Thesis
Fluctuations in Charge Ordering Phenomena

Kazuyoshi Yoshimi
Department of Physics
Graduate School of Science
University of Tokyo
December, 2008

Acknowledgments
I would like to express my sincerest gratitude to Professor Takeo Katoand Hideaki Maebashi for continuous encouragement and invaluable discussions during the course of the present study. I also would like to Prof. Hatsumi Mori for stimulating and helpful discussions.
I express my thanks to all the members of Kato, Mori, Takada and Ueda
group for their various help.
Finally, I acknowledge my parents, brother and sisters for their continual
encouragement and ﬁnancial supports.

Abstract
In this thesis, we study charge-ordering phenomena by a two-dimensional
extended Hubbard model with nearest-neighbor Coulomb interaction. We
ﬁrst formulated a non-skeleton conserving approximation and applied it to
study how charge ﬂuctuation aﬀects charge and spin susceptibilities. We
ﬁnd enhancement of the spin susceptibility toward charge-ordering transition. We also calculated response functions by a self-consistent conserving
approximation. The charge compressibility diverges just before the chargeordering transition, and becomes even negative after divergence. This negative charge compressibility indicates instability of a uniform electron states,
and may provide a key to understand inhomogeneous charge disproportion observed in several two-dimensional organic conductors like θ-(BEDTTTF)2 RbZn(SCN)4 and β-(meso-DMBEDT-TTF)2 PF6 .

Contents
1 Introduction

1

2 Overview of charge-ordering in organic conductors
5
2.1 Electronic properties of charge-ordering phenomena . . . . . . 5
2.1.1 Experimental review . . . . . . . . . . . . . . . . . . . 6
2.1.2 Theoretical review . . . . . . . . . . . . . . . . . . . . 10
3 Enhancement of spin susceptibility toward charge-orderingtransition
13
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Exact relations for Green’s function . . . . . . . . . . . 15
3.2.2 Mean-ﬁeld and random-phase approximations . . . . . 18
3.2.3 Inclusion of vertex corrections . . . . . . . . . . . . . . 19
3.3 Numerical results and discussions . . . . . . . . . . . . . . . . 23
3.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.2 Phase diagram . . . . . . . . . . . . . . . . . . . . . . 24
3.3.3 Static response functions . . . . . . . . . . . . . . . . . 26
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.1 Relation to the Fermi liquid theory . . . . . . . . . . . 31
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Negative compressibility induced by large charge ﬂuctuations
35
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2

Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Baym-Kadanoﬀ conserving approximation . . . . . . .
4.2.2 Shielded interaction approximation . . . . . . . . . . .
4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Transition point determined by RRPA . . . . . . . . .
4.3.2 Response functions . . . . . . . . . . . . . . . . . . . .
4.4 Summary and discussions . . . . . . . . . . . . . . . . . . . .

37
37
39
42
42
43
48

5 Summary

51

A Non-skeleton conserving approximation

53

Chapter 1
Introduction
Low-dimensional organic conductors are known to exhibit a variety of interesting electronic properties such as superconductivity, magnetism andcharge-ordering etc. [1] Regardless of apparent complexity of crystal structures, many classes of organic conductors can be understood by a singleband model starting with the HOMO or LUMO molecular orbits. Uniqueelectronic properties observed in molecular solids originate from the natureof these HOMO(LUMO) molecular orbits. Because of the large molecularorbits, organic conductors possess the following peculiar features. (1) Softness. The weak van der Waals bonding between molecules leads to large
deformation with respect to external pressure or chemical pressure inducedby substitution of cations or anions. (2) Cleanness. The crystal of molecularsolids includes little impurity like defects due to the nature of van der Waals
bonding between large molecules. (3) Long-range Coulomb interaction. The
eﬀective on-site interaction is relatively small compared with the inter-site
Coulomb interaction because screening is weak due to the extended natureof molecular orbits.
Among a variety of phase transitions appearing in organic conductors,
charge-ordering (CO) phenomena, which are a main target in this thesis,
have recently attracted much attention in strongly correlated electron systems. One possible mechanism of charge-ordering is the Wigner crystallization due to long-range Coulomb repulsion. Charge disproportion in the COphase has ﬁrst been reported by NMR measurement in quasi-one dimensional
1

material (DI-DCNQI)2 Ag [2]. After this discovery, a number of experimental studies has been performed to clarify the CO in quasi-one-dimensional
material such as (TMTTF)2 X and quasi-two-dimensional material such asα-(BEDT-TTF)2 I3 , θ-(BEDT-TTF)2 M M ′ (SCN)4 (M = Rb, Cs, M ′ = Zn,
CO) and β-(meso-DMBEDT-TTF)2 PF6 [3]. The experiments have shown
clear evidence of long-range CO accompanied with charge disproportion in
low-temperature insulator phases of these materials. These experiments,
however, also have provided several unsolved questions on CO phenomena
especially in quasi-two-dimensional systems.
As a typical example, we consider the quasi-two-dimensional system θ(BEDT-TTF)2 RbZn(SCN)4 (abbreviated as the RbZn-system in the following). This system shows a metal-insulator transition characterized by rapid
exponential increase of resistance below TMI = 195K [4]. The spin susceptibility shows a broad peak around the metal-insulator transition with noanomaly. A paramagnetic phase is kept down to 30K where a spin-gap behavior indicating the spin-Peierls transition appears. This behavior of theuniform spin susceptibility above 30K is analyzed by the Bonner-Fisher curveof the Heisenberg model, which is an eﬀective model of localized electrons.
Although this analysis may be justiﬁed in the insulator phase, it is nontrivial to apply it to the metallic phase above TMI . It is also clear that thiseﬀective spin model cannot treat the eﬀect of charge disproportion and its
ﬂuctuation properly. In the mean-ﬁeld calculation, which is a frequently usedapproximation for the CO phenomena, the spin susceptibility is expected toshow a temperature-independent Pauli paramagnetism in the metallic phase.
Therefore, one may consider the spin susceptibility from the weak-coupling
perturbative approach starting with the metallic state and incorporate charge
ﬂuctuation into it. This approach has not been studied well in the previous
theoretical studies of the CO phenomena.
Another question is related to the charge disproportion around TMI . In
the low-temperature insulator phase of the RbZn-system, stripe-type longrange CO is observed by the X-ray diﬀraction measurement [5, 6]. 13 C-NMR
measurement has also shown a large charge disproportion in the CO phase [7].
However, anomalous broadening of 13 C-NMR spectrum observed above TMI
revealed a new feature of CO in this system. Such a broadening is unusual2since the metal-insulator transition is of ﬁrst order. By detailed analysis ofboth transverse and longitudinal relaxation rates, this broadening has attributed to large ﬂuctuation of charge disproportion in space, indicating thatthe ﬂuctuation should be extremely slow (< 6KHz) [8]. Similar inhomogeneous charge disproportion above TMI have been reported by infrared andRaman spectrum [9] and dielectric measurement [10]. This strange state isoften called as simply ”short-range charge-ordering” without deﬁning it exactly. By theoretical works based on the atomic limit [11], the Hartree-Fockapproximation [12], the exact diagonalization [13] and the variational Monte
Carlo method [14], it is suggested that several CO phases with diﬀerent patterns becomes almost degenerate. This degeneracy, which indicates domain
formation with diﬀerent CO states, may be relevant to the so-called shortrange CO. It, however, seems that essential discussion on domain formationof diﬀerent CO states in a clean system is lacked.
Motivated by the above experimental results of CO in organic conductors, we investigate the eﬀect of charge ﬂuctuations near CO instability by the
weak-coupling approach equipping a sophisticated diagrammatic expansion.
Starting with an extended Hubbard model including the nearest-neighbor
Coulomb interaction, we calculate the spin and charge susceptibilities bycareful consideration of the vertex corrections in order to satisfy various conservation laws through the Ward identity. We show that enhancement of
the spin susceptibility toward the CO transition may be understood by leading contribution of the vertex corrections. We also show that the uniform
charge compressibility becomes negative just before CO transition by selfconsistent treatment of large charge ﬂuctuations. The latter result, whichindicate instability of the electronic system, may give one possible explanation of inhomogeneous charge disproportion observed in organic conductors.
This thesis is organized as follows. In Chapter 2, we describe the experimental and theoretical reviews on CO in quasi-two-dimensional organic conductors. In Chapter 3, we ﬁrst formulate systematic inclusion of the vertex
corrections in the response functions. Then, we study the leading contribution of the vertex corrections in order to study the behavior of the spin andcharge susceptibility near CO transition. In order to consider the feedback
eﬀect of critical charge ﬂuctuation onto the Green’s function, we formulate3the self-consistent treatment based on the shielded interaction approximation(SIA) in Chapter 4. We discuss the CO transition by the response function
for all the wave numbers by neglecting the vertex correction. We further
calculate the uniform charge compressibility including all the vertex corrections within SIA by calculating derivative of the charge number with respect
to the chemical potential, and show that the charge compressibility diverges
and even becomes negative just before the CO instability. The results are
summarized in Chapter 5.

4Chapter 2
Overview of charge-ordering inorganic conductors
2.1

Electronic properties of charge-ordering
phenomena

Low-dimensional organic conductors exhibit various electronic properties suchas superconductivity, magnetic and charge-ordering. In particular, chargeordering phenomena have recently attracted attention to understand eﬀects
of the long-range Coulomb interaction on strongly correlated electron systems[1,3]. Since the molecular orbital of organic conductors is large, the wave function is extended and the on-site Coulomb repulsion U is expected to be
suppressed. Therefore, the inter-site Coulomb repulsion V is expected tobe comparable to U in organic conductors. It is thought that V is a key
to understand the charge-ordering transition. Therefore, the extended Hubbard model (EHM) with inter-site Coulomb potential is often used to analyze
charge-ordering phenomena.
In this section, we ﬁrst review typical materials which exhibit chargeordering transition in (BEDT-TTF)2 X salts ( abbreviated as (ET)2 X ) where
X represents diﬀerent atoms or molecules. Next, we review several theoreticalstudies based on EHM.
5

θ

κ

α

β

λ

Figure 2.1: Schematic ﬁgure of ﬁve patterns of ET salts. The oval represents
an ET molecule. The dotted and solid line represent the transfer integral
between molecules. The circle around two molecules in κ and λ represent a
dimer unit.

