不交叉近似法於雙渠道贗能隙安德森模型之應用:量子臨界與統一標度律

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國立交通大學碩士論文

Two Channel Kondo Pseuodap

Anderson Model: Quantum

Criticality and Universal Scaling

Student: 巫展霈

Advisor: 仲崇厚

Committee:

牟中瑜

陳柏中

朱仲夏

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不交叉近似法於雙渠道贗能隙單雜質安德森模型之應用:

量子相變與統一標度律

學生: 巫展霈指導教授: 仲崇厚

國立交通大學電子物理系碩士班理論物理組摘要

所謂的量子相變 (quantum phase transition) 是指在絕對零度經由調整某些參數所以引起的基態的連續相變。然而這些相變是因為量子擾動而非熱擾動。另一方面_, 近藤效應 _(Kondo effect) 也是凝態物理中的重要課題, 它是一個磁性雜質被電子自旋屏蔽而引起的物理現象。因為奈米科技的進步,近藤效應在量子點(quantum dot) 系統的想法得以實現,也因此,與近藤效應破滅相關的量子臨界現象(quantum criticality) 也變成了重要的課題。如果我們可以讓磁性雜質同時和兩個獨立的電子庫耦合, 雙渠道的近藤效應 (two-channel Kondo) 就能被實現。雙渠道的近藤效應會造成非費米液體行為 (non-Fermi liquid), 這是有別於一般費米金屬的行為。本論文中,我們使用的是雙渠道贗能隙安德森雜質模型(two-channel pseudogap single

impu-rity Anderson model), 所謂的贗能隙能態密度 (pseudogap density of states) 就是能態密度

(ρ(ω) ∼ |ω|r_{, 0 ≤ (r ≤ 1)} _在接近_{w = 0}_{時會呈現冪次方的消逝。然而贗能隙能態密度的指數} _r

也分別對應不同的磁性物質參雜材料, 例如加入磁性雜質的石墨稀(doped graphene)就是對應

r=1, 而常數能態密度則對應 r=0。如果指數 r過於大, 能態密度會消失過快而沒有足夠的電子去屏蔽雜質,因此雙渠道的近藤效應會破滅,取而代之的是矩限態(local moment, LM).只要r

夠小的話, 系統就會停留在雙渠道的近藤效應基態 (two-channel Kondo ground state)。我們研究雙渠道贗能隙安德森雜質模型 _{(0 ≤ (r ≤ 1)} 在化學能為零 (µ0 = 0) 的情況下, 靠著調整

指數 r來觀察雙渠道近藤效應和矩限態的量子相變。我們只用slave-boson大N 近似法(large

N approach) 去自洽的解雜質電子的格林函數,其中我們忽略所有有不交叉的費曼圖,所以又稱

不交叉近似法 (NCA)。從磁性雜質的能態密度, 我們可以觀察到量子臨界點 (r = rc), 更進一

步的,同時在平衡態和非平衡的導電度 (conductance) 上找出統一標度律 (universal scaling)。本論文可以提供未來研究非費米液體及量子臨界行為的理論基礎。關鍵字: 近藤效應、雙渠道近藤效應贗能隙、安德森雜質模型量子相變、統一標度律不交叉近似法

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Non-crossing Approximation Approach To Two Channel Kondo Pseudogap Anderson Model:

Quantum Criticality And Universal Scaling

Student: Jan-Pei Wu Advisor: Prof. Chung hou Chung

Institute of Electric Physics Department National Chiao Tung University

ABSTRACT

Quantum phase transition (QPT) are the continuous phase transition of ground states by tuning couplings in the quantum system. They are due to zero-temperature quantum fluctuations, not thermal fluctuations. Meanwhile, Kondo effect is an important phe-nomenon in condensed matter systems, which is an effect describing the screening of magnetic impurity by the spin of conduction electrons in magnetical doped metals. Due the advances in nano-technology, Kondo effect in quantum dots (QD) have been realized in single electron tunneling transistor (SET), therefore QPT associated with the broken down of the Kondo effect becomes an interesting subject. If two independent electron reservoirs exist, two-channel Kondo effect (2CK) becomes possible. It leads to non-Fermi liquid (NFL) behavior, which shows different electric transport from Fermi liquid metals. In our study, we use the 2CK pseudogap Anderson impurity model to describe the system where the single impurity is coupled to 2CK pseudogap electron bath, where its density

of states (ρ(ω)) vanishes in a power law fashion (ρ(ω) ∼ |ω|r_{, 0 ≤ r ≤ 1) for ω → 0. The}

exponent r of pseudogap density of states varies with different materials. The magnetical doped graphene (r=1) is an example of 2CK pseudogap Anderson single impurity model system, and two-channel quantum dot system with constant density of states corresponds to r = 0 case. If r is too large, there is no sufficient electron density of states to screen the impurity spin, 2CK state is broken down, resulting in unscreened local moment (LM) ground state. Let r be small enough, two-channel Kondo effect becomes possible. We study QPT in 2CK pseudogap single impurity Anderson model at zero-chemical potential by tuning r (0 < r < 1) both of equilibrium and out of equilibrium. We use slave-boson large-N approach to self-consistently solve Green’s functions of electron on the dot by including all non-crossing diagrams, so-called non-crossing approximation (NCA). We

extract the quantum critical point (rc) from impurity density of states, and find the

uni-versal scaling both in equilibrium and non-equilibrium conductances near rc. This thesis

provides theoretical basis for further study in Kondo break down, quantum criticality and non-Fermi liquid behavior in condensed matter systems.

Keywords: Quantum Criticality, Quantum Phase Transition, Kondo, two-channel Kondo, Pseudogap, Anderson Model, Universal Scaling

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誌謝: 修讀碩士的年頭,首先要感謝指導仲崇厚教授,願意教授學生,傳授知識。另外也要感謝李宗翰學長留下 NCA 數值計算的基礎,讓我得以繼續研究相關課題。在編寫論文的過程中, 也特別感謝張永業學長對我用 Latex 編寫論文的指導, 還有教授場論基礎。研究室的每一位學長學弟學妹對我的許多幫助,我都十分感激。要感謝大家在我碩士生涯的幫助, 還有最重要的是我的父母把我養大,支持我唸研究所,支付我從小到大的學費,感恩的心,到死都不敢忘。最後感謝國立交通大學對我的栽培,以及完善的教學資源。

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List of Figures

1.1 (a) Redline shows resistivity in metals contained magnetic impurities: at low temperature, using third order perturbation theory, Kondo found that this scattering process leads to a lnT behavior in resistivity. Blueline shows resistivity in normal metals. (b) In quantum dot system, Coulomb blockade influence conductance. At low temperature, when temperature decreases, conductance increases (decreases) if electron number is odd (even). Kondo effect appears only odd number, Kondo effect leads to conductance

in-creased at low temperature in QD system. Adapted from [1, 7]. . . 2

1.2 (a) Kondo effect happened because of impurity screened by spin one half

of electrons. Adapted from [58]. (b) Electron-impurity spin-flip scattering. 3

1.3 (a) Scanning electron micrograph of SET device. The top and down elec-trodes on the left side and the electrode on the right side are used to operate the barrier of quantum dot. The middle electrode on the left side are used to tune the energy level of QD relative to 2DEG [13, 14]. (b) Schematic

SET device [21]. . . 5

1.4 (a) Conductance has two different behaviors at even and odd electron

num-ber. (b) The zero bias differential conductance anomaly at VSD ∼ 0, where

VSD is the voltage difference between source and drain. [1, 13, 14]. . . 6

1.5 The illustration of Coulomb blockade. The first order tunneling is blocked by Coulomb potential U, the cotunneling (2nd) solution is solvable. The

additional energy for N to N + 1 state is E2_{/C + ∆E. The voltage spacing}

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LIST OF FIGURES

1.6 (a) Spin-flip cotunneling process of Kondo effect. (b) The density of states(DOS) of quantum dot, The Kondo resonance lies at the Fermi energy, Kondo

ef-fect occurs when the temperature is below the Kondo temperature Tk.

