VOLUME 87, NUMBER 18

29 OCTOBER 2001

PHYSICAL REVIEW LETTERS

Interaction-Induced Restoration of Phase Coherence
A. A. Clerk, P. W. Brouwer, and V. Ambegaokar
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853(Received 15 May 2001; published 10 October 2001)
We study the conductance of a quantum “T junction” coupled to two electron reservoirs and a quantum dot. In the absence of electron-electron interactions, the conductance g is sensitive to interference
between trajectories which enter the dot and those which bypass it. We show that including an intradot
charging interaction has a marked influence: it can enforce a coherent response from the dot at temperatures much larger than the single-particle level spacing D. The result is large oscillations of g as afunction of the voltage applied to a gate that is capacitively coupled to the dot. Without interactions, theconductance has only a weak interference signature when T . D.
DOI: 10.1103/PhysRevLett.87.186801

How do interactions affect the phase coherence of electrons traveling through a quantum dot? This question is ofinterest both because of the fundamental issues it raises andbecause of its relevance to recent experiments. Groundbreaking experiments in which a Coulomb blockaded
quantum dot is embedded in an Aharonov-Bohm ring [1]have demonstrated that transport through such dots is atleast partially coherent, despite strong interactions. Complementary studies which observed Fano resonances in the
conductance of such dots also substantiate this conclusion[2], while the observation of weak localization and conductance fluctuations has demonstrated coherence in open (i.e.,
not Coulomb blockaded) dots [3]. In general, electronelectron interactions are expected to degrade the coherenceof transport through a dot — interference phenomena, suchas the amplitude of Aharonov-Bohm oscillations in the
conductance are suppressed compared to the noninteracting case [4]. In this Letter we study a system in which the
opposite phenomenon occurs: the presence of a charging
interaction in a quantum dot significantly enhances interference phenomena compared to the situation without
interactions [5].
Motivated by the experiments mentioned above, we consider a T-junction system which is sensitive to the coherence of electrons reflected from a quantum dot. The T
junction consists of three coincident single-mode quantum point contacts coupled to source and drain reservoirs
and to a quantum dot (Fig. 1). Without interactions, thesource-drain conductance g is sensitive to constructive ordestructive interference between trajectories which bypass
the dot, and those which enter it. If electron motion in the
dot is fully coherent, g (in units of e 2 兾h兲 is [6]µ∂Zdfsin2 关d 1 a共´兲兴 .
(1)g 苷 gmax d´ 2
d´

PACS numbers: 73.21. – b, 73.23.Hk, 73.40.Gk

S of the T junction without dot [8]. This matrix changes
appreciably only over energies comparable to the Fermi
energy EF , and can be treated as constant over the smallerscales we focus on. At zero temperature, the conductance(1) exhibits full constructive and destructive interference
as a function of the phases d and a, resulting in oscillations of g between 0 and its maximum value gmax . Whiled is a property of the T junction and cannot be tuned, adepends on the dot and can be varied, e.g., by changing
EF . The typical energy scale for variations of a is D, thesingle-particle level spacing of the dot.
In this Letter, we focus on the temperature regimeT ¿ D. Here the signatures of interference in Eq. (1)are washed out by thermal smearing, resulting in g 苷
1
2 gmax 共1 2 jrj cos2d兲. We ask how this now changes
when the effects of intradot electron-electron interactions
are included. We consider the capacitive interaction
HC 苷 EC 共ndot 2 N 兲2 ,(2)

where EC ø EF is the charging energy, ndot the electron
number on the dot, and N the dimensionless voltage ofa gate electrode capacitively coupled to the dot. Although
we consider T ¿ D, we also require T ø EC , so thatcharged dot excitations are suppressed. Our main result isthat in this regime transport through the T junction is more
coherent with interactions than without — electrons scatter from the dot with a well defined N -dependent phase,resulting in interference and hence an N dependent conductance. The origin of the resulting oscillations is entirely

1

|r|
3

2

(a)

(b)

Here d is the transmission phase shift associated with direct trajectories bypassing the dot (Fig. 1a), a共´兲 is thephase shift for scattering from the dot through an effective
tunnel junction with reflection probability jrj2 (Fig. 1b)[7], and f is the Fermi function. The parameters d, jrj,
and gmax are determined by the 3 3 3 scattering matrix

FIG. 1. (a) Schematic of the T-junction plus dot system, showing a direct trajectory; such scattering events are described by
the phase d. The effective tunnel junction in the entrance to thedot has a reflection probability jrj2 . (b) Same as (a), showing ascattering event which involves the dot.

