Thesis

Transport Phenomena and Quantum Phase Transitionsin Mesoscopic Systems

Yuji Hamamoto
Department of Physics, University of Tokyo
December, 2009

Abstract
In this thesis, we study theoretically transport properties of various mesoscopic
systems. Our main goal is to reveal dynamic response of a mesoscopic capacitor
composed of a quantum dot and a lead, where a unique quantization phenomenonof charge relaxation resistance has been experimentally observed recently. In order to perform nonperturbative analyses of mesoscopic systemst, we demonstrate
the path-integral Monte Carlo (PIMC) simulations using the cluster algorithm.
Three target systems are considered. The ﬁrst one is the single impurity problem
in a Tomonaga-Luttinger liquid with spin degeneracy, to which we demonstrated
the performance of the PIMC method. By analyzing the low-temperature behavior of the charge and spin conductances, we identify the phase diagram for anintermediate impurity strength. We compare our results with the previous resultsof the standard perturbative renormalization group for weak and strong impurity
backscatterings. Next we consider a large quantum dot connected to a bulk leadby a single mode point contact. We extend the cluster algorithm so that it takes
into account the charging eﬀect and analyze the Coulomb blockade phenomena in
the quantum dot. By means of the PIMC method, we verify the theoretical prediction that the capacitance diverges logarithmically due to the two-channel Kondo
eﬀect at charge degeneracy points for arbitrary dot-lead connection. Finally, wetackle the dynamic response of a mesoscopic capacitor, where both transport andCoulomb blockade are important for the quantization of the charge relaxation resistance. By means of perturbation theory, renormalization group technique, andthe path-integral Monte Carlo method, the charge relaxation resistance is evaluated in the whole range of dot-lead coupling. We predict that the charge relaxation resistance is universal even in the presence of strong Coulomb blockade forthe Luttinger parameter K > 1/2, while the Kosterlitz-Thouless transition dramatically inﬂuences the universal nature of the charge relaxation resistance forK < 1/2.

1

Acknowledgements
I am deeply grateful to Prof. Takeo Kato whose comments and suggestions wereinnumerably valuable throughout the course of my study. I would also like tothank Takeshi Matsuo who helped me to learn the path-integral Monte Carlo
method. Special thanks go to Ken-Ichiro Imura who taught me the impurity problems in Tomonaga-Luttinger liquids and also gave me warm encouragements. I
am also indebt to Yasuhiro Utsumi whose comments made enormous contribution
to my understanding of Coulomb blockade in open quantum dots. I would liketo give heartful thanks to Prof. Kensuke Kobayashi who introduced me to thefertile ﬁeld in mesoscopic physics, i.e., dynamical transport in mesoscopic systems. I also wish to acknowledge to Prof. Thierry Martin and Thibaut Jonckheere
for their long-term collaborations and invaluable discussions in the course of ourstudy on dynamic response of mesoscopic capacitors. I am sincerely gratefulto Tatsuya Fujii for insightful comments on the renormalization group approach.
My thanks go to all the people of Institute for Solid State Physics, Universityof Tokyo. In particular, I would like to thank my colleagues, Tomohiro Otsuka,
Yasuyuki Kato, Toshihiro Sato, and Shunsuke Kirino, and the members of KatoGroup, Kazuyoshi Yoshimi, Hiroshi Shinaoka, Itofumi Takeuchi, Emi Yukawa,
Eiki Iyoda, and Noritaka Tamura for great discussion on physics and fun time. I
am also indebt to Hiroko Eguchi whose eﬀorts in cumbersome paperwork enabled
me to focus on my study. I would also like to express my gratitude to my family
for their moral support and warm encouragements.
Finally, I acknowledge the ﬁnancial support of the Japan Society for the Promotion of Science (JSPS) that made it possible to complete my thesis. The computation in this thesis was done using the facilities of the Supercomputer Center,
Institute for Solid State Physics, University of Tokyo.

3

Contents
1

Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 ac transport in a mesoscopic capacitor . . . . . . . . . . . . . . .
1.3 Electron interactions in mesoscopic systems . . . . . . . . . . . .
1.3.1 Tomonaga-Luttinger liquid . . . . . . . . . . . . . . . . .
1.3.2 The Coulomb blockade eﬀect . . . . . . . . . . . . . . .
1.3.3 The Kondo eﬀect . . . . . . . . . . . . . . . . . . . . . .
1.4 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
7
9
11
11
13
16
18

2

Path-Integral Monte Carlo Method
2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Single impurity problem in a spinless TLL . . . . . . . . . . . . .
2.2.1 Local update in the Fourier space . . . . . . . . . . . . .
2.2.2 Global update based on the cluster algorithm . . . . . . .
2.3 Single impurity problem in a spinful TLL . . . . . . . . . . . . .
2.3.1 Perturbative renormalization group . . . . . . . . . . . .
2.3.2 The path-integral Monte Carlo method . . . . . . . . . . .
2.3.3 Local update . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Global update . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Linear conductance . . . . . . . . . . . . . . . . . . . . .
2.3.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.7 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . .
2.3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19
19
21
21
22
22
23
26
26
26
28
28
32
32
34

3

Coulomb Blockade in an Open Quantum Dot
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The path-integral Monte Carlo method . . . . . . . . . . . . . . .
3.3.1 Local update . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Global update . . . . . . . . . . . . . . . . . . . . . . . .

35
35
36
38
38
39

5

CONTENTS

6
3.4

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Coulomb oscillation of the dot charge . . . . . . . . . . .
3.4.2 Logarithmic divergence of the capacitance . . . . . . . . .
3.4.3 Eﬀect of channel anisotropy . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40
40
41
45
46

4 Charge Relaxation Resistance
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Universal charge relaxation resistance . . . . . . . . . . . . . . .
4.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Bosonization . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Linear admittance . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Mean ﬁeld theory . . . . . . . . . . . . . . . . . . . . . .
4.4 The path-integral Monte Carlo method . . . . . . . . . . . . . . .
4.4.1 Local update . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Global update . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Weak barrier limit: Perturbation theory . . . . . . . . . .
4.5.2 Strong barrier limit: Renormalization group . . . . . . . .
4.5.3 Intermediate region: The PIMC method . . . . . . . . . .
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47
47
48
51
51
53
54
55
56
56
57
57
62
65
69

A Chiral Luttinger liquid in a mesoscopic capacitor
A.1 Eﬀective action . . . . . . . . . . . . . . . . . . . . . . . . . . .

75
76

3.5

Chapter 1
Introduction
1.1

Motivation

Recent nanofabrication techniques allow researchers to conﬁne electrons in lowdimensional structures with nano scale such as quantum wires and quantum dots.
These nanostructures are called mesoscopic systems and have been studied energetically both by experimentalists and theorists ever since 1980’s. In early studies,
quantum coherence of electrons and quantum-classical crossover at intermediate
length scale were of central interest [1, 2, 3]. Nowadays, one can design nanostructures as needed and actively control the quantum-mechanical states of inside
electrons. Indeed there are a number of experiments which demonstrate coherent manipulation of electronic states in various systems [4]. Reﬂecting the highcontrollability, there is growing interest in the application of the nanostructurs toquantum information in recent years [5].
Among the nanostructures experimentally realizable, quantum dot systems
provide attractive playgrounds for researchers, since they have several advantages
in application to quantum information technologies. For example, the quantum
bit (qubit) is an essential building block for quantum computation and can be realized as superposition of, e.g., diﬀerent charge (or spin) states in a quantum dot.
One can manipulate dynamically the qubits thus obtained by controlling parameters such as a gate voltage, a source-drain bias, and an external magnetic ﬁeld.
Recently one-qubit [6] and two-qubit [7] operations have been demonstrated indouble quantum dot systems. From the viewpoint of quantum communication,on the other hand, the coherent single-electron source fabricated in a quantum
dot system [8] is another promissing candidate for the application. The creationof a single-electron source is expected to be simpler than a single-photon source
because of Fermi statistics and Coulomb interaction.
Toward future developments in rapid manipulation of quantum dot systems, it
7

8

CHAPTER 1. INTRODUCTION

is highly demanded to reveal dynamical properties of coherent transport phenomena. Despite the long history of mesoscopic physics, it is only quite recent thatexperiments on dynamical transport in the coherent regime have become possible.
One of the simplest examples is the measurement of GHz-frequency response ofa mesoscopic capacitor, which is composed of a quantum dot and a lead [9]. Surprisingly, dynamical resistance through a quantum point contact (QPC) is quantized at h/2e2 , which is independent of transparency of the QPC. This behavior
is utterly diﬀerent from that for the dc resistance determined by the LandauerBüttiker formula [10, 11, 12], which explicitly depends on the transmission probability. Thus, in the coherent regime, even the simplest case provides a challenging
problem leading to a counterintuitive result.
To understand the dynamical properties of the mesoscopic systems, one should
also take into account the fact that transport phenomena in nanostructures are frequently aﬀected by interactions between electrons. For example, in a small quantum dot, the cost of charging energy suppresses electron transmission and ensures
a well-deﬁned electron number in the dot, which is called Coulomb blockade [13].
On the other hand, an interacting electron system conﬁned one-dimensionally is
a Tomonaga-Luttinger liquid at low temperature and exhibits characteristic transport properties [14, 15, 16, 17]. It is also well-known that in a quantum dot system
conduction electrons in the lead and a localized electron in the dot form a spinsinglet ground state below a characteristic temperature, which is called the Kondo
eﬀect. So far, these many-body eﬀects have been investigated mainly in the staticor dc case. However little is known about how electron interactions modify the
dynamical transport in the coherent regime.
Motivated by the circumstances mentioned above, we in this thesis aim to reveal the dynamical properties of a mesoscopic capacitor in the presence of strong
electron interactions. Especially, we are interested in the regime with suﬃciently
large transmission, where electronic transport is expected to be phase-coherent.
In order to investigate the intermediate regime, which is in general diﬃcult totreat analytically, we employ the path-Integral Monte Carlo (PIMC) method. The
PIMC method has become applicable to such an intermediate region thanks to thecluster algorithm, which has been recently developed in the context of resistively
shunted Josephson junctions by Werner and Troyer [18]. In Chap. 3, we extend
their algorithm to an incoherent quantum dot with spinful electrons and analyze
the static property, or the capacitance. In Chap. 4, we apply the PIMC simulation
to dynamic response in a mesoscopic capacitor and display our main results.
In the rest of this chapter, we ﬁrst review theoretical studies of dynamical resistance in the coherent regime, as well as the recent experiment of the ac responses
in a quantum dot in Sec. 1.2. Problems left unsolved on the dynamical responses
in nanostructures are also summarized. In Sec. 1.3, we brieﬂy review the known
many-body eﬀects. Finally, we describe the contents of this thesis in Sec. 1.4.

1.2. AC TRANSPORT IN A MESOSCOPIC CAPACITOR

1.2

9

ac transport in a mesoscopic capacitor

To make the problem clearer, let us ﬁrst review prior studies on the dynamical response of a mesoscopic system. The system we have in mind is a coherent quantum dot, also referred to as a mesosocpic capacitor, depicted in the left panel ofFig. 1.1. The dot is connected to the lead via a M-channel quantum point contact(QPC) and is capacitively coupled to a gate voltage. One way of characterizing
the transport properties of such a system is to apply an oscillating gate voltage
and measuring the resistance through the QPC. One can introduce RC time τRCby analogy to the classical RC circuit. If one denotes the resistance through the
QPC by Rq and the capacitance of the dot by Cµ , the charge relaxation time of
the mesoscopic capacitor reads τRC = 1/RqCµ . It is clear from the circuit equationthat the admittance of the mesoscopic capacitor can be expanded at low-frequency
regime ω  1/τRC as()−1
1
G(ω) = Rq +' −iωCµ + ω2Cµ 2 Rq + O(ω3 ).
(1.1)−iωnCµThe quantum resistance Rq appearing in the quadratic term in ω is referred to as
charge relaxation resistance. In their pioneering works [19, 20], Büttiker et al.
have applied the self-consistent scattering theory to the mesoscopic capacitor andderived a low-frequency expansion similar to Eq. (1.1). The charging eﬀect inthe dot is taken into account through an internal eﬀective potential determined
self-consistently. The charge relaxation resistance thus obtained is represented as∫dE(−d f /dE)Tr[S† (E)∂S(E)/∂E]2h
Rq = 2 ∫,(1.2)
2e { dE(−d f /dE)Tr[S† (E)∂S(E)/∂E]}2
where S denotes a M × M scattering matrix. Surprisingly enough, the charge relaxation resistance for a single channel (M = 1) is quantized at Rq = h/2e2 atFigure 1.1: (left) Schematic ﬁgure of a mesoscopic capacitor. (right) Charge relaxation resistance as a function of the gate voltage on the point contact VG . Taken
from Ref. [9].

10CHAPTER 1. INTRODUCTION

zero temperature. The quantization of the charge relaxation resistance essentiallydiﬀers from the quantized dc resistance in quantum wires and (integer) quantum
Hall edges in two points. The quantized value of Rq per channel equals half aresistance quantum h/2e2 ; clearly, the factor 1/2 in the quantized value has nothing to do with spin degeneracy. More importantly, the quantized resistance isachieved irrespective of transmission through the QPC or the gate voltage, due tocancellation of the scattering matrices at T = 0. These two peculiarities suggest
breakdown of the Kirchhoﬀ’s law in the quantum RC circuit at low frequenciesω  1/τRC in the sense that the impedance of the quantum resistor, i.e., QPC is
not given by its dc resistance.
Although the above theory was presented more than a decade ago, recently
there has been a renewed interest in the quantized charge relaxation resistance owing to the progress in experimental techniques in the ac regime. Gabelli et al. has
experimentally conﬁrmed the theoretical prediction by measuring charge relaxation resistance of a mesoscopic capacitor shown in the left panel of Fig. 1.1 [9].
The mesoscopic capacitor is realized electrostatically in a two-dimensional electron gas (2DEG). A magnetic ﬁeld B = 1.3 T is applied perpendicular to the2DEG, so that integer Hall edge states are realized along the edge of the system.
The QPC voltage VG controls transmission through the QPC and in the present
case only a single mode is transmitted. The gate voltage in the dot oscillates withfrequency ω ∼ 1 GHz, which is suﬃciently small compared to the inverse RC
time 1/τRC ∼ 100 GHz. Thus one can expect that charge relaxation conditionω  1/τRC is guaranteed in this frequency regime. Note that the frequency ωshould be larger than the temperature experimentally achievable ∼ 1 GHz, otherwise the observed quantity would be smeared out by thermal ﬂuctuations. It isclear from the circuit equation Z = Rq + 1/(−iωCµ ) that in experiments Rq can beobtained as the real part of impedance Z. In the right panel of Fig. 1.1, the charge
relaxation resistance, or Re(Z), is shown as a function the QPC gate voltage VG .
The data is roughly equal to h/2e2 in consistency with the theory, at least for large
VG (weak reﬂection). Note that the abrupt increase in Rq for small VG indicates
pinch-oﬀ of the transmission channel.
Stimulated by the experimental realization of the quantized charge relaxation
resistance, Nigg et al. have developed a more realistic theory based on a tunnel
Hamiltonian [21]. In their theory, the charging eﬀect in the quantum dot is treatedwithin the Hartree-Fock approximation. Rq calculated in the low-frequency limitω  1/τRC is again quantized at h/2e2 . The charge relaxation resistance seems to
be robust even in the presence of the charging eﬀect, judging from the results in
Refs. [19, 21]. However, it should be noted that these results are obtained basedon mean-ﬁeld-type approximations and are valid only when the charge ﬂuctuationis small. This suggests that on the degeneracy points, where two neighboring
charge states, e.g., Q = ne and Q = (n + 1)e (with an integer n) are degenerate,

1.3. ELECTRON INTERACTIONS IN MESOSCOPIC SYSTEMS

11

large charge ﬂuctuations may modify the dynamical properties of the mesoscopic
capacitor. Indeed, it has been predicted that in incoherent quantum dots the groundstate on the degeneracy points are drastically aﬀected by the Kondo eﬀect due to
pseudo spin both in the weak and strong coupling limits [22, 23]. Then a question
arises; is the charge relaxation resistance still quantized even in the presence ofthe Kondo physics? This is the topic in Chap. 4 and we see that the Kondo physics
is the key to understanding the dynamic resistance on the degeneracy points.

1.3

Electron interactions in mesoscopic systems

Electrons conﬁned in low-dimensional structures with nano scale are subject tostrong electron interactions. In this section, we give a few examples of interaction
related to the problems in Chap. 3 and Chap. 4, and describe how each interactioninﬂuences the transport phenomena.

1.3.1

Tomonaga-Luttinger liquid

In one-dimensional (1D) electron systems the usual Fermi liquid behavior is destroyed by the interaction between electrons. As a consequence, low-energy excitations in the interacting 1D electron system are density ﬂuctuations of charge andspin modes, and the ground state is called a Tomonaga-Luttinger liquid (TLL) [14,15, 16, 17]. The peculiar behaviors of a TLL are experimentally observed in fractional quantum Hall edges [24, 25, 26] and single-wall carbon nanotubes [27, 28,29]. Conventionally, a TLL is formulated with the bosonization method as follows. For simplicity, we here consider the spinless case. If the dispersion relation
is linearized around the Fermi level at low temperatures, the Hamiltonian for freeelectrons reads∑∑
Hfree = vF
(rk − kF )cr,k † cr,k ,
(1.3)
r=±

k

where c±,k is the annihilation operator for an electron with momentum k moving in
the ±x direction. Due to the two branches of linear dispersion in Eq. (1.3), charge
and current ﬂuctuations behave as bosonic operators. Then, one can express the
Hamiltonian for the interacting 1D electron system in the quadratic form
 ( )2( )2 ∫∂θ 
dx  u ∂φ

 ,
+ uK
H0 =
(1.4)
2π K ∂x∂x
where φ and θ are canonically conjugated bosonic ﬁelds satisfying the commutation relationiπ(1.5)[φ(x), θ(x0 )] = sgn(x − x0 ).
2

CHAPTER 1. INTRODUCTION12u ≡ vF /K is the sound velocity, and K is a positive parameter describing (shortrange) interactions between electrons:
K < 1 repulsiveK = 1 noninteracting
(1.6)



 K > 1 attractive.
Eq. (1.4) is obviously the Hamiltonian for an elastic string, which allows us tocalculate exactly various correlation functions. For example, we can calculate thetwo point correlation function hφ(x1 , τ)φ(x2 , 0)i in the imaginary-time representation to estimate dc conductance between x1 and x2∫
e2 2|ωn | βe2
G(x1 , x2 ) = limdτhφ(x1 , τ)φ(x2 , 0)ieiωn τ = K.
(1.7)
iωn →0 hπh
0Thus the interaction parameter K can be also regarded as dimensionless conductance of a homogeneous TLL. Note that the factor K reﬂects interactions in theleads (x < x1 , x2 < x) not in the wire (x1 < x < x2 ). This can be seen by replacing
the TLL leads (K , 1) with Fermi-liquid leads (K = 1), i.e.,{
KL = 1 x < x1 , x2 < x
K(x) =(1.8)
K W , 1 x1 < x < x2 .
The conductance for the inhomogeneous interactions (1.8) is calculated [30] as
G=

e2
e2
KL =
h
h

(1.9)

and recovers the Landauer’s formula without scatterers.
The transport property of a TLL is dramatically modiﬁed by introducing scatterers such as nonmagnetic impurities or single-mode point contacts [31, 32, 33,34]. For example, if a local scatter with barrier height V0 is introduced at the origin x = 0 [31], the Hamiltonian describing an electron reﬂected by the scatterer
reads
HV = −V0 [ψ+ † (x = 0)ψ− (x = 0) + H.c.] = −V cos[2φ(x = 0)].