Adapted from [21]. . . 9

1.7 (a) The first order phase transition. Though gc, the ground state becomes

B state from A state with level crossing. (b) The higher order phase tran-sition, avoiding level crossing. The phase transition whose ground state is

form A state to B state as g exceeds gc is a continuous process. Ref. [36] . 10

1.8 (a) The diagram is obtained via the scanning electronic micrograph (STM) in experiment of N. J. Craig et. al. [37]. (b) Conductance of left quantum dot. When odd number of electrons on the quantum dot and RKKY anti-ferromagnetic interaction is stronger than Kondo coupling, Kondo effect will be suppressed by RKKY. Adapted from [37]. . . 11 1.9 The quantum phase transition coupling K is RKKY coupling, and quantum

critical point (QCP) with criticality at zero temperature is located at the

Kc. Adapted from [39]. . . 12

1.10 Quantum phase transition diagram of two-channel Kondo, where H2CK ∼

J1S1(r) · S + J2S2(r) · S, and J1, J2 are Kondo coupling of each channel,

respectively. Blue and red phase correspond to blue and red channel in

Fig.(1.11), and J1 (J2) is the Kondo coupling of red (blue) channel. At zero

temperature, one can obtain 2CK state for symmetric coupling. There is still the 2CK state if coupling are some imbalance at finite temperature. The quantum critical region exhibits 2CK non-Fermi liquid behavior at

finite temperature. [38] . . . 13

1.11 Two independent electron reservoirs couple to a QD in SET device. Two blue leads and a finite red reservoir represent two independent channel, respectively. Here, red reservoir has to be much large than QD, therefore, there are sufficient electrons to compose a conduction electron band. Two

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LIST OF FIGURES

1.12 2CK conductance data in 2CK and 1CK scaling, where 1CK scaling follows

Fermi liquid (eV /kBT )2behavior. 2CK universal scaling follows (eV /kBT )1/2

behavior, not Fermi liquid (eV /kBT )2, so-called NFL behavior. Adapted

from [6, 38]. . . 16 1.13 The RG flow phase diagram of particle-hole symmetric single impurity

pseudogap Anderson model. The details are in text. Adapted from [43]. . . 18 1.14 The RG flow phase diagram of particle-hole asymmetric single impurity

pseudogap Anderson model. The details are in text. Adapted from [43]. . . 20 1.15 The RG flow phase diagram of 2CK particle-hole asymmetric single

impu-rity pseudogap Anderson model. The details are in text. Adapted from [44]. . . 21 1.16 The RG flow phase diagram of 2CK particle-hole asymmetric single

im-purity pseudogap Kondo model. The details are in text. Adapted from [44]. . . 23 2.1 The set sup of 2CK pseudogap Anderson model out of equilibrium with

strong Coulomb potential. There are two 2CK leads which couple to sin-gle QD out of equilibrium, where 2CK leads with pseudogap conduction electron DOS are in thermal equilibrium, respectively. . . 27 2.2 At lowest order, boson self-energy involves the fermion propagator, and

fermion self-energy involves boson propagator. Adapted from [57]. . . 31 2.3 Diagrammatic self-consistent Dyson’s equations, Eq.(2.38)-Eq.(2.41). Adapted

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LIST OF FIGURES

2.4 The impurity DOS ρσ(w) for the r=1 2CK pseudogap Anderson model both

in equilibrium and non-equilibrium system, where the magnetic impurity symmetrically coupled to leads of Lorentzian bandwidth 2W and chemical

potential µR and µR. Note that: in our study, we use D to replace W to

be half-bandwidth. Here, all energy is units of Γ, the coupling constant of leads and dot, the half-bandwidth at half-maximum is W = 100 and temperature T = 0.005. The dashed cure represents out of equilibrium impurity DOS, Kondo peak splits into two suppressed peaks. The p-h symmetric impurity DOS is shown as the solid line, where single Kondo peak is at w = 0. Adapted from Ref. [57]. . . 37

2.5 The scaling is characteristic of a Fermi liquid when (eV /KBTk)2 plot. The

conductance does not follow a linear behavior, when (eV /kBTk)0.5 plot.

Therefore, rule out spurious effects in the real 2CK behavior for type ”π” Co impurities on graphene. Adapted from [6]. . . 38 2.6 The quantum phase transition in graphene can be investigated by

control-ling the chemical potential µ0. When µ = 0, there is no 2CK state, always

in LM state. If µ 6= 0, the quantum phase transition between LM state

and 2CK can be observed. Adapted from [41]. . . 39

2.7 (a) The phase diagram for 2CK-LM crossover QPT, parameter j represents

coupling Γ or chemical potential µ, and j∗ _{is the crossover scale for a fixed}

temperature T0 = 5 × 10−7D. All parameters in units of half-bandwidth

D=1, and Tk and T0 are energy scales associated with the 2CK universal

scaling. (b) A spin one half impurity couples symmetrically to the two sub-lattices of two graphene leads out of equilibrium, where two graphene leads are in thermal equilibrium, respectively. (c) Doped graphene in momentum space. Adapted from [60]. . . 40

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LIST OF FIGURES

2.8 The impurity DOS via NCA v.s. different chemical potential µ in units of D. Dirac point is located at w/D = 0, and Kondo peaks are pinned near each µ. Different levels w/D with different chemical potentials µ correspond to

different Fermi levels, respectively. The parameter are T0 = 5 × 10−7D,

Γ = 0.2D, ǫd = −0.2D, where D is half-bandwidth. T0 is temperature of

the system, Γ is coupling constant of leads and dot, and ǫd is dot level.

Adapted from [60]. . . 42 2.9 (a) The linear conductance in equilibrium system. It show square root

behavior at temperature which is lower than Kondo temperature. (b) There is an additional power law behavior at high temperature in equilibrium

conductance scaling. It shows T∗ _{∼ |Γ−Γ}∗_|1/µ_{, where Γ}∗_{is a small number,}

0.05D and µ ∼ 0.1. (c) Nonlinear conductance at Γ = 0.2D. The fixed

parameters are T0 = 5 × 10−7D, µ = −0.1D, ǫd = −0.2D, where D = 1 is

half-bandwidth. (d) The (eV /kBT )2 behavior arises as (eV /kBT ) << T .

(e) The (eV /kBT )

1

2 behavior arises as (eV /k_BT ) is between T and T_K.

Adapted from [60]. . . 43 3.1 The phase diagram of equilibrium and non-equilibrium conductance. The

phase diagrams delineated by T∗ _{and V}∗_{, which is crossover scale in}

equi-librium and out of equiequi-librium, respectively. The inverse of crossover scale is correlation length, ξ, which is diverge in quantum critical region, and

νT = 4, νV = 0.5 are universal factor called as correlation length exponent.

Due to νT, νV are different, the universal scaling in equilibrium is different

from out if equilibrium. Here, critical point is at and near rc = 0.115, even

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LIST OF FIGURES

3.2 The r=0.05 density of states of impurity non-equilibrium system with Γ =

0.3D, ǫd= −0.3D, T = T0 = 5 × 10−7D, where D = 1. The impurity DOS

exhibits asymmetry graph, where no peak at d + U due to particle-hole

asymmetry. The non-equilibrium density of states with bias, V = 0.038D, Kondo peak divided into two peaks, because the Fermi energy of left and

right leads are different, the width between two peaks is equal to bias. . . 46

3.3 The Kondo peak of impurity spectral function with Γ = 0.28D5, ǫd =

−0.2D, and T0 = 5 × 10−7D by varying exponent r. (a) The full impurity

DOS with p-h asymmetry. (b) Kondo peaks for different r in Fig.(3.3.a). . 48 3.4 The conductance v.s. T with varying r in equilibrium system , where µ = 0,

Γ = 0.28D, ǫd= −0.2D are fixed, and the half-bandwidth D=1. . . 49

3.5 To illustrate G(0, T ) − G(0, 0) = BkT

1

2 scaling for 0 < r < 1, but we can’t

get a reasonable result. The √T behavior of differential conductance is

obvious for small r, but it gradually disappears as r increase. Here, Td is

the non-universal factor, and µ = 0, Γ = 0.28D, ǫd = −0.2D, D = 1. . . 50

3.6 To illustrate σT v.s. |r − rc| plot with µ = 0, Γ = 0.28D, ǫd = −0.2D,

whereD = 1. . . 51

3.7 The universal scaling in G(0,T) is given by Eq.(3.1) with Γ = 0.28D,

ǫd= −0.2D and µ = 0, whereD = 1. Setr = 0.115 is the critical point. . . . 52

3.8 Crossover scale T∗ _{v.s. r − r}

c plot with Γ = 0.28D, ǫd= −0.2D and µ = 0,

whereD = 1. . . 53

3.9 Non-equilibrium conductance of 2CK pseudogap Anderson model. G(V,T)

saturates at a constant value for V << T , other parameters are T = T0 =

5 × 10−7_{D, ǫ}

d = −0.3D and Γ = 0.3D, D = 1. The bias voltage V is in

unit of half-bandwidth D. . . 54 3.10 Scaling of non-equilibrium conductance, G(V,T) follows Eq.(2.53). The

parameters are set as T = T0 = 5×10−7D, ǫd = −0.3D and Γ = 0.3D, D =

1. The scaling exhibits the square root and square behavior of eV /kBT ,

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LIST OF FIGURES

3.11 σV v.s. |r − rc| plot with µ = 0, Γ = 0.3D and ǫd= −0.3D. We set D=1. . 56

3.12 The non-equilibrium universal scaling in conductance for different r at low

temperature, T = 5 × 10−7_{D. we set dot level ǫ}

e = −0.3D, Γ = 0.3D, and

D=1. . . 56

3.13 Crossover scale V∗ _{v.s. |r − r}

c| plot with µ = 0, Γ = 0.3D and ǫd = −0.3D.

