Subm. to Phys.Rev.Lett.; November 9, 1993

A Simple Formula for the Persistent Current in
Disordered 1D Rings: Parity and Interaction Eﬀects

arXiv:cond-mat/9311026v1 11 Nov 1993

Alexander O.Gogolin†
Landau Institute for Theoretical Physics, Russian Academy of Sciences,
Kosygina str.2, Moscow, 117940, Russia
Nikolai V. Prokof’ev††
Physics Department, University of British Columbia, 6224 agricultural RD.,
Vancouver, BC, Canada V6T 1Z1

The general expression for the persistent current of 1D noninteracting electrons
in a disorder potential with smooth scattering data is derived for zero temperature.
On the basis of this expression the parity eﬀects are discussed. It is shown that
the electron-electron interaction may give rise to an unusual size-dependence of the
current amplitude due to renormalized backscattering from the disorder potential.

†

present address: Institute Laue-Langevin, B.P.156, 38042 Grenoble, Cedex 9,
France; Email: gogolin@gaston.ill.fr
††

permanent address: Kurchatov Institute, Moscow, 123182, Russia
1

The persistent current in 1D mesoscopic rings, being a manifestation of the
Aharonov-Bohm eﬀect for many-body systems, addresses physical questions of principle. Although such an eﬀect was predicted several decades ago [1], in normal state
systems it was only recently discovered experimentally. The experiments were performed on ensembles of many mesoscopic rings [2] as well as on single (or several) rings
[3]. Generally, the theory predicts a magnitude of the current, which is much less then
one experimentally observed [4], although there are some cases for which the theory
and experiment seem to agree [5]. To the best of our knowledge there are no such
studies of the interplay between the disorder and electron-electron interaction in 1D
which would take into account nonperturbative eﬀects in the persistent current (on the
other hand, several numerical investigations have been curried out [5]).
The aim of this paper is to derive a simple formula for the persistent current of
noninteracting electrons in a single-channel mesoscopic ring with a disorder potential
and to study the interplay between the disorder and electron-electron interaction in
1D.
For N spinless electrons on the ring of the length L under Aharonov-Bohm conditions the persistent current is given by the formula:
j(ϕ) = j0 (ϕ) + j par (ϕ) + O(1/L2);
√
vF
TF sin ϕ
√
j0 (ϕ) =
T (p) cos ϕ ,
(1)
F (pF , ϕ); F (p, ϕ) = cos−1
2ϕ
πL 1 − TF cos
√
vF
TF sin ϕ
par
j (ϕ) = − √
, (N = even);
j par (ϕ) = 0, (N = odd) ,
L 1 − TF cos2 ϕ
where ϕ = 2πΦ/Φ0 (Φ is the magnetic ﬂux passing through the ring) and TF ≡ T (pF )
2

is the transmission coeﬃcient of the ring at the Fermi energy. The above formula
explicitly shows that the persistent current is a purely Fermi surface eﬀect. Moreover,
it is parametrized by a single number - TF (i.e. the only characteristic of the potential,
which determines the ﬂux dependence of the current, is TF ) [6].
Derivation. The single electron Schr¨dinger equation
o
{ε0 (−i∂x ) + V (x) − ε} ψ = 0,

(2)

where V (x) is the disorder potential and ε0 (p) is the dispersion law, should be solved
under the twisted boundary condition: ψ(x + L) = eiϕ ψ(x). It was recognized already
in the pioneer papers [1] that in this Bloch functions problem the whole ring plays the
role of the elementary cell and ϕ - the role of the quasimomentum. The ground state
energy as a function of the ﬂux is given by the sum E0 (ϕ) =

ελ (ϕ) over N lowest

values of the band spectrum at ﬁxed ϕ.
Suppose, for clarity, that the potential V (x) is localized within the region of a radius
a < L (actually, the relation between a and L will be shown to be unimportant). The
wave-function ψ(x) can be then written in the form ψ(x) = Aeipx + Be−ipx from the
left of the potential and in the form ψ(x) = Ceipx + De−ipx from the right of it.
The coeﬃcients C and D can be expressed through A and B by making use of the
scattering data of V (x) (transfer matrix). The twisted boundary condition gives then
(after elementary algebra) the equations for the spectrum:
pL = Φ+ (p, ϕ) (n = 0);

pL = 2πn + Φ± (p, ϕ) (n = 1, 2, ...);

(3)

where Φ± (p, ϕ) = δ(p) ± F (p, ϕ) and δ(p) is the forward scattering phase deﬁned by
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the scattering solution: if A = 1 and D = 0, then C =

T (p)eiδ(p) .