2.1.1

Experimental review

ET salts have the alternative layers of X − with the closed shell and ET+1/2
with 3/4 ﬁlled π-band. There are ﬁve patterns classiﬁed by arranging patterns of molecules labeled by the Greek characters α, β, θ ,κ and λ as shownin Fig. 2.1[15]. Among them, we focus on θ-type ET salts to review chargeordering transition.
The electronic property of θ-(ET)2 X salts is related to the dihedral angle φ between ET molecules[4]. Fig. 2.2 shows the schematic phase diagram
experimentally obtained. The materials in the region of large φ such as θ(ET)2 RbX(SCN)4 ( X = Zn, CO) exhibit metal-insulator (MI) transition
around TCO ≈ 200 K. This transition accompanies with the lattice modulation. The uniform spin susceptibility χs changes only slowly increases
and has a broad hump at TCO [16]. With decreasing the temperatures below TCO , χs behaves like a low-dimensional localized spin system and a spin
6

CsZn RbZn
X = I3 CsCo RbCo Cu2(CN)[N(CN)2]2
200 K
CO
Metal

Insulator

φ

50 K

5K

Spin gap

SC
100o

Pressure140oφFigure 2.2: Schematic phase diagram of θ-(ET)2 X salts on temperature T
and dihedral angle φ[4]. CO and SC represent charge ordered and superconductivity state, respectively.
gapped behavior is observed below around 10 K[7]. It is notable that theexistence of the short-range charge order above T > TCO is suggested by Xray scattering[5]. An anomalous broadening occurs is also observed by NMR
for these temperatures[8]. On the other hand, the materials in the region ofsmaller φ such as θ-(ET)2 CsX(SCN)4 (X=Zn, CO), a sharp MI transition
is not observed. However, with decreasing temperatures, the resistivity ρ
gradually increases below around 50 K, while χs changes the behavior from
Pauli paramagnetic to Curie-like behavior at this temperature. With thefurther decrease of φ, the charge-ordering transition does not occur and thesuperconducting state appears below about 5 K for θ-(ET)2 I3 [17].
The similar kind of ET salt β-(meso-DMBEDT-TTF)2 PF6 also occurs the7Temperature ( K )

100

LR

SR

Metal

10 CO CO
SC

1
0

2
4
6
Pressure (kbar)

Figure 2.3: Schematic phase diagram of β-(meso-DMBEDT-TTF)2 PF6 on
temperature and pressure[21]. SRCO, LRCO and SC represent short-range
charge ordered, long-range charge ordered and superconductivity state, respectively.

charge-ordering transition[18, 19, 20]. Fig. 2.3 shows the (P, T ) phase diagram for β-(meso-DMBEDT-TTF)2 PF6 . At ambient pressure, the resistivity
ρ shows a metallic behavior above 120 K. With decreasing temperatures, ρgradually increases below 120 K and MI transition occurs at TM I ≈ 70 K.
On the other hand, the magnetic susceptibility has a broad hump around
TMI . The X-ray measurements shows that the intensity of superstructure reﬂections (h, k/2, l/2) gradually increases below 100 K and suddenly increases
at TM I with decreasing temperatures. From the superstructure reﬂections,
it is found that the insulating state exhibits the checkerboard-type chargeordering where the charge disproportion occurs within the dimer unit. β(meso-DMBEDT-TTF)2 PF6 exhibit SC state next to the CO state with thetransition temperature Tc ≈ 4.6 K under the pressure P ≈ 0.6 kbar [21].
The pressure dependence of the SC state is observed below 5.2 kbar and8have found that the SC state is suppressed with increasing pressure. It is
important to note that both ρ and the intensity of superstructure reﬂections
gradually increase before the long range order transition occurs. It is proposed by Mori that this strange behavior is due to the growth of short rangeorder and is related to the giant non-linear conductivity[22].

92.1.2

Theoretical review

To investigate the charge-ordering transition, there are several theoreticalworks based on the extended Hubbard model (EHM) in which the eﬀectof the long-range Coulomb interaction is represented only by the nearestneighbour repulsive interaction[23]H=∑〈ij〉,σtij c†i,σ cj,σ +∑U∑niσ niσ̄ +Vij ni nj ,
2 i,σ

(2.1)

〈ij〉

where 〈ij〉 is the summation over neighboring pairs, tij is a transfer integral
between neighboring sites. c†iσ (ciσ ) is the creation (annihilation) operator ofan electron on i site with a spin σ (= +(−) for ↑ (↓)). U and Vij are interaction strength for the on-site and inter-site Coulomb repulsion, respectively.
The ground state of the EHM on quarter ﬁlled square lattice was studied
by a numerical Lanczos Exact Diagonalization(LED) method for cluster size
up to L = 16 by Ohta et al [24]. They have shown that the checker-board
type CO (CBCO) state is realized for some parameter regions. To investigate
the electronic property of CBCO state, Calandra et al evaluated the Drude
weight by a numerical LED method for cluster size up to L = 20 [25]. They
found that the CO metallic state exists between the uniform metallic and the
CO insulator state. The existence of the CO metallic state at ﬁnite temperatures has also been suggested by a dynamical mean ﬁeld approximation[26]and correlator projection method[27]. However, there is a possibility that theﬁnite size eﬀect may be important in these calculations, since a lattice size
is strongly restricted in the case of LED method. Therefore, it is expectedthat the charge ﬂuctuations are not fully taken into account. Furthermore,
it is diﬃcult to investigate the various spatial patterns of CO states for LED
method.
To investigate the spatial patterns of the CO states, mean-ﬁeld approximation (MFA) was applied [28, 12] and succeeds to obtain various spatial
patterns of the CO states. A phase diagram was also determined in the MFA
to the thermodynamic potential[29, 30, 31]. In the MFA phase diagram, thestatic charge response function in the random-phase approximation (RPA)
diverges at some ordering wave number along the second-order transition
10

line. An eﬀect of ﬂuctuations due to this divergence of the RPA charge response function for superconductivity is also investigated[29, 30, 31]. It has
been shown that charge ﬂuctuations favor to stabilize d-wave SC state. Thistendency has also been found by Fluctuation Exchange (FLEX) study[32].
An eﬀect of ﬂuctuations due to this divergence of the RPA charge response
function on the self-energy was discussed to ﬁnd an anomalous T -dependenceof the eﬀective mass[33]. Eﬀects of charge ﬂuctuations on the uniform spin
susceptibility have studied by auxiliary-ﬁeld Monte Calro method for lattice sizes up to N = 122 [34]. In this method, the uniform spin susceptibility
seems to follow the Curie-Weiss law and have no singularity at CO transition.
However, the quantum ﬂuctuations may be not fully treated in this method,since the low temperature region cannot be accessed due to the negative sign
problem.
In the theoretical studies above mentioned, eﬀects of charge ﬂuctuations
on the response functions have not been studied well at low temperatures.
Generally, in calculating these response functions, not only the self-energy
but also the vertex corrections are indispensable for satisfying some exact
relations such as conservation laws. In this thesis, we investigate the vertex
corrections to the charge and spin response functions in the 2D EHM at the
3/4-ﬁlling.

11

Chapter 3
Enhancement of spin
susceptibility towardcharge-ordering transition
3.1

Introduction

The vertex corrections to the charge and spin response functions were studied
for the two-dimensional (2D) Hubbard model in the absence of the inter-site
Coulomb interaction, corresponding to the limiting case of V = 0 in theextended Hubbard model (EHM). In this limiting case, there is no chargeordering transition, but anti-ferromagnetic spin ﬂuctuations are developed
near the half-ﬁlling. It has been shown that the uniform charge susceptibility χc is strongly enhanced due to the nested anti-ferromagnetic spin ﬂuctuations through the so-called Aslamazov-Larkin type vertex corrections[35, 36].
Likewise, it is expected that charge ﬂuctuations have some contribution to
the spin susceptibility near charge-ordering transition, though the spin ﬂuctuations is weak at 3/4-ﬁlling.
In this chapter, we investigate the vertex corrections to the charge andspin response functions in the 2D EHM at the 3/4-ﬁlling in the presenceof the nearest-neighbor Coulomb interaction. By calculating the static response functions at an arbitrary wave number vector, we consider eﬀects
of the vertex corrections on the charge-ordering temperature TCO and the13uniform charge (spin) susceptibility χc (χs ). Note that there exists a celebrated Baym-Kadanoﬀ (BK) scheme for a conserving approximation, where
the thermodynamic potential, the self-energy and the response functions canbe constructed in a self-consistent way[38, 39]. It is, however, diﬃcult toobtain the static response functions for all the wave numbers in the BK
scheme for practical reasons. In this chapter, we introduce another scheme
to formulate systematic inclusion of the vertex corrections which enables us
to make actual calculations (the details are shown in Appendix). We apply
this scheme to the EHM on the 2D square lattice. After determining thecharge-ordering transition line in a T -V phase diagram, we obtain χc and χsin the uniform metallic phase near this transition line. Next, we explain the
results of χc and χs in terms of the Landau parameters in the Fermi liquid
theory. Finally, the results are summarized.

14

3.2

Formulation

3.2.1

Exact relations for Green’s function

In this section, we consider a generic single-band Hamiltonian H on a latticewith band dispersion εk and electron-electron interaction vσσ′ (q) to formulate
systematic inclusion of vertex corrections:∑H=(ϵk − µσ )c†k,σ ck,σ
k,σ

+

1 ∑∑
v ′ (q)c†k+q,σ c†k′ −q,σ′ ck′ ,σ′ ck,σ ,2Ω σ,σ′ k,k′ ,q σσ(3.1)

where Ω is the volume of the lattice system, c†kσ (ckσ ) is the creation (annihilation) operator of an electron with a wave number vector k and a spinσ (= +(−) for ↑ (↓)). Because of usefulness for describing the exact relations,
we have introduced µσ = µ + σh with µ and h being a chemical potential
and an external magnetic ﬁeld, respectively.
The charge and spin densities for the Hamiltonian H are given by n =∑∑σ nσ and m =σ σnσ , respectively. Here, nσ is the electron number
density for a spin σ; nσ can be related to the single-particle Green’s function
Gσ (k) by∫nσ = eiϵl η Gσ (k),
(3.2)
∫

∑

k

∑ ∑
where k denotes (T /Ω) k = (T /Ω) k l with k being a combined notation of a wave number vector k and a fermionic Matsubara frequencyiϵl = (2l + 1)πiT with an integer l, and η is a positive inﬁnitesimal. We
can write Gσ (k) as
Gσ (k) =1,
iϵl + µ̃σ − εk − Σ̃σ (k)(3.3)

where µ̃σ absorbs the chemical potential shift due to the mean-ﬁeld (Hartree)
term in the self-energy Σσ (k) as∑µ̃σ = µσ −vσσ′ (0)nσ′ ,
(3.4)
σ′

15

and Σ̃σ (k) is the excess self-energy, which is calculated by subtracting themean-ﬁeld term from the total self-energy Σσ (k).
The charge and spin response functions are related to the density-density
response function χσσ′ (q) as1∑
χσσ′ (q),2 σ,σ′1∑ ′χSz Sz (q) =σσ χσσ′ (q),2 σ,σ′
χN N (q) =(3.5a)(3.5b)

where q represents a combined notation of a wave number vector q and abosonic Matsubara frequency iωl = 2lπiT with an integer l; χσσ′ (q) can be
written in terms of the one-interaction irreducible part χ̃σσ′ (q) as∑
χσσ′ (q) = χ̃σσ′ (q) −χ̃σσ1 (q)vσ1 σ2 (q)χσ2 σ′ (q),(3.6a)
σ1 ,σ2∫χ̃σσ′ (q) = −Gσ (k + q)Gσ (k)Λσσ′ (k; q).

(3.6b)k

Here, Λσσ′ (k; q) is the vertex function, including the vertex corrections to the
response functions.
The uniform charge and spin susceptibilities near charge-ordering are ofcentral interest in this thesis. Here, we deﬁne these quantities (per spin) from∑the electron number density as χc ≡ (1/2)(∂n/∂µ) = (1/2) σ,σ′ (∂nσ /∂µσ′ )∑and χs ≡ (1/2)(∂m/∂h) = (1/2) σ,σ′ σσ ′ (∂nσ /∂µσ′ ). Then, the isothermal
susceptibilities χc and χs should be equal to the so-called q-limit of the charge
and spin response functions, respectively:
1 ∂n= χN N (0),
2 ∂µ
1 ∂m
χs ≡
= χSz Sz (0).
2 ∂h
χc ≡

(3.7a)(3.7b)

These relations hold when both the response functions and the isothermal
susceptibilities are exactly calculated. If one takes some approximation, however, these equations are not necessarily guaranteed to hold. In this chapter,we calculate the response function in an approximate form, and determine
16the charge-ordering transition point by the static response functions suchthat
χN N (q, 0) = ∞ at some ﬁnite q.

