MAGNETOCONDUCTANCE OSCILLATIONS IN METALLIC RINGS &
DECOHERENCE DUE TO ELECTRON-ELECTRON INTERACTION
Christophe Texier†,‡ and Gilles Montambaux‡
†Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626, CNRS,Université Paris-Sud,
F-91405 Orsay Cedex, France.
‡Laboratoire de Physique des Solides, UMR 8502, CNRS, Université Paris-Sud, F-91405 Orsay.
We study weak localization in chains of metallic rings. We show than nonlocality of quantum
transport can drastically affect the behaviour of the harmonics of magnetoconductance oscil-
lations. Two different geometries are considered : the case of rings separated by long wires
compared to the phase coherence length and the case of contacted rings. In a second part we
discuss the role of decoherence due to electron-electron interaction in these two geometries.
1 Introduction
At low temperature, quantum interferences of reversed electronic trajectories are responsible
for a small reduction of the averaged conductivity called the weak localization (WL) correction.
This correction is a manifestation of quantum coherence which is always limited over a certain
length scale, named the phase coherence length Lϕ. A way to extract this important length
scale in experiments is to use the magnetic field sensitivity of the WL. For example the WL
correction of an infinitely long wire of rectangular section of width W and area S submitted to
a perpendicular magnetic field B is 1 〈∆σ〉 = −2e2
)2]−1/2 (in the following we
will forget the 1/S factor). The width of the magnetoconductance (MC) curve provides a direct
determination of Lϕ.
... ...
(b) ......
Figure 1: Chains of rings. If we consider the regime b ≫ Lϕ ≫ L the rings can be considered as independent in
case (a) but not in case (b).
Another possibility to extract phase coherence length is to study arrays of rings whose
MC present oscillations as a function of the flux φ per ring with period half the quantum flux
φ0 = h/e. These are the famous Al’tshuler-Aronov-Spivak oscillations
2 (AAS), observed in
many experiments3,4,5,6. In order to extract the phase coherence length from AAS oscillations,
a precise theoretical prediction for the behaviour of the AAS harmonics with the phase coherence
length is needed. Harmonics of the oscillations are defined as 〈∆σ(φ)〉 =
n∆σne
4πinφ/φ0 . A
well-known expression has been derived in Ref. 1 for an isolated ring of perimeter L :
∆σn = −
−|n|L/Lϕ (1)
However, in a real experiment where the ring is connected to wires, or embedded in a larger
network, this expression can only be relevant in the regime Lϕ ≪ L. At the lowest tempera-
tures, when Lϕ & L, the AAS harmonics are strongly affected by the surrounding wires since
trajectories can expand outside the ring over distances larger than the perimeter. It is the aim
of this paper to discuss the behaviour of AAS harmonics in chains of rings when Lϕ & L. We
will consider two cases represented on figure 1 : in the first situation the rings are separated
http://arxiv.org/abs/0704.0742v1
by a distance b ≫ Lϕ and can therefore be considered as independent (however the connecting
wires will affect the AAS harmonics). In the second case rings are in contact and harmonics can
involve trajectories winding around several neighbouring rings. In section 3, we will see that
when electron-electron interaction is the dominant process for decoherence, eq. (1) cannot be
used even in the regime L ≪ Lϕ.
2 Nonlocality of quantum transport in chain of rings
We consider an array of rings all pierced by the same flux φ. The n-th harmonic of the WL
correction at a given point x of a network can be expressed as
∆σn(x) = −
dtPn(x, x; t) e−t/τϕ (2)
where D is the diffusion constant and τϕ = L
ϕ/D the phase coherence time. The factor 2 stands
for spin degeneracy. Pn(x, x; t) is the probability that a particle diffusing into the network
comes back to its initial point x in a time t, after having encircled a flux nφ. For example,
in an isolated ring Pn(x, x; t) = 1√
−(nL)2/4Dt which immediatly gives eq. (1). Except in
translation invariant systems, ∆σn(x) depends on x and expression (2) must be averaged over
the network in a proper way described in Ref. 7.