To make progress with the Eq.(3), the idea is to expand the solution in 1/L. This
is simplest when a << L (e.g. V (x) represents a single scatterer) and ε0 (p) = p2 /2m.
Then one can write:
pn = k n +

1
1
∂Φ± (kn )
1
Φ± (kn ) + 2 Φ± (kn )
+O
;
L
L
∂k
L3

kn = 2πn/L

(4)

For the ground state energy we therefore have:
E0 =

1
m

k2 +
n

1 ∂
2
k
kδ(k) + 2
L
2L ∂k

Φ2 (ϕ, k) + O
±
±

1
L3

(5)
k=kn

Here the ﬁrst term is the ground state energy in the absence of the potential and
ﬂux (it is proportional to the volume: ∼ L). The second term (∼ 1) is the energy
diﬀerence due the scattering potential (in agreement with Fumi’s theorem [7]), which
is ﬂux independent. Thus, the eﬀect of the ﬂux is of order 1/L and is given by:
∆E0 (ϕ) = E0 (ϕ) − E0 (0) =

n

1 ∂
k
2L2 ∂k

±

1
v
= F F 2 (p, ϕ) − F 2 (p, 0) + O
2πL
L2

Φ2 (ϕ, k) − Φ2 (0, k) + O
±
±
.

1
L3

=
k=kn

(6)

For N = even one additional particle on top of the spectrum contributes to Eq.(5) the
term:
par
∆E0 (ϕ) = −

vF
{F (p, ϕ) − F (p, 0)}
L

(7)

In fact, each particle contributes to the ﬂux dependence of the energy a term ∼ 1/L,
but the contributions of the particles with quantum numbers (n, +) and (n, −), which
would correspond to the momenta +p and −p for V = 0, almost cancel each other,
4

and the entire contribution of the Fermi sea is again of the order of 1/L and converges
actually just at the Fermi surface (Eq.(6)). For even N, the particle on the top, i.e. in
the state (N/2, −), does not have a partner in the state (N/2, +), so the contribution
of this single particle is 1/L; that gives rise to the parity eﬀect (Eq.(7)).
The formula Eq.(1) follows from Eqs.(6,7) by a standard deﬁnition of the current
j(ϕ) = −∂E0 (ϕ)/∂ϕ. The diﬀusion coeﬃcient D = − 1 (∂ 2 E0 (ϕ)/∂ 2 ϕ)|ϕ=0 is equal to
2
D0 for odd N and to D0 + D par for even N, where
D0 =

vF
2πL

TF
tan−1
1 − TF

1 − TF
;
TF

D par = −

vF
2L

TF
.
1 − TF

(8)