(3.8)

It is, then, desirable that the susceptibilities is calculated by the values at
q = 0 in the static response functions that determine the transition point in
an approximation scheme, requiring that eqs. (3.7) hold. In order to construct
such an approximation, it is crucial to satisfy the q-limit Ward identity
Λσσ′ (k; 0) = δσσ′ −∂ Σ̃σ (k).
∂ µ̃σ′(3.9)

We can show that the exact relations (3.7) hold automatically in arbitrary
approximation satisfying the Ward identity (3.9) as follows. By eqs. (3.2),(3.3) and (3.4), it is not hard to see that ∂nσ /∂µσ′ satisﬁes the following
equation:
(
)
∫
∑
∂nσ
∂ µ̃σ1∂Σ̃(k)σ= − Gσ (k)2
δσ,σ1 −∂µσ′
∂ µ̃σ1
∂µσ′
k
σ1
)
(
∫
∑
∂ Σ̃σ (k)2
= − Gσ (k)δσ,σ1 −∂ µ̃σ1
k
σ1
(
)
∑
∂nσ2
× δσ1 ,σ′ −vσ1 σ2 (0).
(3.10)∂µσ′σ
2After using the Ward identity (3.9) with eqs. (3.6), we obtain the relations(3.7).
Before closing this subsection, we give some comments on a set of basic
equations in the quantum many-body theory[40]. In addition to eqs. (3.6),
the following equations make up this set:∑∫Σ̃σ (k) = −Gσ (k + q)Vσσ′ (q)Λσσ′ (k; q) ,(3.11a)σ′qwhere Vσσ′ (q) represents the eﬀective electron-electron interaction given by∑
Vσσ′ (q) = vσσ′ (q) −vσσ1 (q)χ̃σ1 σ2 (q)Vσ2 σ′ (q) .
(3.11b)σ1 ,σ217

If vertex corrections are neglected, i.e. the vertex function is approximatedby Λσσ′ (k; q) = δσ,σ′ in eqs. (3.6) and (3.11), one gets renormalized RPA. It
is, however, noted that neither eqs. (3.9) nor (3.7) hold in the renormalized
RPA. Generic algorithm to include vertex corrections, preserving eqs. (3.9)
and therefore eqs. (3.7), was proposed by Baym and Kadanoﬀ (BK)[38, 39]
on the basis of the skeleton-expansion diagrammatic analysis with respect
to the dressed Green’s function. In principle, the exact self-energy can beobtained from eqs. (3.6) and (3.11) combined with this BK algorithm[41].
In the following subsections, we introduce two approximations, which
satisfy the Ward identity (3.9). For details of a systematic procedure forgenerating a series of such approximations, see Appendix A.

3.2.2

Mean-ﬁeld and random-phase approximations

When the excess self-energy is neglected, the Ward identity (3.9) gives no(0)vertex correction, i.e., Λσ,σ′ (k; q) = δσσ′ . This leads to MFA for the singleparticle Green’s function and (unrenormalized) RPA for the response functions. From eq. (3.3), the single-particle Green’s function in MFA is givenbyG(0)σ (k) =

1.
iϵl + µ̃σ − εk(3.12)On the other hand, from eq. (3.6b), the one-interaction irreducible response
function in RPA is given by∫(0)χ̃σσ′ (q) ≡ χ̃0 (q)δσ,σ′ = −

(0)
G(0)
σ (k + q)Gσ (k)δσ,σ ′ .

(3.13)kThis approximation is, however, not suitable to the present study of the static
susceptibilities; The uniform (q = 0) response functions are not aﬀected at
all by charge ﬂuctuation developed at a ﬁnite wave number q = Q∗ , because
the response function is calculated in a closed form for a given q, and has nomode coupling.
18

Figure 3.1: The Feynman diagram (a) for the eﬀective electron-electron in(0)teraction Vσσ′ (q), (b) for the approximate single-particle Green’s function(1)(1)
Gσ (k), and (c) for the approximate vertex function Λσσ′ . The second termof r.h.s. in (c) corresponds to the Maki-Thompson-type (MT) vertex correction, while the sum of the last two terms corresponds to the AslamazovAL,1
AL,2
Larkin-type (AL) vertex correction (ΛALσσ ′ = Λσσ ′ + Λσσ ′ ).

3.2.3

Inclusion of vertex corrections

We start with the eﬀective interaction of RPA as
∑
(0)
(0)
Vσσ′ (q) = vσσ′ (q) −vσσ1 (q)χ̃σ1 σ2 (q)Vσ2 σ′ (q) ,(3.14)

σ1 ,σ2

which is expressed by the diagram in Fig. 3.1 (a). By using the eﬀective
interaction, we approximate the Green’s function as−1(k)]−1 − Σ̃(1)= [G(0)[G(1)σ (k).
σ (k)]
∫ σ
(0)
Σ̃(1)
G(0)
σ (k) = −σ (k + q)Vσσ (q) ,qwhere the diagram is shown in Fig. 3.1 (b).
19

(3.15)(3.16)

In the present approximation, it is requested that the Ward identity (3.9)
holds as discussed in § 3.2.1. For this purpose, it is essential to utilize the concept of the Hedin’s formalism constructed by the skeleton diagrams, but we
employ the non-skeleton diagrammatic formalism [40]. The vertex function(1)
Λσσ′ (k; q) carrying a momentum q is constructed by replacing the internal
(0)
(1)
(0)
(0)
Gσ′ (k ′ ) line in Σσ (k) by Gσ′ (k ′ + q)Gσ′ (k ′ ) in all the possible ways. Here,note that all the internal momenta k ′ are replaced so as to conserve thewave numbers and the Matsubara frequencies at all the internal and external vertices. The diagram for the vertex function thus obtained is shownin Fig. 3.1 (c). By this procedure, we can prove various types of the Wardidentity, one of which is the Ward identity (3.9), i.e.,(1)∂ Σ̃σ (k)(1)Λσσ′ (k; 0) = δσσ −.
∂ µ̃σ′(3.17)

This relation is proved by seeing the fact that the diﬀerential operation in
the second term on the right hand side is equivalent to an operation in which(0)(1)
one ﬁrst plucks out the internal Gσ′ (k ′ ) line in Σ̃σ (k) in all the possible(0)
ways and then replaces it with Gσ′ (p)2 ; this operation is nothing but thevertex insertion described above for q = 0.
The vertex corrections can be written as the sum of the two contributions
as(1)

Λσσ′ (k; q) = δσ,σ′ + ΛMT (k; q)δσ,σ′ + ΛAL
σσ ′ (k; q) .

(3.18a)The Maki-Thompson (MT) type vertex correction ΛMT (k; q) is written as
∫
MT
(0)
(0)
Λ (k; q) =Vσσ(q1 )G(0)(3.18b)σ (k + q1 + q)Gσ (k + q1 ) ,q1

while the Aslamazov-Larkin (AL) type vertex correction ΛALσσ ′ (k; q) is written
as
∫
(0)
(0)
AL
Λσσ′ (k; q) =G(0)σ (k − q1 )Vσσ ′ (q1 )Vσσ ′ (q1 + q)
q1

× [ γσ′ (q1 ; q) + c.c. ] ,20(3.18c)Figure 3.2: The Feynman diagrams for the one-particle irreducible response
function (a) without vertex corrections, (b) with the Maki-Thompson typevertex corrections and (c) with the Aslamazov-Larkin type vertex corrections.
The thin and thick solid lines with arrows represent the bare and dressed(1)(0)
Green’s functions, Gσ (k) and Gσ (k), respectively. The thick wavy lines(0)
represent the eﬀective electron-electron interaction Vσσ′ (q).
where γσ (q1 ; q) is the fermion loop with three vertex insertions carrying momenta q1 , q and −q1 − q given by
∫
(0)
(0)
γσ (q1 ; q) = − G(0)(3.18d)σ (p)Gσ (p + q1 )Gσ (p + q1 + q) .
p
(1)

The one-interaction irreducible part χ̃σσ′ (q) in the present approximation
is given as
∫
(1)
(1)
(1)
χ̃σσ′ (q) = − G(1)(3.19)σ (k + q)Gσ (k)Λσσ ′ (k; q) .
k
(1)

There are three contributions in χ̃σσ′ (q):(1)χ̃σσ′ (q) = χ̃MF (q)δσ,σ′ + χ̃MT (q)δσ,σ′ + χ̃ALσσ ′ (q) .

(3.20)The Feynman diagram for each component is shown in Fig. 3.2. Note thatthese diagrams are a little diﬀerent from the usual ones appearing in previous
literature; in both MT and AL correction, two of the Fermion lines are dressed21

Green’s functions, and the other are bare Green’s functions. In the nonskeleton diagrammatic formalism, only this combination guarantees the exact
relation (3.7) between the isothermal susceptibilities and the q-limit of the
response functions through the Ward identity (3.9). The approximate spinand charge response functions are then calculated by1∑
χσσ′ (q),2 σ,σ′1∑ ′χSz Sz (q) =σσ χσσ′ (q),2 σ,σ′
χN N (q) =(3.21a)(3.21b)

These response functions contain a leading contribution of nontrivial vertex
corrections and introduce mode coupling between charge and spin ﬂuctuations with diﬀerent wave numbers. Since the MT and AL type vertex corrections has a crucial role near a second-order transition point [35], it is expectedthat physically comprehensive results are obtained at this level of approximation. In the next section, we study a role of these vertex corrections near
the charge-ordering transition based on this approximation.
Finally, we mention extension of the present approximation. We canconstruct the next level of approximation by making the next self-energy fromthe lastly obtained vertex function. Performing this procedure iteratively,
exact series of diagrams for all the correlation functions are generated [41, 45],
although it is practically diﬃcult to sum up them. Details of this formal
extension are given in Appendix.

22

3.3

Numerical results and discussions

3.3.1

Model

In this section, we consider a single-band EHM with the nearest-neighbor
Coulomb potential for a 2D square lattice for a simple demonstration of
charge ﬂuctuation eﬀect on the susceptibilities. The Hamiltonian is given as
HEHM =∑(εk − µ)c†kσ ckσ +∑U∑niσ niσ̄ + V
ni nj .
2 i,σ(3.22)〈ij〉k,σThe band dispersion is given as εk = 2t(cos(kx a) + cos(ky a)), where t is atransfer integral between nearest-neighbor sites and a is a lattice constant. U
and V are coupling constants for the on-site and nearest-neighbor repulsive∑
Coulomb interactions, respectively. µ is determined to ﬁx n = σ nσ = 3/2
by using Eq. (3.2) and σ̄ denotes the opposite spin of σ. In the following, we
set t = 1, a = 1 and kB = 1.
In this model, the bare Coulomb interactions are represented asvσσ (q) = V (q),(3.23a)vσσ̄ (q) = U + V (q),(3.23b)where V (q) = 2V (cos qx + cos qy ). In our approximation described in 3.2.3,
we obtain static charge and spin susceptibilities asχ̃N N (q),1 + vc (q)χ̃N N (q)χ̃Sz Sz (q)χSz Sz (q) =,1 − U χ̃Sz Sz (q)χN N (q) =(3.24a)(3.24b)where vc (q) = U + 2V (q) has been introduced for usefulness. The oneparticle irreducible response functions χ̃N N (q) and χ̃Sz Sz (q) are calculated(0)(1)
from χ̃σσ′ (q) = χ̃0 (q)δσ,σ′ for RPA and from χ̃σσ′ (q) = χ̃SC (q)δσ,σ′ +χ̃MT (q)δσσ′ +χ̃AL
σσ ′ (q) for the approximation with the leading vertex corrections. In the restof this thesis, the latter approximation, which is abbreviated as ”the 1-st levelof non-skeleton conserving approximation (1NSCA) ”, is used if not speciﬁed
23

1.4

RPA
1NSCA

1.2
1

T 0.8
0.6

CO

0.4
0.2
1

1.2

1.4

1.6

1.8

2

V
Figure 3.3: Phase diagram on the plane of T and V at U = 3. The dashedline and solid line represent the CO transition line determined by RPA and1NSCA, respectively.
in the text. In order to calculate the convolution form, we use the FastFourier-Transform (FFT) algorithm. The ﬁrst Brillouin zone is divided into
64 × 64 meshes. The frequency sum is terminated at ϵc whose value is about
40 times as large as the band-width W = 8t for T = 0.1. In this section, we
ﬁrst determine the phase diagram on the (V, T ) plane. Next, we calculate
the V -dependence for uniform spin and charge susceptibilities.