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PHYSICAL REVIEW LETTERS

different from that of standard Coulomb blockade oscillations, as there is no “blockade” here — electrons travelingfrom source to drain are not forced to pass through thedot. We find this result remarkable, as one usually expects
that electron-electron interactions degrade, rather than enhance, coherence, due to the possibility of creating lowenergy particle-hole excitations in the dot.
The underlying reason for this enhanced coherence isa subtle consequence of the charging interaction first discussed by Matveev [9]. Using the convention
N 苷 N0 1 1兾2 1 x,

jxj # 1兾2 ,(3)where N0 is an integer, for T ø EC only the states ndot 苷
N0 and ndot 苷 N0 1 1 are dynamically significant. Thesestates may be regarded as being the # and " states, respectively, of a fictitious impurity spin. If we also assign lead (dot) electrons a fictitious " 共#兲 spin, then thetunneling Hamiltonian between lead and dot takes theform of spin-flip scattering off an impurity. In this way,
the Coulomb blockade problem can be mapped onto ananisotropic Kondo model; for spinless electrons [10] thisis a single-channel Kondo (1CK) model, while with spin
it is a two-channel Kondo (2CK) model. In this analogy,
jtj plays the role of the dimensionless exchange constantJr0 , and x the role of a local impurity magnetic field. The
coherent reflection from the dot results from this effective
Kondo physics, as we will now discuss.
For a quantitative description, we write the Hamiltonianof the T junction
P and dot as Hy 苷 HD 1 HL 1 HS 1 HC ,dbs is the Hamiltonian ofwhere HD 苷 ab,s Hab das
P RPy
the closed dot, HL 苷 j苷1,2 s dk ´共k兲cjs 共k兲cjs 共k兲
is the kinetic energy of electrons in leads 1 and 2, and HC
is given in Eq. (2). Scattering in the T junction is described
"
by
2 Z
X
X Z
y
dk
dk 0 Wij cis 共k兲cjs 共k 0 兲HS 苷s,i苷1,2j苷1#
Xy关Wi3cis 共k兲das 1 H.c.兴
1aXy
W33 dbs das ,
(4)
1
s,a,b

where the 3 3 3 Hermitian matrix W describes a potential
corresponding to the scattering matrix S共EFP兲. The dotyelectron number ndot in Eq. (2) reads ndot 苷 dasdas .
We start with the case of a weakly coupled dot 共jtj2 ø
1兲. Using an approach similar to that of Ref. [11], we express the source-drain conductance in terms of the singleR
particle retarded Green function Gab
共v兲 of the dot. As
the dot is coupled to only a single point contact, and as
G R is diagonal in spin, one can obtain an exact expression
involving only G R evaluated at the contact [6]:
XZ
df R
g
G
d´
G 共´兲 , (5)苷 sin2 d 1 Ime2id
gmax
4
de s
s
1
where G 苷 2p jtj2 D. In the regime of weak tunneling and
T . G, it is possible to do a lowest order calculation in jtj2
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29 OCTOBER 2001

by using GR for an uncoupled dot, which can be obtained
exactly [12]. Averaging over fluctuations of the dot wave
functions, and treating dot occupation factors in the same
way as the rate-equations approach [13], we find to lowest
order in jtj2 , when D ø T ø EC ,∑∏jtj2
EC x兾T1g
1
2 cos共2d兲
2
苷
gmax
2
2
4 sinh共2EC x兾T兲
∏
∑ 2
jtj
Y共x兲 .
(6)
1 sin共2d兲
8Near resonance jxj ø 1, Y共x兲 is approximately
∏
∑
2
2EC 1
Y 共x兲 苷tanh共EC x兾T兲 log min
,
.
p
T jxj

(7)