Thus, for the case of spinless fermions, the ground state is always diamagnetic for
odd N and paramagnetic for even N. In the absence of spin-orbit coupling the spin
eﬀect can be straightforwardly taken into account. The eﬀect of the Fermi sea, i.e. the
quantities j0 (ϕ), ∆E0 (ϕ) and D0 , should be just multiplied by the factor 2. The parity
eﬀect, however, changes a little bit: the diamagnetic ground state (D > 0) is observed
only if N = 4k + 2. So, the deﬁnite conclusion concerning a dia- or paramagnetic
nature of the ground state can be indeed drawn from the parity arguments only, even
in the presence of a disorder potential [8].
The transfer matrix method has already been used in the theory of persistent current
[9]. However, it is a combination of the transfer matrix method and the 1/L - expansion,
which leads to the new results presented above. For a single point scatterer V (x) =
εδ(x) the transmission coeﬃcient is TF = p2 /(p2 + m2 ε2 ). Expanding our formula
F
F
Eq.(1) either in the parameter mε/pF (weak potential) or in the parameter pF /mε
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(strong potential) we reproduce the results for these limiting cases, which were obtained
previously [9]. The condition a << L assumed above for simplicity is unimportant.
Actually, in the case of a ∼ L the derivation is essentially the same, except of the
modiﬁcation due to the fact that the forward scattering phase δ(p) in Eq.(3) is now
∼ L. So, one should redeﬁne the momentum according to pL = 2πn + δ(p) ﬁrst and
only then expand the eﬀect of the ﬂux in 1/L: the results Eq.(1,6-8) will not change
at all. Simple algebra also shows that the results are valid for arbitrary dispersion
law ε0 (p) (TF is then understood to be deﬁned accordingly to the dispersion law).
The only important condition for the potential V (x), which was in fact assumed in
the above derivation, is the requirement for the transmission coeﬃcient TF (p) to be
a smooth function of the momentum on the scale 1/L (otherwise dTF (p)/dp becomes
∼ L and the 1/L - expansion obviously breaks down). Thus Eq.(1) works for the
case of arbitrary single scatterer (or several ones) and, even under strong localization
conditions (mean free path l ≪ L), gives a correct order of magnitude for the current
[9] and predicts also a ∼ sin ϕ shape of its ﬂux dependence, which was observed in
numerical simulations for strong disorder [5].
In what follows we consider the eﬀects due to electron-electron interactions (we
consider the case of a single scatterer; note however, that in the long wave-length limit
q → 0 corresponding to the Luttinger Liquid ﬁxed point we need only the disorder
potential to be conﬁned within the distance 1/q). We make use of Haldane’s [10]
concept of topological excitations in the LL ground state in the boson representation.
Due to the linear dispersion law near the Fermi points bosonic excitations do not
6

contribute to the current, and j is entirely deﬁned by the topological number J which
describes the diﬀerence between the number of right- and left-moving electrons
j=−

∂E0
v
ϕ
= J (J − ) ,
∂ϕ
L
π

(9)

where E0 (ϕ) is the ground state energy of the Hamiltonian [10]
H=

ϕ
πvJ
(J − )2 .
2L
π

(10)

In this formulation the problem is a classical one: to ﬁnd the minimum of the potential
energy Eq.(10) with an integer J satisfying the parity condition (−1)J = (−1)N +1 .
In the boson representation the backscattering Hamiltonian has a form
+∞

Hbsc = Vb
J=−∞

ˆ

ˆ

a++2 aJ eiθ + a+−2 aJ e−iθ ,
J
J

ˆ √
θ= g
q

2π
(b+ + bq ) ,
L|q| q

(11)

where we introduce a+ - the creation operator of the quantum number J; b+ is the
q
J
boson creation operator, and g is the dimensionless electron-electron coupling constant
(g = 1 for the noninteracting system; g > 1 for the attractive interaction, and g < 1
for the case of repulsion).
Every backscattering event changes the topological number J by ±2 with a simultaneous excitation of the bosonic environment. Now J is no longer a conserved quantity
but has a tendency toward delocalization into a ﬁnite-size band. As we show below, the
delocalization of J over the scale < J 2 >= R >> 1 leads to an exponential suppression
of the current ln j ∼ −R. One might consider the eﬀect of delocalization as equivalent
to heating the perfect system up to the temperature T ∼ R 2πvJ /L. The crucial point
is that the parabolic potential Eq.(10) is very weak (of order of 1/L) as compared with
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the bare ”hopping” rate Vb . If the dissipation eﬀects are neglected (g → 0) then in
the ground state one ﬁnds the result R ∼

√

L. However this is never the case because

bosonic environment plays the dominant role in the formation of the ground state.
The Hamiltonian Eq.(11) is well known in the theory of quantum coherence for
so-called Ohmic dissipative environments (see, e.g. [12]). The eﬀective bandwidth (or
kinetic energy) corresponding to Eq.(11) is deﬁned from the self-consistent equation
∆ ≈ Vb