3.3.2

Phase diagram

In the following, we ﬁx U = 3, at which a spin density wave (SDW) instability is not observed for T ≥ 0.05. The phase diagram on the (V, T )plane at U = 3 is shown by a solid line in Fig. 3.4, where a charge-orderingtransition is determined by the divergence of charge susceptibility, i.e., by
1 + vc (q)χ̃N N (q) = 0. At high temperatures, the Fermi surface is obscure, and q-dependence of χ̃N N (q) is weak. As a result, checkerboard-typecharge-ordering with the wave number q = Qcb = (π, π) occurs mainlyby q-dependence of vc (q). With decreasing temperatures, the Fermi sur24

χN N (q, 0)

3

2(T, V) = (0.05, 1.7)
(0.1, 1.7)
(0.2, 1.6)
(0.3, 1.5)

10(0, 0)(π, 0)(π, π)(0, 0)q
Figure 3.4: Charge susceptibilities obtained by 1NSCA for (T, V ) =(0.05, 1.7), (0.1, 1.7), (0.2, 1.6), (0.3, 1.5) at U = 3. The peak of charge susceptibility continuously shifts from Qcb = (π, π) to Qic ≈ (3π/4, 3π/4) for
T < 0.3.

face becomes clear, and q-dependence of χ̃N N (q) becomes strong. Below
T ∼ 0.2, the wave number of divergence of the charge susceptibility changes
from q = Qcb to q = Qic ≈ (3π/4, 3π/4), where the wave number Qic ofincommensurate charge-ordering is determined by both vc (q) and χ̃N N (q).
We show the charge susceptibility in the metallic phase near charge-orderingtransition in Fig. 3.4. In the ﬁgure, it can be seen that the peak of thecharge susceptibility shifts from q = Qcb to q = Qic ≈ (3π/4, 3π/4) as thetemperature decreases. In the phase diagram, reentrant transition (metallic→ CO → metallic with decreasing temperatures) is observed. This is due tononmonotonic temperature-dependence of χN N (Qcb ). Because Qcb is close to
Qic , χN N (Qcb ) ﬁrst increases by broadening of the peak of χ̃N N (q) at q = Qic
with increasing temperatures till T ≈ 0.5, and decreases with increasing temperatures more. As shown by the dashed line in Fig. 3.3, the overall feature ofdivergence of χN N (q) including reentrant behavior are also observed in RPA[29, 33]. It is noted that even considering the vertex-corrections, the reentrant charge-ordering behavior remains[46]. The vertex corrections suppress25

2

χc,s /χ0 (0)

1.6
1.2

χs /χ0 (0)χc /χ0 (0)

0.8

χRPA
/χ0 (0)c
RPAχs /χ0 (0)

0.4
0
0

0.4

0.8

1.2

1.6

V
Figure 3.5: Static charge and spin susceptibility for T = 0.1 at U = 3.
The dotted line and solid line represent static charge susceptibility and spin
susceptibility, respectively. The incommensurate charge-ordering transition
occurs at V = 1.788.
the CO transition especially at high temperatures because electron-electron
scattering becomes strong due to obscureness of the Fermi surface. It is
expected that this suppression of the CO transition may be weaken when
the higher-order vertex corrections are considered. To include these terms,
we have to go beyond the present approximation by, e.g., a higher level ofnon-skeleton conserving approximation (NSCA) discussed in Appendix A.

3.3.3

Static response functions

Let us move on the main result of this chapter. The uniform susceptibilities
χs and χc calculated by RPA and the 1NSCA are shown in Fig. 3.5, where thecharge-ordering transition occurs at V = 1.788 in the 1NSCA for T = 0.1
and U = 3. As shown in Fig. 3.5, χs calculated by the 1NSCA increases
towards the charge-ordering transition. It is noted that χs is independent of
V in RPA as shown in Fig. 3.5 since χ̃Sz Sz (q) = χ0 (q) in Eq. (3.24b) does
not include V . Therefore, this characteristic enhancement of χs is caused by26

the vertex corrections. On the other hand, χc depends on V in both RPA
and the 1NSCA as shown in Fig. 3.5, where the eﬀect of vertex correction
is small as indicated by slight change between two approximations. In thefollowing subsections, we discuss the eﬀect of vertex correction on χs and χcby investigating the V -dependence.
Uniform spin susceptibility
In the 1NSCA, the uniform spin susceptibility is calculated as
χs =

1
,
χ̃Sz Sz (0)−1 − U(3.25)

MT
AL
where χ̃Sz Sz (0) = χ̃SC
s + χ̃s + χ̃s is obtained as a sum of three contributions,from χ̃SC , χ̃MT , and χ̃ALσσ ′ . From eq. (3.25), it is clear that the V -dependenceof χs comes only from χ̃Sz Sz (0) for ﬁxed U . Therefore, χs is enhanced toward the charge-ordering transition by increase of χ̃Sz Sz (0) for positive U .
MTALIn Fig. 3.6, the V -dependence of three components, χ̃SC
ares , χ̃s , and χ̃sshown by subtracting their values for V = 0, respectively. It is found that,with increasing V , χ̃ALand χ̃MT
increase, while χ̃SC
decreases due to thesssself-energy eﬀects.
We ﬁrst discuss the behavior of χ̃MT
s . If we consider the leading correction proportional to V and replace the dressed Green’s function by the bareGreen’s function, χ̃MT
is obtained by the integral of the interaction on thes
Fermi surface SF :∫MT2χ̃s ≈ ρdkdk′ V (k − k′ )SF= ρ 〈V (q)〉SF .
2(3.26a)The sign of this correction depends on both V (q) and the shape of the Fermi
surface. In Fig. 3.7, we show the change of the Fermi surface and the corresponding averaged interaction 〈V (q)〉 when the ﬁlling factor n is changed.
In the present simple model with nearest-neighbor Coulomb interaction on asquare lattice, 〈V (q)〉SF is positive for an arbitrary ﬁlling factor n as shown inthe ﬁgure. This is because the largest contribution comes from q = k−k′ ∼ 027δ χ̃c,s (V ) = χ̃c,s (V ) − χ̃c,s (0)0.03SCδ χ̃SCs (V ) = δ χ̃c (V )MTδ χ̃MTs (V ) = δ χ̃c (V )δ χ̃ALc (V )δ χ̃ALs (V )

0.02
0.01
0
-0.01
-0.02
0

0.4

0.8

1.2

1.6

V
Figure 3.6: The V -dependence of irreducible static response functions isshown. for T = 0.1 at U = 3. The charge-ordering transition occurs at
V = 1.788 in the 1NSCA.
in the integral of eq. (3.26a). Therefore, χ̃MT
is expected in general to ins
crease as V increases; this behavior is consistent with the result in Fig. 3.6.
Next, we discuss the behavior of χ̃ALin terms of the eﬀective interaction
s
(0)
RPA
Vσσ′ (q) = Vσσ′ (q). Near charge-ordering transition, the static componentof the eﬀective interaction is dominant. Therefore, χ̃AL
is approximatelys
proportional to the eﬀective interaction,
1 ∑ ′ RPA ∗ 2σσ Vσσ′ (Q ) = VcRPA (Q∗ )VsRPA (Q∗ ).
(3.27)
2 σ,σ′
Here, VcRPA (q) and VsRPA (q) are calculated asvc (q),
1 + vc (q)χ̃0 (q, iωn = 0)−U
VsRPA (q) =,1 − U χ̃0 (q, iωn = 0)
VcRPA (q) =(3.28a)(3.28b)and Q∗ represents the peak momentum of VcRPA (q). It is remarkable that
VcRPA (Q∗ ) near the CO state is negative because of vc (Q∗ ) < 0, while
28

(a)
π

(b)
0.015

n=1
n = 1.2
n = 1.4
n = 1.6
n = 1.8ky π/20.01<V(q)>SF
0.005

0

π/2
kx

0

π

1

1.2

1.4

1.6

1.8

2

n

Figure 3.7: (a) The n dependence of the Fermi surface for the non-interacting
system. (b) The n dependence of 〈V (q)〉SF . With increasing n, the valueof 〈V (q)〉SF increases till n ≈ 1.9. The sharp decrease of 〈V (q)〉SF aboven ≈ 1.9 is due to the fact that the Fermi surface becomes much small. Notethat the value of 〈V (q)〉 for 0 ≤ n ≤ 1 is the same as the one for 1 − n.
VsRPA (Q∗ ) is always negative for positive U in the absence of magnetic instability. Therefore, χ̃AL
increases toward the charge-ordering transition; Thissbehavior is consistent with the result in Fig. 3.6.
In conclusion, the charge ﬂuctuation developed near charge-ordering transition increases the spin susceptibility through the vertex corrections; thebehavior of χs in Fig. 3.5 can be understood qualitatively by increase of χ̃MTs
ALand χ̃s .
Uniform charge susceptibility
The uniform charge susceptibility is calculated in the 1NSCA as
χc =

1
,
χ̃N N (0)−1 + U + 8V

(3.29)

MT
AL
where χ̃N N (0) = χ̃SC
c + χ̃c + χ̃c in the 1NSCA. In contrast to χs , the charge
susceptibility is suppressed by V appearing in the denominator even in the29

RPA level (χ̃N N (0) = χ̃0 (0)). In the present calculation, V -dependence of χcthrough the vertex correction is weak as indicated from the result in Fig. 3.5.
There remains, however, a possibility that the vertex correction due to thecharge ﬂuctuations aﬀects the behavior of χc . Since χ̃MTis equal to χ̃MTcs , weAL
focus on the behavior of χ̃c . By using the same approximation as for χ̃ALsin the previous subsection, χ̃AL
is
found
to
be
proportionaltoasquareoftheceﬀective interaction
1 ∑ RPA ∗ 2 [ RPA ∗ ]2 [ RPA ∗ ]2
(3.30)V ′ (Q ) = Vc (Q ) + Vs (Q ) .
2 σ,σ′ σσ
As a result, both χ̃MTand χ̃AL
increase as V increases toward the chargeccordering transition. Therefore, χc is expected to increase when the vertex
correction changes considerably to overwhelm the increase of 8V in the denominator in eq. (3.29). In order to study this possibility in details, thecharge ﬂuctuation should be fully treated beyond the present approximation. This problem will be studied in the next chapter.

