Dynamical Coherent-Potential Approximation and Tight-Binding Linear

Muffintin Orbital Approach to Correlated Electron System

Yoshiro KAKEHASHI�, Takafumi SHIMABUKURO, Toshihito TAMASHIRO, and Tetsuro NAKAMURA

Department of Physics and Earth Sciences, Faculty of Science, University of the Ryukyus,

1 Senbaru, Nishihara, Okinawa 903-0213

(Received May 18, 2008; accepted July 1, 2008; published August 25, 2008)

Dynamical coherent-potential approximation (CPA) to correlated electrons has been extended to asystem with realistic Hamiltonian which consists of the first-principles tight-binding linear muffintinorbital (LMTO) bands and intraatomic Coulomb interactions. Thermodynamic potential and self-consistent equations for Green function are obtained on the basis of the functional integral method and theharmonic approximation which neglects the mode–mode couplings between the dynamical potentials withdifferent frequency. Numerical calculations have been performed for Fe and Ni within the second-orderdynamical corrections to the static approximation. The band narrowing of the quasiparticle states and the6 eV satellite are obtained for Ni at finite temperatures. The theory leads to the Curie–Weiss law for bothFe and Ni. Calculated effective Bohr magneton numbers are 3.0 �B for Fe and 1.2 �B for Ni, explainingthe experimental data. But calculated Curie temperatures are 2020 K for Fe and 1260 K for Ni, being stilloverestimated by a factor of two as compared with the experimental ones. Dynamical effects on electronicand magnetic properties are discussed by comparing with those in the static approximations.

KEYWORDS: dynamical CPA, correlated electrons, tight-binding linear muffintin orbital method, iron, nickel, Curietemperature, effective Bohr magneton number, quasiparticle states

DOI: 10.1143/JPSJ.77.094706

1. Introduction

Understanding of electronic and magnetic properties ofthe system with intermediate strength of Coulomb inter-actions has been a challenging problem over half a centuryin condensed matter physics, because simple theoreticalapproaches are not applicable to the system in spite of thefact that many intriguing phenomena are found there.1–4)

Iron and nickel are considered to be an example of suchsystems. These metals in fact show the properties of both theweakly- and the strongly-correlated electrons. Photoemis-sion data, for example, show the existence of metallic d

bands5,6) and the Sommerfeld coefficients in the T-linearspecific heats show rather large values [5 – 7 mJ/(K2�mol)]as compared with those of the noble metal systems.7) Thequasiparticle band widths however are found to be narrowerthan the results of usual band calculations and a satellitepeak is observed at 6 eV below the Fermi level in Ni,8) whichare not able to be explained by a simple band theory,9)

suggesting rather strong electron correlations in thesesystems. The same features are found in the magneticproperties. Noninteger values of the ground-state magnet-ization in Fe and Ni are well explained by the bandtheory,10) while the paramagnetic susceptibilities of thesesystems follow the Curie–Weiss law and their effective Bohrmagneton numbers are close to those expected from thelocal-moment model.7) Large specific heats near the Curietemperature TC are also well explained by the same model.

The magnetic and electronic properties in the intermediateregime of Coulomb interactions have been traditionallyexplained by interpolation theories between the weak andstrong Coulomb interaction limits. Cyrot11) proposed aninterpolation theory on the basis of the functional integralmethod which transforms an interacting electron system intoan independent electron system with time-dependent random

charge and exchange fields. He showed that the staticand saddle-point approximations to the functional integralscheme can explain the local-moment vs itinerant behaviorin magnetism of transition metals, as well as the metal–insulator transition.

Hubbard12) and Hasegawa13) independently developed asingle-site spin fluctuation theory. They adopted a hightemperature approximation (i.e., the static approximation),and treated the random charge and exchange potentials bymaking use of the coherent-potential approximation (CPA).The theory qualitatively described the magnetization vstemperature curves, the Curie temperature, as well as theCurie–Weiss susceptibility in Fe and Ni. The theory,however, reduces to the Hartree–Fock approximation atzero temperature because it relies on the static approxima-tion. This means that the theory does not take into accountthe ground-state electron correlations as discussed byGutzwiller,14) Hubbard,15) and Kanamori.16) Furthermore,the quasiparticle bands and the satellite peak do not appearin the theory using the static approximation.

We proposed the dynamical CPA which fully takes intoaccount the electron correlations within the single-siteapproximation, and clarified the qualitative features ofdynamical effects using a Monte-Carlo sampling method.17)

More recently, we developed analytic method to thedynamical CPA,18) adopting the harmonic approximation(HA).19) The latter is based on the neglect of the mode–modecouplings between dynamical potentials in solving animpurity problem in an effective medium. The HA inter-polates between the weak Coulomb interaction limit and theatomic limit. Especially it describes the Kondo behaviorquantitatively in the strong correlation limit.20) We showedwithin the single band model that the dynamical CPA+HAyields the band narrowing of quasiparticle states and thesatellite peak in Fe and Ni, which were not explained by theearly theories with use of the static approximation. Thetheory was however based on the single-band Hubbard�E-mail: yok@sci.u-ryukyu.ac.jp

Journal of the Physical Society of Japan

Vol. 77, No. 9, September, 2008, 094706

#2008 The Physical Society of Japan

094706-1

model. Quantitative calculations of transition metals andalloys with use of the realistic Hamiltonian have not yet beenmade even within the single-site approximation.