(1.10)

where the reﬂection strength V ∝ V0 . Noticing the fact that the total Hamiltonian
H0 + HV is quadratic in φ(x , 0)’s, one can integrate out these ﬁelds away fromthe origin to derive an eﬀective action
∫
1 ∑
2
S =
|ωn ||φ̃(ωn )| − V dτ cos[2φ(τ)],(1.11)πKβ ωn1.3. ELECTRON INTERACTIONS IN MESOSCOPIC SYSTEMS

13

where φ(τ) ≡ φ(x = 0, τ) and φ̃(ωn ) is the Fourier transform of φ(τ). One canalso start from a fractional quantum Hall edge state and derive the same model,
where K is replaced with the ﬁlling factor ν = 1/m with an odd integer m. The
perturbative renormalization group (RG) [35] is a useful technique to analyze theground state of the single-impurity problem (1.11) for small V. By integrating out
high-frequency modes, one obtains the scaling equations
dVn= (1 − n2 K)Vn
(V1 ≡ V),(1.12)
dl
where Vn denotes reﬂection strength associated with the nth-order scattering process, and dl = −dD/D with the bandwidth D. It is clear from Eq. (1.12) that,as the energy is lowered, at least V1 grows if K < 1, while all the Vn ’s decay tozero if K > 1. A similar RG treatment is possible by taking the strong scatterer
limit as the starting point. In this case, two semi-inﬁnite TLLs are connected via a
weak link with tunneling strength t, and the RG equation for nth-order tunnelingprocess reads
(
)
dtn
n2
= 1−
tn
(t1 ≡ t).
(1.13)
dl
K
This equation means that, as the energy is lowered, at least t1 grows if K > 1,
while all the tn ’s decay to zero if K < 1. The results of the RG equations (1.12)
and (1.13) are summarized in the left panel of Fig. 1.2. For repulsive interactions
K < 1, electrons are perfectly reﬂected by the scatterer at T = 0 (insulator phase);for attractive interactions K > 1, electrons are perfectly transmitted through thescatterer at T = 0 (conductor phase).
Although the RG equations (1.12) and (1.13) are applicable only in the limits V → 0 and t → 0, respectively, the above discussion has been veriﬁed in
the nonperturbative regime based on the path-integral Monte Carlo method [36].
Moreover, H0 + HV is equivalent to the boundary sine-Gordon model and therefore
is exactly solvable by means of the thermodynamic Bethe Ansatz [37]. Experiments in the fractional quantum Hall regime at ﬁlling ν = 1/3 have also turned
out to support the above theory as shown in the right panel of Fig. 1.2.

1.3.2

The Coulomb blockade eﬀect

In a quantum dot system with small geometrical capacitance Cg , tunneling ofelectrons between the dot and a lead is in general exponentially suppressed by the
charging energy at low temperatures. This phenomenon is called the Coulomb
blockade. Since the electrostatic energy of the quantum dot system is given by
HC =Q2+ QVg ,
2Cg

(1.14)

14

CHAPTER 1. INTRODUCTION

Figure 1.2: (left) Schematic ﬂow diagram for the single-impurity problem in aninteracting 1D electron system. The ground state is an insulator for repulsive
interactions (K < 1), while it is a perfect conductor for attractive interactions (K >1). (right) Temperature dependence of conductance through a point contact in thefractional quantum Hall regime with ﬁlling ν = 1/3. The scaling behavior G ∝ T 4
is in good agreement with theoretical prediction G ∝ T 2/ν−2 [See Eq. 1.13]. Taken
from Ref. [24].

where Q is the dot charge and Vg is the gate voltage, the charge state in the dot
is controlled by tuning Vg . Especially, when Vg is tuned to a point where twoneighboring charge states Q = en and Q = e(n + 1) are degenerate, the costof the electrostatic energy vanishes. Therefore, one observes periodic peaks of
conductance through the quantum dot as a function of Vg .
Electron transport in the Coulomb blockade regime has been studied theoretically in many literatures, most of which have dealt with weak tunneling of electrons between leads and a quantum dot [38]. Electron transport is then describedby a classical master equation with tunneling rates for various processes, where
quantum coherence of electrons is not maintained during the tunneling events. In
this sense, such a transport is regarded as incoherent, although the transition rate
itself is determined by quantum tunneling through a quantum point contact (QPC)
between a lead and a dot.
If it is possible to suppress backward scattering on the QPC adjusting its gate
voltage, one needs to consider strong quantum ﬂuctuation due to electron transfer
between a lead and a dot. Such a situation, called the strong tunneling regime,
cannot be treated with the conventional theory based on the rate equation. Strong
tunneling has been studied theoretically by Matveev [23] based on a simple modelof a quantum dot coupled to a lead through a point contact as shown in the left
panel of Fig. 1.3. Since electron transport in this system is governed mainly bya single transmission channel near the point contact, Matveev has considered asimple one-dimensional model. For a spinless electron system, the Hamiltonianis written as1.3. ELECTRON INTERACTIONS IN MESOSCOPIC SYSTEMS

H = Hkin + HV + HC
∫ L
Hkin = vF
dx[ψ+ † (x)(i∇ − kF )ψ+ (x) + ψ− † (x)(−i∇ − kF )φ− (x)],
−∞
∫ L
HV =
dxV(x)[ψ+ † (x) + ψ− † (x)][ψ+ (x) + ψ− (x)],−∞∫ L(Q − eNg )2, Q=edx[ψ+ † (x)ψ+ (x) + ψ− † (x)ψ− (x)].
HC =
2Cg
0

15

(1.15)(1.16)(1.17)(1.18)

Here, Hkin describes a kinetic energy for the right-(left-)going electrons denotedby ψ+ (x) [ψ− (x)]. The potential energy term HV describes the eﬀect of the energy
barrier V(x) near the point contact, and HC is the charging energy for a quantum dot with a size L. In this model, short-range interactions are neglected byassuming that they are well screened by other electrons outside the transmission
channel; only the long-range Coulomb interaction is taken into account throughthe charging energy HC . For further analysis, we take a delta-function-type potential (V(x) = V0 δ(x)) at the origin, and apply the bosonization method to thepresent model. The bosonized Hamiltonian for a suﬃciently large quantum dot(L → ∞) is given by ( )2( )2 ∫∂θ 
dx  ∂φu ,Hkin =+u(1.19)
4π∂x∂x
HV = V cos[2φ(x = 0)],1
HC = 2 EC [φ(x = 0) − πNg ]2 ,π

(1.20)(1.21)

where V = V0 D/πvF , and D is the bandwidth. Thus, the bononization approach
has an advantage that the charging energy is easily formulated in terms of bosonicﬁelds.

Figure 1.3: (left) Simple quantum dot system composed a dot and a lead. (right)
Coulomb blockade of the dot charge Q.

CHAPTER 1. INTRODUCTION

16

In Ref. [23], Matveev has studied the behavior of a quantum dot with strong
tunneling by means of the perturbative calculation with respect to the potential
barrier V. His result is summarized as follows. For V = 0, the charge Q = ehφi/π
is proportional to the gate voltage Ng as shown with the straight line in the rightpanel of Fig. 1.3. As V increases gradually from zero, the Vg -Q curve starts to oscillate slightly around the straight line. For larger V, electron transfer is governedby weak tunneling processes, and the well-known step-like dependence called the
Coulomb staircase is recovered. In Chap. 4 of this thesis, the ac response of aquantum dot system with a ﬁnite length L is studied by formulating the eﬀective
action in a similar way based on the bosonization method.
In addition to the spinless case, Matveev has also considered the spin degreeof freedom both in the weak- and strong-tunneling limits and has found that thelow-energy transport properties are governed by the two-channel Kondo eﬀect due
to a pseudo spin [22, 23]. In Chap. 3 of this thesis, we consider the same problem
in the intermediate region by means of the path-integral Monte Carlo method.

1.3.3

The Kondo eﬀect

How localized electrons interact with delocalized electrons is a central question
to many problems in solid-state physics. The simplest manifestation of this situation is the Kondo eﬀect, which occurs when a magnetic impurity is placed in a
metal [39, 40]. Nowadays, it is well known that a spin-singlet state is formed between the localized electron with a spin and conduction electrons near the Fermi
energy, below a characteristic temperature called the Kondo temperature. The formation of the Kondo singlet state was theoretically predicted also for quantum dot
systems [41, 42], and has been examined experimentally through electron transport [43, 44]. The coherent ac transport of quantum dots in presence of the Kondo
eﬀect for real spins is an attracting problem, although it is out of the scope of thisthesis.
The essence of the Kondo eﬀect can be studied with the Kondo Hamiltonian
written as
HK =

∑
k,σ

vF kc†k,σ ck,σ +

)
Jk
J⊥ (
σz (c↑ † c↑ − c↓ † c↓ ) +σ+ c↓ † c↑ + σ− c↑ † c↓ .
42(1.22)

Here, vF is the Fermi velocity, ck,σ is an annihilation operator of a conduction∑
electron, and cσ = L−1/2 k ck,σ is the Wannier operator of the conduction electron
at the impurity site. The conduction electrons interact with the localized spin
S = σ/2, through the exchange couplings, Jk and J⊥ . The Jk term describes thescattering process where the spin polarization is unchanged, while the J⊥ term
describes spin-ﬂip scattering. The Kondo eﬀect for real spins corresponds to the1.3. ELECTRON INTERACTIONS IN MESOSCOPIC SYSTEMS

17

isotropic case (J⊥ = Jk ) due to the SU(2) symmetry, while the charge-Kondo eﬀect
is described by the above Hamiltonian with spin anisotropy (J⊥ , Jk ).
A simple approach to examine the formation of the spin-singlet state in the
Kondo model is the poor-man’s scaling [45]. Here, we review a more sophisticated scaling approach, which gives the same renormalization group equations [46,47]. The partition function for the impurity in the instanton-gas representation isderived from the Kondo model as)2m ∫ β
∫ τ2m
∫ τ2
∞ (
∑
ρJ⊥
ZK =
dτ2m
dτ2m−1 · · ·
dτ1
2τ
c
0
0
0
m=0


2m
∑


τ
−
τ
j
k
 ,
× exp (2 − ε̄) (−1) j+k log(1.23)
τcj>kwhere the condition τ j+1 − τ j > τc with the short-time cutoﬀ τc is assumed. The
phase factor ε̄ is related to Jk as ε̄ = 2ρJk up to the ﬁrst order in Jk , where ρ =(2πvF )−1 is the density of states of conduction electrons. If we change the cutoﬀ
from τc to τc + dτc , i.e., if we carry out a partial integration of high-energy degrees
of freedom such that the the partition function remains invariant, we obtain a setof diﬀerential equations describing the scaling behavior of each parameter. Up tosecond order in ρJk these scaling equations are given bydε̄= (2 − ε̄)(ρJ⊥ )2 ,dldρJ⊥ ε̄= ρJ⊥ ,dl
2

(1.24)(1.25)

where dl = −dD/D with the bandwidth D. The behavior of these equations areunderstood by plotting the scaling trajectories in the ε̄-(ρJ⊥ ) plane as shown in
Fig. 1.4. The trajectories are classiﬁed into two types of ﬂow; one is toward theweak coupling J⊥ → 0, and the other is toward the strong coupling J⊥ → ∞.
The former reaches one of the ferromagnetic Kondo ﬁxed points (line) at zerotemperature, while the latter to the antiferromagnetic Kondo ﬁxed point. In theformer case, the ground state is doublet due to the spin degeneracy, and the spinﬂip process is completely suppressed at zero temperature. In the latter case, theKondo singlet between the localized spin and the conduction electrons is formed
below the Kondo temperature T K ∼ τ−1
c exp(−1/ρJ⊥ ). In Chap. 4, we will seethat the diﬀerence between the ferro- and antiferro-magnetic Kondo ﬁxed points
is essential to the understanding of interaction eﬀect on the dynamic response ofa mesoscopic capacitor.
We note that the above discussion is closely related to the problem of dissipative quantum eﬀects. For the dissipative two-state system, the corresponding

CHAPTER 1. INTRODUCTION

18

Figure 1.4: The scaling trajectories of (ε̄, ρJ⊥ ).

partition function can be written in the form similar to Eq. (1.23), and the localization transition of a quantum Brownian particle subject to a double-well potential
has been discussed theoretically in Refs [48, 49, 50]. Dynamics in this system has
been studied theoretically with the Monte Carlo method [51, 52], the numerical
renormalization group calculation [53, 54], and an analytical approach [55].

1.4 Contents
This thesis is organized as follows. In Chap. 2, we ﬁrst explain in detail thepath-integral Monte Carlo method employed in this thesis. We demonstrate its
performance by applying it to a simple example of the dc transport through an
impurity located in a spinful Tomonaga-Luttinger liquid. In Chap. 3, we extend
the PIMC method to investigate the Coulomb blockade in quantum dot systems.
In Chap. 4, which is the main part of this thesis, we discuss dynamic response ofa mesoscopic capacitor in the presence of electron interactions.

Chapter 2
Path-Integral Monte Carlo Method
In order to study the non-perturbative regions of the models treated in this thesis,
we perform numerical simulations based on the path-integral Monte Carlo method(PIMC). For eﬃcient Monte Carlo sampling, it is important to adopt intelligent
algorithms optimized for each model. In this chapter, we ﬁrst illustrate our update scheme taking the impurity problem in a spinless Tomonaga-Luttinger liquid(TLL) as an example. To demonstrate performance of the PIMC method, we actually apply the PIMC simulation to the impurity problem in a spinful TLL. By
analyzing the low-temperature behavior of the conductances, we identiﬁed four
distinct phases characterized by either perfect transmission or reﬂection of charge
and spin channels for an intermediate scattering strength. We compare our resultswith the previous results of the standard perturbative renormalization group (RG)
for weak and strong impurity backscattering.
This chapter is organized as follows. We explain fundamentals of the PIMC
method in Sec. 2.1, and state details of update schemes of the PIMC simulation
for the impurity problem in a spinless TLL in Sec. 2.2. Actual simulation for an
impurity in spinful TLLs is performed in Sec. 2.3, where we present the zerotemperature phase diagram deduced from our data, and then make a comparison
with the known RG result.

2.1

Fundamentals

In general ﬁeld-theoretic problems, we are interested in the average of some observable O:∫
Dφ O[φ] e−S [φ]hOi = ∫,
(2.1)
Dφ e−S [φ]1920

CHAPTER 2. PATH-INTEGRAL MONTE CARLO METHOD

where S [φ] is the action describing the system. For the numerical evaluation of
the average, the path integral over an inﬁnite number of smooth paths is replacedwith the sum over a ﬁnite number of discretized paths:∑−S [φ]φ O[φ] ehOi ' ∑ −S [φ] .
(2.2)
φeIn path-integral Monte Carlo simulations, this sum is evaluated with paths generated stochastically. However, it is ineﬃcient to generate uniformly the paths sincemost of the paths thus generated scarcely contribute to the sum due to the Boltzman factor e−S [φ] . One way of improving the sampling eﬃciency is to generate a(i)path φ(i) according to an equilibrium distribution with probability p(i) = e−S [φ ] .
Such a strategy is called importance sampling, and the thermal average is thenreplaced with the arithmetic mean∑i.s. O[φ]hOi ' ∑.
(2.3)
i.s. 1
In actual simulations, such an path is generated sequentially, φ(1) → φ(2) → φ(3) →· · · , so that transition of the path satisﬁes the condition of detailed balanceW(i → j)p(i) = W( j → i)p( j),(2.4)

where W(i → j) denotes the probability of transition φ(i) → φ( j) . The choiceof W(i → j) is arbitrary as long as it satisﬁes Eq. (2.4). Here let us adopt theMetropolis algorithm; the transition φ(i) → φ( j) is always accepted if the energy is(i)( j)reduced S [φ( j) ] − S [φ(i) ] < 0, otherwise it is accepted with probability eS [φ ]−S [φ ] .
For short
(i)

( j)

W(i → j) = min{1, eS [φ ]−S [φ ] }.

(2.5)

Note that W(i → j) identiﬁes the accept probability for a given transition, not thetransition itself; we need to design update schemes which generate the conﬁgurations of φ ergodically.
In a typical routine of the Metropolis algorithm, one repeats the following
three steps (the operation “⇐” denotes substitution of right hand side into the left
hand side):
1. φ(old) ⇐ φ(new) , and generate a new conﬁguration φ(new) randomly.
2. Accept φ(new) with probability p = min{1, eS [φOtherwise, φ(new) ⇐ φ(old) .

(old) ]−S [φ(new) ]};3. Measure O[φ(new) ] and add it to the numerator of Eq. (2.3).
We note that before measurement one needs thermalization process, which is realized by repeating only the steps 1 and 2 so that the conﬁguration of the pathrelaxes into an equilibrium one.

2.2. SINGLE IMPURITY PROBLEM IN A SPINLESS TLL

2.2

21

Single impurity problem in a spinless TLL

To be speciﬁc, we here describe the application of the PIMC method to the impurity problem in a spinless Tomonaga-Luttinger liquid [36]. As shown in theprevious chapter [see Eq. (1.11)], this problem is described by the eﬀective action
S = S 0 + S V,

S0 ≡

1 ∑
|ωn ||φ̃(ωn )|2 ,
πKβ ω

SV ≡

(2.6)

∫ β
dτV(φ(τ)),

(2.7)

0

n

where φ̃(ωn ) is the Fourier component of the path φ(τ), and V(φ) = −V cos[2φ(τ)]
is the potential. Hereafter, we consider a general potential form V(φ). We discretize the imaginary time into J time steps as τ j = jτc ( j = 0, 1, · · · , J − 1) withthe short time cutoﬀ τc ≡ β/J. For the discretized path deﬁned as
φ j ≡ φ(τ j ),J−1φ̃(ωk ) ∑2πiφ̃k ≡φ j e J jk ,
=
τc
j=0

(2.8)

the action reads
J/2
J−1
∑
∑
1
2
S =
V(φ j ),
|φ̃k | + τc
2σk 2
j=0
k=1

σk 2 =J2 K,
8k

σ J/2 2 =

JK
,
2

(2.9)

where φ̃ J−k = φ̃k ∗ and σk 2 denotes standard deviation. In this thesis, we employ
two types of update schemes for the PIMC simulation as explained in the following subsections.

2.2.1

Local update in the Fourier space

Since S 0 have a quadratic form in the Fourier space [56, 57], we modify the transition probability asW(old → new) = e−S 0 [φ̃(new) ]

min{1, eS V [φ(old) ]−SV [φ(new) ]},

(2.10)and constitute the Metropolis algorithm (2.5) by accepting a new φ̃k,0 generated
according to the normal distribution with standard deviation σ2k with probability(old) ]−Sp = min{1, eS V [φ(new) ]V [φ}.

(2.11)
x2

It is well-known that a random number x normally distributed as ∝ e− 2σ2 can begenerated by means of the Box-Muller transformation√√(2.12)
x = σ −2 log(1 − r1 ) cos 2πr2 or σ −2 log(1 − r1 ) sin 2πr2 ,22

CHAPTER 2. PATH-INTEGRAL MONTE CARLO METHOD

where r1 and r2 are random numbers uniformly distributed in the range of [0, 1).
Note that the Fourier component at k = 0, i.e., φ̃0 (new) is randomly chosen from a
uniform distribution ranging over [0, π) and is accepted with probability (2.11).

2.2.2 Global update based on the cluster algorithm
Rewriting S 0 with φ j in Eq. (2.9), we obtain the action in the imaginary-time
representation:
∑
∑
0
0
S =−
κ j j φ j φ j + τc
V(φ j ),
(2.13)j< j0

κ j j0 = −

j
J/2
∑

4
2πi
0
|k|e− J k( j− j ) ,2
K J k=−J/2+1(2.14)

where the interaction kernel κ j j0 decays as ∼ | j − j0 |−2 at low temperatures J → ∞.
If V(φ) is a potential with plural minima, such as cosine potential and double-wellpotential, tunneling between the minima is strongly suppressed by the long-range
interaction. As a result, a typical path stays most of time in the vicinity of potential
minima and kink structures appears between the minima once in a while. Suchpaths can be generated eﬃciently by means of the cluster algorithm [58, 18]. First,
a mirror axis φ = φaxis is randomly chosen on one of the symmetry axes of V(φ j ),and the displacement from it is deﬁned as ϕ j ≡ φ j − φaxis . When one deﬁnes spin
variables by the sign of the displacement of the ﬁeld as s j = ϕ j /|ϕ j |, the action(2.14) is rewritten into the long-range Ising model∑
K j j0 s j s0j , K j j0 = κ j j0 |ϕ j ||ϕ0j |.
(2.15)
S0 = −
j< j0

For this Ising model, we can apply the Swedsen-Wang cluster algorithm as follows. Each pair of sites ( j, j0 ) is connected with bond probability
p j j0 = max{0, 1 − e−2K j j0 },(2.16)and the cluster of connected sites is ﬂipped against the mirror, i.e., φ j (new) = 2φaxis −φ j (old) for each j included in the cluster. Note that one can build a cluster of sites
in time O(J log J) using binary search [59].

