′ ) approaches zero.
PACS numbers: 73.23.-b, 73.61.Ey

I.

INTRODUCTION

A clean one-dimensional (1D) wire biased by contacts
with negligible backscattering is known to exhibit the
conductance quantized as an integer multiple of e2 /h.
The effect can be explained in a Fermi-liquid model of
non-interacting quasi-particles.1,2 In fact, a clean 1D system is not a Fermi liquid due to the electron-electron (e-e)
interaction. Away from the charge-density-wave instability, the system is a correlated Luttinger liquid with collective bosonic elementary excitations which contrast with
independent fermionic quasi-particles of ordinary Fermi
liquids3 . Nevertheless, the Luttinger-liquid model gives
for a clean 1D wire the same conductance (e2 /h per spin)
as the Fermi-liquid model4 .
If a localized scatterer is introduced into the wire,
the conductance quantization breaks down. For noninteracting electrons, the Landauer formula1,2 expresses
the conductance per spin as (e2 /h)|tkF |2 , where tkF is
the transmission amplitude through the scatterer at the
Fermi level.
In the Luttinger liquid model5,6 , the infinite wire which
contains a single structureless scatterer, exhibits the conductance varying with temperature as ∝ T 2α , where α
depends only on the e-e interaction. Thus, for α > 0 (repulsive e-e interaction) the wire is impenetrable at T → 0
regardless the strength of the scatterer. A similar power
law exhibits at T → 0 the differential conductance as a

function of the bias voltage (∝ U 2α ) and conductance
versus the wire length (∝ L−2α ). These power laws are a
sign of the Luttinger liquid5,6,7 . The T 2α and U 2α laws
were successfully measured7 .
In this work we deal with similar power laws exhibited by interacting electrons in an isolated mesoscopic 1D
ring. In particular, magnetic flux φ piercing the opening
of the mesoscopic ring gives rise to the persistent electron
current circulating along the ring1 . This persistent current is given at T = 0K as I = −∂E0 (φ)/∂φ, where E0 is
the energy of the many-body groundstate. If the ring is
clean and the single-particle dispersion law is parabolic,
the e-e interaction does not affect the persistent current
due to the Galilean invariance of the system8 .
However, if a single scatterer is introduced, the noninteracting and interacting result differ fundamentally.
For non-interacting spinless electrons in the 1D ring containing a single scatterer with transmission probability
˜
|tkF |2 ≪ 1, the persistent current at T = 0K depends on
the magnetic flux and ring length (L) as9
˜
I = (evF /2L) |tkF | sin(2πφ/φ0 ),

(1)

where φ0 = h/e is the flux quantum, kF is the Fermi
wave vector, and vF is the Fermi velocity. For a spinless
Luttinger liquid the persistent current follows the power
law I ∝ L−α−1 . More precisely,9
I ∝ L−α−1 sin(2πφ/φ0 ),

(2)

2
where the power α depends only on the e-e interaction,
not on the properties of the scatterer. The formulae (1)
and (2) were derived9 assuming large L.
The formula (2) can also be obtained heuristically9 as
follows. Matveev et al.10 analyzed, how the e-e interac˜
tion renormalizes the bare transmission amplitude tkF of
a single scatterer inside the 1D wire with two contacts.
They derived the renormalized amplitude tkF by using
the renormalization-group (RG) approach suitable for a
weakly-interacting electron gas. If the system length (L)
is large, their result can be expressed in the form
˜
r
tkF ≃ (tkF /|˜kF |)(d/L)α ,

α > 0,

(3)

˜
where |˜kF |2 = 1 − |tkF |2 , d is the spatial range of the
r
e-e interaction V (x − x′ ), and the power α is given (for
spinless electrons) by the expression
αRG = [V (0) − V (2kF )]/2π vF ,

(4)

with V (q) being the Fourier transform of V (x − x′ ). We
note that the formulae (3) and (4) were derived assuming
1 ≪ ln(l/d) ≪ αRG −1 ,

(5)

where l is a properly chosen scale of the RG theory10 .
Moreover, α = αRG only for αRG ≪ 1 (weak e-e interaction). For strong e-e interaction (say αRG ≃ 0.5) the
theory3,11 predicts the more general result,
1/2

α = (1 + 2αRG )

− 1.

(6)

This result is believed11 to hold for any V (x − x′ ) with
range d which is finite but which can in principle be quite
large (in comparison with 1/kF ). Finally, if we replace in
˜
equation (1) the bare amplitude tkF by the renormalized
amplitude (3), we recover the power law (2), where α is
now given by the microscopic formulae (6) and (4).
In the Luttinger-liquid model, the physics of the lowenergy excitations is mapped onto an effective field theory using Bosonization9, where terms expected to be negligible at low energies are omitted. Within this model,
the asymptotic dependence (2) was obtained by using
the analogy to the problem of quantum coherence in dissipative environment9 . To avoid this analogy as well as
Bosonization, in Ref.12 the persistent current was calculated by solving the 1D lattice model with nearestneighbor hopping and interaction. Applying numerical
RG methods, the formula (2) was confirmed for long
chains and strong scatterers.12
However, insofar it has not been verified whether the
RG formula (4) holds in a microscopic model which does
not rely on the RG approach. We present such microscopic many-body model in this work. Dealing with a
continuous model, we can vary the range of the e-e interaction in order to test the robustness of the formula (4)
against various shapes of V (x − x′ ). This point was not
addressed in the lattice-model-based studies as the range
of the e-e interaction was fixed to the nearest-neighborsite interaction.

Further, derivations9,10 of the formulae (2), (3) and
(4) rely on the large number of particles limit. Here we
directly address the interesting question what is the minimum number of electrons which exhibits the onset of the
Luttinger-liquid dependence I ∝ L−α−1 . As we will illustrate, such behavior can be identified from system sizes
of the order of ten electron.
We study the persistent current of correlated electrons
in a continuous 1D ring containing the strongly-reflecting
scatterer, because strong backscattering is known9,10,12
to reduce the system size necessary to achieve the L−α−1
asymptotics. Using advanced configuration-interaction
(CI) and diffusion Monte Carlo (DMC) methods, we solve
the continuous many-body Schrodinger equation for several tens of electrons interacting via the e-e interaction
V (x − x′ ) = V0 exp(− |x − x′ | /d).

(7)

Interaction (7) emulates screening (say by the metallic
gates) and allows us to compare our results with the results of the correlated models9,10,12 which also assume
the e-e interaction of finite range. Our CI and DMC calculations do not rely on the RG techniques and solve the
continuous model, unlike the RG studies focused largely
on the lattice models12,13,14,15 . Our numerical data show
that the formulae (6) and (4) hold only if the range of
the e-e interaction is small (d 1/2kF ). For strong e-e
interaction (αRG 0.3) our CI data show the power law
I ∝ L−1−α already for rings with only ten electrons. In
other words, ten electrons are already sufficient to show
the Luttinger-liquid behavior.
It is known that the fermion sign problem causes
an exponential inefficiency of the DMC method16 unless it is circumvented by the so-called fixed-node or
the fixed-phase approximation for inherently complex
wave functions17 . The accuracy of our DMC results for
αRG 0.3 is therefore limited by the quality of the phase
in the Hartree-Fock trial wave function which is employed
in the fixed-phase approximation. Since Hartree-Fock is
the simplest possible trial wave function and does not
capture the Luttinger-liquid correlation effects it is not
surprising that the fixed-phase bias becomes pronounced.
We analyze these findings later in detail. (Our DMC
should not be confused with the path-integral MonteCarlo methods18,19,20,21 , used to study the tunnelling
conductance of the Luttinger liquid in the bosonised
model.)
Finally, we treat the e-e interaction in the selfconsistent Hartree-Fock calculation. We find for large
L the exponentially decaying I(L) instead of the power
law. However, the slope of log(I(L)) still depends solely
on the parameter αRG given by the RG formula (4), as
long as the range of V (x − x′ ) approaches zero.
In section II.A we start with the single-particle approach to the 1D ring. In section II.B we define our
interacting many-body model. In section II.C we outline
how we solve the many-body model in the Hartree-Fock
approximation. In sections II.D and II.E we describe
how we obtain the fully-correlated many-body solution

3
by means of the DMC method and CI method. Our results are discussed in section III and a summary is given
in section IV. Technical details are in the appendices.

II.
A.

THEORY

Single-particle model

We consider the circular 1D ring threaded by magnetic
flux φ = BS = AL, where S is the area of the ring, B
is the magnetic field (constant and perpendicular to the
ring area), and A is the magnitude of the resulting vector
potential (circulating along the ring circumference). In
this section we discuss the single-particle states in such
ring. In general, the single-electron wave functions ψn (x)
in the 1D ring obey the Sch¨dinger equation
o

where T0 is the transfer matrix2
T0 =

1
t∗
k
k
− rk
t

r∗

k
− t∗

1
tk

k

,

(16)

with tk and rk being the transmission and reflection amplitudes of the electron impinging the region −L/2, L/2
from the left. At the boundaries we express ϕk (−L/2)
and ϕk (L/2) by using equations (13) and (14). To come
back to the ring threaded by magnetic flux, we relate
ϕk (−L/2) and ϕk (L/2) through the boundary condition
(12). Combining the obtained relation with the equations
(15) and (16) we obtain the equation
T

a
b

φ
)
φ0

a
b

,

(17)

k
− t∗ eikL
k
1 −ikL
tk e

.

(18)

= exp(i2π

where
2

2m

1 ∂
2π φ
+
i ∂x
L φ0

2

+ U (x) ψn (x) = εn ψn (x) (8)

1 ikL
t∗ e
k
rk −ikL
− tk e

T =

with the cyclic boundary condition
ψn (x + L) = ψn (x) ,

(9)

where m is the electron effective mass, x is the electron
coordinate along the ring, and U (x) is an external singleparticle potential.
We introduce the wave functions ϕn (x) by substitution
ψn (x) = ϕn (x) exp −i

2π φ
x .
L φ0

(10)

If we set (10) into (8) and (9), we obtain the equation
2

−

d2
+ U (x) ϕn (x) = εn ϕn (x)
2m dx2

(11)

with the boundary condition
ϕn (x + L) = exp i2π

φ
φ0

ϕn (x) .