A ring with two arms.– The case of a ring connected to two arms has been studied in detail
in Ref. 8 where it has been shown that Pn(x, x; t) ≃
2(Dt)3/4
(Dt)1/4
) for time scales t ≫ L2/D
with x inside the ring (the precise form of the dimensionless function Ψ(ξ) is inessential for the
present discussion). Compared with the isolated ring case, where the typical number of winding
scales with time as nt ∼ t1/2, diffusion around the ring is slowed down as nt ∼ t1/4 due to the
time spent in the arms. As a consequence the harmonics of the conductance of a ring are given
by 8 : ∆σn ∝ Lϕ3/2e−|n|
2L/Lϕ (note that the scaling n ∼ L1/2ϕ is analogous to the scaling of
winding with time nt ∼ t1/4 since Lϕ ∼ t1/2).
The chain of distant rings.– The same argument holds for the chain of rings separated by
a distance b ≫ Lϕ (figure 1.a). In this case, averaging properly ∆σn(x) inside the chain of Nr
rings, one finds that the harmonics of the dimensionless conductance read :
∆gn ≃ −
1/2Lϕ
2 [(Nr + 1)b]2
2L/Lϕ for b ≫ Lϕ ≫ L (3)
The chain of attached rings.– If we now consider the network of figure 1.b, we can show
that the probability reads 9 Pn(x, x; t) ≃ L8πDte
−(nL)2/4Dt for t ≫ L2/D. The AAS harmonics
are given in this case by 9 :
∆gn ≃ −
[ln(2Lϕ/|n|L) + bn] for Lϕ/L ≫ |n| (4)
−|n|L/Lϕ
|n|L/Lϕ
for |n| ≫ Lϕ/L ≫ 1 (5)
where bn depends weakly on n (b∞ = −C, the Euler constant).
3 Decoherence due to electron-electron interaction
The above results rely on the fact that, in eq. (2), the long times have been cut off with an
exponential damping e−t/τϕ . However it has been shown recently that this simple modelization
does not account correctly for the decoherence due to electron-electron interaction, which is the
dominant one at low temperature a. In this case, an alternative description was proposed by
Al’tshuler, Aronov & Khmel’nitskii (AAK) 12 but it is only recently that the consequences for
AAS oscillations have been understood13,14,15.
The model of AAK.– The length scale characterizing the efficiency of electron-electron inter-
action to suppress phase coherence in wires is known as the Nyquist length LN = (ν0D
2/T )1/3
where ν0 is the density of states, D the diffusion constant and T the temperature (~ = kB = 1).
In the model of AAK, the random phase accumulated by an electron moving in the fluctuat-
ing electric potential due to other electrons is included in the calculation of the WL. The pair
of reversed interfering trajectories picks a phase eiΦ[C], where C designates a closed diffusive
trajectory, and the harmonics of WL are given by
∆σn ∼ −
〈eiΦ[Cn]〉V = −
〈Φ[Cn]2〉V (6)
The sum runs over all closed trajectories with winding n (a proper formulation of eq. (6) requires
a path integral). Gaussian fluctuations of the electric potential are given by the fluctuation-
dissipation theorem 〈V (~r, t)V (~r ′, t′)〉V = 2e
Tδ(t − t′)Pd(~r,~r ′) (written here in the classical
limit T ≪ ω), where σ0 is the classical Drude conductivity. Pd is solution of the diffusion
equation −∆Pd(~r,~r ′) = δ(~r−~r ′) and therefore depends on the topology of the system. Then14
〈Φ[x(τ)]2〉V =
dτ [Pd(x(τ), x(τ)) − Pd(x(τ), x(t− τ))] (7)
where C ≡ (x(τ), 0 6 τ 6 t |x(0) = x(t)) is a closed diffusive path. The crucial point is that
the simple exponential damping of eq. (2) is replaced in eq. (6) by a functional of the trajectory
〈Φ[Cn]2〉V . Therefore decoherence is now network-dependent and a priori sensitive to the
nature of trajectories (in particular whether they do enclose a magnetic flux or not).
The limit LN ≪ L.– The model described above was applied to the case of a single
ring 13,14,15. The result for an isolated ring is relevant to describe arrays of rings in the limit
LN ≪ L where winding trajectories hardly exit from a ring, which makes rings independent
from each other. For the chain of distant rings (figure 1.a) we have
∆gn ∼ −
[(Nr + 1)b]2
−|n|π
(L/LN )
3/2 ∼
−nL3/2T 1/2
T 1/3
for LN ≪ L ≪ b (8)
(for the case of the chain of rings in contact (figure 1.b), (Nr + 1)b in the denominator is
replaced by NrL/4). Whereas the time characterizing efficiency of electron-electron interaction
to suppress phase coherence in a wire is the Nyquist time τN = L
N/D ∝ T−2/3, it was shown in
Refs. 13,14 that the behaviour (8) is related to a new time scale characterizing decoherence for
winding trajectories : τc = τ
L ∝ T
−1, where τL = L
2/D is the Thouless time of the ring.