We set D=1. . . 57 4.1 This figure depicts the equilibrium thermal transport and Fermi function.

Red line represents T 6= 0 Fermi function, where particles may be excited by thermal energy. . . 60 4.2 The non-equilibrium transport in our system depends on current. The

Fermi function for non-equilibrium case is different from the one in equi-librium because of the difference between chemical potential of the left and right leads. Fermi function of the left and right are different. The Fermi

function of non-equilibrium system shows a jump fL−fR across the impurity. 62

4.3 The three dimensional phase diagram to describe QPT between LM and

2CK both in equilibrium (G(0,T)) and out of equilibrium (G(V,T0)). The

quantum criticality can be accessed either by G(0,T) in equilibrium and

or by G(V,T0) out of equilibrium. The equilibrium and non-equilibrium

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Chapter 1 Introduction

1.1 Kondo Effect

In 1930s, there was an important discovery in solid state physics, where the resistivity in some non-magnetic metals with magnetic impurities manifest itself a minimum at a certain temperature shown as Fig.(1.1.a). This important phenomenon was known as Kondo effect. There are many review papers, books, thesis and related information about Kondo effect [1, 2, 3, 4, 5, 6]. Due the advances in nano-technology, Kondo effect was applied in quantum dot systems, therefore Kondo effect shows different characteristic in bulk system compared to quantum dot (QD) system. The conductance in a QD system as shown in Fig.(1.1.b). We will illustrate the phenomenon of the Kondo effect in bulk system below [1, 5, 7], then introduce Kondo effect in QD system in Section 1.1.2. The original Kondo phenomenon cannot be explained by the scattering theory between electron and phonon. J.Kondo successfully explained this phenomenon in 1964 by spin-flip scattering as shown in Fig.(1.2) [8]. Theoretically, a magnetic impurity is screened by the spin of nearby conduction electrons, leading to a spin singlet, so-called Kondo

singlet. The Kondo singlet appears when T < Tk, Tk is defined as Kondo temperature.

The size of Kondo cloud made of conduction electrons defined as ξ0

k ∼ khνBTFk, where νF is

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1.1. KONDO EFFECT

Figure 1.1: (a) Redline shows resistivity in metals contained magnetic impurities: at low temperature, using third order perturbation theory, Kondo found that this scattering pro-cess leads to a lnT behavior in resistivity. Blueline shows resistivity in normal metals. (b) In quantum dot system, Coulomb blockade influence conductance. At low tempera-ture, when temperature decreases, conductance increases (decreases) if electron number is odd (even). Kondo effect appears only odd number, Kondo effect leads to conductance increased at low temperature in QD system. Adapted from [1, 7].

Kondo effect can be conveniently described by the s-d model proposed by Zener [9], where

magnetic moments carry spin Sd coupling to conduction electrons via Js · Sd, where the

exchange interaction J was called as Kondo coupling (J > 0), Sd is the impurity spin,

and s is spin of conduction electrons. The s-d model can be related to Anderson model at certain parameters regime, presented by P.W. Anderson [5, 6, 11]. Coqblin-Schrieffer transformation can transform Anderson model to Coqblin-Schrieffer model, also called as Kondo model [5, 10, 12]. Note that perturbation theory used to explain Kondo effect

only for T > TK. It breaks down for T < TK, one needs Numerical Renormalization

Group (NRG) to resolve this issue. More theoretical calculations will be introduced in this chapter.

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1.1. KONDO EFFECT

Figure 1.2: (a) Kondo effect happened because of impurity screened by spin one half of electrons. Adapted from [58]. (b) Electron-impurity spin-flip scattering.

1.1.1 Kondo Hamiltonian

From previous section, we know that the magnetic moment of impurity is screened by conduction electrons, leading to Kondo effect. In this section, we provide a

mathemat-ical description of the Kondo effect. We start from a single impurity S = 1₂ Anderson

Hamiltonian [5, 6]. H =X σ ǫdd†σdσ + X kσ ǫ(k)c†_kσckσ+ Un↑n↓+ X kσ (Vkσd†σckσ+ H.C.) (1.1)

The first term in Eq.(1.1) describes the local moment state of impurity with energy ǫd,

where σ is the spin index. The second term describes conduction electrons, where k is the momentum space of Fermi sea. The third and fourth term are the on-site Coulomb repul-sion and the hopping between the leads and the dot. The Coulomb repulrepul-sion potential U is the energy cost for the localized state occupied by two electrons (of opposite spins). From

Anderson model, If ǫd< ǫF (which is the Fermi energy of the metal) and ǫd+ U > ǫF, the

single occupied site will have a net spin-1/2. By using Coqblin-Schrieffer transformation, Anderson model Hamiltonian can be mapped onto the Kondo Hamiltonian in U >> V regime [5, 12]: H ≃X kσ ǫ(k)c†_kσckσ+ X k,k′ Jkk′s_kk′ · S_d, (1.2)

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1.2. KONDO EFFECT IN QUANTUM DOT SYSTEM

where Jk,k′ represents the anti-ferromagnetic coupling which is given by:

Jkk′ = V_kσV_k∗′_σ[ 1 ǫd− ǫk+ U + 1 ǫ′ k− ǫd ], (1.3)

where ǫk is energy level of conduction electrons, ǫd is energy level of quantum dot, skk′ is

the spin of conduction electrons, and Sdis an impurity spin. Because the Kondo resonance

lies exactly at the Fermi energy, the contribution of exchange process is only from states

around Fermi energy(ǫF). So we set that ǫd ≡ ǫd−ǫF, ǫk ≡ ǫk−ǫF and ǫk ∼ 0. Since ǫd< 0

and the Coulomb potential U is larger than ǫd, Jkk′(∼ J_k_F_k_F) is the anti-ferromagnetic

coupling(J_kk′ > 0). For U >> |ǫ_d|, the expression reduces to:

JkFkF ≡ J ∼ − |V2 kσ| ǫd = − |V2 kσ| ǫd− ǫkF > 0. (1.4)

The second term of Eq.(1.2) is precisely the s-d interaction term written as Hex = J(r) ·

S. It describes the spin exchange between an impurity and the surrounding conduction electrons. The distance r is measured from the impurity site to conduction electrons. This simple model explains the Kondo problem that resistivity of metals in magnetic impurity bulk system at low temperature will increase logarithmically. We will discuss Kondo effect in QD system in Section 1.1.2 and two-channel Kondo in Section 1.1.4.

1.2 Kondo Effect In Quantum Dot System

Due to the progress of science and technology, scientist can fabricate semiconductor struc-ture under nano-mater scale. The advance of micro-fabrication and cooling technology make a chance that we can research Kondo effect in nano-size system. Kondo physics can be realized in a tunable quantum dot (TQD). TQD is made by the single electron tunneling transistor (SET) with two dimensional electron gas (2DEG) heterostructure, as shown in Fig.(1.3) [1, 18, 21]. SET device has GaAs/AlGaAs layer and multiple elec-trodes, where three gate electrodes on left and the other one on right in the picture. Then GaAs layer confines 2DEG repelled by electrodes, and induces two tunneling junc-tions under and above it. A metallic island is confined between two tunneling juncjunc-tions called ”quantum dot (QD)”. One of early experiments of Kondo effect in QD system was

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Figure 1.3: (a) Scanning electron micrograph of SET device. The top and down electrodes on the left side and the electrode on the right side are used to operate the barrier of quantum dot. The middle electrode on the left side are used to tune the energy level of QD relative to 2DEG [13, 14]. (b) Schematic SET device [21].

made by D. Gordhrber-Golen et.al in 1998 [13, 14, 17]. The conductance in QD systems is quantized by Coulomb blockade (CB) oscillations, leading to the difference of Kondo effect between QD systems and bulk systems.

1.2.1 Coulomb Blockade Oscillations With Kondo Effect

The original Kondo physics (in bulk system) was introduced in last section, where Kondo effect induce resistivity to be enhanced. This section, we will discuss the Kondo effect in QD system, which is very different from bulk system. In the single electron tunneling transistor (SET) device, the Coulomb blockade (CB) oscillations affects the conductivity [21, 22] as shown in Fig.(1.4). In Fig.(1.4.a), it exhibits conductance increasing as odd

number in quantum dot (blue line), and there are peaks at VSD = 0 in Fig.(1.4.b). The

peaks at VSD = 0 are temperature dependent, as temperature is lower, peak is higher.

These peaks are due to Kondo effect, so-call Kondo peaks. We will introduce CB and and briefly discuss the results in theory and experiment, then discuss Kondo effect in SET.

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Figure 1.4: (a) Conductance has two different behaviors at even and odd electron number.