ϕk (x) = ce

+ de

−ikx

cos 2π

, x ≥ L/2 .

(13)

φ
φ0

= Re

exp(−ikL)
.
tk

(19)

The numerical solution of equation (19) has to be combined with numerical computation of the transmission
amplitude tk (the algorithm for computation of tk and rk
is described in the Appendix A). The solution of equation (19) gives us the dependence kn (φ) and eventually
2
the single-particle eigenenergy εn (φ) = 2 kn (φ)/2m.
For each kn (φ) we can also calculate the wave function
ϕn (x). We proceed as follows. By means of equations
(17) and (18) we express the amplitude a in the form
a=

ϕk (x) = aeikx + be−ikx , x ≤ −L/2 ,
ikx

Thus exp(i2πφ/φ0 ) is the eigenvalue of the matrix T .
The product of the eigenvalues of this matrix is given by
its determinant which is unity. The second eigenvalue is
thus exp(−i2πφ/φ0 ). Their sum is equal to the matrix
trace2 , which gives the equation for the spectrum22 ,

(12)

Equations (11) and (12) can be solved for an arbitrary
potential U (x) numerically. Consider first the ring region
x ∈ −L/2, L/2 as a straight-line segment of an infinite
1D wire. Inside the segment the potential is U (x), outside
we keep it zero. Therefore, the wave function outside is

r∗

1
tk
− n ei(2πφ/φ0 +kn L) b ,
rkn
rkn

(20)

where the amplitude b can be obtained by normalizing
the wave function. Then we express from equation (13)
the boundary conditions ϕn (−L/2) and dϕn (−L/2)/dx.
Finally, we combine these boundary conditions with numerical solution of equation (11) in the discrete form23
ϕn (xj+1 ) =

(14)

2+

2m
2

(U (xj ) − εn ) ∆2 ϕn (xj ) − ϕn (xj−1 ) ,

(21)

The amplitudes a and b are related to c and d by
c
d

= T0

a
b

,

(15)

where ∆ is the step, xj = j∆, and j = 0, ±1, ±2, . . . .
Once the wave functions ϕn (x) are known, the wave functions ψn (x) can be obtained by means of the relation (10).

4
B.

The wave functions ψn (x) obey the Hartree-Fock equation

Interacting many-body model

We now consider the ring with N interacting 1D electrons. This system is described by the Hamiltonian

2

−i

2m
N

2

ˆ
H =

2m

j=1

+

1
2

1 ∂
2π φ
+
i ∂xj
L φ0

2

+ γδ(x)

2

+ γδ(xj )

+ UH (x) + UF (n, x) ψn (x) = εn ψn (x)

N

i,j=1
i=j

∂
2π φ
+
∂x
L φ0

V (xj − xi ) ,

(22)

ˆ
HΨ = EΨ

(23)

with the cyclic boundary condition
Ψ(x1 , . . . , xi + L, . . . , xN ) = Ψ(x1 , . . . , xi , . . . , xN ) (24)
for i = 1, 2, . . . , N .
If the system is in the groundstate with eigenfunction Ψ0 (x1 , x2 , . . . , xN ) and eigenenergy E0 , the persistent current can be expressed as
ˆ
I = Ψ0 | I |Ψ0

with the boundary condition
ψn (x + L) = ψn (x) ,

where xj is the coordinate of the j-th electron, γδ(x) is
the potential of the scatterer, and V (xj − xi ) is the e-e
interaction (7). The eigenfunction Ψ(x1 , x2 , . . . , xN ) and
eigenenergy E obey the Sch¨dinger equation
o

(31)

where
UH (x) =
n′

dx′ V (x − x′ )|ψn′ (x′ )|2

(32)

is the Hartree potential and
UF (n, x) =
1
−
ψn (x) ′
n

∗
dx′ V (x − x′ )ψn (x′ )ψn′ (x′ )ψn′ (x)

(33)

is the Fock nonlocal exchange term (expressed as an effective potencial for further convenience). In the equations
(32) and (33) we sum over all occupied states n′ .
If we use again the substitution

(25)

where

(30)

ψn (x) = ϕn (x) exp −i

2π φ
x ,
L φ0

(34)

the equations (30)-(33) give the Hartree-Fock equation
e
ˆ
I =−
mL

N
j=1

∂
eφ
+
i ∂xj
L

(26)

is the N -particle current operator. One can calculate I
directly from the relation (25), or one can rewrite (25)
by means of the Hellman-Feynman theorem
ˆ
∂E
∂H
= Ψ0 |
|Ψ0 .
∂φ
∂φ

∂
E0 (φ) .
∂φ

d2
+ γδ(x) + UH (x) + UF (n, x) ϕn (x)
2m dx2
= εn ϕn (x)

(28)

ϕn (x + L) = exp i2π

φ
φ0

ϕn (x) ,

(36)

where the potentials UH (x) and UF (n, x) are still given
by equations (32) and (33), but with ψn replaced by ϕn .
From equations (22)-(36) the groundstate energy E0 =
Ψ0 |H|Ψ0 can be expressed as
E0 =
n

C.

(35)

with the boundary condition

(27)

Using (27), (26), and (22) one gets the formula1,24
I=−

2

−

εn −

1
ϕn |UH (x) + UF (n, x)| ϕn
2

.

(37)

Hartree-Fock approximation

In the Hartree-Fock model, the ground-state wave
function Ψ0 is approximated by the Slater determinant
1
Ψ0 (x1 , . . . , xN ) = √
N!

ψ1 (x1 ) · · · ψ1 (xN )
.
.
..
.
.
. (29)
.
.
.
ψN (x1 ) · · · ψN (xN )

The Hartree-Fock equation (35) can be solved by the
same procedure as the single-particle equation (11) assuming that the potential U (x) ≡ γδ(x) + UH (x) +
UF (n, x) is known. We apply this procedure iteratively in
order to obtain the self-consistent Hartree-Fock solution.
In the first iteration step we solve equation (35) for the
non-interacting gas, i.e., for UH (x) = 0 and UF (n, x) = 0.
The resulting ϕn (x) is used to evaluate the Hartree and

5
Fock potentials, where the term UF (n, x) has to be evaluated for each n separately. These potentials are used in
the second iteration step to obtain new ϕn (x) and new
potentials UH (x) and UF (n, x), etc., until the energies
εn and ground-state energy (37) do not change anymore.
In the Appendix B the iteration procedure is described
including a few nontrivial details. Setting the resulting
ground-state energy (37) into (28) we obtain the persistent current. Of course, this Hartree-Fock calculation
does not include the many-body correlations. To include
the correlations we use the DMC and CI techniques described in the next two sections.
D.

Diffusion Monte Carlo (DMC) model

Consider first the N -electron 1D Sch¨dinger equation
o
ˆ
HΨ(X) = EΨ(X)
2

2m

N
i=1

∂2
+ V (X),
∂x2
i

(39)

(40)

where ET is a trial energy. One can write (40) in the
integral form16
Ψ(X, t + τ ) =

dX’ GX←X’ (τ ) Ψ(X’, t),

(41)

ˆ
where GX←X’ (τ ) = X| exp(−(H − ET )τ )|X’ is the
Green’s function and τ is a time step. Equation (41)
allows to project the ground state Ψ0 (X) from any trial
wave function ΨT (X) which has a nonzero overlap with
Ψ0 (X). Inserting any trial ΨT (X) and ET into (41) gives
ΨT (X, τ → ∞) = Ψ0 (X) Ψ0 |ΨT

lim exp[−τ (E0 −ET )].

τ →∞

(42)
By adjusting ET to equal E0 one can make the exponential factor constant. It is also important that the Green’s
function GX←X’ (τ ) can be expressed16 for τ → 0 as

(X − X ′ )2
2τ
′
× exp (−τ [V (X) + V (X ) − 2ET ] /2) . (43)

GX←X ′ (τ ) ≈ (2πτ )−3N/2 exp −

(44)

one can split the equation (23) into the real part and
imaginary part. The real part reads

where X = (x1 , x2 , . . . , xN ), the potential energy V (X)
incorporates all single-electron interactions and all pair
electron-electron interactions, and the wave-function
Ψ(X) is assumed to obey the cyclic condition (24). If
Ψ(X) is a real function, the DMC method is capable to
find the exact ground-state solution of equation (38). We
briefly summarize how the DMC works16 .
Instead of directly solving the equation (38), the DMC
solves the time-dependent diffusion problem16
ˆ
− ∂Ψ(X, t)/∂t = H − ET Ψ(X, t),

Ψ(X) = |Ψ(X)| eiΦ(X)

(38)

with Hamiltonian
ˆ
H=−

In the DMC algorithm the equation (41) is solved
stochastically based on the action of the projection opˆ
erator exp[−τ (H − ET )]16 . At time t = 0 one chooses a
proper trial wave function ΨT (X). A set of walkers (sampling points in the N -dimensional electron configuration
space) is generated according to the distribution ΨT (X).
A small timestep τ is chosen. The walkers are propagated from X to X’ according to the kernel GX←X’ (τ )
in the form (43). In the new sample point X’, the expectation values are calculated and the starting value of
ET is adjusted. Eventually, the ground state expectation
values are projected and sampled.
The problem is that the wave function Ψ(X) obeying the many-body equation (23) is complex because the
Hamiltonian (22) contains the magnetic flux. The problem can be partly eliminated by means of the fixed-phase
approximation17. Using

ˆ
Heff |Ψ(X)| = E |Ψ(X)| ,

(45)

where
ˆ
Heff = −

N

2

2m

i=1

∂2
1
2 + 2m
∂xi
N

N
i=1

1
δ(xi ) +
+γ
2
i=1

∂
eφ
Φ+
∂xi
L

2

N

i,j=1
i=j

V (xi − xj ), (46)

and the imaginary part is
N
i=1

∂
|Ψ|2
∂xi

∂
eφ
Φ+
∂xi
L

= 0.