The chain of distant rings.– If we consider a ring connected to long arms, winding trajectories
spend most of the time in the arms8 and the decoherence mostly occurs in the arms. Therefore
decoherence occurs on a time scale τN , like in a wire. The function 〈eiΦ[C]〉V for a wire was
a The exponential damping gives the correct shape of a MC of a wire 10,11 with Lϕ →
2LN (see below
for definition of LN ), however this simple substitution gives an incorrect result for AAS harmonics as explained
below.
studied in Ref. 16. Using this result and the winding properties recalled in section 2 leads to 14
∆gn ∼ −
[(Nr + 1)b]2
for n2 ≪ LN/L (9)
[(Nr + 1)b]
)7/12
−κ2|n|
L/LN ∼ e
−nL1/2T 1/6
T 11/36
for n2 ≫ LN/L , (10)
where κ2 =
2|u1|1/4 ≃ 1.421.
The chain of attached rings.– In this case, the nature of decoherence was shown to be closely
related to the one of a wire since diffusion along the chain is reminiscent of a 1d diffusion and
again occurs on time scale τN
∆gn ≃ −
ln(LN/|n|L) + cste for |n| ≪ LN/L (11)
≃ − 1
Nr|u1|3/2
−κ3|n|L/LN ∼ e−nLT 1/3 for |n| ≫ LN/L (12)
where κ3 = 2
−1/3|u1|1/2 ≃ 0.801.
4 Conclusion
We have considered networks of connected rings, made of weakly disordered wires. We have first
shown that geometrical effects can strongly modify the exponential behaviour of AAS harmonics
well-known for an isolated ring, since trajectories can now explore the network around each
ring. In the second part we have shown that decoherence due to electron-electron interaction
is sensitive to geometry, a second reason that modifies the simple AAS result. An interesting
experiment would be to compare precisely AAS oscillations for the two networks of figure 1 in
the low temperature regime LN ≫ L.
References
1. B. L. Al’tshuler and A. G. Aronov, JETP Lett. 33(10), 499 (1981).
2. B. L. Al’tshuler, A. G. Aronov, and B. Z. Spivak, JETP Lett. 33(2), 94 (1981).
3. B. Pannetier, J. Chaussy, R. Rammal, and P. Gandit, Phys. Rev. B 31(5), 3209 (1985).
4. G. J. Dolan, J. C. Licini, and D. J. Bishop, Phys. Rev. Lett. 56(14), 1493 (1986).
5. M. Ferrier, L. Angers, A. C. H. Rowe, S. Guéron, H. Bouchiat, C. Texier, G. Montambaux,
and D. Mailly, Phys. Rev. Lett. 93, 246804 (2004).
6. F. Schopfer, F. Mallet, D. Mailly, C. Texier, G. Montambaux, L. Saminadayar, and
C. Bäuerle, Phys. Rev. Lett. 98, 026807 (2007).
7. C. Texier and G. Montambaux, Phys. Rev. Lett. 92, 186801 (2004).
8. C. Texier and G. Montambaux, J. Phys. A: Math. Gen. 38, 3455–3471 (2005).
9. C. Texier and G. Montambaux, in preparation (2007).
10. F. Pierre, A. B. Gougam, A. Anthore, H. Pothier, D. Esteve, and N. O. Birge, Phys.
Rev. B 68, 085413 (2003).
11. É. Akkermans and G. Montambaux, Physique mésoscopique des électrons et des photons,
EDP Sciences, CNRS éditions, 2004. Mesoscopic physics of electrons and photons, Cam-
bridge University Press, 2007.
12. B. L. Altshuler, A. G. Aronov, and D. E. Khmelnitsky, J. Phys. C: Solid St. Phys. 15,
7367 (1982).
13. T. Ludwig and A. D. Mirlin, Phys. Rev. B 69, 193306 (2004).
14. C. Texier and G. Montambaux, Phys. Rev. B 72, 115327 (2005) ; ibid 74, 209902(E)
(2006).