The first jtj2 correction to the conductance in Eq. (6) is
proportional to ImG R , and is identical to the conductance through a Coulomb blockaded dot coupled to twoleads when D ø T ø EC [13]. This term is always
small 共⬃jtj2 ø 1兲. In contrast, the second correction
term to the conductance, arising from ReG R , gives rise
to a low-temperature logarithmic divergence near resonance. Its origin is a partial cancellation between electron-like processes 共ndot 苷 N0 ! N0 1 1兲 and holelike
processes 共ndot 苷 N0 1 1 ! N0 兲, which is identical to
how a logarithmic divergence arises in the conventional
Kondo problem. This is not surprising, given the analogy
already discussed. The logarithm in Eq. (7) is cut off byx, consistent with x playing the role of a magnetic field in
the Kondo analogy.
To summarize, we see that the lowest order in jtj
conductance calculation indicates an instability which
enhances the N dependence of g at low temperatures.
As the only way an N dependence can arise in the Tjunction geometry is via interference, this breakdown suggests an enhancement of coherent scattering from the dot.
In terms of the Kondo analogy, the failure of perturbation
theory results from the instability of the weak-coupling
fixed point. The effective Kondo temperature whichcharacterizes this instability is TK ⬃ EC e2c兾jtj (with c aconstant) [9], consistent with the fact that jtj is analogous
to a dimensionless exchange coupling.
To investigate the regime T, EC x # TK , where Eq. (6)fails and Kondo physics becomes dominant, we now
present results of a calculation for the opposite situationof a strongly coupled quantum dot 共jtj ⯝ 1兲. For strong
coupling, TK ! EC , and the condition T , EC ensuresthat we will be in a regime dominated by Kondo physics
for all values of N . To deal with a strongly coupled dot
at D ø T ø EC , we use the approach of Flensberg [14]and Matveev [15–17]. In this approach the limit D ! 0
is taken, and electron dynamics near the T junction is
described using a one-dimensional model for each point
contact. The interaction is treated exactly using bosonization, while the effects of weak backscattering 共jrj ø 1兲are dealt with perturbatively [6]. In what follows, we
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VOLUME 87, NUMBER 18

PHYSICAL REVIEW LETTERS

discuss the case of spinless electrons and electrons withspin separately.
Without spin [10], our system corresponds to the singlechannel Kondo model, which is well known to have a
Fermi liquid (FL) ground state in which the magnetic impurity acts as a potential scatterer. We thus expect the
T , TK ⬃ EC properties of the open dot system to also
conform to a Fermi liquid state. Indeed, a rigorous calculation gives to order jrj2 :g 苷 gmax sin2 共d 1 pndot 兲 1 O共T 兾EC 兲2 ,

(8)

where
ndot 苷 N 2 共eC jrj兾p兲 sin2pN
1 l2 共eC jrj兾p兲2 sin4pN 1 O共jrj兲3 .

(9)Here C is Euler’s constant, and l2 ⯝ 1.9. Equation (8)indicates that despite being at T ¿ D, a regime where
the noninteracting system is essentially incoherent, reflection from the interacting dot is fully coherent — as N is
tuned, g exhibits full constructive and destructive interference. The scattering phase shift a 苷 pndot obeys theFriedel sum rule [18], as expected for the FL ground stateof the single-channel Kondo model. Equation (8) confirmsthat in the spinless case the breakdown of perturbation theory in Eq. (6) indeed signals coherent scattering from thedot. The fact that a Fermi liquid picture holds in the spinless case (near jrj 苷 0) was first noted by Aleiner andGlazman [19].
A heuristic phase diagram describing the N dependence of the conductance for a fixed temperature D ø
T , EC is given in Fig. 2a. For small jrj, T , TK for
all N , and the coherent expression of Eq. (8) holds for
all N (i.e., one is always in region I). At jrj ⯝ 0,
ndot ⯝ N , and g共N 兲 has a sinusoidal form, while, forlarger jrj, ndot will change rapidly by 1 near resonance[15], implying a corresponding rapid change in the phasea by p, and hence a Fano-type line shape [20]. As we approach the weak-coupling regime 共jrj ! 1兲, we will have
TK , EC , meaning that this Kondo induced coherence will
occur only at sufficiently low temperatures T , TK andclose to resonance, jxj , TK 兾EC . We still expect a narrow Fano line shape in this regime, as all the interesting
phase behavior occurs near resonance. For T $ TK , temperature cuts off scaling to the strong-coupling fixed point,and consequently there will be no enhancement of coherent scattering; the N dependence of g will remain weak,
being described by Eq. (6).
Even though we are at T ¿ D, including spin changes
the behavior of the stub considerably. The analogy is now
to a two-channel Kondo model (the two spin projections
act as the two conserved channels), which is markedly
different from the one-channel case. At zero magnetic field(i.e., x 苷 0 in our system), the low-temperature propertiesof the 2CK model are described by a non-Fermi liquid(NFL) fixed point which corresponds to a dimensionlessexchange constant of order unity (i.e., jtj ⯝ 1). A nonzero
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29 OCTOBER 2001