(47)

Equation (45) is the effective Sch¨dinger equation for the
o
ˆ
modul |Ψ(X)|, with the Hamiltonian Heff depending on
ˆ
the phase Φ(X). Since |Ψ(X)| is real and Heff has the
same form as (39), the equation (45) can be solved by
means of the DMC if the phase Φ(X) is given. In this
work we choose the trial wave function ΨT (X) to be equal
to the Slater determinant (29) of the self-consistently determined Hartree-Fock ground state and we fix the phase
Φ(X) to the phase of this Slater determinant. The DMC
thus gives the lowest possible ground-state energy E0
within the chosen phase17 . Eventually, we obtain the
persistent current from (28) by finite differences evaluated by correlated sampling25 . Our preliminary DMC
results are briefly discussed in26,27 .
To go beyond the fixed phase approximation one has
to couple the DMC solution of equation (45) with the
numerical solution of equation (47). We do not attempt
to do so. However, in section III we check the fixed phase
approximation by comparing the DMC results with the
CI calculations which are free of such approximation.

E.

Configuration-interaction (CI) model

Before starting with the CI procedure we need to solve
the single-electron problem
2

1 ∂
2π φ
+
i ∂x
L φ0

2m

2

+ γδ(x) ψn (x) = εn ψn (x)

(48)
with the cyclic condition ψn (x + L) = ψn (x). We do so
by using the method described in Section II.A. We obtain
the wave functions ψn (x) and energy levels εn not only
for the N lowest energy levels but also for the infinite
ladder of excited states, of course, with an upper cutoff.
Consider now the non-interacting many-body problem
N

2

single−electron
levels

6
6
5
4
3
2
1
0 1

2 3 4 5

6

7

8 9

10 11 12 13 14 15 16 17 18 19

quantum number n of Slater determinant χ n

FIG. 1: Schematic sketch of how the determinants χn are
created for n = 0, 1, . . . in the three-electron system with the
ladder of single-particle levels restricted to six lowest levels.
The occupied levels constituting the state χn are labelled by
circles. In the CI method the many-body state Ψ of the interacting system is expanded as Ψ = c0 χ0 + c1 χ1 + c2 χ2 + . . . .

where n = 0, 1, . . . , M . This system determines the
+ γδ(xj ) χn (X) = En χn (X), eigenvalues El and eigenvectors (cl , cl , . . . , cl ) for l =
0 1
M
2m
j=1
0, 1, . . . , M . We obtain the groundstate energy El=0 and
(49)
groundstate wave function Ψl=0 by solving the system
where X = (x1 , x2 , . . . , xN ). Clearly, the eigenenergies
with ARPACK, or alternatively with LAPACK. Once E0
En are given as
and Ψ0 are known, the persistent current can be obtained
from (28) or (25).
(50)
En = εn1 + · · · + εnN
The problem of the CI method is, that for Nmax singleand the wave functions χn are the Slater determinants
particle levels the expansion (52) still contains Nmax
N
ψn1 (x1 ) . . . ψnN (x1 )
Slater determinants. For a reasonably chosen cutoff, the
1
.
.
..
solution of the equations (55) is not feasible already for
.
.
,
(51)
χn = √
.
.
.
N!
systems with about ten electrons, as the memory requireψn1 (xN ) . . . ψnN (xN )
ments are too large. The question is which determinants
have to be kept in the expansion (52) and which can be
where the quantum numbers n1 , . . . , nN label the state of
omitted without damaging the final results. This probthe first, . . . , N -th electron, respectively, and the quanlem is known in the quantum chemistry28,29 where the
tum number n labels the many-body state correspondCI is applied to calculate the energy spectra of molecules.
ing to the specific set of N occupied single-electron levels
We cannot apply here directly the tricks from the quanεn1 , . . . , εnN . For instance, the figure 1 depicts creation
tum chemistry because our problem is rather special; we
of all Slater determinants in the three-electron system,
search for the persistent current in the 1D ring pierced by
when only six lowest single-electron levels are considered.
magnetic flux. We prefer the following two approaches.
To solve the interacting many-body problem (23), the
CI method28,29 relies on the expansion
Our first CI approach, below referred as the FCI, conceptually resembles the full CI and can be understood
Ψ = c0 χ0 + c1 χ1 + c2 χ2 + . . . .
(52)
by means of the sketch in the figure 1. After choosing
ˆ
the cutoff for the single-particle energy we add into the
Using this expansion and equation χn |H|Ψ = χn |E|Ψ
expansion (52) first the Slater determinant of the ground
we obtain from (23) the infinite set of equations
state, then all determinants with a single excited electron
∞
(the single-excitations), all determinants with two ex(Ej δnj + Vnj ) cj = Ecn , n = 0, 1, . . . , ∞, (53)
cited electrons (the double-excitations), all determinants
j=0
with three excited electrons (the triple-excitations), etc.
If the results do not change, we cut the expansion by
where
stopping to increase the number of the excited electrons.
N
1
Some preliminary FCI results are briefly discussed in27 .
Vnj = χn |
V (xk − xl )|χj .
(54)
2
Our second CI approach, introduced by two of us in
k,l=1
k=l
Ref.30 , is called the bucket-brigade CI method (BBCI).
After choosing the cutoff for the single-particle energy,
We reduce the infinite number of single-energy levels
we need to asses importance of the remaining Slater deεj to the finite one by introducing a proper upper energy
terminants in the expansion (52). In principle, one can
cutoff. This reduces the infinite number of equations (53)
ˆ
to a certain finite number M +1. We get the finite system
compute for each determinant χn the energy χn | H |χn ,
ˆ is the Hamiltonian (22). The obtained numerwhere H
M
ˆ
ical values χn | H |χn can be ordered increasingly and
(Ej δnj + Vnj ) cj = Ecn ,
(55)
we can assume that this orders the determinants χn acj=0
2

1 ∂
2π φ
+
i ∂xj
L φ0

7

III.
A.

RESULTS

Luttinger-liquid scaling in CI and DMC models

Our calculations are performed for parameters typical
of a GaAs ring. We use the electron effective mass m =
0.067m0 and electron density ne = N/L = 5 × 107 m−1 .
The strength V0 and range d of our e-e interaction (7) are
properly varied to demonstrate various e-e interaction
effects. In practice, screening by metallic gates can be
used to vary d while V0 can be varied by varying the
(finite) cross-section of the 1D ring. We study the rings
˜
containing a strong scatterer with |tkF |2 ≪ 1, because
in such case the persistent current reaches asymptotic
dependence on L for relatively small L. Our conclusions
hold for any 1D system, no matter what material is used.
In figure 2 the persistent current I in the GaAs ring
with one strong scatterer is calculated as a function of
the ring length L for magnetic flux φ = 0.25φ0 . The
˜
scatterer is the δ barrier with transmission |tkF |2 = 0.03.
−α
The power law LI ∝ L
decays in the log scale linearly
with slope −α, the same decay show for large L the CI
and DMC data. Agreement of the FCI and BBCI data
is very good, the DMC data are close to the CI data.
Let us check whether our numerical values of α agree
with the formula (6). The Fourier transform of our e-e
interaction (7) is V (q) = 2V0 d/(1 + q 2 d2 ). Using this
expression we can write the RG formula (4) in the form
αRG =

4V0 mkF d3
.
π 2 (1 + 4kF 2 d2 )

(56)

The dashed lines in the figure 2 show the power law
LI ∝ L−α , with α given by the formula (6) and αRG calculated from the formula (56). The proportionality factor
(specified later on) causes in the log scale only a vertical

N
8

4

16

32

64

0.10

Persistent current L I / e vF

cording to their decreasing importance. We can order the
determinants in the expansion (52) according to this importance criterion (a-priori measure of importance) and
we can truncate the expansion after reaching convergence
of the final results. This is the basic idea of the BBCI
method, the term ”bucket brigade” is related to the technical implementation30 reviewed in the Appendix C.
In the next section, reliability of our importance criterion is supported by a successful convergence of the
BBCI results. We cannot prove the criterion exactly, but
we can give a simple intuitive motivation. We search for
the best minimum of the ground-state energy. Therefore,
ˆ
the larger the energy χn | H |χn the smaller should be
the weight of the given χn in the ground-state expansion
(52). It is important that the BBCI allows us to treat
more electrons than the FCI, as it reduces the expansion
(52) more efficiently. Another CI algorithm, more efficient than the FCI, is the CI approach of Ref. 31, where
the determinants are selected by means of a proper Monte
Carlo algorithm.

non-interacting ele

ctrons

0.09

αRG = 0.0

0.08

277

0.085

0.07

5

DMC
FCI
BBCI
Luttinger liquid

0.06

0.1

71

0.2

56

5

0.05

0.08

0.32

0.16

0.64

1.28

L (µm)
FIG. 2: Persistent current LI/evF versus the ring length L for
the GaAs ring with a single scatterer. The transmission of the
˜
scatterer is |tkF |2 = 0.03, magnetic flux is φ = 0.25φ0 . The
upper horizontal axis shows the electron number N = ne L,
where ne = 5 × 107 m−1 . The range of the e-e interaction
is fixed to d = 3nm while the magnitude V0 is varied. The
full curve is our numerical result for V0 = 0 (non-interacting
electrons). This curve saturates for large L exactly at the
˜
value 0.5|tkF |, predicted by the asymptotic formula (1). The
symbols show our FCI, BBCI and DMC results for various
V0 shown in table I, the dotted lines connecting the BBCI
data are a guide for eye. The dashed lines show the Luttinger
liquid asymptotics LI ∝ L−α for α = (1 + 2αRG )1/2 − 1,
where the values of αRG (listed in the figure) are calculated
from the formula (56). The input parameters V0 and d and
the resulting αRG and α are summarized in the table I.

shift of the linear law log(LI) = −α log(L)+const. We
thus can conclude that the CI and DMC data in the figure 2 exhibit the L−α decay with the power α in good
agreement with the theoretical formulae (6) and (56).
In figure 3 we compare the persistent currents for two
˜
very different transmissions |tkF |2 in order to demonstrate that the power α is universal - independent on the
˜
choice of tkF . Indeed, the CI and DMC data in figure 3

V0 (meV) d (nm)
11
34
68
102

3
3
3
3

αRG

α

0.0277
0.0855
0.171
0.2565

0.0273
0.0821
0.158
0.230

TABLE I: The input parameters V0 and d used in the CI and
DMC calculations of the figure 2. Also shown are the resulting
theoretical values of αRG and α.