2.3 Single impurity problem in a spinful TLL
In this section, we actually perform the PIMC simulation to solve the impurity
problem in a spinful TLL [60]. The impurity problem in TLLs has been originally treated with perturbative renormalization group (RG) methods in the weak-2.3. SINGLE IMPURITY PROBLEM IN A SPINFUL TLL

23and strong-impurity limits [31, 32, 33]. Whether a conduction channel becomes
perfectly transmitting or insulating at low temperatures is determined by the relevance of the corresponding backscattering or tunneling process. In contrast to the
spinless case, in which RG analyses in the weak- and strong-backscattering limits
seem to be smoothly connected, the phase diagrams of the spinful case show clear
inconsistency in the two opposite limits, i.e., mismatch of RG ﬂow, suggesting theexistence of an intermediate (unstable) ﬁxed point. From the standard perturbative RG approach, we can obtain little information about the phase diagram for an
impurity with intermediate strength. In this section, we study the phase transitionin the intermediate region of impurity strength using the PIMC simulation.
The single impurity in the spinful TLLs can be formulated in a similar way as
the previous section. The eﬀective action is written in terms of two bosonic ﬁelds
corresponding charge and spin modes, φρ (τ) and φσ (τ) as [31, 32, 33]
S = S 0 + S 1,
∑ 1 ∑
S0 ≡
|ωn ||φ̃ν (ωn )|2 ,2πKν β ω
ν=ρ,σ
n
∫ β
S1 ≡ Vdτ cos φρ (τ) cos φσ (τ),(2.17)(2.18)(2.19)

0where φ̃ν (ωn ) denotes the Fourier component of φν (τ). S 0 is rewritten in a nonlocal
form along the imaginary time as∑ 1 ∫ βφν (τ)φν (τ0 )0,(2.20)dτdτ
S0 = −
2Kν β2 0
sin2 [(π/β)(τ − τ0 )]ν
which is used in the cluster algorithm explained in Sec. 2.2.2.

2.3.1

Perturbative renormalization group

In order to allow for a comparison of our numerical results with the known analytic viewpoints, here we brieﬂy review the renormalization group (RG) picture
presented in Refs. [31, 33]. In the weak backscattering limit, integration of thehigh-energy shell produces the eﬀective higher-order scattering processes, whose
amplitudes scale as
[
]
1
dV(l)
= 1 − (Kρ + Kσ ) V(l),(2.21)dl2dVρ (l)dVσ (l)= (1 − 2Kρ )Vρ (l),= (1 − 2Kσ )Vσ (l).
(2.22)dldl

24

CHAPTER 2. PATH-INTEGRAL MONTE CARLO METHOD

Figure 2.1: Phase diagram of the ground state in the Kρ -Kσ plain in the weak(left) and strong backscattering limits. (I) Both charge and spin are insulating; (II)Charge is conducting, while spin is insulating; (III) Charge is insulating, while
spin is conducting; (IV) Both charge and spin are conducting.

Here the bandwidth is renormalized as D0 = De−l . V(l) reﬂects a single electron,i.e., ψ+ † ψ− +H.c., while Vρ (l) [Vσ (l)] reﬂects two charges (spins), i.e., ψ†+,↑ ψ−,↑ ψ†+,↓ ψ−,↓ +H.c. (ψ†+,↑ ψ−,↑ ψ†−,↓ ψ+,↓ +H.c.). From these scaling equations, the phase diagram forweak backscattering are deduced as shown in the left panel of Fig. 2.1, where thephase boundaries are Kρ + Kσ = 2, Kρ = 1/2, and Kσ = 1/2. In region I, V grows
toward zero temperature (l → ∞), which means perfect reﬂection of electrons at
the potential barrier, hence an insulating phase. On the other hand, in region IV all
the scattering amplitudes vanish toward zero temperature, so electrons can freely
go through the barrier, hence a conducting phase. In region II (III), only Vσ (Vρ )grows at low temperatures while the other scattering amplitudes vanish.
In the opposite limit V → ∞, the scaling equations are derived with help of
the duality transformation as
[
(
)]
dt(l)
1 1
1
= 1−
+
t(l),
(2.23)
dl
2 Kρ Kσ()()
dtρ (l)2dtσ (l)2= 1−tρ (l),= 1−tσ (l),(2.24)
dl
Kρdl
Kσwhere t, tρ , and tσ are amplitudes of the single-electron tunneling, the (twoelectron) charge tunneling, and the (two-electron) spin tunneling. From these scaling equations, the phase diagram for strong backscattering is obtained as shownin the right panel of Fig. 2.1, where the phase boundaries are 1/Kρ + 1/Kσ = 2,
Kρ = 2, and Kσ = 2.
The phase diagrams for the two limiting cases are thus obtained. Then, it isnatural to ask what happens for an intermediate value of the scattering potential V.
In order to illustrate this question visually, the three-dimensional phase diagram

2.3. SINGLE IMPURITY PROBLEM IN A SPINFUL TLL

25

Figure 2.2: Left panel: Three-dimensional phase diagram in the (Kρ , Kσ , V)-space.
The phase boundaries are analytically obtained in the following two limits: for aweak impurity (V → 0), the phase boundaries are three straight lines Kρ + Kσ = 2,
Kρ = 1/2, and Kσ = 1/2; for a strong impurity (V → ∞), the phase boundaries
are a hyperbola Kρ−1 + Kσ−1 = 2 and two straight lines Kρ = 2 and Kσ = 2. The
dotted lines connect the three points in the V = 0 and V = ∞ planes, at which thephase boundaries coincide in the both planes. Right panel: Expected RG ﬂows on
the intersection Kσ = 2Kρ .

in the (Kρ , Kσ )-plane is shown in the left panel of Fig. 2.2. The results of theperturbative RG for V → 0 and V → ∞ indicate that the four phases are smoothly
connected between the two limiting values of V. Because the phase boundaries
are diﬀerent between the two limiting cases, they should vary as a function of
V. The right panel of Fig. 2.2 shows the expected RG ﬂows on the intersection
Kσ = 2Kρ . Here, we assumed that Kρ and Kσ are not renormalized because theimpurity should not change the bulk properties of the 1D electron system. We
ﬁnd that the phase boundary between region I and IV is shifted from Kρ = 4/3
to Kρ = 3/2 as V increases. This indicates that there exists a phase transition
characterized by an unstable ﬁxed point at an intermediate value of V in the range
4/3 < Kρ < 3/2 on this intersection.
While the RG approach with the help of duality transformation revealed theRG ﬂow in the limit of weak- and strong-backscattering barriers, there is little
hope to obtain further information on the RG ﬂow in the whole parameter space,most of which belongs to the non-perturbative regime. In the rest of this chapter,
we instead appeal to a numerical method, i.e., a PIMC simulation for the eﬀective
action S given by Eq. (2.17), and examine transport properties of a spinful TLL
with an impurity for non-perturbative backscattering potential barrier.

26

CHAPTER 2. PATH-INTEGRAL MONTE CARLO METHOD

2.3.2 The path-integral Monte Carlo method
The basic strategy for construction of the path-integral Monte Carlo method adopted
here is the same as the example explained in Sec. 2.2. Here, we brieﬂy explain
the PIMC method by emphasizing some points speciﬁc to the spinful model. The
discretized action of the present problem is given by
S = S0 + S1
J/2
∑∑
S0 =

(2.25)

1|φ̃νk |2 ,22σνk
ν=ρ,σ k=1

S 1 = Vτc
{
σνk ≡
2

J−1
∑

(2.26)

cos φρ j cos φσ j ,(2.27)

j=0

Kν J 2 /4k k = 1, 2, · · · , J/2 − 1,Kν J
k = J/2(2.28)∑
2πi
where τc ≡ β/J is the short-time cutoﬀ, and φ̃νk = φ̃∗ν,J−k = j φν j e J jk . For thepresent model, we consider three types of update schemes as explained below.

2.3.3 Local update
In a local update in the Fourier space, a new value of φ̃νk (k > 0) is randomly
generated from a normal distribution with the variance σνk 2 given in (2.28) bymeans of the Box-Müller method. This update is accepted with a probability
p = min{1, e−(S 1new −S old )1},(2.29)

where S 1 new − S 1 old is the variation of the potential term (2.27). For the k = 0
component φ̃ν0 , a new value is generated from a uniform distribution ranged from−π to π, and the update is accepted again with a probability (2.29).

2.3.4 Global update
A cluster update should be designed so that optimized paths for a given potential
are eﬃciently generated. In the case of the double-cosine potential (2.27), an optimized path near the phase transition typically spends most of the time in potential
minima, and also has some kink structures connecting adjacent potential minima,i.e., (φρ , φσ ) = (nρ π, nσ π) with integers nρ and nσ such that nρ + nσ = odd for
V > 0. In order to generate such kinks, we introduce two relative ﬁeld variablesϕν j ≡ φν j − φν mirror for ν = ρ, σ as shown in Fig. 2.3, where the reference φν mirror isappropriately chosen as described later. When one deﬁnes a spin variable by the2.3. SINGLE IMPURITY PROBLEM IN A SPINFUL TLL

27

Figure 2.3: Cluster updates of ﬁeld variables φρ j and φσ j in a φρ -φσ plane, whereϕν j ≡ φν j − φν mirror . The empty circles represent the potential minima, and thedashed lines represent the reference ﬁelds φρ mirror and φσ mirror . Only the ﬁeld variables at the j-th time step in a cluster are shown. (a) During one period of adouble-ﬁeld cluster update, a point (ϕρ j , ϕσ j ) is subject to mirror reﬂection twice,i.e., once with respect to φρ j = φρ mirror , and subsequently to φσ j = φσ mirror . (Thewhole process is (ϕρ j , ϕσ j ) → (−ϕρ j , ϕσ j ) → (−ϕρ j , −ϕσ j ).) (b) As for a chargeﬁeld cluster update, ϕν j is subject to mirror reﬂection with respect to φρ = φρ mirror ,i.e., (ϕρ j , ϕρ j ) → (−ϕρ j , ϕρ j ). Similarly, ϕσ j in a spin-ﬁeld cluster is updated as(ϕρ j , ϕσ j ) → (ϕρ j , −ϕσ j ).

sign of the relative ﬁeld as sν j ≡ ϕν j /|ϕν j |, the nonlocal term (2.20) is written in a
discrete form as∑∑
S0 = −
κν j j0 sν j sν j0 , .
(2.30)
νj< j0Taking the bandwidth cutoﬀ into account explicitly, the coupling coeﬃcient κν j j0
is evaluated as
J/22|ϕν j ||ϕν j0 | ∑2πi
0κν j j0 ' −|k|e J ( j− j )k .
2
Kν J
k=−J/2+1(2.31)

A cluster of spins is built by connecting sites with the bond probability determined by the exchange couplings κν j j0 , and all the spins in the cluster is ﬂippedwith no rejection. If φν mirror is appropriately chosen so that the potential term
S 1 remains unchanged after the cluster is ﬂipped. In Fig. 2.3, two kinds of suchrejection-free cluster update implemented in this chapter, named double-ﬁeld cluster update and single-ﬁeld cluster update, are illustrated.
In the double-ﬁeld update illustrated in the left panel (a), one horizontal mirror
and one vertical mirror are located along φρ mirror = (nρ + 1/2)π and φσ mirror =28

CHAPTER 2. PATH-INTEGRAL MONTE CARLO METHOD(nσ + 1/2)π, where nρ and nσ are integers. The Ising spins are connected withbond probability∑

p j j0 = max{0, 1 − e−2 ν κν j j0 sν j sν j0 }.

(2.32)

Both the ﬁelds ϕρ j and ϕσ j in the cluster are then reﬂected with respect to the twomirrors, i.e., (ϕρ j , ϕσ j ) → (−ϕρ j , ϕσ j ) → (−ϕρ j , −ϕσ j ). Note that the connected
spins sν j and sν j0 are not necessarily parallel, which is diﬀerent from the original
Swendsen-Wang algorithm. By means of this double-ﬁeld cluster update, kinkstructures connecting nearest-neighbor potential minima are inserted eﬃciently.
In the single-ﬁeld update is illustrated in the right panel (b) in Fig. 2.3, only avertical mirror for the charge degree of freedom is located at φρ mirror = nρ π withan integer nρ . A cluster is constructed with bond probabilitypρ j j0 = max{0, 1 − e−2κρ j j0 sρ j sρ j0 }.

(2.33)The relative ﬁelds ϕρ j in the cluster are then reﬂected with respect to the verticalmirror, i.e., (ϕρ j , ϕσ j ) → (−ϕρ j , ϕσ j ), while the spin ﬁelds are left unchanged. A
similar cluster update is performed also for a horizontal mirror φσ mirror = nσ π withan integer nσ . These single-ﬁeld cluster updates insert kink structures between
next-nearest-neighbor potential minima.

2.3.5

Linear conductanceIn the PIMC simulation, we study zero-bias conductances of charge and spin channels to obtain the zero-temperature phase diagram for an intermediate value of V.
In the linear response regime, they are calculated from analytic continuationGν = lim Gν (iωn ),iωn →0(2.34)

where the conductance at a Matsubara frequency is calculated from a correlation
function given as∫
2e2 |ωn | βGν (iωn ) =dτhφν (τ)φν (0)ieiωn τ .
(2.35)h π 0
In the PIMC simulation, the spin and charge conductances are evaluated with
Monte Carlo sampling for the correlation function in Eq. (2.35).

2.3.6 Results
This subsection is devoted to presenting our PIMC results. Phase transitions on
the line of Kσ = 1 and 1.8 are deduced from numerical data in Sec. 2.3.6 and2.3. SINGLE IMPURITY PROBLEM IN A SPINFUL TLL
0.98
Gσ(iωn) / (2e2Kσ / h)Gρ(iωn) / (2e2Kρ / h)

0.96
0.94
0.92
0.9
0.88
J=400
200
100
50

0.86
0.84
0.82
0

0.2

0.4
0.6
4ωnτc

0.8

1

0.98
0.96
0.94
0.92
0.9
0.88
0.86
0.84
0.82
0.8
0.78

29

J=400
200
100
50
0

0.2

0.4
0.6
4ωnτc

0.8

1

Figure 2.4: Charge conductance Gρ (iωn ) (left) and spin conductance Gσ (iωn )(right) for diﬀerent values of Kρ , plotted as a function of ωn . Kσ is ﬁxed at
Kσ = 1.0 Symmetric coupling case (Kρ ' Kσ ). J = 50, 100, 200, and 400, andonly the ﬁrst ten points are shown for each Trotter number J. From top to bottom,
the values of Kρ are 1.2, 1.1, 1.05, 1.025, 1.0, 0.975, 0.95, 0.9, and 0.8. Conductance curves (of both charge and spin) show an upward bend with decreasing ωnwhen Kρ > 1.025, whereas they are bent downward when Kρ < 0.975.

Sec. 2.3.6, respectively. The whole phase diagram is discussed in Sec. 2.3.7 incomparison to the perturbative RG picture. In the present simulation, we ﬁx thebackscattering strength as Vτc = 1, and the temperature is varied from J = 50 toJ = 400, where J is inversely proportional to temperature as J = β/τc .
Phase transition at Kσ = 1
In Fig. 2.4, we show the result on the Kσ = 1 line (SU(2) symmetric case), andplot the ﬁrst ten points of Gν (iωn ) as a function of ωn for diﬀerent values of Kρnear the phase boundary. For a given Kρ , results for diﬀerent J, i.e., for diﬀerent
temperatures are superposed to form a bundle of curves. For both Gρ (iωn ) andGσ (iω), one can see that the curves for Kρ > 1.025 are bent upward with decreasing ωn (in the limit of ωn → 0), while the curves for Kρ < 0.975 are bent
downward. Note also that for a given Kρ , the slope of diﬀerent curves composing
the same bundle always becomes steeper with decreasing temperature (increasing
J). When the data shows such a monotonic dependence on temperature, one candetermine the phase boundaries by simply identifying a turning point at which
the bend of Gν (iωn ) changes from upward to downward with decreasing ωn . In
the present case, the phase transition appears to be located around Kρ = 1 from
Fig. 2.4.
To obtain the phase diagram, transition points are actually determined by trac-

CHAPTER 2. PATH-INTEGRAL MONTE CARLO METHOD
1
0.98
0.96
0.94
0.92
0.9
0.88
0.86
0.84
0.82
0.8

1
0.95
Kρ=1.200
1.100
1.050
1.025
1.000
0.975
0.950
0.900
0.800

Gσ / (2e2Kσ / h)Gρ / (2e2Kρ / h)30Kρ=1.200
1.100
1.050
1.025
1.000
0.975
0.950
0.900
0.800

0.9
0.85
0.8
0.75

100

1000
J

100

1000
J

Figure 2.5: DC conductances of charge (left) and spin (right) channels for diﬀerent values of Kρ . Kσ is ﬁxed at Kσ = 1.0. Symmetric coupling case (Kρ ' Kσ ).
Conductances are plotted as a function of the inverse temperature J. As expectedfrom Fig. 3, both charge and spin conductances increase with decreasing temperature (increasing J) for Kρ > 1.025, whereas they decrease for Kρ < 0.975.

ing temperature dependence of dc conductances following Refs. [18, 61]. In
Fig. 2.5, we plot the dc conductances as a function of the inverse temperature J,
which are obtained by extrapolating the ﬁrst ﬁve points on each curve in Fig. 2.4
to ωn → 0. With decreasing temperature, the dc conductance (of both spin andcharge) increases monotonically for Kρ > 1.025, whereas it decreases monotonically for Kρ < 0.975. Here, the charge and spin channels show a simultaneous transition from conducting to insulating phase in consistent with the RG results. Recall that in the RG picture (see Fig. 2.1) the IV and I phases touch at(Kρ , Kσ ) = (1, 1) both in the weak- and strong-backscattering regimes, which suggests that the phase boundary at that point is a straight line independent of V inthe (Kρ , Kσ , V)-space.
Phase transition at Kσ = 1.8
We plot the results of Gρ (iωn ) and Gσ (iωn ) for the Kσ = 1.8 in Fig. 2.6. One
can ﬁrst observe that the charge and spin channels behave quite diﬀerently as aclear consequence of the anisotropy between Kρ and Kσ . The curves for charge
conductance Gρ (iωn ) are bent upward for Kρ > 0.5 with decreasing ωn , while thesame curves for Kρ < 0.5 are bent downward. In contrast to the case of Kσ = 1
in the previous subsection, the sign of the slope of the diﬀerent curves composing
the same bundle may change.
We show the corresponding dc conductances as a function of temperature in
Fig. 2.7. The temperature dependence of the (charge) conductance Gρ shows a

1
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
0.9

31

0.98
0.96
Gσ(iωn) / (2e2Kσ / h)Gρ(iωn) / (2e2Kρ / h)2.3. SINGLE IMPURITY PROBLEM IN A SPINFUL TLL

J=400
200
100
50

0.94
0.92
0.9
0.88
J=400
200
100
50

0.86
0.84
0.82

0

0.2

0.4
0.6
4ωnτc

0.8

1

0

0.2

0.4
0.6
4ωnτc

0.8

1

Figure 2.6: Charge conductance Gρ (iωn ) (left) and spin conductance Gσ (iωn )(right) for diﬀerent values of Kρ , plotted as a function of ωn . Kσ is ﬁxed at
Kσ = 1.8. Asymmetric coupling case (Kρ  Kσ ). J = 50, 100, 200, and 400, andonly the ﬁrst ten points are shown for each Trotter number J. From top to bottom, the values of Kρ in the upper panel are 0.6, 0.55, 0.525, 0.5, 0.475, 0.45, and 0.4,
while those in the lower panel are 0.5, 0.45, 0.425, 0.4, 0.375, 0.35, and 0.3. In
this parameter regime, charge and spin channels behave quite diﬀerently.

monotonic behavior similar to the case of Kσ = 1. On the other hand, the temperature dependence of the spin conductance Gσ is more peculiar: For example,
if one focuses on the conductance curves for Kρ = 0.35, their slopes are upwardat higher temperatures, J =50 or 100, whereas the same curves have an opposite slope at lower temperatures, J = 200 or 400. Such non-monotonic behaviors
may be more clearly seen, if we look into the charge conductance in Fig. 2.7 forKρ = 0.350, 0.375, and 0.400. Similar crossover behaviors are observed wheneveranalyzing the boundary between phases I and III.
These monotonic temperature dependences are expected to reﬂect the nonmonotonic ﬂows of V near the intermediate unstable ﬁxed point. We note that the
non-monotonic temperature dependences have already been reported in study ofdissipative Josephson junctions in Ref. [61]. Since the precise location of sucha non-trivial ﬁxed point is unknown, one cannot immediately conclude that the
spin channel is in the conducting phase, even if the conductance, e.g. for Kρ =0.450 or 0.500, tends to increase monotonically toward low temperatures up to
J = 400. Possibly, it might turn insulating at a certain lower temperature whichis numerically inaccessible. Since it is diﬃcult to locate true phase boundaries at
T = 0 in the presence of intermediate unstable ﬁxed points, we instead identify
the phase boundary at T , 0 by observing the temperature dependence near thelowest temperature J ' 400.