In the present paper, we extend the dynamical CPA to themulti-band case adopting the first-principles tight-binding(TB) linear muffintin orbitals (LMTO) method.21,22) Themodern band theory is based on the density functional theory(DFT) which allows us to express the ground-state energy asa functional of the spin and charge densities of the system.23)

In the local density approximation (LDA),24) one approx-imates the energy functional with the energy function of thedensity. The LDA exchange-correlation potentials obtainedfrom the electron gas system have much simplified theelectronic band-structure calculations in solids. The TB-LMTO method allows us to construct the first-principlestight-binding one electron Hamiltonian, and to calculate theLDA band structure. We adopt the TB-LMTO Hamiltonianto describe the noninteracting part of the Hamiltonian, andtake into account the intraatomic Coulomb and exchangeinteractions between d electrons which are dominant amongelectron–electron interactions.

Similar theoretical approach has been developed in theproblem of the metal–insulator transition in infinite dimen-sions.25) The approach called the dynamical mean fieldtheory (DMFT) is equivalent to the dynamical CPA, as wehave shown recently.26) The present theory therefore shouldbe equivalent in principle to the DMFT combined with theLDA+U scheme in the band theory.27) The merits of thepresent approach may be summarized as follows. (1) Thedynamical CPA can treat the transverse spin fluctuations forarbitrary d electron number at finite temperatures, while thestandard DMFT combined with the quantum Monte-Carlomethod (QMC) cannot treat them because it is based on theIsing-type Hubbard–Stratonovich transformation.28) Becauseof the reason, the DMFT calculations for Fe and Ni havebeen performed so far without taking into account thetransverse spin fluctuations at finite temperatures.29) (2) TheHA which we adopted to solve the impurity problem is ananalytic approach from the high-temperature limit. Theapproach is suitable for understanding the finite-temperaturemagnetism because the zeroth approximation to the HAdescribes the magnetic properties much better than theHartree–Fock one. There is no corresponding approach inthe DMFT. (3) Because of the analytic theory, we cancalculate the excitation spectra up to the temperatures muchlower than those calculated by the QMC, using the Padenumerical analytic continuation method.30)

In the following section, we introduce a TB-LMTOHamiltonian with intraatomic Coulomb interactions. In§3, we formulate the dynamical CPA to the realisticHamiltonian on the basis of the functional integral tech-nique.31) Applying a generalized Hubbard–Stratonovichtransformation32) to the free energy, we transform theinteracting electrons into an independent electron systemwith time dependent random fields. Introducing an effectivemedium into the time dependent Hamiltonian, we will makea single-site approximation. We determine the mediumsolving a self-consistent equation, called the CPA equa-tion.33) In §4, we adopt the HA to calculate the dynamicalpart of the free energy, and derive the analytic expressions ofthe free energy, the dynamical CPA equation, and other

thermodynamic quantities. In §5, we present the numericalresults of calculation for Fe and Ni. The calculations havebeen performed by using the second-order dynamical CPA(i.e., the dynamical CPA+HA within the second-orderdynamical corrections). We explain the band narrowing ofthe quasiparticle states, the incoherent satellite peak at 6 eVbelow the Fermi level in Ni. We also present the results ofcalculations for the magnetization vs temperature curve, theparamagnetic susceptibility following the Curie–Weiss law,and the amplitude of local moments. We clarify thequantitative aspects of the theory comparing with theexperimental data, and examine the dynamical effects onvarious quantities comparing the dynamical results withthose in the static approximation. The last section 6 isdevoted to summarize the dynamical CPA and TB-LMTOHamiltonian approach, as well as the dynamical effects in Feand Ni.

2. TB-LMTO Hamiltonian

We adopt in the present paper the first-principles TB-LMTO method22) to construct a realistic many-bodyHamiltonian. In this case, atomic basis function with orbitalL on site i, �iLðr� RiÞ, are constructed from a muffintinatomic orbital ’iLðr� RiÞ on site i with an atomic level E�iL,and a tail function outside the muffintin potential h�jL0iL as