∆, ωmin
EF

g

,

(12)

where EF is the Fermi energy, and ωmin ≈ 2πvS /L is the minimal boson energy. In
the simplest case g > 1 we immediately ﬁnd that the kinetic energy is proportional
to ∆ ∼ 1/Lg and can be neglected as compared with the potential term (10). As
expected, we recover the known result that backscattering is irrelevant for the case of
attractive interaction between the electrons, in this picture, due to localization of J.
For g = 1 it follows from (12) that ∆ is proportional to 1/L. Again, J is localized
(in a sense that ﬂuctuations of J are of order 1), but this time the solution depends
on the details of the potential, electron energy spectrum etc. in agreement with the
above discussion for the noninteracting electrons.
In the most intriguing case g < 1 the coherence is restored between the neighbouring
values of J at frequencies below
∆ ≈ EF

1/(1−g)

Vb
EF

,

(13)

with delocalization of J in the ground state. Let us start with the extreme case of g = 0.
In the ”momentum” representation the Hamiltonian (10)-(11) can be conveniently
8

rewritten in the form (one might identify the momentum p with the canonical conjugate
phase ﬁeld θJ ; [J, θJ ] = i, see Ref. [10]):
H=−

2πvJ ∂ 2
+ 2∆ cos p ,
L ∂p2

(14)

with twisted periodic condition ψ(p + 2π) = eiϕ ψ(p) (we assumed J = even here;
the case of J = odd will correspond to ϕ → ϕ + π). Since the eﬀective mass in the
momentum representation is proportional to 1/L we can consider this Hamiltonian as
a standard tight-binding model with the energy of the lowest state being deﬁned as
E0 (ϕ) = const+D(1−cos ϕ), and the amplitude D being calculated in the semiclassical
approach
D≈

8vJ ∆
L∆
exp −8
πL
2πvJ

.

(15)

Thus, for the case g = 0, the persistent current, j ≈ D(−1)N sin ϕ, would be exponentially suppressed.
With g > 0, however, the concept of the eﬀective mass for J completely fails even
in the ground state. The dynamics of J is overdamped as follows from the constant
mobility in the zero-temperature limit [13], µ = 1/(2πg). In the presence of the
parabolic potential Eq.(10) the delocalization of J in the ground state depends on L
only logarithmically [14], < J 2 >=

µ
π

ln (L∆/(2πvJ µ2 )), and one might expect a power

dependence J(L). It is easy to show [15], that in the momentum representation the
problem can be reduced to the study of the eﬀective action
Sef f =

β/2
−β/2

dτ 2∆ cos p(τ ) + β
n

Dp exp(−Sef f )

2
ω n | pn | 2
;
2(4πvJ /L + 2πg | ωn |)

9

(β → ∞) ,

(16)

where ωn = 2πn/β, and pn is the Fourie-transform of p(τ ). Except very low frequencies
ωn ≤ 2vJ /gL the eﬀective action is governed by the dissipative term

β
4πg

| ω || pω |2 .

The quasiclassical solution for the overdamped motion in the cos-barrier was studied
in Ref. [16], so we can readily write down the expression for the current amplitude D
in the leading logarithmic approximation
L∆
1
D ∼ g∆ exp − ln
g
2πvJ

.

(17)

Thus our basic result for the persistent current in the case of repulsion between the
electrons has a form
j ∼ g∆

2πvJ
L∆

1/g

(−1)N sin ϕ .

(18)

The suppression of the current to higher order in 1/L is closely related to the
result of Ref. [11] that in the limit E → EF backscattering renormalizes into a perfect
reﬂection from the potential. Eﬀectively the transmission coeﬃcient near the Fermi
points has a power law dependence [11]
T (E) ∼ T0

E − EF
∆

2(1/g−1)

,

(19)

leading to TF ∼ T0 L−2(1/g−1) . If we substitute this result into the formula Eq.(??),
then in the limit TF ≪ 1 we easily obtain the same scaling behavior Eq.(18) [17].
To summarize, we derive for the ﬁrst time an explicit analytic solution for the
persistent current of noninteracting electrons in disordered 1D ring under quite general
assumptions about the disorder potential. We ﬁnd that the transmission coeﬃcient of
the ring at the Fermi energy is the only relevant parameter in the problem. The case of
10