8

1/N

N
16

32

0.09
0.08
~
|tk | 2

Persistent current L I /e vF

0.07

F

= 0.

03

0.06
0.05

BBCI
V0

0.04

DMC

d

V0

102 meV, 3.0 nm
1602 meV, 1.0 nm
11957 meV, 0.5 nm

d

102 meV, 3.0 nm
254 meV, 2.0 nm

0.03

~
|tk | 2
F

1.10

0

1/10

1/5

BBCI

1.07

V0

DMC

d

V0

102 meV, 3.0 nm
1602 meV, 1.0 nm
11957 meV, 0.5 nm

1.05

d

102 meV, 3.0 nm
254 meV, 2.0 nm

1.03
1.00
0.98
0.95

0

10

5

1/L (1/µm)
= 0.

003

FIG. 4: Data from figure 3 plotted as the ratio (59), where
˜
˜
˜
I(tkF ,1 ) and I(tkF ,2 ) are the persistent currents for |tkF ,1 |2 =
˜
0.03 and |tkF ,2 |2 = 0.003, respectively.

0.02

0.16

Normalized ratio of persistent currents

0.10

8

0.32

0.64

L (µm)
FIG. 3: Persistent current LI/evF versus the ring length L for
the ring with a single scatterer in the regime LI ∝ L−α . Magnetic flux is φ = 0.25φ0 , the electron density is N/L = 5 × 107
m−1 . We compare the results for a scatterer with two various
˜
transmissions |tkF |2 . We also make comparison for various
e-e interactions chosen so that the RG formula (56) predicts
for each considered interaction the same αRG . Specifically,
each set (V0 , d) listed in the figure is chosen so that the RG
formula predicts αRG (V0 , d) = 0.2565. The dashed lines show
the formula LI ∝ L−α , where α = (1 + 2αRG )1/2 − 1 and
the proportionality factor is specified in the text. The CI and
DMC data are shown by symbols.

exhibit the same α for both transmissions.
One has to note that the formulae (4) and (6) are robust against the choice of the e-e interaction in the sense,
that they give the same α for all e-e interactions with the
same value of [V (0) − V (2kF )]/vF . Obviously, this can
be the case for many various choices of V (x − x′ ).
To see whether such robustness exists in our manybody model, in figure 3 we also make comparison for various e-e interactions chosen so that the formula (56) gives
for various sets (V0 , d) the value αRG (V0 , d) = 0.2565.
Our CI and DMC data indeed show for all considered
1/2
(V0 , d) the decay LI ∝ L−α , where α = (1 + 2αRG ) −1
and αRG = 0.2565. In summary, our methods seem to
give the same α for various sets (V0 , d) fulfilling the equation αRG (V0 , d) =const. Thus, our methods seem to confirm the above discussed robustness of the formulae (4)
and (6) against the choice of the e-e interaction.
However, we will show later on that this robustness

in fact holds only for the e-e interactions which are very
short-ranged [all sets (V0 , d) in figure 3 actually belong
to this special limit]. We know that we are in this limit
as long as our methods confirm the formulae (4) and (6).
This is still the case in the following discussion.
We specify the proportionality factor in the formula
LI ∝ L−α . Following the Introduction, we replace in the
˜
single-particle formula (1) the bare amplitude tkF by the
˜
r
renormalized amplitude tkF ∝ (tkF /|˜kF |)L−α . We get
I=C

˜
evF |tkF | −α
L sin(2πφ/φ0 ),
2L |˜kF |
r

α > 0,

(57)

where C is the proportionality factor. Since α is universal
˜
(independent on tkF ), the formula (57) implies that
˜
˜
I(tkF ,1 )
|tk ,1 |/|˜kF ,1 |
r
= F
˜
˜
I(tkF ,2 )
|tkF ,2 |/|˜kF ,2 |
r

(58)

˜
as long as C does not depend on tkF . To see that the
CI and DMC data in figure 3 fulfill the equality (58), we
show these data again in figure 4 in terms of the ratio
˜
˜
|tkF ,2 |/|˜kF ,2 | I(tkF ,1 )
r
,
˜
˜
|tkF ,1 |/|˜kF ,1 | I(tkF ,2 )
r

(59)

which should equal unity for L → ∞. One sees that it indeed approaches unity for the CI as well as DMC results.
More precisely, the CI data show a few percent deviation
from unity which tends to disappear with increasing L.
A small deviation from unity show also the DMC data
where convergence towards unity is not clear due to the
stochastic noise of the Monte Carlo method.
Thus, if we ignore a small finite-size effect in figure 4,
our data in figure 3 are in accord with equation (58).

9

N

1.8
BBCI
d = 0.5 nm
d = 1.0 nm
d = 3.0 nm

1.4

DMC
d = 2 nm
d = 3 nm

1.2
1.0

0

0.1

16

32
~
| tkF |2 = 0.03

0.2

0.3

0.4

0.5

α
FIG. 5: Dependence C0 (α) obtained by fitting the formula
(60) to our CI and DMC data for LI/evF . Symbols show the
fitted values of C0 . The dashed line is the function C0 (α) =
1 + 1.66α which fits the presented data points.

Therefore, we can fit our data in the L−α regime by the
asymptotic formula (57). For the purpose of fitting it is
useful to define C ≡ ne −α C0 . Setting this definition into
the formula (57) we obtain
˜
LI
1
|tkF | −α
= C0 (α)
N sin(2πφ/φ0 ),
evF
2
|˜kF |
r

(60)

where C0 depends solely on the parameter α (see below).
Instead of (57) we use (60) and we fit C0 instead of C.
The dashed lines in figures 2 and 3 show the dependence (60) fitted to the BBCI data in these figures.
Specifically, C0 is fitted and the function N −α in (60)
1/2
is evaluated by using α = (1 + 2αRG )
− 1, where
αRG (V0 , d) is given by the formula (56). The resulting
values of C0 are shown by open symbols in the figure 5.
The full symbols in the figure 5 show the values of C0
obtained when we fit by means of (60) the DMC data
in figures 2 and 3. Also shown are the results obtained
in the same way for another values of α. Note that for
all (V0 , d) giving the same α(V0 , d) the obtained values of
C0 are essentially the same. This means that C0 depends
solely on the parameter α. We also note that C0 does not
˜
depend on tkF , as has been documented in the figure 4.
The linear fit in the figure 5,
C0 (α) = 1 + 1.66α,

(61)

should be viewed as a first estimate of the analytical dependence. To extract a very precise formula for C0 (α),
it would be desirable to simulate larger systems in order
to better suppress the finite size effect in figure 4.
We conclude that the right-hand side of (60) depends
on a single parameter α via the function C0 (α)N −α ,
which is single-valued for various (V0 , d) giving the same
α(V0 , d). Our results confirm the decay N −α , where α is
given by the formulae (6) and (4).
However, the formulae (6) and (4) in fact hold only for
very small d. In the figure 6 we compare the BBCI data
from the figure 3 with the BBCI data obtained for another two sets (V0 , d) with d as large as 12nm and 24nm.

Persistent current L I /e vF

1.6

C0

0.09

8

0.08

0.07

V0

0.06

0.05

d

6.1 meV, 24.0 nm
12.8 meV, 12.0 nm
102 meV, 3.0 nm
1602 meV, 1.0 nm
11957 meV, 0.5 nm

0.16

0.32

0.64

L (µm)
FIG. 6: The circles, squares and diamonds are the BBCI data
from the figure 3, the triangles and inverted triangles are the
BBCI data for another two sets (V0 , d). For all considered
(V0 , d), the formula (56) gives αRG (V0 , d) = 0.2565 and the
formula (6) predicts α = (1 + 2αRG )1/2 − 1 = 0.23. However,
the dependence LI ∝ L−0.23 (the dashed curves with different
offset) can be seen to fit only the BBCI data for d ≤ 3nm. The
BBCI data for d = 12nm are excellently fitted by the curve
LI ∝ L−0.222 [the full line] and the BBCI data for d = 24nm
are fitted by LI ∝ L−0.207 [the dotted line]. This shows that
the formulae (4) and (6) hold only for the e-e interactions
which are very short-ranged (see the text).

All considered (V0 , d) are chosen so that the formulae
(56) and (6) give for each (V0 , d) the same value of α,
in particular α = 0.23. However, it can be seen, that
the dependence LI ∝ L−0.23 fits only the BBCI data for
d ≤ 3nm. The BBCI data for d = 12nm and d = 24nm
are excellently fitted by the curves LI ∝ L−0.222 and
LI ∝ L−0.207 , respectively, i.e., with increasing d our
numerically obtained α decreases albeit the value of
αRG (V0 , d) is fixed. This is a strong indication that the
formulae (6) and (4) hold only for small d. We support
this numerical finding by the following analytical proof.
In the appendix D we prove analytically that the matrix elements of the e-e interaction are independent on d
for d → 0 at a fixed value of αRG (V0 , d). In other words,
for d → 0 the matrix elements depend on a single parameter αRG , otherwise they depend on two parameters
V0 and d. According to the appendix, the limit d → 0
2
means 4kF d2 ≪ 1. In our calculations 1/2kF ≃ 3nm and
the currents in the figure 6 are close to each other for all
d ≤ 3nm. If we increase αRG (for instance as in the table
II and figure 7), we need to take d < 1/2kF to see the
d-independent current convincingly.
Note also another result in figure 6. As d exceeds

10

3
3
3
1
3
1
0.5
1
0.5
1
0.5

0.0277
0.0855
0.171
0.171
0.2565
0.2565
0.2565
0.342
0.342
0.5
0.5

0.0273
0.0821
0.158
0.158
0.230
0.230
0.230
0.298
0.298
0.414
0.414

N

symbol in Fig. 7
◦
◦
◦

8

0.10
0.09

16

64

0.08

◦
△
△
△

TABLE II: The input parameters V0 and d used in the BBCI
calculations of figure 7 and the values of αRG and α resulting
from the formulae (56) and (6). The last column of the table
ascribes a symbol to each set (V0 , d, αRG ). These symbols are
used in figure 7 to show the BBCI data.