30

3.4

Discussion

3.4.1

Relation to the Fermi liquid theory

In this subsection, the spin and charge susceptibilities are studied up to theleading-order correction derived by systematic inclusion of the vertex function. We expect that the present result gives a correct behavior of the susceptibilities toward the charge-ordering transition until the charge ﬂuctuation
is not developed so much. Just near the transition, however, the higherorder vertex correction becomes relevant due to strong charge ﬂuctuation.
Although the treatment of the higher-order correction is beyond the scopeof this chapter, it is feasible to summarize its eﬀect in terms of the Fermi
liquid theory, which is a natural description of the metallic phase. By using
a standard microscopic derivation of the isotropic Fermi liquid (FL), [43, 44]
the irreducible part of the uniform susceptibilities is written as
χ̃N N (0),1 + vc (0)χ̃N N (0)χ̃Sz Sz (0)χs =.
1 + vs (0)χ̃Sz Sz (0)χc =(3.31a)(3.31b)Here, we deﬁne vc (0) and vs (0) as the spin symmetric and spin anti-symmetricparts of the bare Coulomb interaction, respectively. In FL theory, χ̃c and χ̃s
are given asρχ̃N N (0) =,(3.32a)1 + ρf˜0s
ρ,(3.32b)χ̃Sz Sz (0) =1 + ρf˜0awhere f˜0s and f˜0a are the spin symmetric and spin anti-symmetric partsof the interaction for the quasi-particle based on MF and ρ is the density
for the quasi-particle on the Fermi surface. Using Eq. (3.32a) and (3.32b),
Eq. (3.31a) and (3.31b) are rewritten asρρ
=,(3.33a)χc =1 + ρf0s
1 + ρ(vc (0) + f˜0s )
ρ
ρ
χs =
=
,
(3.33b)a˜
1 + ρf0a
1 + ρ(vs (0) + f0 )31

where f0s = vc (0) + f˜0s and f0a = vs (0) + f˜0a are the spin symmetric andspin anti-symmetric parts of the quasi-particle interaction. Our formalism
corresponds to approximate |ρf˜0s |, |ρf˜0a | ≪ 1, where f˜0s = f0MT + f0s,AL andf˜0a = f0MT + f0a,AL . In this limit, χ̃c and χ̃s are approximated as followsχ̃c ≈ ρ(1 − ρf˜0s ) = ρ(1 − ρ(f0MT + f0s,AL )),χ̃s ≈ ρ(1 − ρf˜a ) = ρ(1 − ρ(f MT + f a,AL )).
000(3.34a)(3.34b)From the discussion given in sec. 3.3.3, we ﬁnd that all of f0MT , f0s,AL and
f0a,AL decrease with the increase of charge ﬂuctuations for a single band EHM
for a square lattice model.
In the case of a single-band Hubbard model, a SDW state appears with increasing U . To consider the fact that the contribution of AL term is expected
to dominate in the vicinity of a SDW state, the uniform charge susceptibility
increases near a SDW state, while the spin susceptibility decreases. On theother hand, in the case of a single-band EHM with nearest-neighbor Coulomb
potential, the uniform charge susceptibility decreases near a CO state, while
the spin susceptibility increases as discussed in 3.3.3. However, as discussed
in 3.3.3, it is noted that the charge ﬂuctuation near a CO state is not fully
taken into account and the bare Coulomb potential becomes dominant forcalculating the uniform charge susceptibility in this approximation. Therefore, there is a possibility that the uniform charge susceptibility increases inthe vicinity of a CO state if the charge ﬂuctuation is fully taken into account.

32

3.5

Summary

In this chapter, we have developed a simple approximate method including
vertex corrections which satisfy the identities between the isothermal susceptibilities and the so-called q-limit of the response functions. This method
gives us a systematic phase diagram for the second-order transition and theisothermal susceptibilities by using the same response function as it should
be. We showed that the reentrant charge-ordering transition, which has been
observed in RPA[29], remains even though the vertex corrections are taken
into account.
The notable point for including vertex corrections is that the uniform
spin susceptibility increases toward the charge-ordering transition point. To
interpret this tendency, we roughly evaluate the vertex-correction terms andshow that this enhancement is due to decrease f0a from a viewpoint of Landau’s Fermi Liquid theory. The experimental data such as θ-(BEDT-TTF)2
RbZn4 [4] and β-(meso-DMBEDT-TTF)2 PF6 [20] show that the spin susceptibility has a broad hump at the charge-ordering transition. This tendency
that the spin susceptibility increases toward a CO transition point from a
metallic state is consistent with our results.
On the other hand, the uniform charge susceptibility decreases toward thecharge-ordering transition point. This is because that though f0MT and f0AL
decrease, the bare Coulomb potential becomes dominant and f0s increases
with increasing V . However, there is the possibility that the uniform charge
susceptibility is enhanced, if the charge ﬂuctuation is fully treated.

33

Chapter 4
Negative compressibility
induced by large charge
ﬂuctuations
4.1

Introduction

In Chapter 3, we have constructed a new diagrammatic approach basedon the non-skeleton conserving approximation including both the bare anddressed Green’s function. We applied this method to the EHM to evaluate
the ﬁrst correction of the response function with respect to development ofcharge ﬂuctuations. This correction consists of the self-energy correction andthe vertex corrections of the Maki-Thompson type and the Aslamazov-Larkin
type. This calculation is suitable to study of the eﬀect of charge ﬂuctuationon the response function toward CO transition. This approximation, however, lacks an important feedback eﬀect of charge ﬂuctuations onto the vertex
correction of the response function.
In this chapter, we employ the Baym-Kadanoﬀ conserving approximation (BKCA) [38, 39]. In particular, the following two aspects of the BKCA
should be noted in the present context: (1) The eﬀective interaction Veﬀ (q)and the self energy Σ(k) are self-consistently determined within the skeleton diagrammatic conserving approximation. By using the Green’s function
thus obtained, the response function can be calculated for an arbitrary wave
35

number neglecting vertex corrections. This way of calculation of the response
function includes Renormalized Random-Phase Approximation (RRPA) and
Fluctuation Exchange (FLEX) approximation. (2) For a limited wave number q, we can calculate the response function including all the vertex corrections corresponding to BKCA by derivative of the particle number withrespect to the external potential[39]. Therefore, it is expected that BKCA
is suitable to include the divergent increase of the eﬀective interaction at q
= Q∗ as the feedback eﬀect onto vertex corrections near the charge-ordering
critical point. For example, This aspect of the BKCA was applied to the
problem of the divergent compressibility near the antiferromagnetic criticalpoint for a 2D Hubbard model[36]. The eﬀects of critical ﬂuctuations on
the superconductivity (SC) transition have also been well studied for a 2D
extended Hubbard model by Onari et al. [32] in FLEX approximation. They
have determined a CDW and SDW state by RRPA and discussed SC transition through charge and spin ﬂuctuations. However, the eﬀects of charge
ﬂuctuations on the uniform susceptibilities have not been investigated.

FLEX approximation is suitable to treat a magnetic transition, sinceladder-type diagrams are included and the contribution of longitudinal spin
susceptibilities is equal to that of transverse spin susceptibility. However,
it is expected that the contribution of ladder-type diagrams is weak for the
charge-ordering transition, since no singularity for magnetic property is expected. Actually, it is indicated that the symmetry of the SC state obtainedby FLEX approximation coincides with RPA in the region where charge ﬂuctuation is strong and spin ﬂuctuation is weak [30]. With this observation of
the validity if RPA for the CO transition, we employ RRPA [38] which subtracts ladder-type diagrams from FLEX approximation to determine the COtransition. The uniform (q = 0) charge susceptibility can be calculated byχc = dn/dµ. We call this approximation of the response function as simply
the Shielded Interaction Approximation (SIA). The response function of SIA
satisﬁes the compressibility sum rule dn/dµ = limq→0 χN N (q, 0). This is the
crucial diﬀerence between SIA and RRPA for the calculations of the response
functions. In the succeeding sections, we give the precise deﬁnition of SIA,
and calculate the isothermal spin susceptibility and compressibility in SIA.
36

4.2

Formulation

4.2.1

Baym-Kadanoﬀ conserving approximation

In this subsection, we brieﬂy review BKCA. The advantage of the approximation based on Baym-Kadanoﬀ scheme is to conserve the particle number,energy, momentum and angular momentum[38]. Baym constructed a simple
formalism to procedure BKCA[39]. In this formalism, we ﬁrst choose theLuttinger-Ward functional Φ[G], where G(q) is the dressed Green’s function
given as[ 0 ]−1− Σσ (q).
(4.1)
G−1
σ (q) = Gσ (q)Here, we use the abbreviation q ≡ (q, iωn ), G0σ (q) is the bare Green’s function
and the self-energy Σσ (q) is obtained by the following relation,
Σσ (q) =δΦ[G].
δGσ (q)(4.2)

For convenience, we deﬁne Σ̃σ (q) as the self-energy subtracting the HartreetermΣ̃σ (q) = Σσ (q) − ΣHartree
.
(4.3)
σ
The vertex function Λσ1 σ2 (k1 ; q) is deﬁned as∑
Λσ1 σ2 (k1 ; q) = δσ1 σ2 +K̃σ1 σ3 (k1 , k3 )Gσ3 (k3 )Gσ3 (k3 + q)Λσ3 σ2 (k3 ; q) (4.4)
k3 ,σ3

with the irreducible kernel K̃σσ′ (q1 ; q2 ) which is given byK̃σ1 σ2 (q1 , q2 ) =δ Σ̃σ1 (q1 ).
δGσ2 (q2 )Then, the response function is given as∑
χσσ′ (q) = χ̃σσ′ (q) −χ̃σσ1 (q)vσ1 σ2 (q)χσ2 σ (q),

(4.5)

(4.6)

σ1 ,σ2where vσ1 σ2 (q) represents the electron-electron interaction and χ̃σσ′ (q) is deﬁned as∑χ̃σσ′ (q) = −β −1Gσ (k + q)Gσ (k)Λσσ′ (k; q).
(4.7)
k

37

It is noted that this formulation satisﬁes the Ward identity given as Eq. (3.9).
For convenience, we write this relation again as follows:
Λσ1 σ2 (k1 ; 0) = δσ1 σ2 −∂ Σ̃σ1 (k),∂ µ̃σ2(4.8)

is the chemical potential absorbing the Hartree
where µ̃σ = µ + hσ − ΣHartreeσterm. We can see that Eq. (4.4) satisﬁes the Ward-identity by rewriting∂ Σ̃σ1 (k)in Eq. (4.8) as∂ µ̃σ
2∑ ∂Gσ (k3 ) ∂ Σ̃σ (k)∂ Σ̃σ1 (k)3
1=∂ µ̃σ2∂µ̃∂Gσ2σ3 (k3 )k3 ,σ3
[
]
∑
∂
Σ̃
(k
)
σ3 3
=−
Gσ3 (k3 )2 δσ2 ,σ3 −Kσ1 σ3 (k, k3 )∂µ̃σ2
k3 ,σ3∑K̃σ1 σ3 (k, k3 )Gσ3 (k3 )2 Λσ3 σ2 (k3 ; 0).
=−

(4.9)

k3 ,σ3

Finally, we check that the the compressibility and isothermal spin susceptibility obtained from the thermodynamic potential coincides with thesusceptibilities deﬁned by Eq. (4.6)[36]. Using Φ[G], the thermodynamic
potential Ω is represented asΩ = −β −1

∑

{ [
]
}
eiωn η ln −G−1
σ (k, iωn ) + Gσ (k, iωn )Σσ (k, iωn ) + Φ[G],k,ωn ,σ(4.10)
where η is a positive inﬁnitesimal. The electron number nσ is obtained by
diﬀerentiating Ω with respect to the chemical potential µσ as
nσ = −∑∂Ω= β −1Gσ (k, iωn )eiωn η .
∂µσk,ω

(4.11)

nThen, we can obtain the susceptibility χσσ′ (0) as[]∑∂Σσ (k, iωn )∂nσ−1
iωn η 2
χσσ′ (0) == −βe Gσ′ (k, iωn ) δσ,σ′ −. (4.12)∂µσ′∂µ
σ′k,ω ,σn38

It is seen that Eq. (4.11) and (4.12) coincide with exact relations as describedin sec. 3.2. The isothermal charge and spin susceptibility are deﬁned as
1 ∑ ∂nσ1∑
χc ==χσσ′ (0),(4.13)
2 σ,σ′ ∂µσ′
2 σ,σ′1∑ ′1 ∑ ′ ∂nσ(4.14)σσ=σσ χσσ′ (0),χs =2 σ,σ′∂µσ′
2 σ,σ′
respectively. In this way, we can obtain the compressibility and isothermal
spin susceptibility which satisfy exact relations. It is noted that the vertex
corrections are included through the diﬀerentiating Σ with respect to µ or
δh.