CHAPTER 2. PATH-INTEGRAL MONTE CARLO METHOD

32
1

0.98
0.96

0.96
Kρ=0.600
0.550
0.525
0.500
0.475
0.450
0.400

0.94
0.92
0.9
0.88

Gσ / (2e2Kσ / h)Gρ / (2e2Kρ / h)

0.98

0.94
0.92
Kρ=0.500
0.450
0.425
0.400
0.375
0.350
0.300

0.9
0.88
0.86
0.84
0.82

100

1000
J

100

1000
J

Figure 2.7: DC conductances of charge (upper) and spin (lower) channels fordiﬀerent values of Kρ . Kσ is ﬁxed at Kσ = 1.8. Asymmetric coupling case (Kρ Kσ ). Conductances are plotted as a function of the inverse temperature J. Upper: The temperature dependence for the charge channel shows monotonic increase(Kρ > 0.5) or decrease (Kρ < 0.5) with decreasing temperature. Lower : The
spin conductance behaves non-monotonically for Kρ = 0.35, 0.375, and 0.4, e.g.,
for Kρ = 0.375, the conductance increases from J = 50 to J = 100, whereas it
decreases from J = 200 toward J = 400.

2.3.7 Phase Diagram
Repeating the analyses outlined in the previous subsection for diﬀerent sets of Kρ
and Kσ near the phase transition, we determine the whole phase boundaries inthe (Kρ , Kσ )-plane. In a left panel of Fig. 2.8, we show our phase diagram for aﬁnite impurity strength Vτc = 1 obtained from the PIMC simulations at inverse
temperature J = β/τc = 400. Due to the symmetry of the action (2.17) in termsof ρ and σ, the following discussion also holds true when the charge and spin
degrees of freedom are interchanged.
The phase boundaries, denoted with solid lines, are indeed located between the
two limits, which are shown in right two panels of Fig. 2.8. For nearly isotropicinteractions Kρ ' Kσ , our phase boundary at an intermediate coupling is close to
the one for weak backscattering. The boundary between region III and IV is also
close to the weak-backscattering phase boundary. On the other hand, for strongly
anisotropic interactions Kρ  Kσ , our phase boundary between I and III phases ismuch closer to the strong-backscattering phase boundary.

2.3.8 Discussion
For nearly isotropic interactions Kρ ' Kσ , the system of Vτc = 1 is considered
to be located in the weak-backscattering regime. This is indicated by the ob-2.3. SINGLE IMPURITY PROBLEM IN A SPINFUL TLL

33

2.2

Phase III2

Phase IV

1.8

Kσ

1.6

Phase I

1.4

Charge insulator
Spin insulator
Charge conductor
Spin conductor
Charge criticality
Spin criticality

1.2
1
0.8
0

0.2

0.4

0.6
Kρ

0.8

1

1.2

Figure 2.8: Phase diagram obtained from our PIMC simulations for the impurity
strength Vτc = 1. For comparison, the RG results in the strong and weak barrier
limits are shown in the upper-right and lower-right panels.

tained transition line between region I and IV similar to the weak-backscattering
case, also by the large conductances of both the charge and spin channels (see,e.g., Fig. 2.5). For strongly anisotropic interactions Kρ  Kσ , however, the I-III
boundary resembles the one of the strong-backscattering case, i.e., Kσ = 2. Let us
discuss what this feature indicates. For small Kρ , the charge ﬁeld is almost pinned
around the potential minima. If we assume that this localized charge ﬁeld does
not aﬀect the spin ﬁeld, it is eﬀectively described by a single-ﬁeld action
∫
1 ∑
2
S '
|ωn ||φ̃σ (ωn )| + V dτ cos φσ (τ),
2πKσ β ω(2.36)nwhich leads naturally to the strong-backscattering phase boundary Kσ = 2. The
assumption of decoupling between charge and spin ﬁelds is not trivial, because
the I-III phase boundary is dependent on Kρ for weak barrier. Our result for thephase boundary between region I and III indicates that the coupling between two
degrees of freedom vanishes by the potential barrier more sensitively than for theother perturbations.

34

CHAPTER 2. PATH-INTEGRAL MONTE CARLO METHOD

2.4 Summary
In this chapter, we have explained intelligent update schemes of the PIMC methodby taking impurity problems in TLLs as examples. We actually perform the PIMC
simulation to solve the single impurity problem in a spinful TLL. Measuring thetemperature dependence of the charge and spin conductances, we have obtained
the phase diagram characterized by perfect conduction or insulation of the charge
and spin channels for an intermediate impurity strength. We showed that the phase
diagram obtained from our simulations is reasonably obtained in good consistency
with the results of the perturbative RG analysis.

Chapter 3
Coulomb Blockade in an Open
Quantum Dot
In this chapter, we study the Coulomb blockade in an open quantum dot connected to a bulk lead via a single mode point contact [62]. We extend the PIMC
method introduced in Chap. 2 by taking into account the charging eﬀect in thedot. The Coulomb oscillation of the average charge and capacitance of the dot
is investigated, and is compared with known analytical results and those of numerical renormalization group. At the degeneracy point, we observe logarithmic
divergence of the capacitance for strong backscattering at the point contact. Thisobservation supports the conjecture that the nature of the present system at thedegeneracy point is described by the two-channel Kondo problem for an arbitrarystrength of tunneling.
This chapter is organized as follows. In Sec. 3.2, we state our model for an
open quantum dot. In Sec. 3.33.3, we give details about the path-integral Monte
Carlo method employed here. In Sec. 3.4, we present numerical results, and compare them with analytical results in some limits.

3.1

Introduction

As explained in Sec. 1.3, strong tunneling in a large quantum dot can be treatedwith a model consisting of a quantum dot and a half-inﬁnite lead connected via
a single mode point contact. The Coulomb blockade in such a structure has beenstudied theoretically by several authors [22, 23, 63, 64]. The peculiar nature of
the large dot system is logarithmic divergence of the capacitance characterized
by the analogy to the two-channel Kondo problem [65]; Matveev has revealed
that with decreasing temperature the capacitance diverges logarithmically at thedegeneracy points both in the weak and strong tunneling limits [22, 23]. The con35

36

CHAPTER 3. COULOMB BLOCKADE IN AN OPEN QUANTUM DOT

Figure 3.1: Schematic ﬁgure of an open quantum dot connected to a bulk lead bya single mode point contact. The dot is formed by applying negative voltage to thegates (gray regions). The solid line shows the boundary of the two-dimensional
electron gas. The gate voltage Vg controls the charge in the dot. The width of theconstriction is adjusted so that only a single transverse mode is allowed to tunnel
through the point contact.

sistency in these opposite limits suggests that the Coulomb blockade system at thedegeneracy points can be described eﬀectively by the two-channel Kondo model
for an arbitrary strength of tunneling. One way of conﬁrming this conjecture is tonumerically study the intermediate tunneling regime. Recently, the two-channel
Kondo eﬀect has been studied by Lebanon et al. with the numerical renormalization group (NRG) method based on the tunneling Hamiltonian [64]. They showedthat the two-channel Kondo behavior is observed in a broad range of tunnelingstrengths. Although their method is powerful enough to clarify properties of anopen quantum dot in the intermediate tunneling regime, it is diﬃcult to extend thisapproach to more complicated problems since numerical cost of matrix diagonalization in the NRG calculation becomes exponentially large as the orthogonal setof states in quantum dots increases. In this chapter, we present another numericalapproach to the same problem, and show that our simulation leads to the sameconclusion as in the NRG calculation. This result demonstrates that the PIMC approach is a powerful method to study dynamic response of quantum dot systems
in the presence of electron interactions, which will be discussed in Chap. 4.

3.2 Model
We formulate a quantum dot strongly connected to a bulk lead shown in Fig. 3.1
using the bosonization method following Ref. [23]. The basic strategy is the sameas in the spinless case discussed in Sec. 1.3.2. If the curvature of the constriction near the center of the point contact is smooth, the system is essentially onedimensional (1D) [23] and is described by a Tomonaga-Luttinger liquid (TLL).
The width of the constriction is adjusted so that only a single transverse mode is3.2. MODEL

37

allowed to tunnel through the point contact. Although the lead and the dot outside
the constriction are two-dimensional (2D), transport through the point contact depends mainly on the one-dimensional behavior of electrons near the constriction.
Then we can replace the 2D parts away from the constriction with 1D half-inﬁnite
leads, whose Hamiltonian is given in the bosonized form as( ∂θ )2 ]∑ ∫ dx [ uν ( ∂φν )2νH0 =+ uν Kν,(3.1)
4π Kν ∂x∂x
ν=ρ,σwhere the subscriptions ρ and σ stand for a charge and spin mode, respectively.
Here, uν is the sound velocity, and the bosonic ﬁelds φν and θν satisfy the commutation relation [φν (x), ∂θν0 (x0 )] = [θν (x), ∂φν0 (x0 )] = 2πiδνν0 δ(x − x0 ). In thefollowings, we ignore the short-range interaction (Kρ = Kσ = 1). The long-range
Coulomb interaction is taken into account through the charging energy in the dot
region as [23]
HC =

Q2
1
+ QVg ' 2 EC [φρ (x = 0) − πNg ]2 ,
2C0
π

(3.2)

where Ng ≡ −C0 Vg /e. Note that the ∫dot charge is expressed in terms of the bosonic∞ﬁeld by means of the relation Q = e 0 dx(−∂ x φ/π). We further assume a potential
barrier V0 δ(x) at the center of constriction. The backscattering at the potential
barrier is given by [23]
HV =

2V0
D cos φρ (x = 0) cos φσ (x = 0).
πvF

(3.3)

Here, vF is the Fermi velocity and D is the bandwidth. The total Hamiltonian
H = H0 + HC + HV describes the present problem.
By integrating out the boson ﬁelds in the bulk part (x , 0), the eﬀective action
is derived as
S ≡ S 0 + S C + S V,
1 ∑∑
S0 =
|ωn ||φ̃ν (ωn )|2 ,2πβ ν=ρ,σ ωn∫S C = U dτ [φρ (τ) − πNg ]2 ,∫S V = V dτ cos φρ (τ) cos φσ (τ),

(3.4)
(3.5)
(3.6)
(3.7)

where the spatial coordinate x is omitted, and φ̃ν denotes the Fourier componentof φν . We have deﬁned U ≡ EC /π2 and V ≡ 2V0 D/πvF .

38

CHAPTER 3. COULOMB BLOCKADE IN AN OPEN QUANTUM DOT

If an in-plane magnetic ﬁeld is applied to the system as in Ref. [63], electrons are spin-polarized and the barrier height relatively shifts to V↑ = V0 (1 − δ)[V↓ = V0 (1 + δ)] for electrons with up (down) spins, where δ is proportional to the
Zeeman energy. Then, S V is replaced with∫
SV = V

3.3dτ[cos φρ (τ) cos φσ (τ) + δ sin φρ (τ) sin φσ (τ)].

(3.8)

The path-integral Monte Carlo method

For simplicity, here we consider the case without the Zeeman eﬀect (δ = 0).
Discretizing the imaginary time into J steps, we deﬁne the jth step on a dis∑cretized path as φν j ≡ φν ( jβ/J), and its Fourier transform as φ̃νk ≡ j φν j e(2πi/J) jk −πJNg δνρ δk0 . We thus obtain the discretized eﬀective action as
S0 + SC =

J/2
∑∑

1
|φ̃ |2 ,2 νk2σνk
ν=ρ,σ k=0

S V = VτcJ−1∑

cos φρ j cos φσ j ,(3.9)(3.10)

j=0

where τc = β/J. Note that the bandwidth D is related to the interval of the timeslice as D = 2π/τc . We adopt two kinds of the update scheme described in thefolloing subsections.

3.3.1 Local update
In a local update, for each pair of ν and k a value of φ̃νk is randomly chosen22following a normal distribution ∝ e−|φ̃νk | /(2σνk ) with variance.
J, σσ0 2 = ∞,
2Uτc
1[ 1
Uτc ]−12
σνJ/2 =
+
δνρ ,
2 2J
J
1[ k
Uτc ]−1σνk 2 =+δνρ(k , 0, J/2).
4 J2
Jσρ0 2 =

(3.11)(3.12)(3.13)

Then a new path φν j is obtained from the inverse Fourier transform, and is acnew
oldcepted with probability p = min{1, e−(S V −S V ) }.

3.3. THE PATH-INTEGRAL MONTE CARLO METHOD

39

3.3.2 Global update
The basic idea of the cluster algorithm is the same as that explained in Sec. 2.2.2.
We rewrite the eﬀective action as∑∑
κ j j0 φ ν j φ ν j 0 ,(3.14)
S0 = −
νj< j0∑
S C = Uτc[φρ j − πNg ]2 ,(3.15)j

S V = Vτc∑cos φρ j cos φσ j ,(3.16)jwhere the kernel is given by
J/2
2 ∑
0
κ j j0 = − 2|k|e(2πi/J)( j− j )k .
J k=−J/2+1(3.17)

We can design two types of cluster update as follows.
In a single-ﬁeld cluster update, clusters for the charge and spin ﬁeld are constructed separately with the bond probability between the jth and j0 th sitepν j j0 = max{0, 1 − exp(−2κ j j0 ϕν j ϕν j0 )}.

(3.18)

Here we deﬁne the relative ﬁeld ϕν j ≡ φν j −πMν measured from the mirror locatedat φν = πMν with an integer Mν .
In a double-ﬁeld cluster update, on the other hand, a pair of the jth and j0 thsite for both the charge ﬁeld and the spin ﬁeld is connected using the probability
{
( ∑
)}
0
0
0
p j j = max 0, 1 − exp −2
κ j j ϕν j ϕν j ,(3.19)
νwith a half integer Mν .
Clearly, S V remains unchanged after these global moves, and the strength ofbackscattering does not matter. S C is invariant exceptionally in the single-ﬁeld
case for the spin mode, but it generally changes in the other cases. Then the cluster
updates for the latter cases are accepted with the probability p = min{1, e−∆S C },where the change in S C has the form∑
∆S C = 4πUτc (Ng − Mρ )ϕρ j .
(3.20)
j∈cluster

The kinks, which run from a potential minimum to an adjacent one of the doublecosine potential in Eq. (3.16), are inserted in the path by these two types of cluster
updates.

CHAPTER 3. COULOMB BLOCKADE IN AN OPEN QUANTUM DOT

π (<Q> / e - Ng )

40

0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5

0.5
0.25
0
-0.25
-0.5

0

0.5

1

Vτc=1

J = 10J = 20J = 50J =100J =200J =400

Vτc=2
0

0.25

0.5

0.75

1

Ng

Figure 3.2: Coulomb oscillation of the average charge of the quantum dot for
diﬀerent temperatures J = 10, 20, 50, 100, 200, and 400. A bundle of curves
with smaller amplitude corresponds to Vτc = 1, and the other to Vτc = 2. Inset:
Comparison of the average charge for Vτc = 1 and 2 between our PIMC data at
J = 400 (crosses) and the analytic result for small V at zero temperature (solidline).

3.4 Results and Discussion
In this section, we present our PIMC results. After showing the result of thegate-voltage dependence of the dot charge in Sec. 4.4.1, we examine the logarithmic divergence of the capacitance in Sec. 4.4.2. Finally, we study the channel
anisotropy eﬀect (δ , 0) in Sec 4.4.3. Uτc = 1/4 is ﬁxed, and the temperature
is varied down to J = β/τc = 400. Without losing generality, we can restrict thevalue of Ng in the range 0 ≤ Ng ≤ 1.