αRG = 0.0277

0.07

0.0855

0.06

0.17

1

0.05

0.25

65

0.04

0.03

0.16

˜
|tkF |
|˜kF | x,
r

2LI
We can write (60) as evF C0 =
where x ≡ N −α
and C0 (α) is given by (61). Similarly, the BBCI data in
the asymptotic regime can be normalized as 2LI/evF C0
and plotted as a function of the variable x ≡ N −α . This
is done in inset to the figure 7. Indeed, all BBCI data
˜
|t |
in inset collapse to a single curve f (x) = |˜kF | x. That
rkF
a single linear curve involves the BBCI data for many
various N , V0 and d, is a clear sign of the Luttinger liquid
with e-e interaction depending on one parameter αRG .
It also documents that our calculations are reliable for a
broad range of variables N , V0 and d, the function C0 (α)
is certainly determined with reasonable accuracy.
An interesting finding in figure 7 is that for large αRG
the BBCI data show the LI ∝ L−α decay already for ten
electrons. In other words, the Luttinger-liquid behavior
arises in the system with only ten particles. For comparison, in the lattice models12,13,14,15 the asymptotic power
law is observed for a much larger number of electrons.
If we compare the formulae (60) and (1), we see that
˜
the interaction modifies the amplitude tkF as

(62)

RG

0.3

42

=0

.5

0.08

0.16
1/2kF the persistent current still decays like LI ∝ L ,
but α depends on two parameters V0 and d rather than
on a single one, αRG . This regime is not captured by the
formulae (6) and (4). We have obtained similar results
(not shown) also for other values of αRG .
Consider now a broader range of the e-e interaction
parameters V0 and d, listed in the table II. The resulting
persistent currents for these parameters are shown in the
figure 7. All shown BBCI data asymptotically converge
to the Luttinger-liquid behavior described by the formula
(60), with the power α in accord with the formulae (6)
and (4). We wish to stress the following aspects.

α

0.12

0.4

−α

α
˜
˜
r
r
tkF ≃ (tkF /|˜kF |)N −α = (tkF /|˜kF |)(n−1 /L) .
e

32

non-interacting electrons

2L I / e vF C0

11
34
68
1068
102
1602
11957
15942
15942
3142
23445

α

Persistent current L I / e vF

V0 (meV) d (nm) αRG

0.6

N −α

0.8

0.32

1

L (µm)

0.64

1.28

FIG. 7: Persistent current versus the ring length for the same
ring as in the figure 2, but for a broader range of the interaction parameters V0 and d (table II). The circles, squares and
triangles show the results of the BBCI method for various sets
(V0 , d, αRG ) listed in the table II. The dotted lines are a guide
for eye. The dashed lines show the asymptotic law LI ∝ L−α
plotted in the form (60), where α = (1 + 2αRG )1/2 − 1 and
αRG is obtained from (56). The BBCI data in the asymptotic
regime are selected, normalized as 2LI/evF C0 [with C0 (α)
given by the formula (61)], and plotted in inset as a function
of the variable x ≡ N −α . The full line is the dependence
f (x) =

˜
|t k

F

|˜k
r

F

|
|

x, predicted by the formula (60).

−1
This result scales with the length ne while the RG result (3) scales with d. What is the origin of this difference? The RG result (3) should hold for any d obeying
the inequality (5), but the inequality (5) can in principle be fulfilled also for d
1/2kF , i.e., just for those d
for which we have obtained the formula (62). We think
that the difference between the two results is due to the
different physical models. The formula (62) holds in our
continuous model, with the single-particle energy dispersion being truly parabolic. The RG formula (3) holds
in the model10 , with the energy dispersion linearized in
the interval < εF − vF /d, εF + vF /d > around the
Fermi energy. For small d the band width vF /d becomes comparable with εF and the model10 has a truly
linear dispersion, like the Luttinger-liquid model5 which
also shows scaling by factor (d/L)α . For d ≫ 1/2kF
the linearization is unessential because of vF /d ≪ εF ,
hence both models should give the same results. The
limit d ≫ 1/2kF is however not feasible by our numerical methods for computational reasons.

11

|tk ~ 2
F |
=

Persistent current L I / evF

10

N
100

|tk ~ 2
F |
=

non-local HF
0.0

3

-1

|tk ~ 2
F |
=

100
local HF

0.0

3

|tk ~ 2
F |
=

0.0

03

10

10

1

0.0

03

-2

V0 = 11 meV
V0 = 34 meV
V0 = 68 meV

0.02

0.2
2
L (µm)

0.02

0.2
2
L (µm)

FIG. 8: Persistent current LI/evF versus ring length L for the
ring with single scatterer, calculated in the Hartree-Fock approximation for parameters φ = 0.25φ0 , N/L = 5 × 107 m−1 ,
˜
d = 3nm, and various V0 and |tkF |2 as listed in the figure.
The left panel shows the nonlocal Hartree-Fock results, the
right panel shows the Hartree-Fock results in the local Fock
approximation (see the text). The Hartree-Fock results are
shown by symbols, the dashed lines show the Luttinger liquid
asymptotics LI ∝ L−α , where α = (1 + 2αRG )1/2 − 1 and
αRG is given by the RG formula (56): αRG = 0.0277, 0.0855,
and 0.1710 for V0 = 11, 34, and 68 meV, respectively.

B.

Universal scaling in the Hartree-Fock model

Now we discuss the persistent currents obtained in the
self-consistent Hartree-Fock approximation (Sect.IIC)
which ignores correlations except for the Fock exchange.
In figure 8 we show the I(L) dependence, calculated
in the self-consistent Hartree-Fock approximation for
the ring parameters already encountered in our correlated many-body calculations. The left panel shows
the Hartree-Fock data obtained for the Fock term (33)
treated as is, the right panel shows the Hartree-Fock data
for the Fock term (33) approximated as32,33
UF (x) ≃ −

n′

∗
dx′ V (x − x′ )Re [ψn′ (x′ )ψn′ (x)] . (63)

Unlike the nonlocal interaction (33) the approximation
(63) is local - it does not depend on n. One readily obtains (63) from (33) by applying the ’almost closure re∗
lation’ n′ ψn′ (x′ )ψn′ (x) ≃ δ(x − x′ ). Once this approximation is adopted, the final expression for the current
has to be approximated correspondingly34.
Let us compare our Hartree-Fock calculations with the
Luttinger liquid asymptotics LI ∝ L−α . As can be seen
in the figure 8, both Hartree-Fock approaches show the
I(L) dependence decaying for large L faster than L−1−α .

Normalized ratio of persistent currents

N
10

1

0

1/N
1/4

1/2

0

1/N
1/4

1/2

1.050
V0 = 11 meV
V0 = 34 meV
V0 = 68 meV

1.025
1.000
0.975
0.950

non-local HF

0.925
0

12.5
1/L (1/µm)

local HF

25

0

12.5
1/L (1/µm)

25

FIG. 9: The Hartree-Fock data from the figure 8 plotted as
˜
˜
the ratio (59), where I(tkF ,1 ) and I(tkF ,2 ) are the persistent
˜
currents for the transmission probabilities |tkF ,1 |2 = 0.03 and
˜
|tkF ,2 |2 = 0.003, respectively.

We will see below that for L → ∞ the decay is in fact exponential. Only for the weak e-e interaction the HartreeFock data show the decay L−1−α , reported also in our
previous Hartree-Fock studies35,36 . However, also in this
case we expect exponential decay at very large L.
Now we show that also in the Hartree-Fock approximation the slope of log(I(L)) becomes for L → ∞ universal
- dependent only on the e-e interaction. Following equations (57)-(59), such universality means that the ratio
˜
˜
|tkF ,2 |/|˜kF ,2 | I(tkF ,1 )
r
˜
˜
|tkF ,1 |/|˜kF ,1 | I(tkF ,2 )
r

approaches unity for L → ∞. If we
redraw in terms of this ratio the Hartree-Fock data from
figure 8, we see (in the figure 9), that the ratio converges
with increasing L to unity. The stronger the interaction
the better the convergence in the figure 9, for further improvement a further increase of L is needed. Universality
of the Hartree-Fock results in figure 9 is quite similar to
the universality of the correlated results in figure 4.
Moreover, we would like to show that the Hartree-Fock
model resembles the correlated model also in the following respect. In the limit d → 0 the I(L) curve is determined by a single e-e interaction parameter αRG [i.e.,
I(L) is the same for various (V0 , d) giving the same value
of αRG (V0 , d)]. This is demonstrated in the figure 10.
Figure 10 shows the Hartree-Fock results for the weak
and strong e-e interactions. Also shown are the corresponding correlated results from the figure 2, in particular the BBCI data (full lines) and the Luttinger-liquid
curves (dashed lines). Note the following details.
For weak e-e interaction (αRG = 0.0277) the HartreeFock data are robust against various (V0 , d) obeying the
equation αRG (V0 , d) = 0.0277. This is illustrated by the
agreement of the Hartree-Fock data for (V0 = 173meV,
d = 1nm) and (V0 = 11meV, d = 3nm). Note also that
these Hartree-Fock data almost reproduce the correlated
results at sizes L considered in the figure. To obtain
conclusions valid for any L and any interaction strength,

12
N

N

10

100

1

non-local HF
-1

α RG (V , d)
0
= 0.0277

0
10
77

αR
65

65

25
0.

25
0.