�iLðr� RiÞ ¼ ’iLðr� RiÞ þXjL0

_’’�jL0 ðr� RjÞh�jL0iL; ð1Þ

_’’�iLðr� RiÞ ¼ _’’iLðr� RiÞ þ ’iLðr� RiÞo�iL: ð2Þ

Here the wave function in the interstitial region has beenneglected because of the atomic sphere approximation. Theatomic wave function ’iLðr� RiÞ and its energy deriva-tive _’’iLðrÞ are defined by ’iLðrÞ ¼ �iLðE�iL; rÞYLðrrÞ and_’’iLðrÞ ¼ _��iLðE�iL; rÞYLðrrÞ, where YLðrrÞ is the cubic harmonicswith L ¼ ðl;mÞ, l being the azimuthal quantum number andm being an orbital index for l. �iLðE; rÞ is obtained by solvingthe radial Schrodinger equation with energy E. The energyE�iL is chosen to be the center of gravity below the Fermilevel for each orbital. The atomic orbitals f’iLg arenormalized in the atomic sphere as h’iLj’iL0 i ¼ �LL0 . Thetail coefficients h�jL0iL in eq. (1) are determined in such a waythat the orbital �iL is continuous and differentiable on thesphere boundary at each sphere. The coefficient o�iL in eq. (2)is determined so that the orbital �iL is well localized. Weadopt here the nearly orthogonal representation (i.e.,o�iL ¼ 0), in which the orbitals �iL’s become orthogonal upto second order in hiLjL0 . The TB-LMTO Hamiltonian matrixis then written as

HiLjL0 ¼ h�iLjð�r2 þ vðrÞÞj�jLi ¼ �iL�ij�LL0 þ tiLjL0 : ð3Þ

Here vðrÞ is a LDA potential, �iL is an atomic level, and tiLjL0

is a transfer integral between orbitals �iL and �jL0 .When we adopt the density functional theory, the one-

electron Hamiltonian (3), especially the atomic level �iL,contains the effects of strong intratomic Coulomb interac-tions in general. According to the LDA+U interpretation byAnisimov et al.,34) the atomic level �0iL for noninteractingsystem is obtained from the relation,

�0iL ¼@ELDA

@niL�@EU

LDA

@niL: ð4Þ

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-2

Here niL is the charge density at the ground state, ELDA isthe ground-state energy in the LDA, and EU

LDA is a LDAfunctional to the intraatomic Coulomb interactions. Amongvarious forms of EU

LDA, we adopt the Hartree–Fock typeform35) since we consider here an itinerant electron systemwhere the ratio of the Coulomb interaction to the d bandwidth is not larger than one.

EULDA ¼

1

2

Xj

Xmm0

Unjdnjd

þ1

2

Xj

Xmm0

0X

ðU � JÞnjdnjd: ð5Þ

Here njd ¼P

m njlm=2ð2lþ 1Þ with l ¼ 2. U and J are theorbital-averaged Coulomb and exchange interactions definedby

U ¼1

ð2lþ 1Þ2Xmm0

Umm0 ; ð6Þ

ðU � JÞ ¼1

2lð2lþ 1Þ

Xmm0

0ðUmm0 � Jmm0 Þ; ð7Þ

where Umm0 and Jmm0 are orbital dependent intraatomicCoulomb and exchange integrals for d electrons. Fromeqs. (4) and (5), we obtain the atomic level �0iL for non-interacting system as

�0iL ¼ �iL ��

1�1

2ð2lþ 1Þ

� �U

�1

21�

1

2lþ 1

� �J

�nd�l2: ð8Þ

Note that nd denotes the total d electron number per atom.The Hamiltonian which we consider here can be written

as

HH ¼ H0 þ H1: ð9Þ

The TB Hamiltonian for noninteracting system H0 is givenby

H0 ¼XiL

ð�0iL � �ÞnniL þXiLjL0

tiLjL0 ayiLajL0: ð10Þ

Here we have introduced the chemical potential � for thecalculation of the free energy. ayiL (aiL) is the creation(annihilation) operator for an electron with orbital L and spin on site i, and nniL ¼ ayiLaiL is a charge density operatorfor electrons with orbital L and spin on site i. We haveneglected the change of the transfer integrals due toelectron–electron interactions.

The interacting part H1 in eq. (9) consists of theintraatomic Coulomb interactions between d electrons.

H1 ¼Xi

"Xm

U0 nnilm"nnilm# þXm>m0

U1 �1

2J

� �nnilmnnilm0 �

Xm>m0

Jssilm � ssilm0#: ð11Þ

Here U0 (U1) and J are the intra-orbital (inter-orbital) Coulomb interaction and the exchange interaction, respectively.nnilm (ssilm) with l ¼ 2 is the charge (spin) density operator for d electrons on site i and orbital m, which is defined bynnilm ¼

P nnilm [ssilm ¼

P� ayiL�ð�=2Þ�aiL], � being the Pauli spin matrices.

3. Functional Integral Approach and Dynamical CPA

Thermodynamic properties of the system are calculated from the partition function, which is given by

Z ¼ Tr T exp �Z �

0

ðH0ð�Þ þ H1ð�ÞÞd�� �� �

: ð12Þ

Here � is the inverse temperature, T denotes the time-ordered product (T-product) for operators. H0ð�Þ [H1ð�Þ] is theinteraction representation of Hamiltonian H0 (H1).