15. C. Texier and G. Montambaux, Comment on Ref. 13, submitted (2007).
16. G. Montambaux and E. Akkermans, Phys. Rev. Lett. 95, 016403 (2005).
	Introduction
	Nonlocality of quantum transport in chain of rings
	Decoherence due to electron-electron interaction
	Conclusion

  We study weak localization in chains of metallic rings. We show than
nonlocality of quantum transport can drastically affect the behaviour of the
harmonics of magnetoconductance oscillations. Two different geometries are
considered: the case of rings separated by long wires compared to the phase
coherence length and the case of contacted rings. In a second part we discuss
the role of decoherence due to electron-electron interaction in these two
geometries.

Introduction
At low temperature, quantum interferences of reversed electronic trajectories are responsible
for a small reduction of the averaged conductivity called the weak localization (WL) correction.
This correction is a manifestation of quantum coherence which is always limited over a certain
length scale, named the phase coherence length Lϕ. A way to extract this important length
scale in experiments is to use the magnetic field sensitivity of the WL. For example the WL
correction of an infinitely long wire of rectangular section of width W and area S submitted to
a perpendicular magnetic field B is 1 〈∆σ〉 = −2e2
)2]−1/2 (in the following we
will forget the 1/S factor). The width of the magnetoconductance (MC) curve provides a direct
determination of Lϕ.
... ...
(b) ......
Figure 1: Chains of rings. If we consider the regime b ≫ Lϕ ≫ L the rings can be considered as independent in
case (a) but not in case (b).
Another possibility to extract phase coherence length is to study arrays of rings whose
MC present oscillations as a function of the flux φ per ring with period half the quantum flux
φ0 = h/e. These are the famous Al’tshuler-Aronov-Spivak oscillations
2 (AAS), observed in
many experiments3,4,5,6. In order to extract the phase coherence length from AAS oscillations,
a precise theoretical prediction for the behaviour of the AAS harmonics with the phase coherence
length is needed. Harmonics of the oscillations are defined as 〈∆σ(φ)〉 =
n∆σne
4πinφ/φ0 . A
well-known expression has been derived in Ref. 1 for an isolated ring of perimeter L :
∆σn = −
−|n|L/Lϕ (1)
However, in a real experiment where the ring is connected to wires, or embedded in a larger
network, this expression can only be relevant in the regime Lϕ ≪ L. At the lowest tempera-
tures, when Lϕ & L, the AAS harmonics are strongly affected by the surrounding wires since
trajectories can expand outside the ring over distances larger than the perimeter. It is the aim
of this paper to discuss the behaviour of AAS harmonics in chains of rings when Lϕ & L. We
will consider two cases represented on figure 1 : in the first situation the rings are separated
http://arxiv.org/abs/0704.0742v1
by a distance b ≫ Lϕ and can therefore be considered as independent (however the connecting
wires will affect the AAS harmonics). In the second case rings are in contact and harmonics can
involve trajectories winding around several neighbouring rings. In section 3, we will see that
when electron-electron interaction is the dominant process for decoherence, eq. (1) cannot be
used even in the regime L ≪ Lϕ.
2 Nonlocality of quantum transport in chain of rings
We consider an array of rings all pierced by the same flux φ. The n-th harmonic of the WL
correction at a given point x of a network can be expressed as
∆σn(x) = −
dtPn(x, x; t) e−t/τϕ (2)
where D is the diffusion constant and τϕ = L
ϕ/D the phase coherence time. The factor 2 stands
for spin degeneracy. Pn(x, x; t) is the probability that a particle diffusing into the network
comes back to its initial point x in a time t, after having encircled a flux nφ. For example,
in an isolated ring Pn(x, x; t) = 1√
−(nL)2/4Dt which immediatly gives eq. (1). Except in
translation invariant systems, ∆σn(x) depends on x and expression (2) must be averaged over
the network in a proper way described in Ref. 7.
A ring with two arms.– The case of a ring connected to two arms has been studied in detail
in Ref. 8 where it has been shown that Pn(x, x; t) ≃
2(Dt)3/4
(Dt)1/4
) for time scales t ≫ L2/D
with x inside the ring (the precise form of the dimensionless function Ψ(ξ) is inessential for the
present discussion). Compared with the isolated ring case, where the typical number of winding
scales with time as nt ∼ t1/2, diffusion around the ring is slowed down as nt ∼ t1/4 due to the
time spent in the arms. As a consequence the harmonics of the conductance of a ring are given
by 8 : ∆σn ∝ Lϕ3/2e−|n|
2L/Lϕ (note that the scaling n ∼ L1/2ϕ is analogous to the scaling of
winding with time nt ∼ t1/4 since Lϕ ∼ t1/2).