(b) The zero bias differential conductance anomaly at VSD∼ 0, where VSD is the voltage

difference between source and drain. [1, 13, 14]. Coulomb Blockade

Coulomb blockade oscillation appears due to strong Coulomb potential system, that’s why it called as Coulomb blockade. In the SET device, Coulomb potential affect the electron tunneling between leads and dot, illustrated in Fig.(1.5). The first order tunneling is blocked by the Coulomb blockade, where U is Coulomb potential between the electron of quantum dot and lead. The CB Hamiltonian related to constant interaction model was constructed as [21, 22]:

H_DCB =X

µ

ξµc†µcµ+ E(N), E(N) = ECN2− eVgN (1.5)

, where E(N) is the interaction term of the system, including gate voltage term as eVgN,

where gate voltage leads to a electric field that increases the energy of dot electrons. In general, we have to simultaneously think about source, drain, and gate voltage. Here we

lump all terms in a gate voltage term. The energy Ec = e

2

2C, where C is the capacitance of

a single electron. The bias voltage leads to µL≥ µdot ≥ µR, and the bias voltage difference

between the left lead (source) and the right lead (drain) was defined as VSD. Because of

the inequality of chemical potential, electrons can flow, therefore an added electron from

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Figure 1.5: The illustration of Coulomb blockade. The first order tunneling is blocked by Coulomb potential U, the cotunneling (2nd) solution is solvable. The additional energy

for N to N + 1 state is E2_{/C + ∆E. The voltage spacing between source and drain is}

defined as VSD which is eV here.

drain, leading dot energy from Edot(N) to Edot(N − 1). The tunneling occurs when

αeVg(N) = E(N + 1) − E(N), (1.6)

where α = Cg/Cis the ratio of gate capacitance to total capacitance, called ”gate

cou-pling”. The αeVg(N) is similar to chemical potential of quantum dot, which is

eVg(N) ≡ µdot(N) = Edot(N) − Edot(N − 1) = (n −

1

2)

e2

C − eVg+ EN, (1.7)

the remain terms defined as EN. The additional energy is given by ∆µdot, where

∆µdot ≡ µdot(N) − µdot(N − 1) =

e2

C + EN − EN −1 =

e2

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The irregular spacing of the single electron levels is defined as ∆E. When charging energy

e2

C is much larger than ∆E, CB oscillations is dependent on it. The peak spacing of CB

as a function of gate voltage is given by

∆Vg = ∆µ(N)/eα = (e/c2+ ∆E)/eα, (1.9)

while condition Eq.(1.6) gives the gate voltage of N-th Coulomb peak. We take differential of E(N) in Eq.(1.5) with respect N, we obtain the optimum number of particles,

Nopt= eVg/2Ec (1.10)

When optimum number is

• Integer: There is an energy gap for adding electrons.

• Half-integer: There are two degenerate charge states, then electrons can transit. As shown in Fig.(1.4.a), it exhibits difference results indicated different number of electron on quantum dot, and Fig.(1.4.b) exhibits non-equilibrium differential conductance

with anomaly behavior at VSD ∼ 0 [1]. The electron number can be changed by gate

tuning. The CB oscillation affect the conductance by electron number, then we discuss how Kondo effect can overcome CB in QD.

Kondo Effect In Quantum Dot System

The tunable QD is sometimes similar to individual artificial magnetic impurity, which

leads to Kondo screening for (T < Tk). When optimum number is odd, the QD with

a single electron which is occupying the top-most quantum state, which is similar to a magnetic impurity. In other words, Kondo screening occurs with a single spin-degenerate

energy state ǫd, no Kondo effect when optimum number is even. Kondo effect appears

when (T < Tk), where the Kondo temperature Tk found to be:

TK = [UΓ]

1

2e−π(µ−ǫd)2Γ , (1.11)

where (ǫd) is dot level, (Γ) is a coupling between leads and dot, and chemical

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1.3. QUANTUM PHASE TRANSITIONS (QPT)

Figure 1.6: (a) Spin-flip cotunneling process of Kondo effect. (b) The density of

states(DOS) of quantum dot, The Kondo resonance lies at the Fermi energy, Kondo

effect occurs when the temperature is below the Kondo temperature Tk. Adapted from

[21].

illustrated in Fig.(1.6), the first order tunneling is blocked by the Coulomb blockade, second order tunneling (cotunneling) leads to Kondo screening with spin-flip exchange. A narrow-resonance is seen in the density-of-states (DOS) of the QD. In summary, the system with tunable tunnel coupling to the leads, there is CB oscillation affected

con-ductivity by electron number when T > Tk. When T < Tk, we have to think about the

coupling between Kondo effect and Coulomb blockade. The odd electron number on the

dot provides a single spin-degenerate state ǫd, which is a single electron with spin up or

down. This condition leads to Kondo screening with spin-flip exchange, and enhanced conductance. The conductance of Kondo effect in QD system with even and odd number is shown in Fig.(1.1.b).

1.3 Quantum Phase Transitions (QPT)

Quantum phase transition (QPT) are the continuous phase transition of ground states at T = 0 by tuning the sets of the system parameters in Hamilton [36]. The energy level

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Figure 1.7: (a) The first order phase transition. Though gc, the ground state becomes B

state from A state with level crossing. (b) The higher order phase transition, avoiding level crossing. The phase transition whose ground state is form A state to B state as g

exceeds gc is a continuous process. Ref. [36]

diagrams as a function parameter g are shown in Fig.(1.7). H(g)=H0+gH1, where H0 and

H1commute to each other, and g is the coupling constant. As shown in Fig.(1.7.a), ground

state is state A when the coupling g is below gc, but it becomes state B when g > gc,

where gc is the critical point of the coupling g. This is the first order phase transition,

which is a level crossing. Fig.(1.7.b) shows the 2nd order phase transition, which is a

continuous phase transition from g < gc to g > gc, avoiding level crossing. QPT is the

phase transition as shown in Fig.(1.7.b), continuous process and without level crossing at zero temperature. The quantum critical behaviors of QPT systems exhibit divergence in

correlation length ξ ∼ |g − gc|−ν. Here, the correlation length exponent ν is a universal

factor. This leads to universal power-law scaling behaviors in all thermal dynamical observable. These behaviors in quantum critical regime cannot be described in Fermi liquid theory, so called ”non-Fermi liquid” behaviors. Quantum critical phenomenon is an important subject in condensed matter as they provided universal behaviors at the quantum critical point (QCP). Single impurity Kondo problems have been well-known, we investigate Kondo effect break down for adding other competition ground state in the

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system at T → 0, it leads to QPT and quantum critical phenomenon. Examples for QPT in Kondo system: include the double quantum dots (DQD) system leads to a quantum phase transition between Kondo effect and spin-singlet, two-channel Kondo ground state, and pseudogap Kondo problems, as we will introduce below. In this thesis, we investigate QPT for 2CK pseudogap single impurity Anderson model, the pseudogap Kondo problem and two-channel Kondo physics will be briefly introduced in below subsections. Here, we discuss DQD problem for understanding QPT in Kondo system.

Figure 1.8: (a) The diagram is obtained via the scanning electronic micrograph (STM) in experiment of N. J. Craig et. al. [37]. (b) Conductance of left quantum dot. When odd number of electrons on the quantum dot and RKKY anti-ferromagnetic interaction is stronger than Kondo coupling, Kondo effect will be suppressed by RKKY. Adapted from [37].

The double quantum dot experiment made by N. J. Craig et. al. [37] for under-standing quantum phase transition is shown in Fig.(1.8). The device in Fig.(1.8.a) shows two quantum dots coupled through an open conducting region, which provides exchange

coupling between two dots. The gate voltage VgL (VgR) changes the occupation number

and energy of left (right) dot, and the coupling between the right QD and central region

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of electrons on both dots leads to split zero bias Kondo resonance into two peaks. If the occupation number of one of QD is even, there is no zero bias Kondo resonance. Theo-retically, this double QD Kondo problem was studied via the numerical renormalization group (NRG) [39, 40] and conformal field theory [32, 54], which is relevant for experimen-tal in Ref. [37]. The quantum critical diagram of double quantum dots system is shown in Fig.(1.9). Kondo effect can be observed in double QD system as RKKY coupling K is small. But Kondo effect can be suppressed if RKKY coupling exceeds the critical point

Kc. In this case, two quantum dots are coupled anti-ferromagnetically through an open

conduction region via RKKY effect, inducing a local spin singlet between two spins on the

dots ground state. A quantum critical point located at K=Kc separate Kondo from local

spin-singlet phase. The universal power-law scaling behaviors were identified in quantum critical region. ! " # $ % &' ()&' * +, -. / ' 0 /

Figure 1.9: The quantum phase transition coupling K is RKKY coupling, and quantum

critical point (QCP) with criticality at zero temperature is located at the Kc. Adapted

from [39].