The functional integral method is based on a Gaussian formula for the Bose-type operators fa�g.

expXmm0

amAmm0am0

!¼

ffiffiffiffiffiffiffiffiffiffiffidetA

M

r Z Ym

dxm

" #exp �

Xmm0

ðxmAmm0xm0 � 2amAmm0xm0 Þ

" #: ð13Þ

Here Amm0 is a M �M matrix, and fxmg are auxiliary field variables. Discretizing the integral with respect to time in eq. (12),and applying the formula (13) to the bose-type operators at each time under the T-product, we obtain a functional integralform of the free energy F as

e��F ¼Z YN

i¼1

Y2lþ1

m¼1

��imð�Þ��imð�Þ

" #Z0ð�ð�Þ; �ð�ÞÞ ð14Þ

� exp �1

4

Xi

Xmm0

Z �

0

d� �imð�ÞAimm0�m0 ð�Þ þXxyz�

�im�ð�ÞB�imm0�im0�ð�Þ

!" #;

Z0ð�ð�Þ; �ð�ÞÞ ¼ Tr T exp �Z �

0

Hð�; �;�i�Þ d�� �� �

; ð15Þ

Hð�; �;�i�Þ ¼XiL

�0iL � ��1

2

Xm0

iAimm0�im0 ð�Þ�l2

!nniLð�Þ �

X�

1

2

Xm0

B�imm0�im0�ð�Þ þ h�im

!�l2mm

�iLð�Þ

" #

þXiLjL0

tiLjL0ayiLð�ÞajL0ð�Þ: ð16Þ

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-3

Here N is the number of sites, mmiL ¼ 2ssiL, and �imð�Þ [�imð�Þ] is an auxiliary field being conjugate with inniLð�Þ [mmiLð�Þ] forl ¼ 2. The functional integrals in eq. (14) are, for example, defined by

Z Y2lþ1

m¼1

��imð�Þ

" #¼Z YN 0

n¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2lþ1 detAi

ð4 Þ2lþ1

s Y2lþ1

m¼1

d�imð�nÞffiffiffiffiffiN 0p

" #; ð17Þ

where 2lþ 1 in the square roots denotes the number of d orbitals (i.e., 2lþ 1 ¼ 5). detAi is the determinant of theð2lþ 1Þ � ð2lþ 1Þ matrix Aimm0 . �n denotes the n-th time when the time interval ½0; �� is divided into N 0 segments. Thematrices Aimm0 and B�imm0 (� ¼ x; y; z) are defined as

Aimm0 ¼ U0�mm0 þ ð2U1 � JÞð1� �mm0 Þ; ð18ÞB�imm0 ¼ Jð1� �mm0 Þ ð� ¼ x; yÞ; ð19ÞBzimm0 ¼ U0�mm0 þ Jð1� �mm0 Þ: ð20Þ

Equation (15) is a partition function for a time-dependent Hamiltonian Hð�; �;�i�Þ of an independent particle system. Notethat we have introduced a magnetic field h�im for convenience.

In the Matsubara frequency representation, the free energy F is written as

e��F ¼Z YN

j¼1

Y2lþ1

m¼1

��jm��jm

" #exp ��E½�; ��

� �; ð21Þ

E½�; �� ¼ ���1 ln Trðe��H0Þ � ��1 Sp lnð1� vgÞ

þ1

4

Xin

Xmm0

��imði!nÞAimm0�m0 ði!nÞ þX�

��im�ði!nÞB�imm0�im0�ði!nÞ

" #; ð22Þ

ðvÞiLnjL0n00 ¼ vjL0 ði!n � i!n0 Þ�ij�LL0 ; ð23Þ

viL0 ði!nÞ ¼ �1

2

Xm0

iAimm0�im0 ði!nÞ�l2�0 �X�

1

2

Xm0

B�imm0�im0�ði!nÞ þ h�im

!�l2ð�Þ0 : ð24Þ

The functional integrals in the Fourier representation in eq. (21) is given by

Z Y2lþ1

m¼1

��im

" #¼Z YN 0

m¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2lþ1 detAi

ð4 Þ2lþ1

s Y2lþ1

m¼1

d�imð0ÞY1n¼1

�2lþ1 detAi

ð4 Þ2lþ1d2�imði!nÞ

" #: ð25Þ

Here the field variable �imði!nÞ [�im�ði!nÞ] denotes the n-frequency component of �imð�Þ [�im�ð�Þ], and d2�imði!nÞ ¼dRe �imði!nÞd Im �imði!nÞ. The energy functional E½�; �� ineq. (22) consists of the noninteracting term [the first termat the right-hand-side (r.h.s.)], the scattering term due todynamical potential (the second term), and the Gaussianterm (the third term). Sp in the second term at the r.h.s. ofeq. (22) means a trace over site, orbital, frequency, and spin.g in the second term denotes the temperature Green functionfor noninteracting system H0. The dynamical potential v isdefined by eqs. (23) and (24), and � in eq. (24) denotes the� component of the Pauli spin matrices.

In the effective medium approach,18) we introduce acoherent potential

ð�ÞiLnjL0n00 ¼ �Lði!nÞ�ij�LL0�nn0�0 ; ð26Þ

into the energy functional E½�; ��, and expand it with respectto v�� as

E½�; �� ¼ ~FF þXi

Ei½�i; �i� þ�E: ð27Þ

Here the zeroth order term ~FF is a coherent part of the freeenergy which is defined by

~FF ¼ ���1 ln Trðe��H0 Þ � ��1 Sp lnð1��gÞ: ð28Þ

Note that the coherent part does not depend on the dynam-ical potential.