)=

)=

,d
(V 0

G

,d
(V 0

V0 = 11956 meV, d = 0.5 nm

V0 = 173 meV, d = 1 nm
V0 = 1603 meV, d = 1 nm
V0 = 11 meV, d = 3 nm
V0 = 102 meV, d = 3 nm

0.02

0.2
2
L (µm)

50

100

150

-1

α RG(V , d)
0
= 0.02

G

10

-2

N

100

local HF

αR

Persistent current L I / evF

10

10

Persistent current L I / evF

1

local HF

-2

10

10

-3

V0 = 1.46 keV, d = 0.1 nm
V0 =

12 eV, d = 0.5 nm

V0 = 102 meV, d = 3.0 nm

-4

10

0

1

2

3

L (µm)

0.02

0.2
2
L (µm)

FIG. 10: Length-dependence of the persistent current in the
ring with single scatterer, obtained in the Hartree-Fock approximation. The Hartree-Fock results are shown by symbols. The ring parameters are: φ = 0.25φ0 , N/L = 5 × 107
˜
m−1 , and |tkF |2 = 0.03. The weak and strong e-e interactions are simulated for various (V0 , d) obeying the equations
αRG (V0 , d) = 0.0277 and αRG (V0 , d) = 0.2565, respectively,
where αRG (V0 , d) is given by the RG formula (56). Also shown
are the corresponding BBCI data (full lines) and asymptotic
LI ∝ L−α curves (dashed lines), taken from the figure 2.

we have to look at the stronger e-e interaction.
For strong e-e interaction (α = 0.2565) the HartreeFock data fail to agree with the correlated results. However, the Hartree-Fock data still tend to be robust against
various (V0 , d) obeying the equation αRG (V0 , d) = 0.2565
in the limit d → 0. This universal dependence on a single
e-e interaction parameter αRG is clearly visible both in
the nonlocal and local Hartree-Fock calculation.
Finally, we would like to show that the decay of I(L) in
the Hartree-Fock approximation is exponential for large
L. This exponential decay is demonstrated in the figure
11, where the I(L) dependence is presented in the semilogarithmic scale. We need to consider large L and strong
e-e interaction, which strongly prolongs the time of the
Hartree-Fock computation. Fortunately, the calculation
is feasible in the local approximation (63).
The I(L) curves in figure 11 do not decay equally fast,
albeit the value of αRG (V0 , d) is the same for all considered (V0 , d). It can however be seen that if we keep the
same value of αRG (V0 , d) in the limit d → 0, then the
slope of I(L) is determined solely by the value of αRG .
This is again a clear manifestation of the universal dependence on a single e-e interaction parameter. The limit
d → 0 is well represented by the data for d = 0.1nm.
In summary, our self-consistent Hartree-Fock results
show, that the slope of log(I(L)) in the limit of large L is

FIG. 11: Length-dependence of the persistent current in the
ring with single scatterer, obtained in the local Hartree-Fock
approximation. The Hartree-Fock data are shown by symbols.
The ring parameters are: φ = 0.25φ0 , N/L = 5 × 107 m−1 ,
˜
and |tkF |2 = 0.03. The parameters of the e-e interaction, V0
and d, are listed in the figure. The considered sets (V0 , d)
obey the equation αRG (V0 , d) = 0.2565, where αRG (V0 , d) is
given by the RG formula (56). For large L the current decays
with L linearly, but note that the scale is semi-logarithmic.
The dotted lines fit linearly the Hartree-Fock data at large L.

still universal - dependent solely on the e-e interaction not
on the strength of the scatterer. Moreover, for d → 0 this
dependence on the e-e interaction is determined by a single e-e interaction parameter, specifically by αRG given
by the RG formula (56). These features are very similar
to the universal behavior observed in our correlated calculations. A major difference is that in the Hartree-Fock
approximation the asymptotic decay of I(L) is exponential with L, as we have shown numerically.

C.

Reliability and limitations of the CI and DMC

Both DMC and CI methods in their most rigorous formulations show exponential scaling of the computer time
in the number of particles. In CI this is due the necessity of infinite size of the basis and high level of excitations which are required to make the method size
consistent28,29 . In practice, for not too large systems,
useful and accurate results can be obtained providing a
sufficiently large set of virtual orbitals can be treated.
The question is whether the low-order (doubles, triples,
quadruples) excitations are enough to capture the key
many-body effects. For the DMC method, the exponential scaling originates in the fermion sign problem which
is in practice avoided by the fixed-node or fixed-phase
approximations. The usefulness of the method crucially
relies on accuracy of the trial functions and ability to
efficiently describe the impact of the many-body effects
on accuracy of the nodes or phase16 . The DMC practice

13

Persistent current L I / e vF

0.11
V0 = 68 meV

0.10

V0 = 200 meV

0.09
0.08
0.07
0.06
0.05
0.04
0.03

CI
DMC - Hartree-Fock ΨT
DMC - non-interacting ΨT
Hartree-Fock
non-interacting electrons

0.02
0.01
0.00

0

0.1

0.2

0.3

φ/φ0

0.4

0.5 0

0.1

0.2

0.3

φ/φ0

0.4

0.5

FIG. 12: Persistent current versus magnetic flux for the GaAs
ring with N = 4 and L = N/ne = 80 nm. The transmission
˜
of the scatterer is |tkF |2 = 0.03. The e-e interaction range is
set to d = 3 nm and the e-e interaction magnitude is V0 =
68 meV in the left panel and V0 = 200 meV in the right
panel. The triangles show the results of the fixed-phase DMC
calculation with the trial wave function ΨT (X) equal to the
Slater determinant (29) of the self-consistently determined
Hartree-Fock ground state. The squares show the results of
the fixed-phase DMC calculation with the trial wave function
ΨT (X) equal to the Slater determinant of the non-interacting
ground state. The CI results are shown by circles connected
by the dotted line. For completeness, the results for the noninteracting ground state are shown in a full line and the results
of the self-consistent Hartree-Fock calculation are shown in a
dashed line.

shows that very large systems can be treated nevertheless the fixed-node/phase bias is present and it depends
on phenomena of interest whether it can affect the results.
In subsection IIIA, the largest many-body systems (48
electrons) were simulated by the CI method while only
at most 32 electrons were simulated by the DMC. This
is caused by the fact that the phase of the HartreeFock wave function becomes less accurate with increasing
strength of the interaction and also with increasing size of
the system. This is not too difficult to understand since
in larger and strongly interacting systems the collective
excitations become more complicated and the single determinant wave function becomes very poor representation of the actual ground state. The many-body effects
description built into the trial function thus directly determines the accuracy of the obtained results.
In figure 12 we show the persistent current versus magnetic flux for the ring with four electrons. The range of
the e-e interaction is d = 3nm, the e-e interaction magnitude is V0 = 68meV in the left panel and V0 = 200meV
in the right panel. The triangles represent the DMC calculation with the trial wave function ΨT (X) equal to
the Slater determinant (29) of the Hartree-Fock ground
state. The squares show the DMC calculation with the
trial wave function equal to the Slater determinant of

FIG. 13: Comparison of various phases from the DMC
calculations of figure 12 at φ = 0.25φ0 . The top panel
shows the phase Φ(x1 , x2 , x3 , x4 ) of the non-interacting trial
wave function ΨT (x1 , x2 , x3 , x4 ) evaluated at x3 = −L/6
and x4 = L/6. It is labelled as Φ(0meV) since it holds
for V0 = 0meV. The middle panel shows the phase difference |Φ(68meV) − Φ(0meV)|, where Φ(68meV) is the phase
Φ(x1 , x2 , x3 = −L/6, x4 = L/6) of the trial wave function
ΨT (x1 , x2 , x3 , x4 ) obtained for V0 = 68meV by the HartreeFock method. The bottom panel shows the phase difference
|Φ(200meV) − Φ(0meV)|. We note that the abrupt change of
the phase for xi = xj arises because the Slater determinant
changes abruptly its sign for xi ↔ xj .

the non-interacting ground state. The results of both
DMC calculations are very close when V0 = 68meV, but
they are very different when V0 = 200meV. This suggests
that the DMC results for V0 = 200meV are strongly affected by the phase of the trial wave function. Indeed,
for V0 = 68meV both DMC calculations essentially agree
with the corresponding CI calculation (circles), but for
V0 = 200meV the differences with the CI data are very
clear.
It is interesting to see the origin of the fixed-phase
biases. For illustration, in figure 13 we compare the
phases involved in the DMC calculations of figure 12.

14
0.080
0.0765

Persistent current L I / evF

Persistent current L I / evF

0.080

0.075

0.070
single-excitations
single and double
single, double, and triple
single, double, triple, and quadruple

0.065

0.060
10

15

20

25

30

2x10

4

Persistent current L I / evF

0.080
0.0765

0.075

0.070

0.065

0.0765

0.075

0.070

0.065

0.060
0

Nmax = 16
Nmax = 20
Nmax = 24
Nmax = 32
Nmax = 64
3

4

5x10
10
1.5x10
Number of Slater determinants

4

4

2x10

FIG. 15: Convergence of the persistent current in the BBCI
calculation for various Nmax in dependence on the number of
the Slater determinants involved in the expansion (52). We
recall that the determinants are selected by using the bucketbrigade algorithm of Appendix C and Nmax is the number of
the considered single-electron levels. In this BBCI calculation
ˆ
the current is evaluated by using the formula I = Ψ0 | I |Ψ0
and the parameters of the ring are the same as in the FCI
calculation of the preceding figure. The value 0.0765 is the
average from the (well saturated) data for Nmax ≥ 24.

0.060
10

20
30
15
25
Number of single-electron levels, Nmax
Number of single-electron levels, Nmax

35 4
2x10

FIG. 14: Convergence of the persistent current in the FCI
calculation in dependence on the number of the considered
single-electron levels for the single-excitations, single- and
double-excitations, etc. The top panel shows the currents
ˆ
obtained from the formula I = Ψ0 | I |Ψ0 , the bottom panel
shows the currents obtained in the same FCI calculation from
the formula I = −∂E0 /∂φ. The results are shown by symbols. They were obtained for the ring with parameters N = 8
and L = N/ne = 0.16 µm, penetrated by magnetic flux
˜
φ = 0.25φ0 . The transmission of the scatterer is |tkF |2 = 0.03.
The e-e interaction is given by V0 = 102 meV and d = 3 nm.