3.4.1 Coulomb oscillation of the dot charge
The average charge in the dot is evaluated ase
hQi = hφ̄ρ i,
π

1
φ̄ρ ≡
β

∫ β

dτ φρ (τ),(3.21)

0∫where hOi ≡ Z −1 Dφρ Dφσ Oe−S denotes the expectation value of an observable
O. In Fig. 3.2, we plot hQi for Vτc = 1 and 2 as a function of the gate voltage Ng .
As the temperature decreases (the inverse temperature J increases), the Coulomb
blockade is enhanced. If we focus on the lowest temperature J = 400, the curve
appears to converge except near the degeneracy point Ng = 1/2. We compare the3.4. RESULTS AND DISCUSSION

41

6
J = 10J = 20J = 50J =100J =200J =400C / C0

5
4
3
Vτc=2

2
1

Vτc=1

0
0

0.25

0.5

0.75

1

Ng

Figure 3.3: Coulomb oscillation of the capacitance of the dot for diﬀerent temperatures J = 10, 20, 50, 100, 200, and 400. A bundle of curves with smaller
amplitude corresponds to Vτc = 1, and the other to Vτc = 2.

data of J = 400 with the analytic result for T = 0 in the strong tunneling limit(V → 0). In Ref. [23], the charge ﬁeld φρ is pinned down to πNg due to the large
charging energy T  EC < D, and the backscattering strength V is renormalized
by the charge ﬂuctuation. As a result, the average charge is calculated for small V(corresponding to strong tunneling) as
hQi
2γ|r|2= Ng +
log(e |r|2 cos2 πNg ) sin 2πNg ,2e
π(3.22)

where r = V0 /vF is a reﬂection coeﬃcient, and γ = eC with the Euler’s constantC ' 0.5772. The inset of the Fig. 3.2 shows the comparison for Vτc = 1 and
2 between our data of J = 400 and Eq. (3.22). For Vτc = 1, the analytic result
shown as a solid line is almost superposed on our data points; the expression(3.22) based on the pinning of φρ describes well the Coulomb oscillation of hQi.
For Vτc = 2, on the other hand, our data points away from Ng = 1/2 disagree withthe analytic expression; the system is no longer in the strong tunneling regime forsuch a large value of V. It should be noted that similar deviation from Eq. (3.22)
in the intermediate tunneling regime has been reported in the NRG results. [64]3.4.2

Logarithmic divergence of the capacitance

Near Ng = 1/2 the oscillation curve in Fig. 3.2 depends on J even at lower temperatures. This is related to the singularity derived from the two-channel Kondo
problem hidden in the Coulomb blockade system at the degeneracy point. To see

42

CHAPTER 3. COULOMB BLOCKADE IN AN OPEN QUANTUM DOT
6
Ng=0.500

5C / C04
Ng=0.4753

Ng=0.4502

Ng=0.400

1
0
10

100
J

1000

Figure 3.4: Temperature dependence of the capacitance of the dot for Vτc = 2.
The gate voltage Ng is varied from 0.15 to 0.50 (from bottom to top) at intervalsof 0.05. The curve for Ng = 0.475 is also shown.

the peculiar nature of the system more clearly, we now measure the capacitanceof the dot deﬁned as
C=−

∂
e2
hQi = 2 β[hφ̄ρ 2 i − hφ̄ρ i2 ].
∂Vg
π(3.23)In Fig. 3.3, we plot the capacitance for Vτc = 1 and 2 as a function of the gate
voltage Ng at low temperatures. The amplitude of the Coulomb oscillation of
the capacitance grows larger as J and V increase. The temperature dependence
strongly depends on the gate voltage; away from Ng = 1/2, the capacitance isconvergent at low temperatures whereas the peak at Ng = 1/2 keeps growing andsharpening up to the lowest temperature J = 400.
In Fig. 3.4, we show the capacitance for intermediate tunneling Vτc = 2 as afunction of the inverse temperature J for diﬀerent values of the gate voltage Ng . In
this ﬁgure, the temperature dependence of the capacitance is classiﬁed into three
diﬀerent types: When the temperature decreases, (I) for Ng ≤ 0.40, the capacitance decreases monotonically; (II) for Ng = 0.45 and 0.475, the capacitance once
increases, and decreases below a crossover temperature; (III) for Ng = 0.50, thecapacitance increases monotonically. In Ref. [23], it is suggested from analytic
results that the low-energy property of the Coulomb blockade system near the degeneracy point (Ng = 1/2) is governed by the physics of the two-channel Kondo
problem for an arbitrary strength of tunneling. This prediction is supported by thelogarithmic divergence of the capacitance for the type III. The transient increasein the capacitance for the type II is also a sign of the two-channel Kondo physics.
For Ng , 1/2, an intrinsic low energy cutoﬀ in the strong tunneling limit (V → 0)

3.4. RESULTS AND DISCUSSION
28

4Vτc = 15
13
11
9
7
5
3
1

24
20
C / C0

43

16
12
8
4
0
10

100
J

1000

Figure 3.5: Capacitance at a degeneracy point Ng = 1/2 plotted as a function of
the inverse temperature J ≡ β/τc for various values of V. Logarithmic ﬁt of thedata for each V is shown as a dashed line.

is given by [23]
Γ=

8γ|r|2EC cos2 πNg .
π2(3.24)

At low temperatures below Γ, the logarithmic divergence characteristic of the twochannel Kondo model disappears. One can evaluate the order of the crossover
temperatures of the curves in Fig. 3.4 using the expression (3.24). For example,
the crossover temperatures for Ng = 0.475, 0.45, and 0.4 are estimated as J ' 182,
46, and 12, respectively.
In order to study in detail the logarithmic divergence of the capacitance, weshow the capacitance at the degeneracy point Ng = 1/2 in Fig. 3.5 as a function of
the inverse temperature J. The results for diﬀerent backscattering strengths fromVτc = 1/4 to Vτc = 15/4 are shown. Here, one can estimate the transmission
probability through the barrier using the expression T = 1/[1 + (V0 /vF )2 ] in thenoninteracting case (U = 0); for example, Vτc = 1/4 corresponds to T ' 0.9961(strong tunneling), while Vτc = 15/4 to T ' 0.5322 (intermediate tunneling).
One can see that our data points diverge logarithmically with decreasing temperature for any value of V as far as our simulation is performed.
Next, we compare this logarithmic behavior of our data with the analytic results for the strong and weak tunneling limit. In Fig. 3.6, the coeﬃcient derived
from the logarithmic ﬁt of the capacitance is plotted as a function of V. According to the analytic calculation for the strong tunneling limit (V → 0) [23], theCHAPTER 3. COULOMB BLOCKADE IN AN OPEN QUANTUM DOT

Coefficient of the logarithm

44

1

0.1
Weak tunneling
Strong tunnelingMonte Carlo

0.01
0

2

4

6

8
4Vτc

10

12

14

16

Figure 3.6: Coeﬃcient of logarithm of the capacitance plotted as a function of V.
The analytic result for strong(weak) tunneling is shown by the dashed(solid) line
(see also the text).

capacitance diverges at low temperatures as[( E )]
4γ|r|2C
C = C0 1 +
log
.
π
T

(3.25)For small V, i.e., in the strong tunneling regime, our data points agree well withthe analytic result 4πγ|r|2C0 /e2 ' 0.004429V 2 shown as a solid line. For larger
values of V, on the other hand, they grows faster than the analytic result. Thus,
the analytic expression (3.25) fails for intermediate strength of tunneling. Nevertheless, the logarithmic divergence in Fig. 3.5 in this regime clearly indicates thatthe present system is understood by the physics of the two-channel Kondo model
for any value of V.
In the present calculation, we could not reach the weak tunneling regime, because exponential decay of the Kondo temperature T K with respect to V makes
it diﬃcult to observe the Kondo eﬀect in the ﬁnite-temperature simulation. However, a qualitative discussion is possible by focusing on the asymptotic behavior of
the coeﬃcient of the logarithm. The analytical expression for large V is obtainedby mapping to the two-channel Kondo model as [23, 66]
(T )
( π )
e2
K
C ' 0.050
log, T K = Dtν exp −.
(3.26)TK
T
4tν
Here t is the tunneling matrix element and ν is the density of states in the lead.
We can estimate tν from the barrier height V0 through the transmission probabilityT√= 4(πtν)2 /[1 + (πtν)2 ]2 derived for U = 0. We thus obtain the relation (tν)−1 =π[ (V0 /vF )2 + 1+V0 /vF ]. If we assume that this relation holds also for U > 0, we3.4. RESULTS AND DISCUSSION451

δ = 0.00
δ = 0.25
δ = 0.50
δ = 0.75
δ = 1.00

0.9

π2C / e2

0.8
0.7
0.6
0.5
0.4
0.3
0

0.25

0.5
ε

0.75

1

Figure 3.7: Coulomb oscillation of the capacitance in the presence of the in-planemagnetic ﬁeld for J = 400 and Vτc = 2.

can calculate the coeﬃcient of the logarithm for large V from (3.26). However,
we found that the prefactor of T K given in Eq. (3.26) disagrees with our data
as found in the NRG calculation [64]. There seems to be several reasons forthis inconsistency. For example, the result (3.26) is obtained for the isotropic
Kondo model [66], while the present system is described by the anisotropic Kondo
model [22]; this diﬀerence may aﬀect the prefactor quantitatively. On the other
hand, the exponent of T K deduced from our data is asymptotically consistent with
Eq. (3.26) in the weak tunneling region (V → ∞). In Fig. 3.6, we also show
∝ A exp(π/4tν) as an asymptote (solid line), where its prefactor A is adjusted by
hand.

3.4.3

Eﬀect of channel anisotropy

In this subsection, we discuss the eﬀect of channel anisotropy in the presence of anin-plane magnetic ﬁeld (δ , 0). The channel anisotropy term proportional to δ is,together with the gate-voltage diﬀerence  ≡ Ng − 1/2 from the degeneracy point,a relevant perturbation to the two-channel Kondo ﬁxed point in the context of therenormalization group. Therefore, the divergence of the capacitance is expected tobe suppressed at low temperatures. Actually, the capacitance at zero temperaturefor strong tunneling (V → 0) is calculated analytically as [63]
( E )
]
[
4γ ( V0 )2C
2(1 − δ ) log
cos(2πNg ) .
(3.27)C = C0 1 +π vF
Γ(Ng )The low-energy cutoﬀ takes the form Γ(Ng ) = (8γEC /π2 )(V0 /vF )2 min(sin2 π, cos2 π).
Note that the capacitance (3.27) remains ﬁnite except for the two-channel Kondo

46

CHAPTER 3. COULOMB BLOCKADE IN AN OPEN QUANTUM DOT

ﬁxed point (, δ) = (0, 0). Because the above analytic results are obtained only inthe two limits (  δ and   δ), the whole Coulomb oscillation of the capacitance C has not been calculated so far. We now show the Monte Carlo result for
the gate-dependence of the capacitance for Vτc = 1 at low temperatures (J = 400)
in Fig. 3.7. We see that the Coulomb oscillation of C is smoothly suppressed as δ
increases. As δ approaches 1, one spin-channel shows nearly perfect transmission
while the other shows nearly perfect reﬂection. This is why its capacitance hascosine-like line shape near δ = 1, in good agreement with the spinless case [23].

3.5 Summary
In this chapter, we have demonstrated the PIMC simulation of the Coulomb blockade phenomena in a open quantum dot at low temperatures. The Coulomb oscillation of the average charge and capacitance of the dot are studied in the intermediate tunneling regime, where the analytical results for the two limits of strong andweak tunneling are invalid. The amplitude of oscillation monotonically increases
with decreasing temperature or increasing the backscattering strength. We conﬁrmed that logarithmic divergence of the capacitance appears at the degeneracy
point Ng = 1/2 even in the intermediate tunneling regime. This observation indicates that the present system has a two-channel Kondo nature for an arbitrarystrength of tunneling. We also studied the eﬀect of the channel anisotropy in thepresence of the Zeeman energy.
These conclusions agree well with the numerical renormalization group (NRG)
calculation by Lebanon et al. [64]. The NRG approach based on the tunneling
Hamiltonian has several advantages. For example, NRG can access ultra-low
temperatures because ﬁne energy-meshes are taken near the Fermi energy. However, this method tends to need heavy calculation if one increases the number ofbases of states in quantum dots. Our approach by the PIMC simulation gives analternative way to treat transport in quantum dots, which is applicable to more
complicated models. Our results obtained here indicate that the PIMC method actually provides a powerful method to access low-temperature transport properties.
Based on this success, we consider dynamic response of quantum dot systems in
the next chapter.

Chapter 4
Charge Relaxation ResistanceIn this chapter, we study dynamic response of a mesoscopic capacitor composedof a quantum dot and a lead in the presence of strong electron interactions [67].
By means of perturbation theory, renormalization group technique, and the pathintegral Monte Carlo method, we analyze the charge relaxation resistance in thewhole range of dot-lead coupling. We predict that the charge relaxation resistanceis universal even in the presence of strong Coulomb blockade for the Luttinger parameter K > 1/2, while the Kosterlitz-Thouless transition dramatically inﬂuences
the universal nature of the charge relaxation resistance for K < 1/2.
This chapter is organized as follows. After stating introduction in Sec. 4.1,
we summarize the universal nature of the charge relaxation resistance based on
the scattering theory by Büttiker et al. in Sec. 4.2. In Sec. 4.3, we introduce
our model for a mesoscopic capacitor, and explain its basic property. The update
scheme in the PIMC method is described in Sec. 4.4, and our results are presented
in Sec. 4.5.

4.1

Introduction

There is renewed interest in a dynamical aspect of mesoscopic systems stimulated
by recent experiments observing low-frequency response of a mesoscopic capacitor [9], which is schematically shown in the left panel of Fig. 4.1. The mesoscopic
capacitor is a quantum analog of classical RC circuit as shown in the right panelof the ﬁgure; reﬂection on the point contact between the dot and the lead acts as a
resistor Rq , while capacitive coupling between the dot charge and the gate voltage
plays the role of a capacitor Cµ . In a classical RC circuit, the admittance is givenby the Kirchhoﬀ’s law
G(ω) =

1,
R − (iωC)−1
47

(4.1)

48

CHAPTER 4. CHARGE RELAXATION RESISTANCE

where the ac resistance R equals the dc resistance. However, the latter equality is
not the case in the mesoscopic capacitor. In their pioneering work [19], Büttiker et
al. have extended the Kirchhoﬀ’s law (4.1) to the mesoscopic capacitor, replacing
C and R with their quantum counterparts Cµ and Rq , respectively. In the lowfrequency expansion of the admittanceG(ω) = −iωCµ + ω2Cµ 2 Rq + O(ω3 ),(ω  1/RqCµ ),(4.2)

they have proved that Rq equals half a resistance quantum h/2e2 irrespective oftransmission through the point contact. A recent experiment supports this theoretical prediction [9]. Rq measured at such a low frequency region is named charge
relaxation resistance after the condition ω  1/RqCµ , which clearly guarantees
charge relaxation in the dot. The universal quantization of the charge relaxation
resistance is remarkable because it contradicts a naive conjecture based on the additive nature of impedance in classical ac circuits. Indeed, if the “resistor” and the“capacitor” in the quantum RC circuit were independent building blocks as in theclassical case, the resistance would be described by the Landauer’s formula forthe point contact, explicitly dependent on the transmission probability through theconstriction.
So far the charge relaxation resistance has been studied theoretically within
mean-ﬁeld-type approximation. As the size of the mesoscopic capacitor is reduced, however, we need to examine whether the electron interaction inﬂuences
the universal nature of the charge relaxation resistance. In this chapter, we study
the charge relaxation resistance in the presence of electron interactions, employing both analytical and numerical methods.

4.2

Universal charge relaxation resistanceIn this section, we brieﬂy summarize the self-consistent scattering theory by Büttiker
et al. [19] and derive the universal quantization of the charge relaxation resistance

Figure 4.1: Mesoscopic capacitor and its equivalent RC circuit. Reﬂection on the
point contact corresponds to Rq and capacitive coupling between the dot charge
and the gate voltage to Cµ .

4.2. UNIVERSAL CHARGE RELAXATION RESISTANCE

49

Rq = h/2e2 . The essence of their theory is that the charging eﬀect in the quantum dot is taken into account self-consistently by considering an eﬀective timedependent electric potential inside the dot denoted by Ug (t), which may diﬀer
from the external ac gate voltage Vg (t). For simplicity, the electric potential of thelead is ﬁxed to zero. Then the dot charge can be expressed in three ways. First,
focusing on the capacitive coupling coupling between the gate electrode and thedot, variation in the dot charge reads−dhQi = Cg (dVg − dUg ),(4.3)

where Cg denotes the geometrical capacitance. Next we can also express the
charge using the density of states in the dot D():−dhQi = e2 D()dUg ≡ Cq dUg ,(4.4)

which implies that the dot charge is capatively coupled to the lead by the capacitance Cq ≡ e2 D(). Combining the deﬁnition of the electrochemical capacitanceCµ ≡ −dhQi/dVg , i.e., −dhQi = Cµ dVg with Eqs. (4.3) and (4.4), we arrive at
Cµ dVg = Cg (dVg − dUg ) = Cq dUg ,

∴

1
1
1
=
+
Cµ Cg Cq

(4.5)

Thus the introduction of the eﬀective potential Ug results in the electrochmicalcapacitance Cµ split into Cg and Cq in series as depicted in Fig. 4.2.
In the same way, when the gate voltage is oscillating with frequency ω, thecurrent through the circuit can be expressed in three ways:
I(ω) = −dhQi= G(ω)Vg (ω) = −iωCg [Vg (ω) − Ug (ω)] = g(ω)Ug (ω)
dt
1
1
1
∴
=
+
,
G(ω) g(ω) −iωCg

(4.6)
(4.7)

Figure 4.2: Equivalent circuit of a mesoscpic capacitor in the self-consistent scattering theory. The serial combination of the geometrical capacitance Cg and thequantum capacitance Cq results from the eﬀective electric potential Ug associatedwith the charging eﬀect.

50

CHAPTER 4. CHARGE RELAXATION RESISTANCE

where G(ω) and g(ω) are the total admittance and the admittance through the point
contact, respectively. Note that the scattering theory predicts∫e2
f (E) − f (E + ω)g(ω) =
dE Tr[1 − S† (E)S(E + ω)]
,
(4.8)
h
ω
where S(E) is the scattering matrix of the point contact and f (E) is the Fermi
distribution. By substitute Eq. (4.8) into Eq. (4.7) and expanding G(ω) in ω, we
obtain a low-frequency expression of the total admittance similar to Eq. (4.2).
Then comparison of each term in the the two expressions up to second order in ω
leads to()∫
1
d f † ∂S
Cq =
dE −
S
,
(4.9)
2πi
dE
∂E
∫
dE(−d f /dE)Tr[S† (E)∂S(E)/∂E]2h.
(4.10)
Rq = 2 ∫2e { dE(−d f /dE)Tr[S† (E)∂S(E)/∂E]}2
At ﬁnite temperature, the charge relaxation resistance Rq depends on transmission
probability of the point contact, as well as the gate voltage, through the scatteringmatrix S. At zero temperature, on the other hand, S’s in Eq. (4.10) are canceledout due to −d f /dE = δ(E). As a result, the charge relaxation resistance Rq deﬁned
at low frequencies ω  1/RqCµ is quantized to h/2e2 per channel at T = 0. Thisis the remarkable result Büttiker et al. obtained in their pioneering work [19].
We should note that the two-capacitor picture for the total capacitance (4.5)
breaks down when the charge ﬂuctuation becomes large and the charging eﬀectcannot be well-described with the mean ﬁeld Ug . This can be seen from, e.g., theobvious inequality resulting from Eq. (4.5):
C gC qCµ =≤ Cg .
(4.11)
Cg + CqAt zero temperature, Cq can be estimated from Eq. (4.9) as [9]e2
1 − r2,(4.12)∆ 1 − 2r cos(2π/∆) + r2
where r is the reﬂection coeﬃcient and  denotes the gate voltage relative to adegeneracy point of two neighboring charge states. From Eq. (4.12), Cq oscillates
as a function of the gate voltage Vg and has sharp peaks at degeneracy points. The
peak height should be enhanced inﬁnitely as r → 1, whereas Eq. (4.11) indicatesthat it is saturated at Cg . If the latter were true, we could not any more expect a
sharp Coulomb staircase in the dot charge, which clearly contradicts our intuition.
In this sense, the self-consistent scattering theory cannot describe the degeneracy
points exactly. In the rest of this chapter, we develop a theory beyond the meanﬁeld approximation and show that the eﬀect of electron interaction is prominent
at the degeneracy points.
Cq =4.3. MODEL

4.3

51

Model

In this section, we ﬁrst derive a bosonized model for the mesoscopic capacitor in
the presence of strong electron interaction. Then we explain the way of estimating
the charge relaxation resistance in the Matsubara formalizm. For comparison to
the self-consistent scattering theory [19], the mean-ﬁeld approximation for ourbosonized model is also shown.

4.3.1 Bosonization
The mesoscopic capacitor of interest is shown in the left panel of Fig. 4.3. We ﬁrst
assume that the transport property of the system is mainly determined by the onedimensionality near the center of the constriction. Then it is not necessary to treat
exactly the two-dimensional regions in the quantum dot and the lead, so that we
can consider the whole mesoscopic capacitor as a semi-inﬁnite one-dimensional
system, which is described by the (non-chiral) Tomonaga-Luttinger model. The
present system is formulated by considering the open boundary condition on theclosed end of the quantum dot [68]. Here, we instead derive the eﬀective action ina diﬀerent way; we consider a 1D quantum dot system with two point contacts andpinch oﬀ one of the point contacts later so that the boundary condition is satisﬁed.
The bulk Hamiltonian for an inﬁnite Tomonaga-Luttinger liquid is given by ( )2( )2 
dx  v ∂φ∂θ 
 .

H0 =
+ vK
∂x
−∞ 2π K ∂x
∫ ∞(4.13)

Here the canonically conjugated ﬁelds φ and θ satisfy the bosonic commutation
relation [φ(x), θ(x0 )] = (iπ/2)sgn(x − x0 ), the Luttinger parameter K describes
short-range electron interaction around the constriction, and v ≡ vF /K denotes thesound velocity. We deﬁne a quantum dot in the range of x1 < x < x2 by locating
point contacts at x1 = 0 and x2 = L with barrier height V1 and V2 , respectively.
The electron scattering on the point contacts is expressed by the ﬁeld operators of

Figure 4.3: Schematic illustration of an open quantum dot (left panel) and a quantum Hall edge state (right panel).