1.3.1 Two Channel Kondo (2CK) Physics

We have introduced Kondo effect in single channel QD system in Section 1.2 and 1.3. Here, two independent electron reservoirs are applied to QD system, it leads to

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two-1.3. QUANTUM PHASE TRANSITIONS (QPT)

Figure 1.10: Quantum phase transition diagram of two-channel Kondo, where H2CK ∼

J1S1(r) · S + J2S2(r) · S, and J1, J2 are Kondo coupling of each channel, respectively.

Blue and red phase correspond to blue and red channel in Fig.(1.11), and J1 (J2) is the

Kondo coupling of red (blue) channel. At zero temperature, one can obtain 2CK state for symmetric coupling. There is still the 2CK state if coupling are some imbalance at finite temperature. The quantum critical region exhibits 2CK non-Fermi liquid behavior at finite temperature. [38]

channel Kondo (2CK) effect. Two-channel Kondo model was introduced by Zawadowski and Nozi‘eres et. al. decades ago, where a local spin S is coupled to two independent electron reservoirs [19, 20, 23, 6]. The discussion for 2CK details in theory is in Appendix A. Two independent electron reservoirs all couple to single QD system has been realized in recently years [16]. Compare to single channel Kondo Hamiltonian Eq.(1.2), the 2CK Hamiltonian can be written as

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The S1(r) and S2(r) is a conduction electron spin of each independent channel. The

coupling J1 and J2 of Eq.(1.6) represents anti-ferromagnetic coupling of each independent

channel, respectively, where each reservoir individually attempts to screen the local spin.

The competition between J1 and J2 leads to continuous phase transition of two

competi-tion single channel ground states at zero temperature. The quantum critical region of two competition single channel QPT is as 2CK fixed point and it exhibits non-Fermi liquid

(NFL) behavior as shown in Fig.(1.10). In a symmetric case, the Kondo couplings J1

and J2 are equally coupled to the magnetic impurity at zero temperature, leading to 2CK

ground state [34]. If J1 6= J2 at zero temperature, the localized impurity spin couples to

one of the electron channel, resulting in single channel Kondo effect (1CK). However, we

can observe 2CK effect at finite temperature even if J1 6= J2, the condition is that the

Kondo coupling asymmetry have to be small enough [6]. At finite temperature, we can investigate quantum critical region at 2CK fixed point and find universal scaling [38, 6].

Landau　Fermi liquid theory is a theoretical model of interacting fermions describes

prop-erties of general metals at low temperature. The behavior in 2CK quantum critical region can be not be explained under Fermi liquid theory, so called non-Fermi liquid (NFL) be-havior. NFL behavior appear in heavy fermion materials. Some heavy fermion materials show specific heat anomalies [24, 25, 26, 27]. The specific heat in heavy fermion metals [3, 32, 33]and anomalous shrinkage of zero-bias conductance [30, 31] in 2CK system have been observed. The entropy of impurity spin in 2CK system is anomaly at T → 0, the entropy of impurity can be written as

S = kBln(Ω), (1.13)

Ω is number of states at T → 0. We find 2CK entropy is S = kBln(

√

2), however, Ω = 2 as T → 0. Theoretically, the anomaly corresponds a free ”Majorana fermion”, which is the anti-particle of itself, two Majorana fermions is a complete fermion. Note that:

S = kBln(2) at T → 0 for 1CK.

In experiments, it’s difficult to control 2CK fixed-point stability. One needs fine con-trol of each droplet in electrochemical potential, which can be adjusted near the voltage on the gate electrode [24]. The 2CK experiment was made by Potok and Goldhaber-Gorden

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Figure 1.11: Two independent electron reservoirs couple to a QD in SET device. Two blue leads and a finite red reservoir represent two independent channel, respectively. Here, red reservoir has to be much large than QD, therefore, there are sufficient electrons to compose a conduction electron band. Two blue leads comes a reservoir which is faraway from QD. Adapted from [16].

et. al. [16], where the modified single-electron tunneling transistor (SET) with two spatially-separated sets of confined electrons can help us to understand 2CK on quantum dots. The SET device with two independent electron reservoirs is shown in Fig.(1.11). For avoiding one of electron reservoirs is scattering to the other, the red channel is hold zero conductance at low temperature, on the other hand, blue channel is hold finite con-ductance. There is a experimental scaling analysis can distinguish conductance behavior between 1CK and 2CK as shown in Fig.(1.12), where the experimental results for 2CK out of equilibrium system in conductance scaling.

The single-channel Kondo effect shows T2_{, (eV /k}

BT )2 behavior at T < TK,

theo-retically, it is Fermi liquid with scattering rate that varies as T2_{. But the 2CK data}

in 1CK scaling deviates from (eV /kBT )2. However, the non-Fermi liquid as 2CK fixed

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Figure 1.12: 2CK conductance data in 2CK and 1CK scaling, where 1CK scaling follows

Fermi liquid (eV /kBT )2 behavior. 2CK universal scaling follows (eV /kBT )1/2 behavior,

not Fermi liquid (eV /kBT )2, so-called NFL behavior. Adapted from [6, 38].

(peV /kBT ) behavior. Theoretically, 2CK universal scaling in conductance can be

ana-lyzed by field theory, which will be discussed in Chapter 2, and summarized in chapter 4. In theory, 2CK can be extended to multi channel Kondo (M-channel) (M ≥ 2) with non-trivial solutions [29, 28]. The large N approaches which are feasible theoretically used to solve the multi-channel Kondo problems. The large N approaches will be introduced in section 1.5 and Chapter 2.

1.3.2 The Pseudogap Kondo Problems

Here we define the pseudogap kondo problems. These are special Kondo problems where conduction electron density of states (DOS) vanishes in a power law fashion at ω = 0

(ρc(w) ∼ |w|r, 0 ≤ r ≤ 1). Pseudogap Kondo problems have been extensively studied

in recent years, by RG, NRG [41, 43, 44, 45, 46, 47] ,and slave-boson large N technique [48, 61]. By tuning the exponent r of the pseudogap DOS, ground state of pseudogap

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Kondo systems may undergo a QPT between Kondo and local moment (LM) state. If exponent r is too large, the conduction DOS is not sufficient to Kondo screening, leading

to LM ground state. On the other hand, Kondo screening was observed at T < TK if the

exponent r is small enough that conduction electrons are sufficient to screen the impurity. Pseudogap DOS in single channel and two-channel Kondo systems leads to QPT with particle-hole symmetry and asymmetry. The QPT in pseudogap Kondo can be analyzed by renormalization group (RG) techniques. We briefly introduce the rich ground phase diagrams of pseudogap Kondo system given by Matthias Vojta et. al. via perturbative RG and numerical renormalization group (NRG) approach [41, 43, 44].

Renormalization Group And Numerical Renormalization Group

Renormalization group (RG) approach originally comes from quantum field theory, and has been applied to condensed matter system. Anderson et. al. applied so called ”poor man scaling RG” to Kondo problem [49], where all the leading to logarithmic terms were summed up via perturbation theory. However, perturbation theory breaks down for

T < TK as the system reaches the strong coupling Kondo ground state. K.G. Wilson

use a non-perturbative technique: numerical renormalization group (NRG) approach to

analyze Kondo physics at T < Tk [50, 51, 52]. There is the other non-perturbative method

called Bethe ansatz, confirming Wilson’s NRG calculation [53]. The s-d model was given a definitive result for ground state by NRG calculation. In condensed matter system, NRG can accurately describe magnetic doped metals , while methods can not. The QPT of pseudogap Kondo problems can be studied by using perturbative RG and NRG approach.

RG phase diagrams for 1CK single impurity pseudogap Anderson model In results by Matthisa Vojta et. al. [41, 43, 44]. Pseudogap Kondo problems analyzed by perturbative RG leads to non-trivial fixed point and associated phase transitions. Here, we introduce the single impurity Anderson model coupled to single channel electron reservoir with particle-hole symmetry and asymmetry. We briefly introduce particle-hole symmetry and asymmetry below. The full symmetry of 1CK single impurity pseudogap

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1.3. QUANTUM PHASE TRANSITIONS (QPT) ε = −u/2 v2 −∞ SC LM 0 ∞ ε = −u/2 v2 −∞ FImp 0 ∞ LM ε = −u/2 v2 −∞ 0 ∞ LM´ LM´ a) r = 0 b) 0 < r < 1/2 c) 1/2 ≤ r < 1 ε = −u/2 v2 d) r ≥ 1 −∞ 0 ∞ LM LM´ LM LM´ SSC SCR SCR’ SSC

Figure 1.13: The RG flow phase diagram of particle-hole symmetric single impurity pseu-dogap Anderson model. The details are in text. Adapted from [43].