spinless electrons is solved including electron-electron interactions, and a nontrivial size
dependence of the persistent current with interaction-dependent exponent is obtained
(provided the parameter ∆, Eq. (13), is large enough compared with vJ 2π/L). It
seems interesting to generalize the above approach to the case of spin-orbit coupling
and to the (more realistic) case of multi-channel rings for which the existence of a
formula expressing the persistent current in terms of the transmittance matrix at the
Fermi energy is still expected (although the ﬂux dependence of the current will be more
complicated). These generalizations are now in progress [15].
We are thankful to P.C.E.Stamp, A.G.Aronov, H.Cappelmann, A.Ioselevich,
A.I.Larkin and M.Fabrizio for valuable discussions. One of us (N.V.P.) was supported
by NSERC.

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REFERENCES
[1] N.Byers and C.N.Yang, Phys.Rev.Lett.7, 46(1961); W.Kohn, Phys.Rev. 133,
A171(1964); F.Bloch, Phys.Rev. 21, 1241(1968); L.Gunther and Y.Imry,
Solid State Commun.7 1394(1969);

M.B¨ ttiker, Y.Imry and R.Landauer,
u

Phys.Lett.96A, 365(1983).
[2] L.P.Levy,G.Dolan, J.Dunsmuir and H.Brouchiat, Phys.Rev.Lett. 64, 2074 (1990).
[3] V.Chandrasekhar,

R.A.Webb, M.J.Brady,

M.B.Ketchen,

W.J.Galager and

A.Kleinsasser, Phys.Rev.Lett. 67, 3578 (1991).
[4] Recently the literature on the persistent current problem has been really exploding.
For recent perturbation theory calculations see R.A.Smith and V.Ambegaokar,
Europhys.Lett. 20, 161(1992) and references therein.
[5] M.Abraham and R.Berkovits, Rhys.Rev.Lett. 70, 1509(1993); G.Bouzerar,
D.Poilblanc and G.Montambaux, ”Persistent Current in 1D Disordered Rings of
Interacting Electrons”, preprint(1993).
[6] This formula can be considered as an analog of the famous Landauer formula for
the conductance; R.Landauer, IBM J.Res.Dev. 1, 223(1957).
[7] F.G.Fumi, Philos.Mag. 46, 1007(1955).
[8] That may be considered as a proof of the so-called Leggett conjecture; A.J.Leggett,
in Granular Nanoelectronics, edited by D.K.Ferry, J.R.Barker and C.Jacoboni,
NATO ASI Ser. B251 (Plenum, New York, 1991), p.297; see also D.Loss,
12

Phys.Rev.Lett. 69, 343(1992).
[9] H.F.Cheung, Y.Gefen, E.K.Riedel and W.H.Shih, Phys.Rev.B37, 6050(1988).
[10] F.D.M.Haldane, J.Phys.C, 14, 2585(1981).
[11] C.L.Kane and M.P.A.Fisher, Phys.Rev.Lett. 68, 1220(1992); K.A.Matveev, D.Yue
and L.I.Glazman, ”Conduction of a Weakly Interacting 1D Electron Gas through
a Single Barrier”, preprint(1993).
[12] A.J.Leggett, S.Chakravarty, A.T.Dorsey, M.P.A.Fisher, A.Garg and W.Zwerger,
Rev.Mod.Phys.59, 1(1987).
[13] A.Schmid, Phys.Rev.Lett.51, 1506(1983); S.A.Bulgadaev, Zh.Teor.Eksp.Fiz.90,
634(1986) [Sov.Phys.JETP 63, 369(1986)].
[14] N.V.Prokof’ev, Int.J.Mod.Phys.B 7, 3327(1993).
[15] A.O.Gogolin, N.V.Prokof’ev, to be published.
[16] S.E.Korshunov,

Zh.Eksp.Teor.Fiz.92,

1828(1987)

[Sov.Phys.JETP

65,

1025(1987)].
[17] In fact, such a substitution turns out to be correct for all such ”Fermi surface” eﬀects: e.g., the conductance (Ref. [11]); the X-Ray problem (A.O.Gogolin,
Phys.Rev.Lett.71, 2995(1993); N.V.Prokof’ev, Phys.Rev.B (1993), in press) and,
now, the persistent current.

13