CHAPTER 4. CHARGE RELAXATION RESISTANCE

52

a right(left)-moving electron ψ+(−) as∑
HV = −
Vi [ψ+ † (xi )ψ− (xi ) + H.c.],(4.14)

i=1,2

whose bosonized expression is given by∑ Vi
HV = −
D cos[2{φ(xi ) − kF xi }],πvF
i(4.15)

with the bandwidth D. We take into account charging eﬀect in the dot and capacitive coupling between the dot charge Q and the (static) gate voltage Vg by adding
the Hamiltonian
(
)
Q2
e
HC =
+ QVg
Q ≡ [φ(0) − φ(L)] ,(4.16)
2Cg
πwhere Cg is the geometrical capacitance within x1 < x < x2 . By integratingout the ﬁelds away from x1 and x2 , we arrive at the eﬀective action in terms ofφ± ≡ φ(x1 ) ± φ(x2 ) as in Ref. [33]:[]∫πCg Vg 2
1 ∑∑
|ωn |
1 e2
2
S =
|φ̃ν (ωn )| + 2dτ φ− (τ) −2πKβ ν=± ω 1 + νe−πK|ωn |/∆
π 2Cg
|e|
n
∫
∑ Vi D−dτ cos[2{φ(xi , τ) − kF xi }],(x1 = 0, x2 = L)
(4.17)
πvF
iwhere the Fourier transform of φν reads∫ β1∑−iωn τ
φ± (τ) ≡φ̃± (ωn )e,φ̃ν (ωn ) =dτφν (τ)eiωn τ .
β ω
0(4.18)

n

Now we return to the problem of the mesoscopic capacitor with a single lead bypinching oﬀ the point contact at x = x2 , i.e., V2 → ∞. As a result, φ(x2 , τ) is
pinned at one of the minima of the cosine potential asφ(x2 , τ) = kF x2 + mπ

(m = 0, ±1, ±2, · · · ).

(4.19)We thus obtain the eﬀective action for the mesoscopic capacitor written with a
new ﬁeld φ ≡ φ− + kF x2 = φ− + kF L:
S = S kin + S C + S V
1 ∑
|ωn |
|φ̃(ωn )|2
S kin =
−2πK|ω
|/∆nπKβ ω 1 − e∫n1
S C = 2 EC dτ[φ(τ) − πNg ]2π ∫S V = −V dτ cos[2φ(τ)](4.20)(4.21)(4.22)(4.23)

4.3. MODEL

53

where ∆ ≡ πvF /L and EC ≡ e2 /2Cg are respectively the level spacing and thecharging energy in the quantum dot. V ≡ V1 D/πvF denotes the reﬂection strengthof the point contact. Ng ≡ EC −1 [|e|Vg /2 + (kF L/π)(EC + ∆/2K 2 )] is the dimensionless gate voltage. We note that the above eﬀective action can be also obtainedin fractional quantum Hall edges shown in the right panel of Fig. 4.3 (see also
Appendix A). We have already seen a model similar to Eq. (4.20) in Chapter 4,
where the eﬀective action (3.4) describes a large (∆ = 0) quantum dot systemwith spinful electrons. Here we consider a quantum dot with ﬁnite size (∆ , 0) in
order to ensure coherent transport in the quantum dot, which is necessary for thehalf-quantization of the charge relaxation resistance.
It is worthwhile examining the frequency-dependence of the factor γ(ωn ) ≡[1 − e−2πK|ωn |/∆ ]−1 in S kin , which is approximated in the high- and low-frequency
regions as [33]{11,
|ωn |  ∆γ(ωn ) ≡'(4.24)∆
1−2πK|ω
|/∆n+ 2 , |ωn |  ∆1−e2πK|ωn |For ω1 = 2πT  ∆, discretization of the energy levels in the dot is smearedout by the thermal ﬂuctuations, hence incoherent transport in the dot. Indeed theeﬀective action for γ(ωn ) = 1 takes the same form as that for a large quantum
dot (∆ = 0) with spinless electrons [23]. For ω1  ∆, on the other hand, thelow-frequency behavior of γ(ωn ) gives two eﬀects; shift of the charging energy
EC by ∆/2K 2 and multiplication of the interaction parameter K by 2. It is in sucha low-frequency regime that the half quantization of charge relaxation resistanceis observed. Indeed the doubled interaction parameter is the origin of the prefactor
1/2 in Rq as we see in the following sections.

4.3.2

Linear admittance

We formulate admittance within the linear response theory by applying an oscillating gate voltage Vg + δVg (t) to the quantum dot. By calculating the dynamics ofthe dot charge Q = (e/π)φ− under the oscillating ﬁeld H 0 (t) = QδVg (t), we obtain
the admittance at frequency ωG(ω) = G(iωn → ω + iδ)∫
e2 2|ωn | β
e2 2|ωn |iωn τhφ(−ωn )φ(ωn )i.
G(iωn ) =dτhφ(τ)φ(0)ie
=
h π
h πβ
0

(4.25)(4.26)In analogy to the classical RC circuit, we split the impedance into the real part andthe imaginary part
1
1
= R(ω) +.
G(ω)−iωC(ω)(4.27)

CHAPTER 4. CHARGE RELAXATION RESISTANCE

54

Here we assumed that both the resistance R and the capacitance C in general depend on ω. The RC time deﬁned as τRC (ω) ≡ R(ω)C(ω) is the time scale of charge
relaxation in the dot. If the the condition ω  1/τRC (ω) holds, which is requiredfor charge relaxation at frequency ω, then the admittance can be expanded at low
frequencies:
G(ω) = −iωC(0) + ω2 [{C(0)}2 R(0) − iC 0 (0)] + O(ω3 ).
Comparing this with the expansion (4.2), we ﬁnd the relations[]
C 0 (0)Cµ = C(0),Rq = R(0) 1 − i.
τRC (0)C(0)(4.28)(4.29)

If we assume that the ratio C 0 (ω)/C(ω) diminishes faster than τRC (ω) as ω → 0,
we can estimate the charge relaxation resistance as Rq ' R(ω → 0). In what
follows we adopt this assumption. We should note that the expansion (4.28) is
not always available because it is possible that τRC (ω) diverges faster than 1/ω
with lowering ω. In such a situation, R(ω → 0) can not be interpreted as charge
relaxation resistance in the sense that the gate voltage oscillates faster than charge
relaxation occurs in the dot. Instead let us call R(ω → 0) just low-frequency
resistance so that it is well-deﬁned whether the dot charge relaxes or not. We canestimate the low-frequency resistance using Eq. (4.27), or equivalently analyticalcontinuation from the Matsubara representation:
R(ω → 0) = lim R(iωn )
iωn →0with

R(iωn ) ≡1
1−.
G(iωn ) ωnC(iωn )(4.30)

Here G(iωn ) has been already deﬁned in Eq. (4.26), while we evaluate C(iωn → 0)with the static capacitance, i.e.,
(
)
∫
e2
1 β
2
2
Cµ = C(iωn → 0) = 2 β[hφ̄ i − hφ̄i ]dτφ(τ) .
(4.31)
φ̄ ≡π
β 04.3.3 Mean ﬁeld theory
To conﬁrm that the eﬀective action (4.20) correctly describe the mesoscopic capacitor, we here calculate the electrochemical capacitance Cµ within the Hartree
approximation to reproduce the results obtained in the self-consistent scattering
theory [19, 20]. In the Hartree approximation, it is assumed that ﬂuctuation of thedot charge ∝ φ(τ) − hφi is small enough that the term ∝ [φ(τ) − hφi]2 is negligible.
We thus obtain the mean-ﬁeld eﬀective action (4.20)∫
eUg
dτφ(τ),(4.32)S MF = S kin + S V +π

4.4. THE PATH-INTEGRAL MONTE CARLO METHOD

55

where
(
)
2hφi
2kF L
∆
Ug ≡ Vg +EC − 2 EC +
πe
π
2K 2

(4.33)

is the eﬀective gate voltage coupled to the dot charge hQi = (e/π)hφi. If thedensity of states in the quantum dot is denoted by D(), variation of the dot charge
is given by dhQi = −e2 D()dUg , so that Eq. (4.33) leads to the self-consistentcondition for the eﬀective gate voltage Ug :−e2 D()dUg = Cg (dUg − dVg ),(4.34)

By using this condition, one can easily reproduce the electrochemical capacitancein Refs. [19, 20]
(
)
Cµ 2
∂hQi
Cµ ≡ −= 1−e D()
∂Vg
Cg

∴

1
1
1
=
+ ,
Cµ Cg Cq

(4.35)

where Cq ≡ e2 D(). If K = 1, the present model can be put back to the noninteracting electron system. Then the self-consistent scattering theory [19, 20] withthe eﬀective gate voltage Ug deﬁned in Eq. (4.33) leads to Eqs. (4.9) and (4.10).
The reﬂection coeﬃcient in the point contact is expressed using the parameter inthe bosonized model (4.20) as r = πV/D with the bandwidth D.

4.4

The path-integral Monte Carlo method

The setup of the path-integral Monte Carlo method that we describe here is similar
to that for the impurity problem in a spinless TLL illustrated in Sec. 2.2. Using
the discretized path φ j ≡ φ( jβ/J) ( j = 0, 1, · · · , J − 1) and its Fourier transform∑φ̃k ≡ j φ j e(2πi/J) jk − πJNg (1 + ∆/2K 2 EC ), we obtain the discretized action
J/2
J
∑
∑
1
2
|φ̃k | − Vτccos[2φ j ],
S =
2σk 2
j=0
k=0

where τc ≡ β/J is the short time cutoﬀ.

(4.36)

CHAPTER 4. CHARGE RELAXATION RESISTANCE

56

4.4.1 Local update
In a local update, for each k a value of φ̃k is randomly chosen following a normal
2
2
distribution e−|φ̃k | /2σk with variance
π2 J,2(EC + ∆/2K 2 )τc
[
]−1
1
EC τc
1 1
2
σ J/2 =
+ 2,
2 K J 1 − e−2π2 K/∆τc
π J
[
]−1
1 2k
1
E C τc2σk =+ 2
4 K J 2 1 − e−(2π)2 K|k|/J∆τc
π J
σ0 2 =(4.37)(4.38)(k , 0, J/2),(4.39)

Then a new path φ j is obtained from the inverse Fourier transform and is acceptednewoldwith probability p = {1, e−(S V −S V ) }.

4.4.2 Global update
The basic idea of the cluster algorithm is the same as that explained in Sec. 2.2.2
except that the present model has the charging energy. We rewrite the eﬀective
action in the imaginary-time representation∑
κ j j0 φ j φ j 0 ,(4.40)
S kin = −
j< j0
J−1
∑
1
S C = 2 EC τc[φ j − πNg ]2 ,π
j=0

S V = −Vτcj−1∑

cos[2φ j ],(4.41)(4.42)

j=0

where the interaction kernel κ j j0 is deﬁned as
κ

j j0

J/2
∑
4
|k|
2πi
0
=− 2
e− J k( j− j ) .
2 K|k|/J∆τ−(2π)
c
K J k=−J/2+1 1 − e(4.43)

Each pair of jth and j0 th sites is connected with bond probability
p j j0 = max{0, 1 − e−2κ j j0 (φ j −π/2)(φ j0 −π/2) },(4.44)and then all the sites φ j are ﬂipped with respect to φmirror = π/2, i.e., φ j (new) =π − φ j (old) .

4.5. RESULTS AND DISCUSSION

4.5

57

Results and discussion

In this section, we investigate the dynamic response of the mesoscopic capacitor
described by the eﬀective action (4.20) using analytical and numerical approaches.
In the small barrier limit V → 0, the capacitance Cµ and the charge relaxation resistance Rq is calculated perturbatively in V 1 . In the large barrier limit V → ∞, on
the other hand, we consider the tunneling of an electron through the point contact
as perturbation and discuss scaling of tunneling strength and charge ﬂuctuation.
Finally, we demonstrate the path-integral Monte Carlo simulation to study theintermediate region directly.

4.5.1

Weak barrier limit: Perturbation theory

It is clear from Eq. (4.26) that if we can estimate the correlation function c(ωn ) ≡hφ(−ωn )φ(ωn )i, we can calculate the linear admittance of the mesoscopic capacitorasG(iωn ) =e2 2|ωn |c(ωn ).
h πβ(4.45)

However, estimation of correlation functions is in general not an easy task. In
our case, reﬂection of electrons on the point contact causes a cosine potential
for the ﬁeld φ, which makes it diﬃcult to solve the problem exactly. Here wefocus on the weak barrier limit V → 0 and develop a perturbation theory. We
calculate the admittance up to second order in V and evaluate the electrochemicalcapacitance Cµ = Cµ (0) + Cµ (1) + Cµ (2) and the charge relaxation resistance Rq =Rµ (0) + Rq (1) + Rq (2) .
We begin with the partition function in the path-integral representation∫
Z[ξ] =



∑


1
Dφ exp −S −ξ(−ωn )φ̃(ωn ) ,β(4.46)

ωnwhere S is the eﬀective action (4.20). ξ is the counting ﬁeld and will be set to zeroin the end of the calculation. One can check that the correlation function c(ωn ) isexpressed in terms of the partition function (4.46) asc(ωn ) ≡ hφ(−ωn )φ(ωn )i = β21∂2 Z[ξ]1.
Z[ξ] ∂ξ(ωn )∂ξ(−ωn ) ξ→0(4.47)The results in 4.5.1 are based on calculation by one of the collaborators, T. Jonckheere.

CHAPTER 4. CHARGE RELAXATION RESISTANCE

58

The perturbative expansion of the right hand side in orders of V reads
1∂2 (Z (0) + Z (1) + Z (2) + · · · )Z (0) + Z (1) + Z (2) + · · ·∂ξ(ωn )∂ξ(−ωn )
2 (0)
∂Z
1
= β2 (0)Z ∂ξ(ωn )∂ξ(−ωn )[]∂2 Z (1)Z (1)∂2 Z (0)2 1+ β (0)−Z∂ξ(ωn )∂ξ(−ωn ) Z (0) ∂ξ(ωn )∂ξ(−ωn )[∂2 Z (2)Z (2)∂2 Z (0)2 1+ β (0)− (0)Z∂ξ(ωn )∂ξ(−ωn ) Z ∂ξ(ωn )∂ξ(−ωn )
{
}]
Z (1)
∂2 Z (1)Z (1)∂2 Z (0)− (0)−+ ··· ,
Z
∂ξ(ωn )∂ξ(−ωn ) Z (0) ∂ξ(ωn )∂ξ(−ωn )β2(4.48)(4.49)

where Z (m) denotes mth order contribution to the partition function. From this, we
obtain perturbative contributions to the correlation function up to second order:
1∂2 Z (0)Z (0) ∂ξ(ωn )∂ξ(−ωn )1∂2 Z (1)Z (1)− (0) c(0) (ωn )c(1) (ωn ) ≡ β2 (0)Z ∂ξ(ωn )∂ξ(−ωn ) Z
1∂2 Z (2)
Z (2)
Z (1)
c(2) (ωn ) ≡ β2 (0)− (0) c(0) (ωn ) − (0) c(1) (ωn ).
Z ∂ξ(ωn )∂ξ(−ωn ) Z
Zc(0) (ωn ) ≡ β2

(4.50)(4.51)(4.52)

To calculate each contribution c(m) , we need to evaluate Z (m) and its second derivative ∂2 Z (m) /∂ξ(−ωn )∂ξ(ωn ). For this purpose, let us ﬁrst express the total action in
the following form1∑
ξ(−ωn )φ̃(ωn )(4.53)
β ω
n
[
}
]
{
2Ng δn01∑=
βEC φ̃(ωn )φ̃(−ωn )A(ωn )φ̃(ωn ) + ξ(−ωn ) −
β ω
π
n


∫
∑
 2iσ ∑V−dτ exp φ̃(ωn )e−iωn τ (4.54)2 σ=±β ωS kin + S C + S V +n

withA(ωn ) ≡1|ωn |1+ 2 EC .
−2πK|ω
|/∆nπK 1 − eπ(4.55)

4.5. RESULTS AND DISCUSSION

59

Expanding Z[ξ] in V and performing the path integral in φ, we obtain
Z

(m)

∑∫
1 ( V )m ∑
[ξ] =
···
dτ1 · · · dτmm! 2 σ =± σ =±
[

×e
ξ→0

−→

m

1

]

[

∑
∑
∑
N δ βENg δn0 βEC
11−iωn τliωn τl− β1 ωn 12 ξ(−ωn )− g n0π C −i m−i m
l=1 σl e
l=1 σl e
A(ωn ) 2 ξ(ωn )−π∫
1 ( V )m ∑

m! 2
− β1

×e

∑

[
ωn

]

(4.56)dτ1 · · · dτm

σ1 ···σm][∑∑Ng δn0 βEC
N δ βE+i l σl e−iωn τl A(ω1 n ) g n0π C +i l σl eiωn τlπ].

(4.57)

Then the second derivative of Z (m) [ξ] reads∂2 Z (m) [ξ](4.58)∂ξ(ωn )∂ξ(−ωn )∫Z (m) [ξ]1 ( V )m ∑1dτ1 · · · dτm=+
2βA(ωn ) m! 2 σ ···σ[A(ωn )]2
1
m



∑
∑


 1

Ng δn0
N
δ
1


g
n0
σl e−iωn τl   ξ(ωn ) −σl eiωn τl ×  ξ(−ωn ) −βEC − iβEC − i
2
π
2
π
l
l
[

×e
ξ→0

−→

]

[

∑
∑
Ng δn0 βEC
N δ βE11 ∑−i l σl e−iωn σl A(ω1 n ) 12 ξ(ωn )− g n0π C −i l σl eiωn σlβ ωn 2 ξ(−ωn )−π(m)∫
1 ( V )m ∑Z [0]+2βA(ωn ) m! 2

σ ···σdτ1 · · · dτm](4.59)1[A(ωn )]2

1
m



∑
∑
 Ng δn0  Ng δn0

−iω
τ
iω
τ
n
n
l
l
 
× 
βEC + iσl eβEC + iσl e 
π
π
l
l

[

×e

]

[

∑
∑
Ng δn0Ng δn01
1 ∑−iωn σliωn σlβ ωnπ βEC +i l σl eA(ωn )π βEC +i l σl e].

(4.60)

Substituting Eqs. (4.57) and (4.60) into the the correlation functions in each order(4.50)-(4.52), we can expand perturbatively the admittance (4.45) ase2 |ωn | 1,
h π A(ωn )e2 |ωn | 2V √F+ (0) cos 2πN,G(1) (ωn ) = −h π [A(ωn )]2e2 |ωn | 2V 2
F+ (0)I(ωn ),G(2) (ωn ) =h π [A(ωn )]2
G(0) (ωn ) =(4.61)(4.62)(4.63)

CHAPTER 4. CHARGE RELAXATION RESISTANCE

60

where we have deﬁned two functions 1 ∑ ±2
F± (τ) ≡ exp −(4.64)cos ωn τ ,β ω A(ωn )n∫ β/2I(ωn ) ≡dτ[{1 − F+ (τ)(1 + cos ωn τ)} cos 2πN + {1 − F− (τ)(1 − cos ωn τ)}]0(4.65)
and redeﬁned the dimensionless gate voltage N as
N≡

|e|Vg kF L
EC
Ng =
+
ET
2ET
2

with

E T = EC +

∆
.
2K 2

(4.66)

The fact that Eqs. (4.61)-(4.63) are analytic at ωn → 0 means that one can always
expand the admittance at low frequency as Eq. (4.2) in the small barrier limit V →0. As a result the charge relaxation resistance is well-deﬁned as the factor Rq in theterm proportional to ωn 2 . Performing the Maclaurin expansion for the admittance,
we obtain perturbative contributions up to second order to the electrochemicalcapacitance
e2 π
h ET

(4.67)Cµ (0) = −Cµ (0)(4.68)Cµ (0) =2π2 √V F+ (0) cos 2πNET
2π2 2Cµ (0) = Cµ (0)V F+ (0)I(ωn → 0) cos 4πNET(4.69)and the charge relaxation resistance
Rq (0) =h,
2Ke2

Rq (1) = 0,

Rq (2) = 0.