Anderson model with particle-hole symmetry is SU(2)spin× SU(2)charge, where

particle-hole symmetry is SU(2) pseudospin symmetry. The 1CK single impurity Anderson model as Eq.(1.1) is coupled to pseudogap conduction electron density of states,

H =X σ ǫdd†σdσ+ Z Λ −Λdk|k| r_kc† kσckσ+ Un↑n↓+ X kσ (Vkdd†σckσ+ H.C.). (1.14)

The second term of in Eq.(1.1) was replaced by the bath Hamiltonian of pseudogap host conduction electron DOS. The other terms of Eq.(1.14) are the same as Eq.(1.1). The Λ is the untraviolet (UV) cutoff. In the presence of particle-hole symmetry, the

Coulomb potential is assumed as U0 = −2ǫd. And the Hamiltonian is invariant by below

transformation,

d†_σ _{−→ d}σ,

c†_kσ _{−→ c}−kσ.

On the other hand, if U0 6= −2ǫd, the particle-hole symmetry is broken. For example,

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fixed-point structure changes at r∗_{and r = 1/2 , the relevant case of r=1 case is}

inaccessi-ble from weak coupling. [43, 45] r∗ _{= 0.375 is given by NRG. P-h symmetry is restored for}

0 < r < r∗_{. The RG flows of the particle-hole symmetric 1CK single impurity pseudogap}

Anderson model is shown in Fig.(1.13). The horizontal axis denotes the renormalized dot level ǫ, where U = −2ǫ; the vertical axis is the renormalized hybridization V, hoping of dot and leads. The continuous boundary phase transitions were represented by the thick lines; the full (open) circles are stable (unstable) fixed points. Now, we introduce the fixed points corresponded to the phases,

LM : local moment ground state,

SC : strong coupling as Kondo-screened fixed point, SSC : symmetric strong coupling fixed point,

SCR : symmetry critical region fixed point, FImp : free impurity fixed point.

When r=0, the flow is towards to SC fixed point at any finite U. For 0 < r < 1/2 case, LM fixed points are stable, the SSC fixed point which is stable is located at ǫ = 0, and SCR (SCR’) fixed point control the phase transition between SSC and LM (LM’). The SSC fixed point becomes unstable as ≤ r < 1, and SCR (SCR’) fixed point disappears. The phase transition between LM and LM’ is controlled by SSC fixed point. When r ≥ 1, there is no QPT. The first order phase transition with level crossing between LM and LM’. The FImp fixed point is located U = 0, meaning that no hoping between leads and dot. As particle-hole asymmetric 1CK single impurity pseudogap Anderson model: The RG flow phase diagram was shown in Fig.(1.14). Particle-hole asymmetric Anderson model can be

realized as Coulomb potential is too large, where U0 −→ ∞. The horizontal axis denotes

the on-site dot energy levelǫ; the vertical axis is the fermionic coupling v. The bare on-site

repulsion is fixed at u0 = ∞. Here, we introduce the fixed points corresponded to the

phases,

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LM

ε

v

2 0 ∞

ASC

a) r* < r < 1

ε

v

2 b) r ≥ 1 −∞ 0 ∞

LM

−∞

ASC

VFl

ACR

Figure 1.14: The RG flow phase diagram of particle-hole asymmetric single impurity pseudogap Anderson model. The details are in text. Adapted from [43].

ASC : asymmetry strong coupling fixed point, ACR : asymmetry critical region fixed point, VFI : the valence fluctuation fixed point.

The hybridization V0 which is small leaves the moment unscreened, whereas large V0

directs the flow towards ASC fixed point. When r=0, the constant DOS, where the strong-coupling fixed point is the same as in the p-h symmetric situation. For 0 < r < r∗ case, particle-hole symmetry is restored. The phase transition is controlled by ACR unstable

fixed point as r∗ _{< r < 1. When r ≥ 1, there is no QPT. The first order phase transition}

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1.3. QUANTUM PHASE TRANSITIONS (QPT) j v NFL ∞ a) r = 0 b) 0 < r < rmax c) r > rmax LM j v NFL ∞ LM ACR SCR j v ∞ LM ACR NFL´ LM´ NFL

Figure 1.15: The RG flow phase diagram of 2CK particle-hole asymmetric single impurity pseudogap Anderson model. The details are in text. Adapted from [44].

RG phase diagrams for 2CK single impurity pseudogap Anderson and Kondo model

In the results of Matthisa Vojta et. al. [41, 43, 44], the non-Fermi liquid (NFL) phase

in 2CK Kondo model only survives in 0 < rmax region, where rmax = 0.23. The p-h

asymmetry is irrelevant for r > 0.23. We will discuss pseudogap 2CK quantum phase transition by Kondo model and Anderson model as shown Fig.(1.15) and Fig.(1.16). The full (open) circles in diagrams are stable (unstable) fixed points and LM fixed means that local moment state. NFL represents non-Fermi liquid, ACR (SCR) represent critical p-h asymmetric (symmetric) point. For r=0 case, the lines of NFL fixed point represent non-Fermi liquid (over-screened Kondo effect, 2CK), it shows that flow is always towards NFL fixed point at any finite coupling. In the RG flows of 2CK pseudogap single impurity Kondo model as shown in Fig.(1.1.5): the horizontal axis denotes the renormalized Kondo

coupling j, and renormalized potential scattering v which is representing particle 　hole

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For the metallic case (r=0), all lines flow into NFL fixed point at any finite coupling, 2CK governs the behaviors everywhere. LM is a unstable fixed point in the metallic case. In Fig.(1.15.b), the other non-Fermi liquid fixed point, NFL’ represents a phase at large

couplings and asymmetries. For 0 < r < rmax, p-h asymmetry is irrelevant in non-Fermi

liquid phase, a single p-h asymmetric NFL fixed point which separates NFL and NFL’ fixed points is a asymmetric critical region (ACR). The LM fixed point is stable here, and a critical p-h asymmetric fixed point (SCR) controls the phase transition between

LM and NFL fixed points. For r > rmax = 0.23 case, there is no NFL phase as shown in

Fig.(1.15.b). The other LM phase, LM’ fixed point represents a free local moment. The ACR fixed point controls phase transition between LM and LM’ fixed points. The RG flows of the 2CK pseudogap single impurity Anderson model is shown in Fig.(1.16). The horizontal axis denotes the energy difference between spin and flavor impurity levels, the renormalized hybridization g is the vertical axis. The diagrams represent cuts, taken at v = 0, through the full RG flow. LM and LM’ fixed points represent unscreened spin and free local moment phases, respectively. For r=0 case, the lines accessed to NFL fixed point

at finite coupling ǫ, will access to LM fixed point (unstable) as ǫ → ±∞. For 0 < r < rmax,

LM (LM’) fixed point becomes a stable fixed point. The NFL fixed point in Fig.(1.16.a) is replaced by ACR fixed point here, and two isolated p-h symmetric fixed point are located outside the u = 0 plane. The two SCR fixed points control phase transition between

ACR and LM (LM’) fixed point. For rmax < r < 1, the phase transition between LM and

LM’ can be controlled by ACR fixed point. The flow is towards to ACR fixed point as at ǫ = 0. For r ≥ 1, the transition is a level crossing, in other words, there is no QPT. Free impurity fixed point(FIMP) is located at g = ǫ = 0, and flow is towards to FIMP at ǫ = 0.

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1.3. QUANTUM PHASE TRANSITIONS (QPT) ε g2 −∞ NFL LM 0 ∞ ε g2 −∞ FImp 0 ∞ LM ε g2 −∞ 0 ∞ LM´ LM´ a) r = 0 b) 0 < r < rmax c) rmax < r < 1 ε g2 d) r ≥ 1 −∞ 0 ∞ LM LM´ LM LM´ ACR (NFL) SCR SCR’ ACR

Figure 1.16: The RG flow phase diagram of 2CK particle-hole asymmetric single impurity pseudogap Kondo model. The details are in text. Adapted from [44].

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Chapter 2 Large N Approaches To 2CK

Anderson Model

The ground state and thermodynamic behaviors of s-d model represented by the the non-degenerate Anderson model can be understood by the Fermi liquid theory [5], conformal field theory [32, 54], the Bethe ansatz solutions [55], renormalization group, and numerical renormalization group [51, 56]. The QPT of pseudogap Kondo problems with the N-fold degenerate Anderson model analyzed by RG and NRG have been introduced in Chapter 1 [41, 43, 44]. Here, we introduce one of large N approaches, so called non-crossing approximation (NCA) to solve the N-fold degenerate Anderson model, where N → ∞ is number of different spin flavor of fermions [57, 58, 60, 61]. The N-fold degenerate Anderson model with infinite U Coulomb potential can be solved by salve-boson representation [62]. The NCA approach has been successfully applied to Anderson impurity modes to address quantum field theory and critical phenomena. In this chapter, we will solve two-channel pseudogap Anderson model with infinite U Coulomb potential via NCA and salve-boson representation.