4.2.2

Shielded interaction approximation

In the vicinity of charge-ordering transition, charge density ﬂuctuations become important. To consider this eﬀect, we choose the Luttinger-Ward functional Φ[G] as the sum of all bubble ring-diagrams as shown in Fig. 4.1 (a).
Then, the self-energy Σ(q) in Fig. 4.1 (b) is given by Eq. (4.2) as∑∑
SIAΣσ (q) =vσσ′ (0)nσ′ + β −1Vσσ(k)Gσ (k + q),(4.15)σ′k

SIA
where the shielded interaction Vσσ(k) shown in Fig. 4.1 (c) is deﬁned as
∑
SIA
Vσσ
(q) = −vσσ (q) +vσσ1 (q)χRRPA(4.16)
σ1 σ2 (q)vσ2 σ (q).
σ1 σ2Here, vσσ (q) = V (q) and vσσ̄ (q) = U + V (q) are the wave representationof the Coulomb potential between the same and diﬀerent spins for EHM,
respectively. The susceptibility χRRPA
σσ ′ (q) is given as
∑
SC
RRPA
χRRPA
χ̃SC
(4.17)
σσ ′ (q) = χ̃σσ (q) −σσ1 (q)vσ1 σ2 (q)χσ2 σ ′ (q),σ1 ,σ2where χ̃SC
σσ ′ (q) (SC means the Self-energy Correction) is the irreducible susceptibility for the dressed Green’s function,∑−1
Gσ (k)Gσ (k + q)δσ,σ′ .
(4.18)
χ̃SCσσ ′ (q) = −β
k

39

(a)

ΦSIA [G] =

+

1
2

+

+ ···

1
3

(b)

Σ

=

+

=

+

(c)

(d)

~

K

=

Λ

=

+

+

(e)

+

~

K

Λ

Figure 4.1: The Feynman diagrams of the SIA: (a) the Luttinger-Ward funcSIA, (c) thetional ΦSIA , (b) the self-energy Σ, (c) the shielded interaction Vσσirreducible Kernel K̃ and (d) the vertex functions Λ.

40

In this thesis, the chemical potential µ is determined to ﬁx the electron∑
number n = σ nσ as 3/2 by using Eq. (4.11) within the accuracy |n−nﬁx | ≤10−8 . We solved coupled equations Eq. (4.1) and (4.15) iteratively until
satisfying the convergence condition for each k and σ within the accuracy
(i+1)

|Σσ

( i)

(k) − Σσ (k)|(i+1)
|Σσ (k)|≤ 10−6 ,(4.19)

where i denotes the i-th iteration.
SIA
In SIA, the divergence of eﬀective interaction Vσσ(q) reﬂects on the selfenergy Σ(q). We investigate the inﬂuence of the charge ﬂuctuations through
SIA
this feedback eﬀect. The divergence point of Vσσ(q) is equivalent to that of
RRPA
χN N (q) deﬁned as
χRRPA
(q) =
NN

1 ∑ RRPA
χ ′ (q).
2 σσ′ σσ(4.20)

Hereafter, we deﬁne the charge-ordering transition point as the divergencepoint of χRRPA(q) at some ﬁnite q. It is noted that the sum rule
NN
χc ≡

∂n
= lim χRRPA (q)∂µ q→0 N N(4.21)

does not hold in RRPA, since the vertex corrections required in BKCA are
not taken into account. Uniform charge susceptibility, however, can be calculated by direct derivative of the particle number with respect to the chemical
potential (see l.h.s in Eq. (4.21)). Charge compressibility χc thus obtained
should include not only Maki-Thompson (MT) and Aslamazov-Larkin (AL)type vertex corrections but also higher-order ones. Likewise, χs obtained by
diﬀerentiating m with respect to δh also includes appropriate vertex corrections. In the next section, we investigate the eﬀect of charge ﬂuctuations on
χc and χs in this way.

41

4.3

Numerical results

In this section, we take a lattice size N = 642 and the Matsubara frequency
for fermions −(2Nc −1)πT ≤ ωn ≤ (2Nc −1)πT with Nc = 211 above T = 0.1.
While, we take N = 1282 and Nc = 211 below T = 0.1.

4.3.1

Transition point determined by RRPA

In the case of h = 0, χ̃SC
σσ ′ does not depend on the spin indices and we canSCSCdeﬁne χ̃ (q) ≡ χ̃σσ′ (q). Then, the charge and spin susceptibility obtained
by RRPA at SIA level is rewritten as
RRPA

χN N

(q) =

∑

χRRPA
σσ ′ (q) =

σσ ′

χRRPA
(q) =
Sz Sz

∑

χ̃SC (q),1 + [U + 2V (q)] χ̃SC (q)σσ ′ χRRPA
σσ ′ (q) =σσ ′χ̃SC (q).
1 − U χ̃SC (q)(4.22)(4.23)We determine the charge-ordering transition by the divergence of χRRPA(q, 0)
NNat some ﬁnite q.
First, we see the momentum dependence of χRRPA(q, 0) and specify the
NN
∗
RRPA
wave vector Q , where χN N (q, 0) has maximum value. Fig. 4.2 (a) shows
the momentum dependence of χRRPA(q, 0) and χRRPA(q, 0) at U = 4 and
NN
Sz Sz
RRPA
T = 0.1. Two peaks of χN N (q, 0) are due to the anisotropic wave dependence of V (q) and that of the Fermi surface. The former gives a peak at
q = Qcb = (π, π) and the latter at q = Qinc = ( 43 π, 34 π). It is seen that
χRRPA
(Qcb , 0) is enhanced with increasing V in this case. To determine theNNdivergent point of χRRPA(Qcb , 0), we calculate V -dependence of χRRPA(Qcb , 0)NNNN
as shown in Fig. 4.2 (b) and ﬁnd that charge-ordering transition occurs at
V ≈ 2.066. We also check the V -dependence of the maximum value ofχRRPA(q, 0). With increasing V , χRRPA(q, 0) decreases. Therefore, it is seen
Sz Sz
Sz Sz
that the spin ﬂuctuations are suppressed by the charge ﬂuctuations.

42

(a)

(b)
5

RRPA
(q, 0)
χNN
V=2
V = 2.06
χRRPA(q, 0)S z Sz3

V=2
V = 2.062
10
Max χRRPA(q, 0) NN
RRPA
Max χSz Sz (q, 0)

4

χ(q, 0)χ(q, 0)4

32
1(0, 0)(π, π)(π, 0)(0, 0)

0

1.2

1.4

1.6

1.8

2

V
Figure 4.2: (a) The wave dependence of χRRPA(q, 0) and χRRPA(q, 0) ats
NN[]
U = 4 and T = 0.1. (b) The V dependence of Max χRRPA
(q,
0)
and
NN
[ RRPA]Max χs(q, 0) at U = 4 and T = 0.1. The charge-ordering transition
occurs about V ≈ 2.066.

4.3.2

Response functions

In the previous subsection, we determine the charge-ordering transition point
by RRPA. In this subsection, we investigate the eﬀect of the charge ﬂuctuations on the isothermal spin susceptibility χs and compressibility χc . To
obtain χc , we utilize the relation Eq. (4.12). In this method, χc includes
vertex corrections through ∂Σas explained §4.2.2. Likewise, we calculate χs∂µby diﬀerentiating m with respect to δh to include the vertex corrections.
Isothermal spin susceptibility
To obtain χs , we numerically apply a small magnetic ﬁeld δh and calculate∑magnetization m = σ σnσ by the following relation,∑nσ = β −1Gσ (k, iωn )eiωn η .
(4.24)
k,ωnFig. 4.3 (a) shows h-dependence of m at V = 1 (red line), 2 (green line) and2.05 (blue line). It is seen that m varies linearly near δh ≈ 0 for each V .
43

2.2

(b)

(a)
4 [ 10-4]

8 [ 10 ]-3

V = 1.5
V=2
V = 2.06

V=1
V=2
V = 2.05

δ µ̃ 0

m0

-4
-5

0

5 [ 10 ]
-4

h

-8
-5

0

δn

5[ 10-4]

Figure 4.3: (a) Magnetic ﬁeld h dependence of the magnetization m at U = 4
for T = 0.1. (b) δn dependence of δ µ̃ = δµ − ΣHartree at U = 4 for T = 0.1,
where we deﬁne δn = n − 1.5 and δµ = µ − µﬁx . µﬁx is the chemical potential
determined to ﬁx n = 1.5. The divergence of χc occurs at V ≈ 2.064.

Therefore, we calculate χs by δm/δh, where δh = 10−4 .
The blue line in Fig. 4.4 (a) represents χ0 /χs at U = 4 for T = 0.1,
where χ0 is the compressibility for non-interaction system. With decreasingδV ≡ VCO − V , the charge ﬂuctuations become large, where VCO represents
the charge-ordering transition point. Though the enhancement of charge
ﬂuctuations near VCO is reﬂected on χs through vertex corrections, anomalous
behavior is not observed and χ−1monotonically decreases with increasings
V . This tendency qualitatively coincides with the 1NSCA as discussed in
Chapter 3.
Isothermal charge susceptibility
To obtain χc , we numerically increase the electron number density n = 3/2
by a small amount δn. Then, the chemical potential µ slightly changes fromµ to µ + δµ. Fig. 4.3 (b) shows δn dependence of the chemical potential
absorbing Hartree term δ µ̃ = δµ − δΣHartree for V = 1.5 (red line), 2 (greenline) and 2.06 (blue line). It is seen that δn is proportional to δ µ̃ near δ µ̃ ≈ 0
44

(b)
4

4

3

2

χ0/χc
χ0/χs

2
1
0

χ0/χc

χ0/χc , χ0/χs

(a)

0

0.2

0.4

χ0/χc

0
-2

0.6

0.8

1

-4

0

0.004 0.008 0.012 0.016 0.02

VCO - VVCO - V

Figure 4.4: V -dependence of χ0 /χc and χ0 /χs at U = 4 for T = 0.1 in therange (a) 0 ≤ VCO − V ≤ 1 and (b) 0 ≤ VCO − V ≤ 0.1. Here, VCO ≈ 2.066
is deﬁned as the charge-ordering transition point determined by RRPA. The
red and green line represent χ0 /χc and χ0 /χs , respectively. The divergenceof χc occurs at V ≈ 2.060.