The next term in eq. (27) consists of a sum of the single-site energies Ei½�i; �i�, which are defined by

Ei½�i; �i� ¼ ���1 tr lnð1� �viFiÞ

þ1

4

Xn

Xmm0

½��imði!nÞAimm0�im0 ði!nÞ

þXxyz�

��im�ði!nÞB�imm0�im0�ði!nÞ�: ð29Þ

Here tr means a trace over orbital, frequency, and spin onsite i. �vi ¼ vi ��i, and vi (�i) is the dynamical (coherent)potential on site i. Fi is the site-diagonal component of thecoherent Green function defined by

ðFiÞjLnj0L0n00 ¼ FiLði!nÞ�ij�ij0�LL0�nn0�0 ; ð30ÞFiLði!nÞ ¼ ½ðg�1 ��Þ�1�iLniLn: ð31Þ

The last term in eq. (27) denotes the higher order terms inexpansion.

�E ¼ ���1 Sp lnð1� ~ttF0Þ: ð32Þ

Here ~tt is the single-site t-matrix defined by

~tt ¼ ð1� �viFiÞ�1�vi; ð33Þ

and F0 is the off-diagonal coherent Green function definedby

ðF0ÞiLnjL0n00 ¼ ½ðg�1 ��Þ�1�iLnjL0n00 ð1� �ijÞ�0 : ð34Þ

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-4

The dynamical CPA is a single-site approximation whichneglects the intersite dynamical correlations �E. The freeenergy is then written as

FCPA ¼ ~FF �Xi

��1 ln

Z Ym

��im��im

" #e��Ei½�i ;�i�: ð35Þ

The dynamical coherent potential �iLði!nÞ should bedetermined so that the nonlocal corrections �E vanish inaverage. This means that

h~ttii ¼ 0; ð36Þ

where

hð�Þi ¼

Z Ym

��im��im

" #ð�Þ e��Ei½�i;�i�

Z Ym

��im��im

" #e��Ei½�i ;�i�

: ð37Þ

The above condition called the CPA equation is writtenas

hGðiÞiLði!nÞi ¼ FiLði!nÞ; ð38Þ

GðiÞiLði!nÞ ¼ ½ðF�1i � �viÞ

�1�iLniLn: ð39ÞHere the left-hand-side (l.h.s.) of eq. (38) is a temperatureGreen function for an impurity system in the effectivemedium, whose Hamiltonian is given as follows.

HðiÞð�Þ ¼ ~HHð�Þ þ HðiÞ1 ð�Þ

�Z �

0

d�0XL

ayiLð�Þ�iLð� � �0ÞajLð�0Þ; ð40Þ

~HHð�Þ ¼XiL

ð�0iL � �Þ nniLð�Þ þXiLjL0

tiLjL0 ayiLð�ÞajL0ð�Þ

þXjL

Z �

0

d�0ayjLð�Þ�jLð� � �0ÞajLð�0Þ; ð41Þ

HðiÞ1 ð�Þ ¼Xm

U0 nnilm"ð�Þnnilm#ð�Þ

þXm>m0

U1 �1

2J

� �nnilmð�Þnnilm0 ð�Þ

�Xm>m0

J ssilmð�Þ � ssilm0 ð�Þ: ð42Þ

It should be noted that the CPA equation (38) is equivalentto the following stationary condition.

�FCPA

��iLði!nÞ¼ 0: ð43Þ

4. Harmonic Approximation to the Dynamical CPA

We can rewrite the free energy (35) by means of aneffective potential projected onto the zero frequencyvariables �im ¼ �imð0Þ and �im ¼ �imð0Þ.

FCPA ¼ ~FF � ��1 ln

Z Y�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2lþ1 detB�

ð4 Þ2lþ1

s Ym

d�m�

" # ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2lþ1 detA

ð4 Þ2lþ1

s Ym

d�m

" #e��Eð�;�Þ: ð44Þ

Note that we have redefined FCPA and ~FF by those per site, assuming that all the sites are equivalent to each other.Furthermore we omit here and in the following all the site indices for simplicity.

The effective potential Eð�; �Þ in eq. (44) consists of the static part Estð�; �Þ and the dynamical one Edynð�; �Þ.

Eð�; �Þ ¼ Estð�; �Þ þ Edynð�; �Þ; ð45Þ

Estð�; �Þ ¼ ���1 tr ln½1� �v0Fi� þ1

4

Xmm0

�mAmm0�m0 þXxyz�

�m�B�mm0�im0�

" #; ð46Þ

e��Edynð�;�Þ ¼ D

�Z Y1

n¼1

Y�

�2lþ1 detB�

ð2 Þ2lþ1d2�m�ði!nÞ

" #�2lþ1 detA

ð2 Þ2lþ1

Ym

d2�mði!nÞ

" #

� D exp ��

4

Xn 6¼0

Xmm0

��mði!nÞAmm0�m0 ði!nÞ þX�

��m�ði!nÞB�mm0�m0�ði!nÞ

!" #; ð47Þ

D ¼ det �nn0�LL0�0 �X00

~vvL00 ði!n � i!n0 Þ ~ggL00L00 ði!n0 Þ

!: ð48Þ

Here �v0 in eq. (46) is defined by �v0 ¼ vð0Þ ��:

ð�v0ÞLnL0n00 ¼ ðvL0 ð0Þ ��Lði!nÞ�0 Þ�LL0�nn0 : ð49Þ

vL0 ð0Þ is the static potential, while ~vv in eq. (48) is the dynamical potential without zero frequency part.