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2.1. METHODS TO N-FOLD DEGENERATE ANDERSON MODEL

2.1 Methods To N-fold Degenerate Anderson Model

2.1.1 The Foundation Of Slave Boson Representation

For simplicity, we start from the Hamiltonian for N-fold degenerate infinite U Anderson model out of equilibrium [5],

H =E0|0, 0 >< 0, 0| + X σ E1,σ|1, σ >< 1, σ| + 1/2U X σ,σ′_,σ6=σ′ nσnσ′ + X k,σ ǫkc†_k,σck,σ (2.1) +X k,σ (Vk|1, σ >< 0, 0|ck,σ+ Vk∗Ck,σ|0, 0 >< 1, σ),

where infinite U Coulomb potential leads to no doublely-occupied state. The index m (m=0,1) denotes the spin operator, spin up (m=0) or down (m=1). The diagrammatic perturbation theory does not work here, because U ∼ ∞ goes beyond the validity of perturbation theory. The function for coupling strength function of dot and leads is

Γσ(ω) = 2πX

k,σ

|Vkσ|2δ(ω − ǫkσ), (2.2)

where Γσ_{(ω) = Γρ}

c, Γ = πVkσ2, and ρc is the conduction electron density of states. Now

we express this model by Hubbard operators,

|1, σ >< 0, 0| = χσ,0, |0, 0 >< 1, σ| = χo,σ, (2.3)

|0, 0 >< 0, 0| = χ0,0, |1, σ >< 1, σ| = χσ,σ.

Diagrammatic approaches based on Wick’s theorem can not be applied here, because the

commutation relations, [χp,q, χq′_,p′] = χ_p,p′δ_q,q′ ± χ_q,q′δ_p′_,p. Nevertheless, the slave-boson

representation [62] can solve these problem, where the χ operators are replaced by boson and fermion operators.

χσ,o= f_b†f, χ0,σ = fσb†, (2.4)

χ0,0 = b†b, χσ,σ = fσ†fσ,

where f and b operators are called pseudo-fermion and slave-boson satisfying the

communication relation [b, b†_]

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2.1. METHODS TO N-FOLD DEGENERATE ANDERSON MODEL

respectively. This approach enforce a local constraint to ensure no double occupy on the impurity,

Q = b†b +X

σ

f_σ†fσ = 1. (2.5)

From this representation, the N-fold degenerate Anderson model becomes

H =X σ ǫffσ†fσ + X k,σ (Vkfσ†ck,σb + Vk∗c † k,σfσb†) + X k,σ ǫkc†k,σck,σ. (2.6)

The first two term are quadratic terms, and hybridization Vkcan be a expansion parameter

in perturbation theory. We provide more details on slave-boson representation and apply to study the 2CK infinite U Anderson model out of equilibrium in Section 2.2.

2.1.2 Non crossing approximation (NCA) approach

Non crossing approximation (NCA) approach is one of large N approaches to solve many-body system. Instead of having two flavors (N=2, spin=↑ or ↓). This method assumes N → ∞ flavors of fermions in conduction bath and on the impurity. In the limit of N → ∞, one can self-consistently solve for Green functions of conduction electrons, impurity fermion and slave-boson, by including all self-energy diagrams with no lines crossing each other, so called NCA approach. Solutions via NCA are exact when N → ∞. At a finite N (< ∞), we can systematically calculate the O(1/N) calculations to the large N solutions [58]. In the physical system where N=2 (not N → ∞), large-N approach has been successfully used to provide qualitatively correct Kondo physics in a single-impurity Kondo problem. However, NCA always leads to NFL singular single-impurity DOS

ρσ(w) in any Kondo models. It does not correctly describe Fermi-liquid behavior of

single channel Kondo system. A more accurate numerical approach, NRG can resolve this artifact. Nevertheless, NCA approach is able to correctly describe NFL behavior in multi-channel Kondo problems. In fact, NCA approach has been successfully applied to 2CK equilibrium and out of equilibrium system [57, 58]. The 2CK infinite U single impurity Anderson model out of equilibrium will be solved by NCA in Section 2.2.

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2.2. THE 2CK PSEUDOGAP ANDERSON MODEL OUT OF EQUILIBRIUM

2.2 The 2CK Pseudogap Anderson Model out of

equi-librium

Figure 2.1: The set sup of 2CK pseudogap Anderson model out of equilibrium with strong Coulomb potential. There are two 2CK leads which couple to single QD out of equilibrium, where 2CK leads with pseudogap conduction electron DOS are in thermal equilibrium, respectively.

This section, we will solve non-equilibrium 2CK infinite U pseudogap single impurity Anderson model with SU(M = 2) × SU(N = 2) symmetry, where N and M is the spin degeneracy of impurity and number of electron reservoirs (or Kondo screening channels), respectively. Here, non-equilibrium system arises different coupling strength function of leads and dot different form Eq.(2.2). The pseudogap conduction electron density of states

ρc ∼ |w|r in the 2CK Anderson model corresponds constant density of states Anderson

model for exponent r=0 [44, 61]. The schematic 2CK pseudogap Anderson model out of equilibrium was shown in Fig.(2.1). For simplicity, we start from the non-equilibrium

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2CK single impurity Anderson model with constant DOS [57, 58],

H = X τ,α,σ,k (ǫkσ − µα)cα†_kστcαkστ + X σ ǫσd†σdσ+ 1 2U X σ X σ′_6=σ nσnσ′ + X τ,α,σ,k (V_kσαcα†_kστdσ+ H.C.) (2.7)

There are different chemical potentials µα of left (α=L) and right(α=R) lead in

non-equilibrium system, where α ∈ L, R. Spin flavors are represented by σ = 1, ....N and τ = 1, ....M corresponds to independent electron reservoirs. Here, we set N = M = 2. The

cα†_kστ (cα†_kστ) in Eq.(2.7) is the operator which creates (destroys) an electron in conduction

electron Fermi sea with momentum k of left or right 2CK leads. The second term in Eq.(2.7) describes the spin σ electrons on the quantum dot, and the last two terms represent the electron Coulomb interaction on the quantum dot and leads-dot hopping, respectively. The retarded conduction electron Green function was defined as

Grc(t) = −iθ(t) < {c†kσ(t), ckσ(0)} > . (2.8)

The conduction electron density of states is the imaginary part of the retarded conduction electron Green function,

ρc(w) = ImGrc(t). (2.9)

The coupling strength function (without pseudogap DOS) of dot and leads is ΓL(R)σ (ω) ∼

ρc(w), it can be written as

ΓL(R)_σ (ω) = 2π X

k∈L(R)

|Vkσα|2δ(ω − ǫkσ − µα). (2.10)

The the chemical potential of two-channel non-equilibrium Anderson model is

µα = µ0+ (−)

eVα

2 , α = L(+), α = R(−), (2.11)

where µ0 is chemical potential of conduction baths, µα represents chemical potential with

bias voltage (eVα) of left or right lead. The Fermi function with bias voltage of left side

or right side is

fα(w) = f (w + µα), α = L(+), α = R(−). (2.12)

In non-equilibrium system, different chemical potentials (µL, µR) lead to different Fermi

functions (fL(w), fR(w)). However, µL = µR and fL(w) = fR(w) correspond to

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occupancy on the dot. In this case, our model does not show p-h symmetry, because of

Coulomb repulsion is set to be U → ∞ and ǫd 6= −1/2U. The third term in Eq.(2.7) is

a unperturbed term, where U is the infinite Coulomb interaction of electrons on the dot. Diagrammatic perturbation theory can not be used here as U → ∞. The hybridization

coupling Vα

kσ in Eq.(2.7) seems to be a more useful expansion parameter for

perturba-tion theory. From Secperturba-tion 2.1, the infinite U Anderson model can be made quadratic by a transformation with the slave-boson and pseudo-fermion operators. Here, we

de-fine the slave-boson and pseudo-fermion operators, where b† _{creates a empty state and f}†

σ

(σ =↑ or ↓) creates a singly occupied state. The d†

σ and dσ in Eq.(2.7) can be decomposed

as,

dσ(t) = b†τ(t)fσ, (2.13)

d†_σ(t) = f_σ†bτ(t),

where σ is spin (up or down), and c†

σf † ¯

σ|Ω >= fσ†bτfσ¯†|Ω >= 0 , |Ω > is the vacuum state.