for each V . Therefore, we calculate χ̃c = δn/δ µ̃ for δn = 10−4 and obtain χc−1by χ−1c = χ̃c + U + 2V (0).
The red line in Fig. 4.4 (a) and (b) represents χ0 /χc at U = 4 for T = 0.1.
With decreasing δV from δV = 1, χ−1
c ﬁrst increases till δV ≈ 0.3. This result
coincides with the result of 1NSCA as discussed in Chapter 3. In this region,
the contribution of the bare Coulomb potential V to the compressibility
is dominant and the compressibility decreases. Since the enhancement of
SIA
Vσσ(Qcb , 0) near VCO is taken into account in SIA, it is expected that thecontribution of the vertex corrections to the compressibility is dominant in
the vicinity of charge-ordering transition. This tendency is appeared belowδV ≈ 0.3 and χ−1
c becomes 0 before charge-ordering transition. This indicatesthat the electronic system becomes unstable due to charge ﬂuctuations.
Next, we determine the phase diagram on the V -T plane obtained fromdivergence of the response function χRRPA(Q∗ , 0) responsible for a chargeNNordering transition and the uniform charge compressibility χc . The phase
diagram on the V -T plane for U = 4 is shown in Fig. 4.5. The blue points
45

0.5

χc-1= 00.4
0.3

TCharge Order( RRPA )

Metal
0.2
0.1
0

1.6

1.8

2

2.2

2.4

V
Figure 4.5: The phase diagram in the V − T plane for U = 4 with a uniform metallic and a charge ordered state determined by RRPA. The blue
points represent the charge-ordering transitions. The red points represent
the divergence points of static charge susceptibility. An end point appears
at T ≈ 0.401.
represent the charge-ordering transition points on which χRRPA(Q∗ , 0) diNNverges. This result shows reentrant transition (metallic → CO → metallic
with decreasing the temperature) in a ﬁnite region of V at RRPA level. This
reentrant transition is also observed in both 1NSCA and RPA as shown in
the previous chapter. The characteristic transition temperature obtained
here is, however, lowered from RPA (or 1NSCA). From viewpoint of Fermi
liquid, the self-energy correction included in RRPA is expected to lower thetransition temperature by suppression of the response function at ﬁnite temperatures through both strong electron scattering (the imaginary part of theself-energy) and mass renormalization (the real part of the self-energy). In
order to study this eﬀect more quantitatively, careful analysis of the selfenergy is needed since the response function at q = Q∗ = Qcb should involve
quasi-particle excitation far from the Fermi surface.
The red points in Fig. 4.5 represent Vins (T ) where χc diverges for ﬁxed
T . It is seen that an end point of Vins appears at T ≈ 0.401. Fig. 4.6 shows46

6
T = 0.42
T = 0.41

4

T = 0.4

χc-1 2

T = 0.39

0
-2
1.8

1.82

1.84

1.86

1.88

1.9

V
Figure 4.6: V -dependence of χ−1
c at U = 4. The pink, red, blue and green line
represent the results for T = 0.39, 0.4, 0.41 and 0.42, respectively. Above
T = 0.41, the divergence of χc does not appear.
V -dependence of χ−1
at U = 4 for T = 0.39, 0.4, 0.41 and 0.42. It is seencthat χc has a local minimum point at V = Vmin at ﬁxed T . With increasing
SIA
V , the eﬀective interaction Vσσ(Q∗ , 0) becomes large, while G(q) decreases
since the electron scattering (the self-energy correction) is enhanced. Therefore, it is expected that the decrease of the compressibility for V > Vmin isdue to the eﬀect of G(q). In SIA, χc has higher-order terms of AslamazovLarkin (AL) type vertex corrections which are expected to be important forthe compressibility with increasing charge ﬂuctuations as discussed in theprevious chapter. These terms have λ(QCB = (QCB , 0)) which is deﬁned as∑λ(QCB ) = −G2 (k)G2 (k + QCB ).
(4.25)
kFig. 4.7 (a) and (b) show V -dependence of λ(QCB ) at T = 0.2 and 0.42.
It is seen that λ(QCB ) decreases with increasing V for both temperatures.
However, the ratio of decrease of λ(QCB ) is large and the value of λ(QCB )is small for high temperatures. This tendency coincides with the behavior
of Vins at high temperatures and it is expected that the suppression of G(q)has the key to form the end point.
47

(b) T = 0.42

0.058

0.013

0.057

0.012

λ(QCB )

λ(QCB )(a) T = 0.2

0.056

0.055
1.9

1.91

1.92

0.011

0.01
1.86

V

1.87

1.88

1.89

V

Figure 4.7: V -dependence of λ(QCB )at U = 4 for (a) T = 0.2 and (b)T = 0.42. The charge-ordering transition point determined by RRPA is at
VCO = 1.92 for T = 0.2 and VCO = 1.954 for T = 0.42, respectively.

4.4

Summary and discussions

In this chapter, we have shown divergence of the uniform charge susceptibility χc before the charge-ordering transition in the present model, i.e., 2D
EHM with nearest-neighbour Coulomb interaction. The negative charge susceptibility means instability of electrons with respect to the uniform changeof the electron density. One possible state is a spatially inhomogeneous stateof a uniform metallic state and a charge-ordered state with diﬀerent electron
densities. We call this mixed state as a ”phase separation”. We show aschematic ﬁgure of the phase separation in Fig. 4.8. It is noted that the system is already in the phase separation before divergence of χc . The transition
point into the phase separation should be determined by reconstruction of
the nonconvex free energy by the Maxwell construction law. In the present
calculation, we cannot perform this procedure because of the lack of the freeenergy in the charge ordered phase. The negative compressibility, however,guarantees at least the existence of the phase separation in the present model.
In order to relate this electronic instability to the experimental observation of inhomogeneous charge disproportion in organic conductors, we needcareful consideration of approximation adopted in the model. Especially, the48

long-range Coulomb interaction neglected in the EHM is important since thecharge-rich(poor) phase with imbalance between the electron charge and thebackground positive charge of ions has a macroscopic charge, and therefore
has a large charging energy due to long-range Coulomb interaction. We expect that in the presence of the long-range Coulomb interaction this negative
compressibility drives the system into a ”microscopic” inhomogeneous staterather than the macroscopic phase separation[47]. It is an important question
left for a future problem to clarify the ﬁnal inhomogeneous electronic staterealized after electronic instability of uniform metallic state. We conjecturethat strong electron-phonon coupling may play an important role to relax akind of charge frustration induced by long-range interaction.
In this chapter, we clariﬁed one possible realization of electronic instability starting with a metallic state by perturbative approach using diagrammatic expansion. We emphasize that the existence of the Fermi surface isessential for the instability in the present calculation. We speculate that thenegative compressibility may be understood by the Pomeranchuk instability
with respect to uniform change of electron density in terms of the Fermi liquid theory. Then, it is critical to ask whether the negative compressibility
survives even in the presence of strong electron correlation. This is a diﬃcult
question to be answered, and have to be left for a future problem. We only
note that the phase separation is obtained in the one-dimensional EHM forlarge positive V by exact diagonalization method [48]. Finally, we provide
a complementary result for a hint of consideration of phase separation in
the strongly correlated regime. In Fig. 4.9, we show the distance between
two instabilities of phase separation and CO indicating the size of the phaseseparation region in the phase diagram for T = 0.1. We ﬁnd that the phaseseparation region is suppressed when the on-site interaction U increases. This
result strongly indicates that the phase separation seen in the present calculation disappears in the limit of U → ∞ while ﬁxing V . This suggests thatthe correct strong-coupling limit should be taken to keep a ﬁnite portion of
doublon excitation if one considers phase separation in the strong couplingregion.

49

F (δn)
χ−1
c =0

CO

Metal

Phase
Separation

δnO

Figure 4.8: Schematic particle density (δn = n − 1) dependence of the freeenergy for the phase separation with uniform metal and a CO states. The
concave region appears in the dashed line. This thermodynamically unstableregion is replaced with the phase separated state by Maxwell construction.
0.05
3
CO

VCO-Vins

0.04

2
V
0

0.02

VCO

Metal

1

0.03

0 1 2 3 4 5 6
U

0.01
0
0

1

2

3

4

5

6

U
Figure 4.9: The U dependence of VCO − Vins at T = 0.1. The inset shows thephase diagram on (U, V ) plane at T = 0.1. The charge-ordering transitionline is determined by RRPA.
50

Chapter 5
Summary
In this thesis, electronic properties of organic conductors in the presenceof the inter-site Coulomb interaction are investigated in the viewpoint of
diagrammatic expansion starting from the metallic state.
We ﬁrst studied the spin and charge response functions by using a newscheme of generating a series of non-skeleton conserving approximation(NSCA).
We showed that the reentrant transition observed in RPA remains even in theﬁrst-order NSCA. We also showed that the spin susceptibility is enhanced toward the CO transition. This result is consistent with the measurement of thespin susceptibility of θ-(BEDT-TTF)2 RbZn(SCN)4 and β-(meso-DMBEDTTTF)2 PF6 .
Next, we studied the uniform charge susceptibility by using the shieldedinteraction approximation within the Baym-Kadanoﬀ conserving approximation. The merit of this approximation is that all the vertex corrections canbe fully treated by direct diﬀerential calculation of the response of the systemwith respect to the external potential. This calculation allows us to study
the feedback eﬀect of strong charge ﬂuctuations onto the response function
through the vertex corrections. We ﬁnd that the strong charge ﬂuctuations
lead to a remarkable enhancement of the charge compressibility and evenmake it negative indicating the instability of uniform metallic state near the
CO transition. This may provide a trigger of the inhomogeneous charge
disproportion called as ”short-range charge-ordering” observed in the exper51

iments of θ-(BEDT-TTF)2 RbZn(SCN)4 etc.
The present calculation has been done for a simple square-lattice EHM
with nearest-neighbor Coulomb interaction. In order to compare the present
theory with experiments, it is needed to consider several important eﬀects. In
particular, the long-range Coulomb interaction beyond the nearest-neighborsite and electron-phonon interaction will be important to analyze the experiments. More realistic calculation including these eﬀects will be a future
problem.

52

Appendix A
Non-skeleton conserving
approximation
In this Appendix, we present an iterative approximation scheme for systematic inclusion of vertex corrections to the charge and spin response functions.
(i)
We construct (A) the single-particle Green’s function Gσ (k), (B) the ver(i)(i)(i)
tex function Λσσ′ (k; q), and (C) the response functions χN N (q) and χSz Sz (q).
These functions are constructed so that Ward identity holds at each level ofapproximation assigned by an integer i based on the Kadanoﬀ-Baym theory,and the next level of approximation assigned by i + 1 is constructed from thei th approximation. Our algorithm of generating a series of approximations isalmost the same as the one proposed by Takada [41]. Our scheme diﬀers from
Takada’s one on three points; (1) we start with the Hartree approximation as
an initial approximation, (2) we use the standard diagrammatic analysis for(0)Σ̃σ (k) with respect to the bare or mean-ﬁeld Green’s function Gσ (k) insteadof the skeleton-expansion diagrammatic analysis with respect to the dressed(i)Green’s function Gσ (k), and (3) we simplify iterative procedure by avoiding
the Bethe-Salpeter equations. Our scheme is suitable to study the leading
contribution of the vertex corrections.
(A) The single-particle Green’s function: Suppose the excess self-energy(i)(0)Σ̃σ (k) is given as a functional of the mean-ﬁeld Green’s functions Gσ (σ =53±), i.e.
(0)

(i)
Σ̃(i)
σ (k) ≡ Σ̃σ (k : [G± ]) .