The phase difference |Φ(68meV)− Φ(0meV)| is small and
this is why the corresponding DMC currents in the left
panel of figure 12 almost coincide. The phase difference
|Φ(200meV) − Φ(0meV)| is large and the corresponding
DMC currents in the right panel of figure 12 clearly differ.
We conclude that the main limitation of our DMC results is the Hartree-Fock approximation of the correct
phase, which deteriorates with increase of the e-e interaction and system size. In particular, we see in the figure
12 that the DMC with the Hartree-Fock trial wave function underestimates the persistent current. For instance,
in figure 2 the DMC data for the αRG = 0.2565 exhibit
a weak underestimation which is of the same origin.
Reliability of the CI results depends on how the results converge with increasing the number of the Slater
determinants in the expansion (52).

The figure 14 shows a typical convergence process in
the FCI calculation. We recall that we add into the expansion (52) first the Slater determinant of the ground
state, then all determinants with a single excited electron (the single-excitations), all determinants with two
excited electrons (the double-excitations), etc. The figure 14 demonstrates how the persistent current saturates with increasing the size of the single-electron basis (the number of the considered single-electron levels) for the single-excitations only, for the single- and
double-excitations, etc. Moreover, the top and bottom
panel compare convergence of the persistent currents
ˆ
calculated by means of the formulae I = Ψ0 | I |Ψ0
and I = −∂E0 /∂φ, respectively. It can be seen that
both approaches converge to the same final result when
the single-, double-, triple-, and quadruple-excitations
are considered. However, precision of the results in
the bottom panel is good already for the single- and
double-excitations while in the top panel also the tripleexcitations are needed to achieve a comparable precision.
The FCI calculations with the formula I = −∂E0 /∂φ
therefore consume much less computer time and memory. The FCI data presented in this text were obtained
by means of the formula I = −∂E0 /∂φ and by considering the single- and double-excitations. In a few cases,
sufficiency of the achieved convergence was confirmed
by adding the triple-excitations or even the quadrupleexcitations, with a similar success as in the figure 14.
The figure 15 shows typical convergence of the BBCI
calculation in dependence on the number of the Slater determinants involved in the expansion (52). The persistent

57.5

BBCI

single-excitations
single and double
single, double and triple
single, double, triple and
quadruple

57.0

Nmax = 16
Nmax = 20
Nmax = 24
Nmax = 32
Nmax = 64

56.5

10

15

20

25

30

35

Number of single-electron levels, Nmax

0

3

4

57.6
57.3

HF
FCI
BBCI
DMC

57.0
56.7
56.4
56.1

4

5x10
10 1.5x10 2x10
Number of Slater determinants

4

FIG. 16: The left panel shows how the ground-state energy E0
converges in the FCI calculation of figure 14. The right panel
shows how E0 converges in the BBCI calculation of figure 15.

currents in the BBCI converge to the value 0.0765, in accord with the value reached by the FCI data in figure 14.
The BBCI data in this text were obtained by using the
ˆ
formula I = Ψ0 | I |Ψ0 . The BBCI relying on the formula I = −∂E0 /∂φ gives the same results (not shown),
but the computational time is about twice longer.
In figure 16 we demonstrate convergence of the groundstate energy E0 in the FCI (left panel) and BBCI calculation (right panel). In both cases E0 converges to a certain
minimum. Here E0 is better minimized in the FCI calculation than in the BBCI, but we note that the BBCI result would eventually converge (for a significantly larger
number of determinants) to the same minimum. In both
cases, however, we need to add more determinants to
achieve a perfectly saturated E0 . In contrast to this,
if we look at the convergence of the corresponding persistent currents (figures 14 and 15, respectively), we see
that it is satisfactory finished both in the FCI and BBCI.
Thus, as long as we are interested in the persistent current, it is economical to look at the convergence of the
current rather than at the convergence of E0 .
In figure 17 we show the ground-state energy and persistent current as functions of magnetic flux for the same
parameters as in figures 14, 15, and 16. The CI and
DMC results for the currents are in good agreement.
However, the CI and DMC ground-state energies exhibit
small differences which deserve a comment. In particular, in spite of the fixed-phase approximation the DMC
ground-state energy shows the best minimum. The reasons for this are the following. First, the FCI results in
the figure 17 were obtained by including the single, double, and triple-excitations (see the discussion of figures
14 and 16). The quadruple-excitations, etc., would shift
the FCI ground-state energy to a slightly lower value.
Second, the presented BBCI results were obtained for
Nmax = 64 and for 1.6 × 104 Slater determinants (see figures 15 and 16). Inclusion of more determinants would
shift the BBCI ground-state energy to a lower value.
It is tempting to think that the result showing the lowest ground-state energy is the best one. Such criterion

Persistent current LI /evF

Ground state energy E0 (meV)

FCI

Energy E0 (meV)

15

0.08
0.04
HF
FCI
BBCI
DMC

0.00
-0.04
-0.08
-0.5 -0.4 -0.3 -0.2 -0.1

0.0

φ/φ0

0.1

0.2

0.3

0.4

0.5

FIG. 17: The ground-state energy E0 and persistent current
as functions of magnetic flux for the ring with parameters N =
8 and L = N/ne = 0.16 µm. The transmission of the scatterer
˜
is |tkF |2 = 0.03. The parameters of the e-e interaction are
V0 = 102 meV and d = 3 nm, they give αRG = 0.2565.
The CI, DMC and Hartree-Fock (HF) results are shown by
symbols. The full lines are a guide for eye, the dashed line
shows the formula (60), with α = (1 + 2αRG )1/2 − 1 in the
function N −α and with the value of C0 taken from figure 5.

indeed follows from the variational principle if we compare various methods applied to the same Hamiltonian.
Here the CI methods are applied to the exact Hamiltonian but the DMC is applied to the effective Hamiltonian. If the ground-state energy of the effective Hamiltonian is lower than the ground-state energy of the exact
Hamiltonian, this is not in conflict with the variational
principle. It should also be stressed that the convergence of other properties such as expectation value of
the current operator is not necessarily governed by the
same behavior as the energy convergence. We believe
that our CI results are closer to the true ones due to
the good CI convergence demonstrated above. On the
other hand the crude Hartree-Fock wave functions used in
the fixed-phase approximation are probably not accurate
enough to provide accurate currents for strongly interacting limit. This remains an interesting point for future
studies since more accurate trial wave functions can be
constructed by means of pairing orbitals and pfaffians,
expansions in pfaffians and/or using backflow (dressed)
many-body coordinates37.

IV.

SUMMARY AND CONCLUDING
REMARKS

We have studied the persistent current of the interacting spinless electrons in a continuous 1D ring with
a single strongly reflecting scatterer. We have included
correlations by solving the Schrodinger equation for sev-

eral tens of electrons interacting via the pair interaction
V (x − x′ ) = V0 exp(− |x − x′ | /d). Our aim was to solve
this continuous many-body problem without any physical approximation and to examine microscopically the
power-law behavior of the Luttinger liquid. We have
used advanced CI and DMC methods which, unlike the
Luttinger-liquid model, do not rely on the Bozonization
technique.
In the past, similar studies were performed by numerical RG methods in the lattice model12,13,14,15 . Our methods do not rely on the RG techniques and thus serve as
independent confirmation of such approaches. Moreover,
dealing with the continuous model, we can vary the range
of the e-e interaction in order to test the robustness of
the Luttinger-liquid power laws against various shapes of
V (x − x′ ). This point was not addressed in the latticemodel studies as the range of the interaction was usually
fixed to the on-site interaction and/or to the nearestneighbor-site interaction. Our major findings are:
(i) Our CI and DMC calculations confirm that the persistent current exhibits the asymptotic power-law behavior of the Luttinger liquid, I ∝ L−1−α .
(ii) In our continuous model, the question whether the
power α is determined by a single e-e interaction parameter αRG is addressed by using various shapes of V (x− x′ )
giving the same value of αRG . Our numerical values of
1/2
α confirm the theoretical formula α = (1 + 2αRG ) − 1
with αRG given by the RG expression (4), but only if the
range of V (x − x′ ) is small (d 1/2kF ).
(iii) The CI data for αRG
0.3 show onset of the
asymptotics I ∝ L−1−α already for ten electrons. In
other words, the Luttinger-liquid behavior emerges in the
system with only ten particles. For comparison, a far
much larger number of electrons is needed to observe the
asymptotic power law in the lattice model12,13,14,15 . To
understand origin of this difference, it would be desirable
to study smooth transition from the lattice model to the
continuous model.
(iv) We have treated the e-e interaction in the selfconsistent Hartree-Fock approximation. We observe for
large L the exponentially decaying I(L) instead of the
power law. However, the slope of log(I(L)) still depends
solely on the parameter αRG given by the RG formula
(4), as long as the range of V (x − x′ ) approaches zero.
(v) We have discussed the reliability and limits of our
CI and DMC calculations. The CI methods appear to
provide convergent results for the sizes and strengths of
interactions we have studied. The reliability of the FCI
and BBCI results is easy to analyze in this case and both
methods converge to the same results although the BBCI
can treat larger systems. The main limitation of our
DMC calculations is the accuracy of Hartree-Fock trial
function phase which has been employed in the fixedphase calculations and appears to be a rather poor approximation for strong e-e interaction.
Although our work is not aimed to address experimental aspects, nevertheless, one of our findings might be a
motivation for experimental work. It might be interest-

e-e interaction (meV)

16

40

fit by formula
34×exp(-|x|/3)

30
20

bare

10

Friedel oscillations in detail

4
2
0

0

50

100

screened

0

0

5

10

15

20

25

e-e distance (nm)

FIG. 18: Electron-electron interaction versus distance in the
GaAs 1D system. The dotted line shows the bare e-e ine2
teraction Vbare (x − x′ ) = 4πǫ |x−x1|+x0 for ǫ = 12.5ǫ0 and
′
x0 = 3 nm, where the cutoff x0 mimics the finite wire thickness. The thin full line shows the potential of a single electron screened by the free 1D electron gas, calculated in the
Hartree picture39 . The full line is the best fit by formula
′
V (x − x′ ) = V0 e−|x−x |/d , where d = 3nm essentially coincides with the value of 1/2kF . We expect that external
gates would further increase the screening, i.e., it is indeed
meaningful to focus on the range d
1/2kF as we did in
our work. The Friedel oscillations in the figure are artefact
of static screening and would be further suppressed by the
gates.

ing to observe onset of the Luttinger-liquid behavior in
the 1D system with a small number of strongly interacting electrons, e.g. with only ten electrons as predicts our
work. In this respect we mention, that the screened e-e
interaction (7) with the parameters V0 and d considered
in our calculations is quite realistic. This is documented
in figure 18, where the model interaction (7) is compared
with the microscopic Hartree screening of the bare e-e
interaction.
V.