~vvL0 ði!n � i!n0 Þ ¼ vL0 ði!n � i!n0 Þ � vL0 ð0Þ�nn0 : ð50Þ

Furthermore, ~ggLL00 ði!nÞ in eq. (48) is the Green function in the static approximation defined by

~ggLL00 ði!nÞ ¼ ½ðF�1 � �v0Þ�1�LnL0n0 ; ð51Þ

where the coherent Green function F is defined by eqs. (30) and (31).In the functional integral approach, we first have to calculate the determinant (48), and second have to evaluate the

functional integral in eq. (47). In order to implement these calculations, we expand the determinant (48) with respect to the

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-5

frequency modes of the dynamical potential vL0 as follows.

D ¼ 1þX�

ðD� � 1Þ þXð�;�0ÞðD��0 � D� � D�0 þ 1Þ þ � � � ; ð52Þ

D� ¼ det �LL0�0�nn0 �X00ðvL00 ði!�Þ�n�n0 ;� þ vL00 ði!��Þ�n�n0 ;��Þ ~ggL00L00 ði!n0 Þ

" #; ð53Þ

D��0 ¼ det

"�LL0�0�nn0 �

X00ðvL00 ði!�Þ�n�n0 ;� þ vL00 ði!��Þ�n�n0 ;��Þ ~ggL00L00 ði!n0 Þ

�X00ðvL00 ði!�0 Þ�n�n0;�0 þ vL00 ði!��0 Þ�n�n0 ;��0 Þ ~ggL00L00 ði!n0 Þ

#: ð54Þ

The first term at the r.h.s. of eq. (52) corresponds to the zeroth approximation (i.e., the static approximation) which neglectsdynamical potentials. The second term is a superposition of the independent scattering terms of dynamical potentialvL0 ði!�Þ. Higher order terms describe dynamical mode–mode couplings.

We adopt here the harmonic approximation19) which neglects the mode–mode coupling terms in eq. (52). We have then

Edynð�; �Þ ¼ ���1 ln 1þX�

ðD� � 1Þ

" #: ð55Þ

The approximation yields the result of the second-order perturbation in the weak Coulomb interaction limit, and describes theKondo anomaly in the strong interaction limit.20)

Let us now calculate D� in eq. (55). The determinant D� in the harmonic approximation is written by a product of those ofthe tridiagonal-type matrices as

D� ¼Y��1

k¼0

Y2lþ1

m¼1

D�ðk;mÞ

" #; ð56Þ

D�ðk;mÞ ¼

. ..

1 1 0

a��þkð�;mÞ 1 1

akð�;mÞ 1 1

a�þkð�;mÞ 1 1

0 a2�þkð�;mÞ

. ..

: ð57Þ

Here 1 in the determinant is the 2� 2 unit matrix, anð�;mÞ is a 2� 2 matrix defined by

anð�;mÞ0 ¼X

000000000vL00 ð�Þ ~ggL00000 ðn� �ÞvL0000000 ð��Þ ~ggL00000 ðnÞ: ð58Þ

We assumed in the above expression that the orbitals fLg form an irreducible representation of the point group of the system,so that ~ggLL00 ði!nÞ ¼ ~ggL0 ði!nÞ�LL0 [see eq. (51)]. Furthermore here and in the following, we write the frequencydependence, for example, of ~ggL0 ði!nÞ as ~ggL0 ðnÞ for simplicity.

The determinant D�ðk;mÞ is expanded with respect to the dynamical potentials as follows.

D�ðk;mÞ ¼ 1þ Dð1Þ� ðk;mÞ þ Dð2Þ� ðk;mÞ þ � � � ; ð59Þ

DðnÞ� ðk;mÞ ¼X

�11����nnv�1ð�;mÞv1

ð��;mÞ � � � v�nð�;mÞvnð��;mÞDDðnÞf�gð�; k;mÞ: ð60Þ

Here the subscripts �i and i take 4 values 0, x, y, and z, and

v0ð�;mÞ ¼ �1

2iXm0

Amm0�m0 ð�Þ�l2; ð61Þ

v�ð�;mÞ ¼ �1

2

Xm0

B�mm0�m0�ð�Þ�l2 ð� ¼ x; y; zÞ: ð62Þ

Note that the subscript f�g in eq. (60) denotes a set of ð�11 � � ��nnÞ. The expressions of DDðnÞf�gð�; k;mÞ are given inAppendix A.