FIG. 2. Heuristic phase diagrams for the conductance g of the
T junction, for a fixed T ø EC , without spin (a) and with spin(b). jrj2 is the reflection probability from the dot entrance, and x
is the dimensionless gate voltage (x 苷 0 implies charge degeneracy); cf. Eq. (3). In regions I and II, infrared Kondo fixed points
determine the physics: the solid line indicates EC x 苷 TK 共jrj兲,and the dashed line in (b) indicates EC x 苷 Gc 共jrj兲; see text.
Scattering from the dot is mainly coherent in region I (implying
g depends strongly on x here), whereas it is mainly incoherent
in II and in the unlabeled region outside I. The dot-dashed line
indicates T 苷 TK 共jrj兲.

magnetic field (i.e., jxj . 0) destroys the stability of this
fixed point, and the system flows towards an alternate FL
fixed point. Both these fixed points have an impact on the
conductance, as we now demonstrate.
Perturbation theory in jrj for T ø EC ⬃ TK yields thefollowing form for the conductance:g苷gmax
关1 2 x cos共2d 1 2a兲兴 ,2(10)where, defining Gc 苷 2eC p 22 jrj2 EC sin2共px兲,sµ ∂3兾2Gc
Gc
1O,x共x, T 兲 苷 c1
T
T
µ
∂
p 1
1 x 2 u共x兲 1 O共jrj兲2 .
a共x, T 兲 苷
2 2

(11)

(12)

Here c1 ⯝ 1.8, and we have used Eq. (3) for the gate voltage N ; it follows that g is periodic in N with period 1.
The order jrj2 correction to a is proportional to sin2px,
and diverges only logarithmically at low T.
The first term of Eq. (10) can be interpreted as an incoherent contribution to the conductance, while the second
term represents an interference contribution. At jrj 苷 0
only the former contributes, thus agreeing with what was
found for the noninteracting system, but in stark contrast
to the interacting spinless case; cf. Eq. (8). The complete
incoherence at jrj 苷 0 corresponds to the vanishing probability for single-particle scattering at the NFL fixed pointof the 2CK model [21]. Equivalently, one can think offluctuations in the dot spin as suppressing a coherent
response in this temperature regime [22]. For nonzero
jrj, the second term in Eq. (10) also contributes. This
term has an interference form, with a well-defined,
x-dependent scattering phase a associated with thedot. The weight x of this term is zero on resonance共x 苷 0兲, and grows at low temperatures when offresonance 共x ﬁ 0兲. These features can be understoodwithin the 2CK analogy. A nonzero x makes the NFL
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PHYSICAL REVIEW LETTERS

fixed point unstable, and the resulting renormalizationgroup flow at low temperatures is towards the FL fixed
point, which has a well-defined phase shift for scatteringfrom the dot [23]. This flow manifests itself here as asmall coherent term in the conductance which grows at
low temperatures. The flow is parametrized in Eq. (10) by
the function x共x, T兲; we expect x ! 1 in the vicinity of
the FL fixed point [16].
Note that the scattering phase a in Eq. (12) is not simply
proportional to ndot . (At jrj 苷 0, ndot 苷 N [15].) Near
the FL fixed point [i.e., T ø Gc 共x兲], a can be obtained
within the Kondo analogy using Fermi liquid arguments[24]. First, note that 共ndot 2 1兾2 2 N0 兲 is equivalent to
具Sz 典, the moment of the Kondo impurity spin. This moment will be equal to the bare moment of the impurity plus
a quasiparticle contribution, which may be written in terms
d 共x兲
1 d 共x兲1of phase shifts: 具Sz 典 苷 2 sgn共x兲 1 2 3 2 共 "p 2 #p 兲.
The factor of 2 corresponds to the two equivalent channelsof the model. Finally, as the impurity spin has zero charge,
one has d" 苷 2d# . In the Kondo analogy, " is associatedwith electrons in the lead. Equating a with d" then yieldsp关ndot 2 N0 2 u共x兲兴
关mod p兴 ,(13)a苷2which agrees with Eq. (12) [25].
A heuristic phase diagram is shown in Fig. 2b for a fixed
temperature D ø T ø EC . For small jrj, Gc 共x兲 , T
for all x, and one is always in region II: the incoherent
term in Eq. (10) dominates, and g共x兲 exhibits only weak
oscillations. These will grow as jrj is increased or as T
is lowered, following Eq. (10). For sufficiently large jrj(or low T), tuning x can take one from region II to region I, with the crossover occurring at Gc 共x兲 苷 T (dashedline in Fig. 2b). Near resonance (in II), g is still given by
Eq. (10), but away from resonance (in I), it is given by
the coherent expression g 苷 gmax sin2 共d 1 a兲, where ais given by Eq. (13). We expect large oscillations in g共N 兲
in this regime, with a sharp feature emerging around x 苷 0
as T is lowered due to the rapid jump by p兾2 in a.
The coherence effects discussed here are expected to belargely insensitive to additional sources of dephasing in
the quantum dot (e.g., from external sources or from electron-electron interaction terms we neglect) if jtj (and hence
TK ) is sufficiently large. For strong coupling 共jtj ! 1兲, thetime an electron effectively spends in the dot before beingreflected is ⬃h̄兾EC , which is much shorter than typical dephasing times [3]. Thus, electrons should still
scatter coherently from the dot in this regime. Note thatthe model of Ref. [15] used for the strongly coupled dot already assumes that electron motion in the dot is completely
incoherent.
Finally, our results may have relevance to the experiments in Ref. [1], as they indicate that the relation between ndot and the scattering phase from an interacting dot
may be significantly different than that expected for a noninteracting dot.
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29 OCTOBER 2001