ACKNOLEDGEMENT

The IEE group was supported by the grant APVV51-003505, grant VEGA 2/6101/27, and ESF project
VCITE. L. M. thanks for support from NSF and DOE.
Appendix A: Numerical algorithm for calculation of
amplitudes tk and rk

Consider again the segment −L/2, L/2 embedded between two perfect semi-infinite leads. Outside the segment the electron wave function reads
ϕk (x) = eikx + rk e−ikx , ϕk (x) = tk eikx

(64)

for the electron impinging the segment from the left and
′
ϕ−k (x) = t′ e−ikx , ϕ−k (x) = e−ikx + rk eikx
k

(65)

17
for the electron impinging the segment from the right,
where k > 0, rk is the reflection amplitude, and tk
is the transmission amplitude. Therefore, the boundary conditions for elastic tunneling through the segment
−L/2, L/2 have a standard form
L
L
L
L
L
ϕk (− ) = e−ik 2 + rk eik 2 , ϕk ( ) = tk eik 2 ,
2
2

(66)

L
L
L
L
L
′
ϕ−k (− ) = t′ eik 2 , ϕ−k ( ) = e−ik 2 + rk eik 2 . (67)
k
2
2
To calculate tk and rk , we have to solve equation (11)
as a tunneling problem with boundary conditions (66).
We write equation (11) in the discrete form (21). Consider now the electron with k > 0. It leaves the segment
−L/2, L/2 at x = L/2 as a free wave ∝ eikL/2 . We
can thus initialize the scheme (21) at x = L/2 + ∆ and
x = L/2 by using
ϕnum (L/2 + ∆) = eik(L/2+∆) ,
k

ϕnum (L/2) = eikL/2 .
k

By means of (21) we generate the numerical solution
ϕnum (xj ) starting at x = L/2 − ∆ and ending at x =
k
−L/2. Since ϕk (L/2) = tk ϕnum (L/2), the correctly
k
scaled result is ϕk (x) = tk ϕnum (x), where tk is yet unk
known. We can readily express tk and rk from the boundary condition
tk ϕnum (x) = eikx + rk e−ikx ,
k

L
x=− ,
2

(68)

FIG. 19: Flowchart of a single Hartree-Fock iteration step as
described in the text.

and from the flux continuity equation
d num
d
tk
eikx + rk e−ikx ,
ϕ
(x) =
dx k
dx

L
x = − . (69)
2

A similar procedure can be used to obtain the solution
′
ϕ−k (x) together with the amplitudes t′ and rk .
k
Appendix B: Iteration steps of the Hartree-Fock
calculation

For numerical purposes it is useful38 to replace the
potential γδ(x) + UH (x) + UF (n, x) in the Hartree-Fock
equation (35) by the potential

Unew (n, x) = f [γδ(x) + UH (x) + UF (n, x)]
+ (1 − f )Uold (n, x) .

(70)

In a given iteration step the equation (35) is solved with
the potential Unew (n, x), where Uold is Unew from the
previous iteration step, and f < 1 is a properly chosen
weight. In the first iteration step we set Unew ≡ γδ(x).

The iteration step works as follows. We search for the
Hartree-Fock energies εn by solving the equation (19).
Precisely, we solve the equation fn (k) = 0, where
fn (k) = Re{e−ikL /tk } − cos(2πφ/φ0 )

(71)

and tk is the transmission evaluated for the potential
Unew (n, x). Clearly, the equation fn (k) = 0 is fulfilled
for many different values of the variable k. Among these
values we choose as physically correct only those k =
kn which fulfill simultaneously two conditions. The first
condition is the inequality ε1 < ε2 < · · · < εN , where
ε1 , ε2 , . . . εN are the energies of the N lowest levels, each
2
of them given as εn = 2 kn /2m. The second condition
is that the wave functions ϕn (x) of these N lowest levels
are mutually orthogonal. The flowchart of the iteration
step is shown in the figure 19.
After finishing the iteration step we set the obtained
ϕn (x) into the potentials UH (x) and UF (n, x) and we
obtain Unew (n, x) for the following iteration step. We
repeat the iteration steps until the energies εn do not
change anymore.

18
We note that we calculate the transmission tk for the
given Unew (n, x) by means of the algorithm from Appendix A. The wave function ϕn (x) of the energy level
2
εn = 2 kn /2m is calculated by using the procedure described in the last paragraph of Section II.A.

Appendix C: Implementation of BBCI

Here we outline the BBCI algorithm30 . As mentioned
in section II.E, we determine the single-particle basis
ψj (x) with the ladder of the single-particle-energy levels
εj and we consider only the first Nmax levels by introducing the upper energy cutoff as illustrated in the figure 1. Due to the cutoff, the expansion (52) contains the
finite number of the Slater determinants χn . This number, equal to Nmax by simple combinatorics, is usually
N
huge. The question is how to asses the importance of all
Nmax
determinants and to omit those of them which
N
are unimportant. In principle, the importance criterion
adopted in section II.E requires to compute the energy
ˆ
χn | H |χn for all Nmax determinants and to order
N
ˆ
the obtained numerical values χn | H |χn increasingly.

Since Nmax is huge, such direct approach still conN
sumes enormous amount of the computer memory and it
is also time-consuming to order increasingly a series of
Nmax
numbers. The bucket-brigade algorithm30 elimN
inates these difficulties as follows.
We use the single-particle states ψj (x) to generate the
Slater determinants χn via a recursion algorithm (Fig.
20) which is based on the sequence of buckets SJ,N with
J = 0, · · · , Nmax and N = 0, · · · , Nmax , where N is the
number of the particles in the bucket and J labels the recursion step J → J + 1. Each bucket SJ,N contains only
the Slater determinants of N particles which occupy the
single-particle states ψj (x) with j = 0, · · · , J − 1. In
each recursion step J → J + 1, we first expand the old
buckets SJ,N by adding the Slater determinants from the
buckets SJ,N −1 , but with an extra particle created in
the single-particle state j = J. After this expansion,
the buckets SJ+1,N are truncated by selecting only the
most important Slater determinants in accord with our
importance criterion based on the chosen measure of imˆ
portance, which is chosen as χn | H |χn in this paper.
Owing to this algorithm our importance criterion is applied only to the determinants in the bucket rather than
to all Nmax determinants.
N

FIG. 20: Visualization of the recursion step J → J + 1 discussed in the text: Copy the remaining (after truncation)
Slater determinants from the N -particle bucket in step J into
the N -particle bucket in step J + 1. Copy the remaining (after truncation) Slater determinants from the (N − 1) -particle
bucket in step J into the N -particle bucket in step J + 1, but,
when copying, enlarge each Slater determinant by creating
a particle in the state j = J. Truncate with the help of the
ˆ
measure χn | H |χn the Slater determinants of each bucket in
the step J + 1. Go to the step J + 2. Stop the procedure after
J reaches Nmax . The squares represent the occupied singleparticle states j = 1, 2, . . . , J. The number of the particles in
the bucket is labelled by N .

L/2

∞

vn f (n) (x) ,

dyV (y)f (x + y) =

(72)

n=0

−L/2

where f (n) (x) =

∂n
∂xn f (x)

and
L/2

Appendix D: Matrix elements of electron-electron
interaction

If we expand the function f (x+y) into the Taylor series
around x, we can write

1
vn =
n!

dyV (y)y n .

(73)

−L/2

Setting the short range interaction V (y) = V0 e−|y|/d into
(73) we get

19

L/2

V0
vn =
n!

L/2

dye

V0
y =
[1 + (−1)n ]
n!

−|y|/d n

0

−L/2

Vαβγδ =

dye−y/dy n
(74)

L/2

L/2

V0

and after integration per partes we obtain

dx

∗
dy e−|y|/d ψβ (x + y)ψδ (x + y)

∗
ψα (x)ψγ (x)

−L/2

−L/2

(79)
n

dm (L/2)n−m
.
(n − m)!
m=0
(75)
Setting this vn back into (72), reordering the summations over m and n, identifying the particular Taylor series of f (n) (x ± L/2), and using the periodicity condition
f (n) (x ± L) = f (n) (x), we obtain the relation
vn = V0 d [1 + (−1)n ] dn − e−L/2d

into the form

Vij ≈ V0 d

−|y|/d

Aijn =

f (x + y) =

∞

2V0 d
n=0

d2n f (2n) (x) − e−L/2d f (2n) (x − L/2) . (76)

In the limit L ≫ d this relation simplifies to
L/2

∞

dyV0 e−|y|/df (x + y) ≈ 2V0 d
−L/2

d2n f (2n) (x) . (77)
n=0

Using the relation (77) and substitution z = 2πx/L we
can rewrite the matrix elements

Vij =

1
2

aijαβγδ Vαβγδ

(78)

α,β,γ,δ

with aijαβγδ being one of the values {−1, 0, 1} and with

3
4

5

6
7

aijαβγδ
α,β,γ,δ

−L/2

2

n=0

2πd
L

2n

π

dyV0 e

1

∞

Electronic address: martin.mosko@savba.sk
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8
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