Substituting eq. (59) into eq. (56) and taking the Gaussian average (47), we have

D� ¼X1n¼0

X�11����nn

XP

kmlðk;mÞ¼n

Y2lþ1

m¼1

Yi

v�ið�;mÞvið��;mÞ

" # Y2lþ1

m¼1

Y��1

k¼0

DDðlðk;mÞÞf�g ð�; k;mÞ

" #: ð63Þ

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-6

Here flðk;mÞg ðk ¼ 0; . . . ; �� 1;m ¼ 1; . . . ; 2lþ 1Þ are zero or positive integer, satisfyingP

km lðk;mÞ ¼ n.Qi v�ið�;mÞvi ð��;mÞ are the products of v�ið�;mÞvið��;mÞ belonging to the m-th orbital block. Calculations of the

Gaussian average of the dynamical potentials are given in Appendix B, and we reach the following expression.

D� ¼ 1þ Dð1Þ� þ D

ð2Þ� þ � � � ; ð64Þ

DðnÞ� ¼

1

ð2�ÞnX

Pkm

lðk;mÞ¼n

Xf�jðk;mÞg

XP

Y2lþ1

m¼1

Y��1

k¼0

Ylðk;mÞj¼1

C�jðk;mÞmmp

!DDðlðk;mÞÞf��p�1 gð�; k;mÞ

" #: ð65Þ

Here j denotes the j-th member of the ðk;mÞ block. P denotes a permutation of a set fð j; k;mÞg: Pfð j; k;mÞg ¼ fð jp; kp;mpÞg,�p�1 means an rearrangement of f�jðk;mÞg according to the inverse permutation P�1. Note that �jðk;mÞ takes 4 values 0, x, y,and z. C�mm0 is a Coulomb interaction defined by

C�mm0 ¼�Amm0 (� ¼ 0)

B�mm0 (� ¼ x; y; z).

ð66Þ

Equations (55) and (64) determine the dynamical potential Edynð�; �Þ.The free energy (44) is written alternatively as

FCPA ¼ ~FF � ��1 ln

Z Y�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2lþ1 detB�

ð4 Þ2lþ1

s Ym

d�m

" #e��Eeff ð�Þ: ð67Þ

In the itinerant electron system, spin fluctuations plays an important role, and we may neglect the thermal charge fluctuationsmaking use of the saddle-point approximation to the static charge fields �m. We have then Eeffð�Þ ¼ Eð�; ��Þ. The saddle pointvalue ��m is determined from @Eð�; ��Þ=@�m ¼ 0:

�i��m ¼ ~nnLð�Þ ¼X

~nnLð�Þ; ð68Þ

~nnLð�Þ ¼1

�

Xn

GLðnÞ: ð69Þ

In order to reduce the number of variables, we neglect the out-of-phase thermal spin fluctuations between different orbitalson a site, and take into account their in-phase fluctuations. This can be made by introducing a large variable �� ¼

Pm �m�.

Inserting 1 ¼R½Q� d�� d��� exp½�2 i��ð�� �

Pm �m�Þ� into eq. (67), and replacing variables �m� with ��=ð2lþ 1Þ in the

non-Gaussian terms of Eeffð�Þ, we reach

FCPA ¼ ~FF � ��1 ln

Z Y�

ffiffiffiffiffiffiffiffi� ~JJ�

4

sd��

24

35e��Eeff ð�Þ; ð70Þ

Eeffð�Þ ¼ Estð�Þ þ Edynð�Þ; ð71Þ

Estð�Þ ¼ �1

�

Xmn

ln ð1� �vL"ð0ÞFL"ðnÞÞð1� �vL#ð0ÞFL#ðnÞÞ �1

4~JJ2?�

2?FL"ðnÞFL#ðnÞ

� �

þ1

4�ðU0 � 2U1 þ JÞ

Xm

~nnLð�Þ2 � ð2U1 � JÞ ~nnlð�Þ2 þ ~JJ2?�

2? þ ~JJ2

z �2z

" #: ð72Þ

Here ~JJx ¼ ~JJy ¼ ~JJ? ¼ ½1� 1=ð2lþ 1Þ�J, ~JJz ¼ U0=ð2lþ 1Þ þ~JJ?, �vLð0Þ ¼ vLð0Þ ��LðnÞ, and vLð0Þ ¼ v0ð0;mÞ þvzð0;mÞ. The charge densities, ~nnLð�Þ and ~nnlð�Þ are definedby ~nnLð�Þ ¼

P ~nnLð�Þ and ~nnlð�Þ ¼

Pm ~nnLð�Þ. Furthermore

Edynð�Þ is given by eq. (55) in which �m (�m�) has beenreplaced by i ~nnLð�Þ [��=ð2lþ 1Þ].