We thank L. I. Glazman, C. M. Marcus, K. Matveev,
X. Waintal and X. G. Wen for discussions. This work was
supported by the NSF under Grants No. DMR-0086509
and No. DMR-9805613, by the Sloan Foundation, by the
Olin Foundation, and by the Cornell Center for Materials
Research.

[1] A. Yacoby et al., Phys. Rev. Lett. 74, 4047 (1995); E. Bukset al., Phys. Rev. Lett. 77, 4664 (1996); R. Schuster et al.,Nature (London) 385, 417 (1997); Y. Ji et al., Science 290,
779 (2000).
[2] J. Göres et al., Phys. Rev. B 62, 2188 (2000).
[3] For a review, see C. M. Marcus et al., Chaos Solitons
Fractals 8, 1261 (1997).
[4] Y. Imry, An Introduction to Mesoscopic Physics (Oxford
University Press, Oxford, 1997).
[5] An enhancement of weak localization and conductancefluctuations in open dots is predicted in P. W. Brouwer and
I. L. Aleiner, Phys. Rev. Lett. 82, 390 (1999).
[6] A. A. Clerk, Ph.D. thesis, Cornell University, 2001
(unpublished).
[7] We have e 2ia 苷 e2ia0 共1 1 jrje22ia0 兲兾共1 1 jrje2ia0 兲,
where a0 is the scattering phase of the dot when jrj 苷 0.
[8] Using the labeling in Fig. 1a, one has jrj2 苷 jS33 j2 ⬅ 1 2
jtj2 , gmax 苷 8共jS31 S32 j2 兲兾jtj4 , and d 苷 arg共S31 S32 兲. For
spinless electrons, gmax 苷 4共jS31 S32 j2 兲兾jtj4 .
[9] K. A. Matveev, Zh. Eksp. Teor. Fiz. 99, 1598 (1991) [Sov.
Phys. JETP 72, 892 (1991)].
[10] The case of spinless electrons can be realized experimentally by applying a large in-plane magnetic field to the Tjunction, allowing the transmission of only one spin direction into the dot.
[11] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512
(1992).
[12] See, e.g., G. Hackenbroich, Phys. Rep. 6, 464 (2001).
[13] C. W. J. Beenakker, Phys. Rev. B 44, 1646 (1991).
[14] K. Flensberg, Phys. Rev. B 48, 11 156 (1993).
[15] K. A. Matveev, Phys. Rev. B 51, 1743 (1995).
[16] For a review, see I. L. Aleiner et al., cond-mat/0103008.
[17] An alternate approach to strong coupling is given in
H. Schoeller and G. Schön, Phys. Rev. B 50, 18 436(1994).
[18] J. S. Langer and V. Ambegaokar, Phys. Rev. 121, 1090
(1961).
[19] I. L. Aleiner and L. I. Glazman, Phys. Rev. B 57, 9608
(1998).
[20] J. A. Simpson and U. Fano, Phys. Rev. Lett. 11, 158 (1963).
[21] A. W. W. Ludwig and I. Affleck, Phys. Rev. Lett. 67, 3160
(1991).
[22] Dephasing due to spin fluctuations in the regime T ø D
is discussed in J. König and Y. Gefen, Phys. Rev. Lett. 86,3855 (2001).
[23] I. Affleck et al., Phys. Rev. B 45, 7918 (1992).
[24] P. Nozieres, J. Low Temp. Phys. 17, 31 (1974).
[25] Note that the weak-coupling result of Eq. (6) can be written
in the form of Eq. (10), with the phase a共N 兲 being given
again by Eq. (13).

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