Due to no double occupancy on the dot, we add a local constraint,

Q = b†_τbτ +

X

f_σ†fσ = 1 (2.14)

where Q is the total physical states, it must be equal to unity (Q=1). Now we rewrite the Hamiltonian in the slave-boson representation,

H = X kστ α (ǫk− µα)cα†kστcαkστ + ǫd X σ f_σ†fσ + X kστ α (V_kσα(f_σ†bτcαkστ) + H.C) (2.15)

The first two terms in Eq.(2.13) are the unperturbed quadratic terms, and the last term is

the hybridization. Diagrammatic perturbation theory can be used as hybridization Vα

kσ is

a small expansion parameter. The Green function of pseudo-particles up to lowest order self-energy is shown in Fig.(2.2), where pseudo-fermion self-energy involves slave-boson propagator, and slave-boson self-energy involves pseudo-fermion propagator. Note that the Hamiltonian represented by salve-boson, pseudo-fermion and the local constraint are equivalent to Eq.(2.7). Since we study the non-equilibrium infinite U Anderson model, how to deal with the local constraint is an important problem here. We apply Keldysh diagrammatic perturbation theory to formulate our equations (see Ref. [57].) We start

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from the action Sc in Q=1 ensemble, defined as:

Sc(−∞, −∞) = e−i

H

cdt′H(t′) (2.16)

Next, we write down the non-equilibrium partition function as,

ZQ=n= T r{e−β(H0−µLNL−µRNR)× δQ.nTc[Sc(−∞, −∞)]}. (2.17)

Here n=1, and TC is the operator which orders operators along the Keldysh contour. If

we have an operator ˆO, the expectation value in Q = 1 ensemble is given by

< ˆO >Q=1=

1

ZQ=1T r{e

−β(H0−µLNL−µRNR)_{× δ}

Q.1Tc[Sc(−∞, −∞) ˆO]}. (2.18)

We can rewrite δQ,1 as an integral over a complex chemical potential iλ [63]

δQ,1 = β 2π Z π/β −π/β dλe−iβλ(Q−1). (2.19)

Now we divide both numerator and denominator of < ˆO >Q=1by partition function ZQ=0,

it leads to

< ˆO >Q=1=

ZQ=0

ZQ=1

< ˆO >(1)_iλ . (2.20)

There are two contributions in non-equilibrium expectation value of ˆO in Q=1 ensemble,

< Ô >(1)_iλ and ZQ=0 ZQ=1, where < Ô >iλ= 1 ZQ=1T r{e −β(H0−µLNL−µRNR+iλQ)_{× T} c[Sc(−∞, −∞)]}, (2.21) < Ô >(1)_iλ= β 2π Z π/β −π/β eiβλ < Ô >iλ, and ZQ=0 = T r{e−β(H0−µLNL−µRNR)× δQ.0Tc[Sc(−∞, −∞)]}, (2.22) δQ,0 = β 2π Z π/β −π/β dλe−iβλ(Q)_.

As the trace of < ˆO >iλ divides by ZQ=0, it does not restrict in Q=1 ensemble, and the

normalization ZQ=0 Q=1 can be obtained from identity < Q >Q=1= 1, where

ZQ=1 ZQ=0 =< b†_{b >}(1) iλ + X σ < f† σfσ >(1)iλ . (2.23)

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From these, the constraint Eq.(2.12) can be solved, and expectation value < ˆo >1

iλ can be

obtained diagrammatically. And the impurity retarded Green function is

Gr_σ_{(r) = −iθ(t) < {d}σ(t), d†σ(0)} > . (2.24)

It can be written as Gr_{(t) = θ(t)[G}>_{(t) − G}<_(t)],

Gr<_σ_{(r) = i < {d}σ(o), d†σ(t)} >, (2.25)

Gr>_σ_{(r) = −i < {d}σ(t), d†σ(0)} >, (2.26)

where G<_{(t) is the impurity lesser Green function, and G}>_{(t) is the impurity greater}

Green function.

Figure 2.2: At lowest order, boson self-energy involves the fermion propagator, and fermion self-energy involves boson propagator. Adapted from [57].

The impurity spectral function is

ρσ(w) = − 1 πImG r σ(w), (2.27) where Gr

σ(w) is the Fourier transform of the impurity retarded Green function. Due to

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chemical potential iλ,

ρσ(w) = − ZQ=0 ZQ=1| 1 πImG r(1) σ,iλ(w)| (2.28)

Furthermore, we can write down the current though the dot,

J = e ~ Z dω 2ΓL(w)ΓR(w) ΓL(w) + ΓR(w) ρσ(w, V ) × [f(w + eVα/2 − f(w − eVα/2))] (2.29)

Our goal is solve the current, therefore we have to analyze the impurity retarded Green function first, then density of states can be solved easily. Here, we expand SU(2) × SU(2) Anderson impurity model to SU(N) × SU(M) Anderson impurity model, where N > 2 and M > 2. We apply the large N approach, non-crossing approximation to our model on the Keldysh contour. The impurity retarded Green function in the Q=1 ensemble with iλ can be expressed in terms of pseudo-particles within NCA approach, where neglects vertex relation.

Gr(1)_σ,iλ_{(t) = −iθ(t) < {c}σ(t), c†σ(0)} >

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iλ (2.30)

=N CA _−iθ(t)[D>_(−t)G<_{f σ}_{(t) − D}<_(−t)G>_{f σ}(t)].

Here, we define Green functions of pseudo-fermion:

G>_{f σ} _{≡ −i < f}σ(t)fσ†(0) >0iλ, (2.31) G<_{f σ} _{≡ i < f}_σ†(0)fσ(t) >1iλ, and slave-boson: D>_{≡ −i < b}τ(t)b†τ(0) >0iλ, (2.32) D<_{≡ i < b}† τ(0)bτ(t) >1iλ,

where the natation < and > represents lesser and greater Green function. The fermion and boson propagators within NCA as shown in Fig.(2.3) are self-consistent via Dyson’ equations , where the self-energies are iterated to all orders [57]. The lesser (greater ) Green functions can be written as [59]:

D>(<)(w) = Dr(w)Π>(<)(w)Da(w), (2.33) G>(<)_{f σ} = Gr_{f σ}(w)Σ<(>)_{f σ} (w)Ga_{f σ},

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where Da_{(w) and G}a

f σ is advanced Green function of boson and fermion, respectively. The

advanced Green functions are complex conjugates of the retarded Green functions. The Π>(<)_{(w) is the self-energy of boson greater (lesser) Green function, and Σ}<(>)

f σ (w) is the

self-energy of fermion greater (lesser) Green function. The boson self-energies of lesser and greater Green function are given by (see Ref. [57, 60])

Π<_b_{(w) = (−2i)} Z +∞ −∞ dǫG<_{f σ}(ǫd+ ω)[|VkσL|2]f (−ǫd+ µL)× (2.34) ρL(ǫd− µL− µ0) + [|VkσR|2]f (−ǫd+ µR)ρR(ǫd− µR− µ0), Π>_b(w) = 2i Z +∞ −∞ dǫG>_{f σ}(ǫd+ ω)[|VkσL|2]f (ǫd− µL)× (2.35) ρL(ǫd− µL− µ0) + [|VkσR|2]f (ǫd− µR)ρR(ǫd− µR− µ0),

and fermion self-energies,

Π<_f(w) = 2i Z +∞ −∞ dǫD_{f σ}<(ǫd+ ω)[|VkσL|2]f (−ǫd+ µL)× (2.36) ρL(−ǫd+ µL+ µ0) + [|VkσR|2]f (−ǫd+ µR)ρR(−ǫd+ µR+ µ0), Π>_f(w) = 2i Z +∞ −∞ dǫD>_{f σ}(ǫd+ ω)[|VkσL|2]f (ǫd− µL)× (2.37) ρL(−ǫd+ µL+ µ0) + [|VkσR|2]f (ǫd− µR)ρR(−ǫd+ µR+ µ0),

The relation between greater Green function and retarded Green function are, D>_{(w) =}

2iImDr_{(w), G}>

f σ= 2iImGrf σ, where the self-energies, Π>(w) = 2iImΠr(w) and Σ>f σ(w) =

2iImΣr

f σ(w). The retarded self-energies can be solved by greater self-energies, where

Πr(w) = i 2π Z +∞ −∞ dw′ Π >_(w) w − w′_{− iη} (2.38) Σr(w) = i 2π Z +∞ −∞ dw′ Σ r_(w) w − w′_{− iη} (2.39)

The retarded Greens function for pseudo-fermion is given by

Gr_{(w) = [w − ǫ}d− Σr(w)]−1, (2.40)

and for slave-boson,

不交叉近似法於雙渠道贗能隙安德森模型之應用:量子臨界與統一標度律

國立交通大學碩士論文

Two Channel Kondo Pseuodap

Anderson Model: Quantum

Criticality and Universal Scaling

Student: 巫展霈

Advisor: 仲崇厚

Committee:

牟中瑜

陳柏中

朱仲夏

List of Figures

Contents

Chapter 1

Introduction

1.1

Kondo Effect

1.1.1

Kondo Hamiltonian

1.2

Kondo Effect In Quantum Dot System

1.2.1

Coulomb Blockade Oscillations With Kondo Effect

1.3

Quantum Phase Transitions (QPT)

1.3.1

Two Channel Kondo (2CK) Physics

1.3.2

The Pseudogap Kondo Problems

LM

ε

v

ASC

ε

v

LM

ASC

VFl

ACR

Chapter 2

Large N Approaches To 2CK

Anderson Model

2.1

Methods To N-fold Degenerate Anderson Model

2.1.1

The Foundation Of Slave Boson Representation

2.1.2

Non crossing approximation (NCA) approach

2.2

The 2CK Pseudogap Anderson Model out of

equi-librium