(4.70)

It is clear from Eqs. (4.67)-(4.69) that one Coulomb oscillation occurs in the electrochemical capacitance Cµ (0) + Cµ (1) + Cµ (2) while N is shifted by 1, i.e., theelectrochemical potential eVg is shifted by the expanded level spacing 2ET =2EC + ∆/K 2 . Thus we focus on the interval 0 ≤ N < 1 in the rest of this chapter.
At zero temperature, the sums and the integration of Eqs. (4.64) and (4.65) canbe done analytically in certain cases. For example, in the case of K = 1 one has
F+ (0) = (ET /πD)2 if EC , ∆  D, so that one can show that
Cµ =

e2
λ(N)
2ET

(4.71)

4.5. RESULTS AND DISCUSSION

61

where λ(N) = 1 − 2r cos 2πN + 2r2 cos 4πN is the oscillating factor with thereﬂection coeﬃcient r ' πV/D. Eq. (4.71) suggests that the electrochemical capacitance is the serial combination of two capacitances

Cg λ(N)

and

Cq ≡

e2
λ(N),
∆

(4.72)

both of which similarly oscillate as λ(N). It is instructive to compare this resultwith that of the mean ﬁeld theory [19], where the electrochemical capacitanceCµ is split into a constant geometrical capacitance Cg and an oscillating quantum
capacitance Cq , unlike in Eq. (4.72). Cq in Eq. (4.72) coincides with the development of the noninteracting formula (4.12) in powers of r. Thus the inconsistency
between the perturbation theory and the mean ﬁeld theory is that the geometrical capacitance Cg is multiplied by the oscillating factor λ(N) in the former case.
This reﬂects the fact that the mean ﬁeld theory neglects charge ﬂuctuations due tocharging eﬀect, which is in fact enhanced near the degeneracy point N = 1/2, i.e.,hQi ' e/2. Indeed, in the weak barrier limit r → 0, the deviation of λ(N) from 1
remains small for N , 1/2, whereas it becomes large for N = 1/2.
Eq. (4.70) indicates that the charge relaxation resistance Rq is quantized at
h/2Ke2 per channel irrespective of the barrier height V and the gate voltage Vg (atleast in the weak barrier limit V → 0). In particular at K = 1 we reproduces Rq =h/2e2 predicted in Refs.[19, 21]. Surprisingly, the universal quantization is exact
up to second order in V, which implies that this behavior is preserved in higher
orders. In addition, we can see from Eq. (4.70) that the long-range interaction in
the dot EC has no inﬂuence on Rq , while the bulk short-range interaction describedby K explicitly modiﬁes the quantized value of Rq .

Figure 4.4: Typical path of φ near the degeneracy point hQi ' e/2 in the strong
barrier limit V → ∞. s and t denotes the width and the tunneling coeﬃcient
between the potential minima, respectively.

62

CHAPTER 4. CHARGE RELAXATION RESISTANCE

4.5.2 Strong barrier limit: Renormalization group
In the strong barrier limit, the lead and the quantum dot are almost disconnected,
so that most of electrons incident to the point contact are reﬂected. In this limit,
we can treat the transmission coeﬃcient t perturbatively. Near the degeneracy
point hQi ' e/2 the bosonic ﬁeld φ is subject to a double well potential whose
minima corresponds to the two charge states hQi ' 0 and hQi ' e/2 as shown in
Fig. 4.4. φ spends most of the time in either of the minima and sometimes a kinkstructure between them appears for inﬁnitesimally short time. In such a situation,
we can employ the dilute instanton gas approximation (DIGA) [46] and express
the path with n pairs of instanton and anti-instanton asφ(τ) ' π2n∑π
s j Θ(τ − τ j ) + (1 − s)2
j=1

(4.73)

where Θ(τ − τ j ) is the step function describing a kink structure at the imaginary
time τ j . s is the width between the well minima (0 ≤ s ≤ 1) as shown in Fig. 4.4,
and s j ≡ s(−1) j−1 represents the “charge” of an (anti-)instanton. Note that the expression (4.73) has reduced the degrees of freedom of the path to a set of (s j , τ j )’s,and as a result the perpendicular parts of the path in Fig. 4.4 are of less importance.
We will take into account the shape of an instanton later. Let us derive the eﬀective action in the DIGA representation by substituting Eq. (4.73) to Eq. (4.20).
∑
First, the Fourier transform of the step function Θ(τ) = (i/β) ωn e−iωn τ /(ωn + iη)
leads to the kinetic term written with s j and τ j :S kin =∑π ∑eiωn (τ j −τk )s j sk.
Kβ j,k|ωn |(1 − e−2πK|ωn |/∆ )ω

(4.74)

nSince the long-time property of the system is important at low energy, the high∑frequency terms in ωn contribute little to the sum, hence a high-energy cutoﬀ∼ |τ j − τk |−1 . Then recalling the relation (4.24), we can split the kinetic term into
two parts at low frequencies |ωn |  |τ j − τk |−1  ∆:
)
∑( 1
∆ 1
π ∑
+
eiωn (τ j −τk )(4.75)
S kin 's j sk2
Kβ j,k
2|ω
|
2πK
ω
n
n
ωn
∑
τ j − τk1
∆ ∑=−s jτ j.
(4.76)s j sk log−
2K j,kτc
2K 2 jWith respect to S V and S C , on the other hand, we can make most of the fact thatcos[2φ(τ)] and [φ(τ) − π/2]2 are approximately independent of τ. Therefore S V is

4.5. RESULTS AND DISCUSSION
almost constant, while S C can be rewritten as
S C ' (2Ng − 1)EC63∑

s jτ j,(4.77)jwhere we have neglected some constant terms. Finally, we assume that the detailed shapes of instantons are characterized by a single parameter, i.e., tunneling
amplitude t ∼ e−S inst . We thus complete the DIGA of the eﬀective action for thepath with 2n (anti-)instantons. The partition function of the system is given byZ=)2n ∫ β
∞ (
∑
t

∫ τ2n

∫ τ2

dτ2n
dτ2n−1 · · ·
dτ1
τc
0
0
0


∑
 s2 ∑

τ
−
τ
j
k
j−k
j−1
× exp (−1) log− us (−1) τ j 2K j,kτcj
n=0(4.78)

where u ≡ (2N − 1)ET denotes the small deviation from the degeneracy point.
Note that partition functions similar to Eq. (4.46) also appear in the context of the
Kondo problem [46], one-dimensional long-range Ising chains [47], and dissipative Josephson junctions [49], which suggests a potential phase transition in themesoscopic capacitor. Indeed, if we assume that t is perturbatively small, we canidentify the scaling equations using the renormalization group technique [46, 49]:
(
)
dt
s2
ds2
= 1−
t,
= −4s2 t2 ,(4.79)dl
2K
dl
dus= us(1 − 2t2 )(4.80)dl
which are familiar in the context of a Kosterlitz Thouless transition in the twodimensional XY model.
Let us ﬁrst focus on Eq. (4.80), which suggests that no further arguments areneeded when one deviates from the degeneracy point, i.e., u , 0. The double-wellpotential due to S C + S V approximates near the degeneracy point N ' 1/2 as
1 [π]− u φ(τ) −− V cos[2φ(τ)],(4.81)π2
so that the diﬀerence between the two lowest minima is of the order of u as shownin the left panned of Fig. 4.5. At high temperatures, this diﬀerence is blurred bythermal ﬂuctuations and the two charge states remain almost degenerate. Upondecreasing temperature, however, the diﬀerence between the two minima further increases because Eq. (4.80) predicts that u , 0 is always relevant. Thus,a crossover occurs around some temperature ∼ u, below which excitations from

64

CHAPTER 4. CHARGE RELAXATION RESISTANCE

the lower minimum (φ ∼ π) to the higher (φ ∼ 0) are exponentially suppressed. As
a result, the system reaches a ﬁxed point with an integral number of dot electrons
at T = 0. At the ﬁxed point, φ is trapped in an eﬀective harmonic potential shown
as a dotted curve in Fig. 4.5. This situation is quite similar to the case of zero
backscattering V = 0, which we have already discussed in Sec. 4.5.1. Therefore
we can conclude from Eq. (4.70) that the charge relaxation resistance is always
Rq =

h
2Ke2

(u , 0).

(4.82)

Note that this result is independent of Eqs. (4.79), and only the ﬂow of u is important for its universality. In addition, the eﬀective harmonic potential suppresses the
charge ﬂuctuation as T → 0 and, as a result, always validates the two-capacitor
picture in Fig. 4.2 except for u = 0.
For the charge degenerate case N = 1/2(u = 0), u is marginal and therefore
Eqs. (4.79), called the Kosterlitz equations, determine the low-energy propertiesof the mesoscopic capacitor. This case is related to the Kondo problem in thesense that the neighboring two charge states act as a pseudo spin. It is well-known
that for small B ≡ 1 − s2 /2K, the solution of Eqs. (4.79) is a set of hyperbolic
curves shown in Fig. 4.5:
B2 − 4t2 = const.

(4.83)and the system undergoes the Kosterlitz-Thouless (KT) transition when crossing
the separatrix 2t + B = 0. One can see from Fig. 4.5 that if 2t + B > 0, t → ∞and s2 → 0, while otherwise t → 0 and s2 → const. For K > 1/2, the tunnelingstrength t always grows upon reducing the temperature, and the system reaches the
Kondo ﬁxed point where the dot is strongly coupled to the reservoir. An electron
freely tunnels in and out of the dot irrespective of the initial barrier height V.

Figure 4.5: (left) Double-well potential away from the degeneracy point N = 1/2.
The diﬀerence between the two minima is of the order of u. (right) Flow diagram
for the tunneling amplitude t and B ≡ 1− s2 /2K. The tendency of the ﬂow changes
on the separatrix 2t + B = 0.

4.5. RESULTS AND DISCUSSION

65

We again arrive at the case where V = 0 is eﬀectively achieved as in the nondegenerate case u , 0 2 and therefore
Rq =

h
2Ke2

(u = 0, K > 1/2).

(4.84)

In particular at K = 1, by combining Eq. (4.82) and Eq. (4.84), we can reproduce
the quantized charge relaxation resistance Rq = h/2e2 irrespective of the barrierheight V nor the gate voltage Vg [19, 21]. However we should emphasize that in
the present picture the universal Rq at u = 0 is due to the Kondo eﬀect associatedwith the pseudo spin. In the mean ﬁeld theories [19, 21], on the other hand, theuniversality results from the mean ﬁeld approximation, which is generally invalid
at the degeneracy point u = 0 (N = 1/2) doe to large charge ﬂuctuations. Howeverat K = 1(> 1/2) the Kondo eﬀect also suppresses the charge ﬂuctuation ∼ s2 → 0,
and the mean ﬁeld approximation is validated accidentally. This suggests that if
s2 remains ﬁnite at T = 0, the charge relaxation resistance may not be universal atu = 0. Indeed, for K < 1/2 there is the possibility that at suﬃciently weak transmission t (large V), the RG ﬂow always drives the system into a weak coupling
conﬁguration with speciﬁed charge. Then the charge ﬂuctuation remains ﬁnite, sothat the capacitance diverges as Cµ ∼ βs2 ∝ T −1 . This situation corresponds to
the region 2t + B < 0 in Fig. 4.5. The KT transition of a static occupation hQi/e
in a similar setup has been predicted in Ref. [69]. We will discuss how the KT
transition inﬂuences a dynamic property of the mesoscopic capacitor in the next
section.

4.5.3

Intermediate region: The PIMC method

In the weak barrier limit (Sec. 4.5.1), the perturbation theory has proved that the
charge relaxation resistance is universal (Rq = h/2e2 ) even in the presence of interactions. In the strong barrier limit (Sec. 4.5.2), on the other hand, the RG treatment
indicates that, if K < 1/2, the system undergoes the KT transition at the degeneracy point. To study the non-perturbative regime, we apply the path-integral Monte
Carlo to the action for the discretized path φ(τ = jτc ) ( j = 0, 1, · · · , J − 1) withthe short-time cutoﬀ τc . In what follows, we use the parameters Dτc = 2π andEC τc = ∆τc /2K 2 = (π/2)2 . The top (bottom) row of Fig. 4.6 shows the calculated
capacitance Cµ as a function of V at J = 100 and K = 1 (K = 1/3). The left
and right columns correspond to the non-degenerate case (N = 0) and the charge
degenerate case (N = 1/2), respectively. With increasing V the Coulomb staircase
becomes sharper, which results in the decrease (increase) in Cµ ≡ −∂hQi/∂Vg in
However, the present case is diﬀerent from the case of u , 0 in that the universality is essentially due to the renormalization of t or V, not that of u.
2

CHAPTER 4. CHARGE RELAXATION RESISTANCE

66
0.5

3.5

Cµ / C

0.4

K=1

3

K=1

N=0

2.5

N = 1/2

0.3
2
0.2
1.5
0.1

1

0
0.5

0.5

K = 1/3

0.4

K = 1/3

100

N = 1/2

Cµ / C

N=0
0.3
10

0.2

J = 400
J = 100
J = 50

0.1
1
0
0

1

2

3

4
4Vτc

5

6

7

8

0

1

2

3

4
4Vτc

5

6

7

8

Figure 4.6: Capacitance Cµ as a function of the backscattering strength V obtained
as inverse temperature J = 100 with Monte Carlo computations (dashed lines).
The solid lines show the predictions of the perturbative calculations up to second
order. Temperature dependence is shown in the bottom right panel, where thevertical axis is measured on a logarithmic scale.

the case of N = 0 (N = 1/2). The second-order perturbation theory, shown as
solid lines, displays an excellent agreement for small V. Especially, it is remarkable that only for the case of K = 1/3 and N = 1/2 (the right bottom panel ofFig. 4.6), Cµ exhibits an abrupt increase at a ﬁnite V, signaling the KT transition
mentioned in Sec. 4.5.2. Indeed, one can see that Cµ grows as ∝ 1/T in the large
barrier region, reﬂecting ﬁnite charge ﬂuctuation ∼ s2 .
We now describe the eﬀect of the KT transition on the dynamical properties.
As described in Sec. 4.3.2, the low-frequency resistance R(ω → 0) of the point
contact can be estimated from analytical continuation of the impedance in the
Matsubara representation:R(iωn ) ≡1
1−,
G(iωn ) ωnCµ(4.85)

where G(iωn ) and Cµ are deﬁned in Eqs. (4.26) and (4.31), respectively. To obtainthe value of R(ω → 0) = R(iωn → 0) by analytical continuation, we plot R(iωn )as a function of ωn and extrapolate ﬁrst ﬁve points into ωn → 0. Note that τRC =4.5. RESULTS AND DISCUSSION

6720

K = 1/5
1/3
1/1

5

R(iωn = 0) / (h/Ke2)R(iωn = 0) / (h/Ke2)

6

4
J = 100
3
2
1
0

16

J = 400
J = 100
J = 50

12

K = 1/3

8
4
0

1

2

3

4
4Vτc

5

6

7

5

5.5

6

6.5

7

7.5

8

4Vτc

Figure 4.7: Extrapolated value R(iωn → 0) as a function of V at inverse temperature J = 100 in the degenerate case. The results at J = 100 for K = 1, 1/3, and
1/5 are shown in the left panel, while those at J = 50, 100, and 400 for K = 1/3
are shown in the right panel. The arrows represents the temperature dependence
as T → 0 (J → ∞).
30
N = 0.5000
N = 0.4875
N = 0.4750
N = 0.4500
N = 0.0000

R(iωn=0) / (h/Ke2)

25
20

Figure 4.8: Temperature dependence ofR(iωn → 0) at 4Vτc = 8 and K = 1/3.
The results at the degenerate case N =0.5 and the non-degenerate cases N =0, 0.45, 0.475 and 0.4875 are shown.

15
10
5
0
10

16 20

32

50 70 100
J

200

400

R(0)Cµ . In the left panel of Fig. 4.7, we plot R(0) for K = 1, 1/3 and 1/5 as afunction of V at temperature J = 100. For K = 1, R(0) equals h/2e2 irrespective of
V, in agreement with the universal charge relaxation resistance [19, 21]. For K =1/3 and 1/5, the same behavior is observed in the weak barrier region, whereas
R(0) is abruptly enhanced with increasing V, reﬂecting the RG ﬂow to the weakcoupling regime due to the KT transition. The temperature dependence of R(0) for
K = 1/3 is shown in the right panel of Fig 4.7, which indicates that R(0) diverges
as T → 0 (J → ∞) in the strong barrier region.
As mentioned in the previous subsection, in the non-generate case u , 0, thedivergence of R(0) stops at some temperature and the system crossovers to a ﬁxed
point with an integral number of dot electrons. In Fig. 4.8, we plot temperaturedependence of R(0) for 4Vτc = 8 and K = 1/3 in the non-degenerate cases N =0, 0.45, 0.475 and 0.4875. For comparison, monotonic increase in R(0) in thedegenerate case N = 0.5 is also shown. the large error bars reﬂect the fact that

CHAPTER 4. CHARGE RELAXATION RESISTANCE

68

102

J = 50J = 100
J = 400

J = 50
100
400

0.8

R(0)CµTG(iωn = 0) R(iωn = 0)

1

0.6
0.4

101

0.2
100

0
1

2

3

4

5
4Vτc

6

7

8

5

5.5

6

6.5
4Vτc

7

7.5

8

Figure 4.9: Product G(iωn = 0)R(iωn = 0) and ratio τRC /τcoh = R(iωn = 0)Cµ T
for K = 1/3 as a function of the backscattering strength V, for diﬀerent inverse
temperatures J. The crossing of the curves in the right panel gives a good estimateof the transition point.

the cluster algorithm is not suitable to asymmetric potentials especially near thecorssover temperature. The crossover temperatures can be estimated from T c ∼ u,i.e., Jc ∼ 1/|2N − 1|ET τc = 1/π2 |N − 1/2|. One can see from the data for N =0.45, 0.475 and 0.4875 that the starting points of crossover shifts as ∝ 1/|N − 1/2|
as expected.
The KT transition plays a crucial role in the relevance of the universal charge
relaxation resistance. If 2t + B > 0, the system scales to the weak barrier limit,
where τRC is independent of temperature. If 2t + B < 0, on the other hand, thescaling equations (4.79) predict s2 → const. and t ∝ T −B , so that τRC roughly
scales as ∝ T −1 (T 2B + const.), which grows faster than the (thermal) coherencetime τcoh ∝ 1/T as temperature is lowered. These observations suggest that if
2t + B > 0 coherent transport can be realized by lowering temperature to guarantee τRC < τcoh , while if 2t + B < 0 electronic transport in the dot decoheres before
charge relaxation is achieved. In the latter case, the quantum dot eﬀectively acts
as a reservoir and consequently the dynamical property of the system is governedby transport through the point contact between the two “reservoirs”. Therefore theV-dependent low-frequency resistance observed in the inset of Fig. 4.7 reﬂects therevival of the Landauer-type transport. To see this behavior more clearly, we plot
in Fig. 4.9 the product G(iωn → 0)R(iωn → 0) for K = 1/3 as a function of V.
In the strong barrier region, G(0) is ﬁnite and equal to [R(0)]−1 , which is a familiar property of transport through a point contact. Upon decreasing V, however,
G(0)R(0) is suppressed since G(0) decays to zero because of charging up, althoughR(0) → Rq is ﬁnite. Moreover, we see that the coherent region G(0)R(0) = 0 extends to larger V upon lowering temperature. Finally, we determine the phase4.6. SUMMARY

69

Figure 4.10: Schematic phase diagram in the degenerate case. Vc denotes thecritical backscattering strength for K = 1/3.

boundary of the coherent-incoherent transition by tracing the temperature dependence of the ratio τRC /τcoh = R(0)Cµ T . The above discussion suggests that there
exists a critical backscattering strength Vc , below which τRC /τcoh decays to zero,
while it diverges otherwise. (see Fig. 4.10). From the inset of Fig. 4.9, the criticalvalue is estimated as 4Vc τc ' 7.