The chain of distant rings.– The same argument holds for the chain of rings separated by
a distance b ≫ Lϕ (figure 1.a). In this case, averaging properly ∆σn(x) inside the chain of Nr
rings, one finds that the harmonics of the dimensionless conductance read :
∆gn ≃ −
1/2Lϕ
2 [(Nr + 1)b]2
2L/Lϕ for b ≫ Lϕ ≫ L (3)
The chain of attached rings.– If we now consider the network of figure 1.b, we can show
that the probability reads 9 Pn(x, x; t) ≃ L8πDte
−(nL)2/4Dt for t ≫ L2/D. The AAS harmonics
are given in this case by 9 :
∆gn ≃ −
[ln(2Lϕ/|n|L) + bn] for Lϕ/L ≫ |n| (4)
−|n|L/Lϕ
|n|L/Lϕ
for |n| ≫ Lϕ/L ≫ 1 (5)
where bn depends weakly on n (b∞ = −C, the Euler constant).
3 Decoherence due to electron-electron interaction
The above results rely on the fact that, in eq. (2), the long times have been cut off with an
exponential damping e−t/τϕ . However it has been shown recently that this simple modelization
does not account correctly for the decoherence due to electron-electron interaction, which is the
dominant one at low temperature a. In this case, an alternative description was proposed by
Al’tshuler, Aronov & Khmel’nitskii (AAK) 12 but it is only recently that the consequences for
AAS oscillations have been understood13,14,15.
The model of AAK.– The length scale characterizing the efficiency of electron-electron inter-
action to suppress phase coherence in wires is known as the Nyquist length LN = (ν0D
2/T )1/3
where ν0 is the density of states, D the diffusion constant and T the temperature (~ = kB = 1).
In the model of AAK, the random phase accumulated by an electron moving in the fluctuat-
ing electric potential due to other electrons is included in the calculation of the WL. The pair
of reversed interfering trajectories picks a phase eiΦ[C], where C designates a closed diffusive
trajectory, and the harmonics of WL are given by
∆σn ∼ −
〈eiΦ[Cn]〉V = −
〈Φ[Cn]2〉V (6)
The sum runs over all closed trajectories with winding n (a proper formulation of eq. (6) requires
a path integral). Gaussian fluctuations of the electric potential are given by the fluctuation-
dissipation theorem 〈V (~r, t)V (~r ′, t′)〉V = 2e
Tδ(t − t′)Pd(~r,~r ′) (written here in the classical
limit T ≪ ω), where σ0 is the classical Drude conductivity. Pd is solution of the diffusion
equation −∆Pd(~r,~r ′) = δ(~r−~r ′) and therefore depends on the topology of the system. Then14
〈Φ[x(τ)]2〉V =
dτ [Pd(x(τ), x(τ)) − Pd(x(τ), x(t− τ))] (7)
where C ≡ (x(τ), 0 6 τ 6 t |x(0) = x(t)) is a closed diffusive path. The crucial point is that
the simple exponential damping of eq. (2) is replaced in eq. (6) by a functional of the trajectory
〈Φ[Cn]2〉V . Therefore decoherence is now network-dependent and a priori sensitive to the
nature of trajectories (in particular whether they do enclose a magnetic flux or not).
The limit LN ≪ L.– The model described above was applied to the case of a single
ring 13,14,15. The result for an isolated ring is relevant to describe arrays of rings in the limit
LN ≪ L where winding trajectories hardly exit from a ring, which makes rings independent
from each other. For the chain of distant rings (figure 1.a) we have
∆gn ∼ −
[(Nr + 1)b]2
−|n|π
(L/LN )
3/2 ∼
−nL3/2T 1/2
T 1/3
for LN ≪ L ≪ b (8)
(for the case of the chain of rings in contact (figure 1.b), (Nr + 1)b in the denominator is
replaced by NrL/4). Whereas the time characterizing efficiency of electron-electron interaction
to suppress phase coherence in a wire is the Nyquist time τN = L
N/D ∝ T−2/3, it was shown in
Refs. 13,14 that the behaviour (8) is related to a new time scale characterizing decoherence for
winding trajectories : τc = τ
L ∝ T
−1, where τL = L
2/D is the Thouless time of the ring.