(A.1)

(0)

We take the initial function as Σ̃σ (k) = 0 (the Hartree approximation). Let
the single-particle Green’s function be
(0)
−1
−1
G(i)
− Σ̃(i),σ (k) = [Gσ (k)σ (k)](A.2)(i)

where the corresponding diagram is shown in Fig. A.1 (a). Gσ (k) is a func(i)tion of µ̃σ ; the particle number and chemical potential are calculated as∫(i)nσ = eiϵη G(i)(A.3)
σ (k) ,k∑(i)
µσ(i) = µ̃(i)+vσσ′ (0)nσ′ .
(A.4)
σ
σ′
(i)

In this paper, we ﬁx µ̃σ ≡ µ̃σ during the calculation, and determine µ̃σ by(i)nσ = 3/4 from a µ̃σ -n curve.
(B) The vertex function: Following the Kadanoﬀ-Baym scheme, the ver(i)(i)(0)
tex function Λσσ′ (k; q) ≡ Λσσ′ (k; q : [G± ]) is given as the sum of δσσ′ andall the possible diagrams obtained by inserting the external vertex carrying
a momentum q into an arbitrary G(0) -line with a spin σ ′ in each diagram(0)(i)
for Σ̃σ (k : [G± ]). This procedure is schematically shown by the Feynman
diagrams in Fig. A.1 (b). Note that the wave numbers and the Matsubara
frequencies at all the internal and external vertexes are conserved. The vertex function thus obtained satisﬁes various types of the Ward identity, oneof which is given by
(i)

(i)

Λσσ′ (k; 0) = δσσ′ −∂ Σ̃σ (k).
∂ µ̃σ′(A.5)(i)

This identity is proved by the same way as in § 3.2.3 as follows. Since Σ̃σ (k)(0)(i)
depends on µ̃± only through G± , the diﬀerential operation on Σ̃σ (k) with(0)
respect to µ̃σ corresponds to an operation to pick up one internal Gσ′ (p)(i)(0)
line in Σ̃σ (k) in all the possible ways, and to replaces it with Gσ′ (p)2 . Thisdiﬀerential operation is nothing but the vertex insertion for q = 0 as seen
54

(i)

in Fig. A.1 (b). (C) The response functions: The response function χσσ′ (q)(i)
is written in terms of the one-interaction irreducible part χ̃σσ′ (q) at the i thlevel of approximation as
∑
(i)
(i)
(i)
(A.6a)
χσσ′ (q) = χ̃σσ′ (q) −χ̃(i)σσ1 (q)vσ1 σ2 (q)χσ2 σ ′ (q) ,σ1 ,σ2

∫

(i)

(i)

χ̃σσ′ (q) = −

(i)
G(i)
σ (k + q)Gσ (k)Λσσ ′ (k; q) .

(A.6b)

k(i)(i)

Then, the charge and spin response functions χN N (q) and χSz Sz (q) are calculated as
1 ∑ (i)(i)
χN N (q) =χ ′ (q),(A.7a)
2 σ,σ′ σσ
1 ∑ ′ (i)(i)χSz Sz (q) =σσ χσσ′ (q).
(A.7b)
2 σ,σ′As proved in § 3.2.3, the Ward identity (A.5) leads the following identity between the isothermal susceptibilities and the q-limit of the response functions
at the i th level of approximation:
1 ∂n(i)(i)= χN N (0),2 ∂µ1 ∂m(i)(i)= χSz Sz (0).
χ(i)
≡
s
2 ∂h
χ(i)
c ≡

(A.8a)(A.8b)(D) The self-energy in the next level of approximation: For a systematic improvement in the approximation, we require that the vertex functions should be taken account in a manner consistent with the single-particle
Green’s function. This requirement makes us to select the self-energy in thenext level of approximation as
∑∫
(i)
(i)
(i+1)
(A.9a)G(i)Σ̃σ (k) = −σ (k + q)Vσσ ′ (q)Λσσ ′ (k; q) ,
σ′

q
(i)

where the eﬀective interaction Vσσ′ (q) is given by
∑
(i)
(i)
Vσσ′ (q) = vσσ′ (q) −vσσ1 (q)χ̃(i)σ1 σ2 (q)Vσ2 σ ′ (q) .
σ1 ,σ2

55(A.9b)

The diagrams of these equations are shown in Fig. A.1 (c) and (d). With
(i+1)
(i+1)
(0)
this Σ̃σ (k) ≡ Σ̃σ (k : [G± ]), we can construct the approximation for
(i+1)
(i+1)
χN N (q) and χSz Sz (q) following the processes in (A), (B) and (C).
In the text, we have used two approximations, i.e., the RPA and the approximation with leading vertex corrections. In the iterative approximation
described above, the former corresponds to i = 0, and the latter to i = 1. In
principle, we can continue the iterative process as one hopes, although it isdiﬃcult to continue it for i ≥ 2 practically. We note that response functions
approaches the exact ones in the limit of i → ∞ as [41](i)(A.10a)(i)(A.10b)lim χN N (q) = χN N (q) ,i→∞lim χSz Sz (q) = χSz Sz (q) .

i→∞

56

Figure A.1: The Feynman diagrams of the present iterative approximation:(a) the Dyson equation, (b) construction of the vertex function, (c) the effective interaction, and (d) the next deﬁnition of the self-energy.

57

Bibliography[1] H. Seo, C. Hotta and H. Fukuyama: Chem. Rev. 104 (2004) 5005.
[2] K. Hiraki and K. Kanoda: Phys. Rev. Lett. 80 (1998) 4737.
[3] T. Takahashi, Y. Nogami and K. Yakushi: J. Phys. Soc. Jpn. 75 (2006)
051008.
[4] H. Mori, S. Tanaka and T. Mori: Phys. Rev. B 57 (1998) 12023.
[5] M. Watanabe, Y. Noda, Y. Nogami and H. Mori: J. Phys. Soc. Jpn. 73(2004) 116.
[6] M. Watanabe, Y. Noda, Y. Nogami and H. Mori: J. Phys. Soc. Jpn. 74(2005) 2011.
[7] K. Miyagawa, A. Kawamoto and K. Kanoda: Phys. Rev. B 62 (2000)
R7679.
[8] R. Chiba, K. Hiraki, T. Takahashi, H. M. Yamamoto and T. Nakamura:Phys. Rev. Lett 93 (2004) 216405.
[9] K. Yamamoto, K. Yakushi, K. Miyagawa, K. Kanoda and A. Kawamoto:Phys. Rev. B 65 (2002) 085110.
[10] K. Inagaki, I. Terasaki and H. Mori: Physica B 329 (2003) 1162.
[11] T. Mori: J. Phys. Soc. Jpn. 72 (2003) 1469.
[12] M. Kaneko and M. Ogata: J. Phys. Soc. Jpn. 75 (2006) 014710.
59

[13] S. Nishimoto, M. Shingai, and Y. Ohta: Phys. Rev. B 78 (2008) 035113.
[14] H. Watanabe and M. Ogata: J. Phys. Soc. Jpn. 75 (2006) 063702.
[15] C. Hotta: J. Phys. Soc. Jpn. 72 (2003) 840.
[16] R. Chiba, H. M. Yamamoto, K. Hiraki, T. Nakamura and T. Takahashi:
Synth. Met. 120 (2001) 919.
[17] H. Kobayashi, R. Kato, A. Kobayashi, Y. Nishio, K. Kajita and W.
Sasaki: Chem. Lett. (1986) 833.
[18] S. Kimura, H. Suzuki, T. Maejima, H. Mori, J. Yamaura, T. Kakiuchi,
H. Sawa and H. Moriyama: J. Am. Chem. Soc. 128 (2006) 1456.
[19] S. Kimura, T. Maejima, H. Suzuki, R. Chiba, H. Mori, T. Kawamoto, T.
Mori, H. Moriyama, Y. Nishio and K. Kajita: Chem. Commun. (2004)
2454.
[20] H. Mori: J. Phys. Soc. Jpn. 75 (2006) 051003.
[21] H. Mori: private communication.
[22] S. Niizeki, F. Yoshikane, K. Kohno, K. Takahashi, H. Mori, Y. Bando,
T. Kawamoto and T. Mori: J. Phys. Soc. Jpn. 77 (2008) 024714.
[23] H. Seo, J. Merino, H. Yoshioka and M. Ogata: J. Phys. Soc. Jpn. 75(2006) 051009.
[24] Y. Ohta, K. Tsusui, W. Koshibae and S. Maekawa: Phys. Rev. B 50
(1994) 13594.
[25] M. Calandra, J. Merino and R. H. McKenzie: Phys. Rev. B 66 (2002)
195102.
[26] R. Pieteg, R. Bulla and S. Blawid: Phys. Rev. Lett. 82 (1999) 4046.
[27] K. Hanasaki and M. Imada: J. Phys. Soc. Jpn. 74 (2005) 2769.
[28] H. Seo: J. Phys. Soc. Jpn. 69 (2000) 805.
60

[29] A. Kobayashi, Y. Tanaka, M. Ogata and Y. Suzumura: J. Phys. Soc.
Jpn. 73 (2004) 1115.
[30] Y. Tanaka, Y. Yanase and M. Ogata: J. Phys. Soc. Jpn. 73 (2004) 2053.
[31] K. Kuroki: J. Phys. Soc. Jpn. 75 (2006) 114716.
[32] S. Onari, R. Arita, K. Kuroki and H. Aoki: Phys. Rev. B 73 (2006)
014526.
[33] J. Merino, A. Greco, N. Drichko and M. Dressel: Phys. Rev. Lett. 96(2006) 216402.
[34] Y. Tanaka: Ph.D thesis, The University of Tokyo.
[35] K. Miyake and O. Narikiyo: J. Phys. Soc. Jpn. 63 (1994) 2042.
[36] K. Morita, H. Maebashi and K. Miyake: J. Phys. Soc. Jpn. 72 (2003)
3164.
[37] Y. Fuseya, H. Maebashi, S. Yotsuhashi and K. Miyake: J. Phys. Soc.
Jpn. 69 (2000) 2158.
[38] G. Baym and L. P. Kadanoﬀ: Phys. Rev. 124 (1961) 287.
[39] G. Baym: Phys. Rev. 127 (1962) 1391.
[40] L. Hedin: Phys. Rev. 139 (1965) A796.
[41] Y. Takada: Phys. Rev. B 52 (1995) 12708.
[42] K. Yoshimi, H. Maebashi, and T. Kato: in preparation.
[43] A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (New York, Dover, 1975).
[44] W. Metzner, C. Castellani and C. D. Castro: Adv. Phys. 47 (1998) 317.
61

[45] To obtain the exact relations, inclusion and exactness properties are
required[41]. In our formalism, it is easily seen that inclusion property
is satisﬁed for each iterations. While, the exactness property is satisﬁed
under the condition limi→∞ G(i) = G. In this limit, we can obtain theexact response functions.
[46] In RPA, another reentrant transition (metal → CO with further decreasing temperatures) occurs as shown by a small return of the transitionline at lowest temperatures in Fig. 3.3. This reentrant transition seems
to be suppressed in the 1NSCA, although one needs to calculate lowtemperature regions more to clarify it.
[47] V. J. Emery and S. A. Kivelson: Physica C 209 (1993) 597.
[48] K. Penc and F. Mila: Phys. Rev. B 49 (1994) 9670.

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