4.6

Summary

In this chapter, we have studied dynamic response of a mesoscopic capacitor in
the presence of strong electron interactions. Unlike the prior theories based onmean ﬁeld type approximations, we have employed the bosonization technique totake into account interactions exactly. By combining perturbation theory, renormalization group, and the PIMC simulation, we treat whole parameter range of
dot-lead coupling. Our results show that the relaxation resistance for a dot connected to a Luttinger liquid is universal R = h/(2e2 K) as long as interactions are
suﬃciently weak. Below K < 1/2, this resistance is governed by the strengthof the dot-lead coupling: at the charge degeneracy point, there is a critical coupling strength, governed by a KT type phase transition, below which the dot acts
as an incoherent reservoir and the low-frequency resistance exceeds the universal
value. In this incoherent regime, the charge relaxation resistance cannot be deﬁned anymore due to the divergence of RC time. These behaviors are expected tobe observed experimentally in the same setup as in Ref. [9], if a suﬃciently highmagnetic ﬁeld ∼ 10 T is applied in order to realize the fractional quantum Halleﬀect at ν = 1/3.

SummaryIn this thesis, we have investigated theoretically transport phenomena in mesoscopic systems based on numerical and analytical approaches. Our main goal isto reveal how the charge relaxation resistance in a mesoscopic capacitor, which iscomposed of a quantum dot and a lead, is aﬀected by strong electron interactions.
We have formulated the mesoscopic capacitor using the bosonization technique inorder to treat exactly the interaction eﬀects, which has been so far handled only
approximately based on mean-ﬁeld type methods in prior studies. The bosonizedmodel is not only tractable analytically, but it has an expression suitable to numerical simulation based on the path-integral Monte Carlo (PIMC) method. In
addition, the cluster algorithm recently developed by Werner and Troyer in thecontext of resistively shunted Josephson junctions [18] is extensible to our numerical approach, which makes it possible to analyze the nonperturbative region.
In order to check the performance of the cluster algorithm, we have applied the
PIMC method to the single impurity problem in a spinful Tomonaga-Luttinger
liquid. The phase diagram we obtained at ﬁnite impurity strength has been compared with the results of renormalization group calculations in the weak and strong
impurity limits [32, 33].
To establish the PIMC method in the Coulomb blockade regime, we have investigate the static properties of a large quantum dot with spinful electrons in
Chap. 3. By measuring the capacitance of dot, we have conﬁrmed that the capacitance at the degeneracy point diverges logarithmically with decreasing temperature for arbitrary tunneling between the dot and the lead. This phenomenon is asign of the two-channel Kondo eﬀect and has been predicted by Matveev based on
the analytical calculations in the weak and strong tunneling limits [22, 23]. Our
results are consistent with a prior study with numerical renormalization group by
Lebanon et al. based on a tunneling model [64]. We have also conﬁrmed that
anisotropy between up and down spin channels due to the Zeeman eﬀect blurs thedivergence as analytically predicted by Le Hur [63].
In Chap. 4, which is the main part of this thesis, we have analyzed dynamic response of a mesoscopic capacitor using analytical calculations and the PIMC simulation. We have assumed spinless electrons and taken into account as two types
71

CHAPTER 4. CHARGE RELAXATION RESISTANCE

72

of interaction; short-range interactions described with the Luttinger parameter K
and long-range interactions leading to the charging energy EC in the quantum dot.
In the weak barrier limit, where the dot and the lead are strongly connected, we
have treated reﬂection strength V as perturbation up to second order and shownthat the charge relaxation resistance is quantized at h/2Ke2 irrespective of V, thegate voltage, or the charging energy. We have also shown that the electrochemicalcapacitance Cµ can be expressed as a serial combination of two capacitances asin the self-consistent scattering theory by Büttiker et al [19]. An important difference is that the capacitance associated to EC displays Coulomb oscillation in
our theory reﬂecting the exact treatment of electron interactions, while it does notin the latter. In the strong barrier limit V → ∞, on the other hand, we have performed perturbative renormalization group and shown that, if K < 1/2 and V issuﬃciently large, the charge relaxation time τRC diverges faster than T −1 as T → 0
due to the Kosterlitz-Thouless transition. As a result, one observes Landauer-type
dc resistance dependent on V, instead of the universal charge relaxation resistance.
In the intermediate regime of V, we have applied the PIMC method to the present
problem to conﬁrm the above predictions. For this purpose, we have contrived
a new scheme to measure the charge relaxation resistance in the PIMC method:
Calculates the linear admittance G(iωn ) and the electrochemical capacitance Cµ in
the PIMC simulation and substitute them toR(iωn ) =1
1−.
G(iωn ) ωnCµThen, the extrapolation iωn → 0 gives the quantized charge relaxation resistanceh/2Ke2 as long as the system lies in the normal phase. In the case of K < 1/2,
we have numerically observed abrupt increase in R(iωn → 0) upon increasing V.
We stress that this behavior can be observed experimentally in fractional quantum
Hall edges if the ﬁlling is equal to an inverse odd number and |u| < T .

Outlooks
Finally, we mention some possible extensions of the present work.
• By performing the analytic continuation iωn → ω + iδ numerically with,e.g., the padé approximation or the maximum extropy method, one can obtain the charge relaxation resistance at ﬁnite frequency. It is expected thatω-dependence of the charge relaxation resistance is characterized by K, reﬂecting the orthogonality catastrophe.
• The Kosterlitz-Thouless (KT) transition discussed in Chap. 4 also occurs
due to a noisy gate voltage. This situation can be realized by connecting the4.6. SUMMARY

73

gate to a transmission line with a ﬁnite impedance. Le Hur has discussed
the KT transition in such a quantum dot system from the viewpoint of the
Bose-Fermi Kondo model [70].
• Charge relaxation resistance in the Kondo regime can be investigated byextending the spinful model we have treated in Chap. 3 to a mesoscopic
capacitor with ﬁnite size. The relation between the Kondo eﬀect and spincharge separation has been studied by Glazman et al. [71]Appendix A
Chiral Luttinger liquid in amesoscopic capacitor
In this appendix, we formulate the fractional quantum Hall edge state with ﬁllingg realized along a mesoscopic capacitor in a perpendicular magnetic ﬁeld. For
simplicity, g = 1/m with an odd number m is assumed, which allows us to describe
the edge state in terms of a chiral Luttinger liquid (CLL) [72, 73, 74]. We consider
an edge state along a mesoscopic capacitor dipicted in Fig. A.1 and begin with the
Euclidean action for the bulk CLL∫
1
S 0 [φ] =dxdτ(∂ x φ)(i∂τ φ + v∂ x φ).
(A.1)
4πg
In our notation, the electron density is given by ρ = ∂ x φ/2π, and the ﬁeld operatorof an electron can be written in terms of the bosonic ﬁled φ in Eq. (4.20) as[]
Ψ(x) ∝ exp −iθ(x) + iφ(x) ,(A.2)

where θ(x) is the phase a quasiparticle gains in the range of (−∞, x], i.e., the sum
of kF x and the Peierls phase. On the point contact, we take into account tunnelingof a quasiparticle e∗ (= ge) between x = x1 and x = x2 with the tunneling matrix

Figure A.1: A mesoscopic capacitor and a chiral Luttinger liquid running along its edge. Electrons in the
dot region x1 ≤ x ≤ x2 is capacitively coupled to an
oscillating gate voltage Vg (t). A quasiparticle e∗ = getunnels between x1 and x2 , where the two edges become closest.
75

76APPENDIX A. CHIRAL LUTTINGER LIQUID IN A MESOSCOPIC CAPACITOR
element Γ0 . The action corresponding to this process is given by∫
S V [φ] = Γ0 dτ[Ψ† (x2 , τ)Ψ(x1 , τ) + H.c.]∫= V dτ cos[θ + {φ(x1 , τ) − φ(x2 , τ)}],

(A.3)
(A.4)

where V is a constant proportional to Γ0 and θ ≡ θ(x2 ) − θ(x1 ). Moreover, we add
the terms describing the charging eﬀect and coupling between the gate voltage
and the charges in the region x1 ≤ x ≤ x2 ;
[
]
∫
1
e
2
SC =
dτ
{Q(τ)} + Q(τ)Vg ,Q(τ) =[φ(x2 , τ) − φ(x1 , τ)]. (A.5)
2C
2π
In the rest of this appendix, we derive an eﬀective action from S 0 + S V + S C andshow that it is equivalent to Eq. (4.20).

A.1 Eﬀective action
The total action S 0 + S V + S C is quadratic in φ except for x = x1 and x = x2 .
Thus we can derive an eﬀective action by integrating out φ away from x1 and x2 in
the path integral picture as descried bellow. To perform this integration, we ﬁrst
replace the φ in the argument of S V and S C with local ﬁelds φ1 and φ2 as
∫
(1)
(2)
Z∝
DφDφ1 Dφ2 Dλ1 Dλ2 e−S 0 [φ]−S V [φ1 ,φ2 ]−S C [φ1 ,φ2 ]−S λ [φ,λ1 ,λ2 ]−S λ [φ1 ,φ2 ,λ1 ,λ2 ] .
(A.6)
Here the auxiliary ﬁelds λ1 and λ2 has been introduced to relate φ1 and φ2 indirectly to φ through the additional actions∑∫(1)
S λ [φ, λ1 , λ2 ] = idτλ j (τ)φ(x j , τ),
(A.7)
j=1,2

S λ [φ1 , φ2 , λ1 , λ2 ] = −i(2)∑∫dτλ j (τ)φ j (τ).

(A.8)

j=1,2

Now the routine work to do is to perform the Gaussian integrations in φ, λ1 and
λ2 , which appear in S 0 , S λ (1) and S λ (1) . So here we focus on these actions for awhile and express them in the Fourier space;
φ(x, τ) =1 ∑∑φ̃(k, ωn )eikx−iωn τ ,Lβ k ω
n(A.9)

A.1. EFFECTIVE ACTION

77

1∑
1∑
φ̃ j (ωn )e−iωn τ ,λ j (τ) =λ̃ j (ωn )e−iωn τ ,(A.10)
β ω
β ω
n
n
∫
1
1 ∑∑
S 0 [φ] =dxdτ 2 2k(iωm − vq)φ̃(k, ωn )φ̃(q, ωm )ei(k+q)x−i(ωn +ωm )τ
4πg
L β kq ω ω
φ j (τ) =n==

1 ∑ ∑ k(vk − iωn )Lβ k(,0) ωn∑∑1
Lβ k(,0) ωS λ (1) [φ, λ1 , λ2 ] = i

4πgm(A.11)|φ̃(k, ωn )|2(

Akn |φ̃(k, ωn )|2n∑ 1 ∑∑)k(vk − iωn )Akn == A−k,−n ,4πg

λ̃ j ∗ (ωn )φ̃(k, ωm )eikx j ,Lβ k ωn∑1∑
λ̃ j ∗ (ωn )φ̃ j (ωm ).
S λ (2) [φ1 , φ2 , λ1 , λ2 ] = −iβ ωj(A.12)(A.13)(A.14)

j(A.15)n

For simplicity, we write only the part related to each integration in the followings.
First, the Gaussian integral in φ can be calculated as∫(1)
Dφe−S 0 [φ]−S λ [φ,λ1 ,λ2 ](A.16)∫∑
 1 ∑ ∑ 




∗
2
ikx
j 


 (A.17)
φ̃(k,
ω
)|
λ̃
(ω
)
φ̃(k,
ω
)e
=
Dφ exp −
A
|
+
i


n
j
n
n
kn



Lβ k(,0) ω 
j
n



 1 ∑ ∑ 1|λ̃1 (ωn )eikx1 + λ̃2 (ωn )eikx2 |2 (A.18)∝ exp −Lβ k(,0) ω 4Aknn 1 ∑ ∑ 1[|λ̃1 (ωn )|2 + |λ̃2 (ωn )|2= exp −Lβ k(,0) ω 4Aknn

+ {λ1 (ωn )λ2 ∗ (ωn ) + λ1 ∗ (ωn )λ2 (ωn )} cos kL]−i{λ1 (ωn )λ2 (ωn ) − λ1 (ωn )λ2 (ωn )} sin kL] ,∗∗and the summations over k are estimated as
2πvg ∑
1 ∑ 1
1
=
2
2
L k(,0) 4Akn
L k>0 v k + ωn 2
∫
g ∞
πg
dk
=
=
,
v 0 k2 + (ωn /v)2 2|ωn |1 ∑ cos kL 2πvg ∑ cos kL=
L k(,0) 4Akn
L k>0 v2 k2 + ωn 2

(A.19)(A.20)(A.21)(A.22)

78APPENDIX A. CHIRAL LUTTINGER LIQUID IN A MESOSCOPIC CAPACITOR∫g ∞cos kLπg −|ωn |L/v
=
e
,
=
dk 2
v 0
k + (ωn /v)2 2|ωn |1 ∑ sin kL 2πigωn ∑sin kL=2
2
L k(,0) 4Akn
Lv
k[k + (ωn /v)2 ]k>0∫
sin kLigωn ∞iπg
dk 2(1 − e−|ωn |L/v ).
= 2
=
2
v
k[k
+
(ω
/v)
]
2ω
n
n
0
Then, the path integral over φ is summarized as∫(1)∴
Dφe−S 0 [φ]−S λ [φ,λ1 ,λ2 ] 1 ∑ πg {= exp −|λ̃1 (ωn )|2 + |λ̃2 (ωn )|2
β ω 2|ωn |(A.23)(A.24)(A.25)(A.26)n+ {λ1 (ωn )λ2 ∗ (ωn ) + λ1 ∗ (ωn )λ2 (ωn )}e−|ωn |L/v}]+{λ1 (ωn )λ2 (ωn ) − λ1 (ωn )λ2 (ωn )}sgn(ωn )(1 − e
)



 1 ∑ πg{|λ̃1 (ωn )|2 + |λ̃2 (ωn )|2 + 2B(ωn )λ̃1 (ωn )λ̃2 ∗ (ωn )} ,= exp −β2|ωn |∗∗−|ωn |L/v(A.27)(A.28)

ωn

where
B(ωn ) = e−|ωn |L/v+ sgn(ωn )(1 − e−|ωn |L/v

)

[L→∞
]
−→ sgn(ωn ) .

(A.29)

Next, the Gaussian integral in λ1 is performed in the same way;∫ i ∑
Dλ1 [Eq (A.28)] exp φ̃1 ∗ (ωn )λ̃1 (ωn )
(A.30)
β ω
n

∫
∑
 1
πg
{|λ̃1 (ωn )|2 + |λ̃2 (ωn )|2 }=Dλ1 exp −β ω 2|ωn |
n

}
{

1 ∑ πg∗∗B(ωn )λ̃2 (ωn ) − iφ̃1 (ωn ) λ̃1 (ωn )
−(A.31)β ω |ωn |n
}∑ |ωn | { πg 1 ∑ πg∗∗2
|λ̃2 (ωn )| +B(ωn )λ̃2 (ωn ) − iφ̃1 (ωn )= exp −β ω 2|ωn |2πgβ |ωn |
ωn
n
{
}]
πg
×
B(−ωn )λ̃2 (ωn ) − iφ̃1 (ωn )(A.32)|ωn |
 1 ∑ πg
= exp −{1 − B(ωn )B(−ωn )} |λ̃2 (ωn )|2
β ω 2|ωn |nA.1. EFFECTIVE ACTION79∑ |ωn |
i∑−
B(−ωn )φ̃1 ∗ (ωn )λ̃2 (ωn ) −|φ̃1 (ωn )|2  .
β ω
2πgβ
ω
n

(A.33)

nFinally, the Gaussian integral in λ2 leads to the Boltzmann factor in φ1 and φ2∫
 i ∑
Dλ2 [Eq. (A.33)] exp φ̃2 ∗ (ωn )λ̃2 (ωn )
(A.34)β ωn∫ 1 ∑ πg=
Dλ2 exp −{1 − B(ωn )B(−ωn )}|λ̃2 (ωn )|2
β ω 2|ωn |ni∑−{B(−ωn )φ̃1 ∗ (ωn ) − φ̃2 ∗ (ωn )}λ̃2 (ωn )
β ω
n


1 ∑
−
|ωn ||φ̃1 (ωn )|2 (A.35)
2πgβ ωn ∑ |ωn |1= exp −2πgβ 1 − B(ωn )B(−ωn )
ωn
{
}{
}
× B(−ωn )φ̃1 ∗ (ωn ) − φ̃2 ∗ (ωn ) B(ωn )φ̃1 (ωn ) − φ̃2 (ωn )


1 ∑
−
|ωn ||φ̃1 (ωn )|2 (A.36)
2πgβ ω
n


|ωn |
1 ∑
{|φ̃1 (ωn )|2= exp −2πgβ ω 1 − B(ωn )B(−ωn )n]−2B(−ωn )φ̃1 ∗ (ωn )φ̃2 (ωn ) + |φ̃2 (ωn )|2 }(A.37)
Recall the fact that the tunneling term S V and the charging term S C couple only
to the asymmetric combination of the local ﬁelds, i.e., φ1 − φ2 , which means thesymmetric combination φ1 + φ2 can be also integrated out. To see this, we redeﬁne
new bosonic ﬁelds and perform the last Gaussian integral in the symmetric ﬁeld;
φ1 = φ s + φa ,φ2 = φ s − φa ,2φa = φ1 − φ2 ,2φ s = φ1 + φ2 ,|φ̃2 (ωn )|2 + |φ̃2 (ωn )|2 = 2[|φ̃ s (ωn )|2 + |φ̃a (ωn )|2 ],φ̃1 (ωn )φ̃2 ∗ (ωn ) = [φ̃ s (ωn ) + φ̃a (ωn )][φ̃ s ∗ (ωn ) − φ̃a ∗ (ωn )](A.38)(A.39)(A.40)= |φ̃ s (ωn )|2 + φ̃ s ∗ (ωn )φ̃a (ωn ) − φ̃ s (ωn )φ̃a ∗ (ωn ) − |φ̃a (ωn )|2 , (A.41)φ̃1 ∗ (ωn )φ̃2 (ωn ) = [φ̃ s ∗ (ωn ) + φ̃a ∗ (ωn )][φ̃ s (ωn ) − φ̃a (ωn )](A.42)= |φ̃ s (ωn )|2 − φ̃ s ∗ (ωn )φ̃a (ωn ) + φ̃ s (ωn )φ̃a ∗ (ωn ) − |φ̃a (ωn )|2 , (A.43)

∫

1 ∑
|ωn |
Z=
Dφ s Dφa exp −(A.44)
2πgβ
1 − B(ωn )B(−ωn )ωn80APPENDIX A. CHIRAL LUTTINGER LIQUID IN A MESOSCOPIC CAPACITOR
× [{2 − B(ωn ) − B(−ωn )}|φ̃ s (ωn )|2 + {2 + B(ωn ) + B(−ωn )}|φ̃a (ωn )|2 (A.45)]+ 2{B(ωn ) − B(−ωn )}φ̃a ∗ (ωn )φ̃ s (ωn )] e−S V [φa ]−S C [φa ](A.46)∫
1 ∑ 2 + B(ωn ) + B(−ωn )=Dφa exp −|ωn ||φ̃a (ωn )|2(A.47)
2πgβ ω 1 − B(ωn )B(−ωn )
n


1 ∑
{B(ωn ) − B(−ωn )}2 |ωn ||φ̃a (ωn )|2 e−S V [φa ]−S C [φa ]−2πgβ ω {2 − B(ωn ) − B(−ωn )}{1 − B(ωn )B(−ωn )}n(A.48)
∫
 1 ∑
2
|ωn ||φ̃a (ωn )|2  e−S V [φa ]−S C [φa ]=Dφa exp −πgβ
2 − B(ωn )B(−ωn )
ωn



∫
 1 ∑

|ω
|
n
2
 e−S V [φa ]−S C [φa ] .
=Dφa exp −
|
φ̃
(ω
)|
a
n

−|ω
|L/v
n
πgβ
1−e

(A.49)(A.50)

ωnThen we arrive at the eﬀective action in terms of φa
∫
|ωn |
1 ∑
2
|φ̃a (ωn )| + V dτ cos[θ + 2φa (τ)]
S eﬀ =
πgβ ω 1 − e−|ωn |L/v
n
[
]
∫
eVg
1 e2
2
+ dτ 2{φa (τ)} −φa (τ) ,π 2C
π(A.51)

which is clearly equivalent to the eﬀective action in the non-chiral case [see Eq. (4.20)].

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