The chain of distant rings.– If we consider a ring connected to long arms, winding trajectories
spend most of the time in the arms8 and the decoherence mostly occurs in the arms. Therefore
decoherence occurs on a time scale τN , like in a wire. The function 〈eiΦ[C]〉V for a wire was
a The exponential damping gives the correct shape of a MC of a wire 10,11 with Lϕ →
2LN (see below
for definition of LN ), however this simple substitution gives an incorrect result for AAS harmonics as explained
below.
studied in Ref. 16. Using this result and the winding properties recalled in section 2 leads to 14
∆gn ∼ −
[(Nr + 1)b]2
for n2 ≪ LN/L (9)
[(Nr + 1)b]
)7/12
−κ2|n|
L/LN ∼ e
−nL1/2T 1/6
T 11/36
for n2 ≫ LN/L , (10)
where κ2 =
2|u1|1/4 ≃ 1.421.
The chain of attached rings.– In this case, the nature of decoherence was shown to be closely
related to the one of a wire since diffusion along the chain is reminiscent of a 1d diffusion and
again occurs on time scale τN
∆gn ≃ −
ln(LN/|n|L) + cste for |n| ≪ LN/L (11)
≃ − 1
Nr|u1|3/2
−κ3|n|L/LN ∼ e−nLT 1/3 for |n| ≫ LN/L (12)
where κ3 = 2
−1/3|u1|1/2 ≃ 0.801.
4 Conclusion
We have considered networks of connected rings, made of weakly disordered wires. We have first
shown that geometrical effects can strongly modify the exponential behaviour of AAS harmonics
well-known for an isolated ring, since trajectories can now explore the network around each
ring. In the second part we have shown that decoherence due to electron-electron interaction
is sensitive to geometry, a second reason that modifies the simple AAS result. An interesting
experiment would be to compare precisely AAS oscillations for the two networks of figure 1 in
the low temperature regime LN ≫ L.
References
1. B. L. Al’tshuler and A. G. Aronov, JETP Lett. 33(10), 499 (1981).
2. B. L. Al’tshuler, A. G. Aronov, and B. Z. Spivak, JETP Lett. 33(2), 94 (1981).
3. B. Pannetier, J. Chaussy, R. Rammal, and P. Gandit, Phys. Rev. B 31(5), 3209 (1985).
4. G. J. Dolan, J. C. Licini, and D. J. Bishop, Phys. Rev. Lett. 56(14), 1493 (1986).
5. M. Ferrier, L. Angers, A. C. H. Rowe, S. Guéron, H. Bouchiat, C. Texier, G. Montambaux,
and D. Mailly, Phys. Rev. Lett. 93, 246804 (2004).
6. F. Schopfer, F. Mallet, D. Mailly, C. Texier, G. Montambaux, L. Saminadayar, and
C. Bäuerle, Phys. Rev. Lett. 98, 026807 (2007).
7. C. Texier and G. Montambaux, Phys. Rev. Lett. 92, 186801 (2004).
8. C. Texier and G. Montambaux, J. Phys. A: Math. Gen. 38, 3455–3471 (2005).
9. C. Texier and G. Montambaux, in preparation (2007).
10. F. Pierre, A. B. Gougam, A. Anthore, H. Pothier, D. Esteve, and N. O. Birge, Phys.
Rev. B 68, 085413 (2003).
11. É. Akkermans and G. Montambaux, Physique mésoscopique des électrons et des photons,
EDP Sciences, CNRS éditions, 2004. Mesoscopic physics of electrons and photons, Cam-
bridge University Press, 2007.
12. B. L. Altshuler, A. G. Aronov, and D. E. Khmelnitsky, J. Phys. C: Solid St. Phys. 15,
7367 (1982).
13. T. Ludwig and A. D. Mirlin, Phys. Rev. B 69, 193306 (2004).
14. C. Texier and G. Montambaux, Phys. Rev. B 72, 115327 (2005) ; ibid 74, 209902(E)
(2006).
15. C. Texier and G. Montambaux, Comment on Ref. 13, submitted (2007).
16. G. Montambaux and E. Akkermans, Phys. Rev. Lett. 95, 016403 (2005).
	Introduction
	Nonlocality of quantum transport in chain of rings
	Decoherence due to electron-electron interaction
	Conclusion