The CPA equation in the HA is obtained from thestationary condition (43) with the free energy (70).

hGLðnÞi ¼ FLðnÞ; ð73Þ

and

hGLðnÞi ¼ ~ggLðnÞ ��

�LðnÞ�Edyn

��LðnÞ

� �: ð74Þ

Here �LðnÞ ¼ 1� FLðnÞ�2HLðnÞ and HLðnÞ ¼ �FLðnÞ=��LðnÞ. The average h�i at the r.h.s. of eq. (74) is nowdefined by a classical average with respect to the effective

potential (71).

h�i ¼

Z Y�

d��

" #ð�Þe��Eeff ð�Þ

Z Y�

d��

" #e��Eeff ð�Þ

: ð75Þ

Substituting eq. (55) into eq. (74), we obtain the expression

hGLðnÞi ¼ ~ggLðnÞ �

X�

f�D�=½�LðnÞ��LðnÞ�g

1þX�

ðD� � 1Þ

* +: ð76Þ

The local charge and magnetic moment are obtained from@FCPA=@�

0L and �@FCPA=@h

�L. Making use of the stationary

conditions of FCPA with respect to ��m and �L , and the CPAequation (73), we reach

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-7

hnnLi ¼1

�

Xn

FLðnÞ; ð77Þ

hmmzLi ¼

1

�

Xn

FLðnÞ: ð78Þ

In particular, the l ¼ 2 components of local charge andmagnetic moment are expressed as

hnnli ¼ h ~nnlð�Þi; ð79Þhmmli ¼ h�i: ð80Þ

The amplitude of charge and local moments for d

electrons are calculated from the formulae.

hnn2l i ¼ hnnli þ 2

Xm

@FCPA

@Umm

þXmm0

0 @FCPA

@Umm0; ð81Þ

hm2l i ¼ 3hnli � 6

Xm

@FCPA

@Umm

�Xmm0

0 @FCPA

@Umm0þ 2

@FCPA

@Jmm0

� �:

ð82Þ

Here we have introduced for convenience orbital-dependentCoulomb and exchange interactions Umm0 and Jmm0 into theinteraction H1 to derive the expressions. Making use of thestationary conditions of FCPA and integrations by parts, weobtain

hnn2l i ¼ h ~nnlð�Þi þ

1

2

Xm

h ~nnLð�Þ2i þXmm0

0h ~nnLð�Þ ~nnL0 ð�Þi �

1

2ð2lþ 1Þh�2z i �

2

� ~JJz

� �

þ 2Xm

@Edyn

@Umm

� �v

� �þXmm0

0 @Edyn

@Umm0

� �v

� �; ð83Þ

hmm2l i ¼ 3h ~nnlð�Þi �

3

2

Xm

h ~nnLð�Þ2i þ3

2ð2lþ 1Þh�2z i �

2

� ~JJz

� �þ 1�

1

2lþ 1

� � X�¼x;y

h�2�i �2

� ~JJ�

� �

� 6Xm

@Edyn

@Umm

� �v

� ��Xmm0

0 @Edyn

@Umm0

� �v

� �þ 2

@Edyn

@Jmm0

� �v

� �� �: ð84Þ

Here ½@Edyn=@Umm0 �v means taking derivative of Edyn with respect to Umm0 fixing the static potentials vL0 ð0Þ. In the HA, thesevalues are obtained from eq. (55) as

@Edyn

@Umm0

� �v

¼ �1

�

X1�¼1

�@D�

@Umm0

�v

1þX1�¼1

ðD� � 1Þ; ð85Þ

@Edyn

@Jmm0

� �v

¼ �1

�

X1�¼1

�@D�

@Jmm0

�v

1þX1�¼1

ðD� � 1Þ: ð86Þ

The entropy is calculated from �2@FCPA=@� as

S ¼ �2 @~FF@�þ �2 @Eeff

@�

� �þ ln

Z Y�

ffiffiffiffiffiffiffiffi� ~JJ�

4

sd��

24

35 exp½��ðEeffð�Þ � hEeffð�ÞiÞ� �

3

2: ð87Þ

Here

�2 @~FF@�¼

1

NSp lnðg�1 ��Þ þ

Xn

XL

FLðnÞ; ð88Þ

�2 @Eeff

@�

� �¼ htr lnð1� �v0FÞi � �hEdyni þ �

@ð�EdynÞ@�

� �!�

� �: ð89Þ

The first term at the r.h.s. of eq. (87) [i.e., eq. (88)] is the contribution from the coherent free energy and reduces to theentropy S0 for noninteracting electrons when �L �! 0:

S0 ¼ �2

Zd!�0ð!Þ½ f ð!Þ ln f ð!Þ þ ð1� f ð!ÞÞ lnð1� f ð!ÞÞ�; ð90Þ

where �0ð!Þ is the total density of states per spin for noninteracting electrons, f ð!Þ is the Fermi distribution function. Thesecond term in eq. (87) [i.e., eq. (89)] is the entropy due to the temperature dependence of the effective potential.½@ð�EdynÞ=@��!� in eq. (89) means to take the derivative with respect to � fixing the frequency i!n and the coherent potential�Lði!nÞ. It is given in the HA as

�@ð�EdynÞ@�

� �!�

¼

X1�¼1

X1n¼1

nDðnÞ�

1þX1�¼1

ðD� � 1Þ: ð91Þ

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-8

The third and fourth terms in eq. (87) produce the magnetic entropy due to thermal spin fluctuations.The thermodynamic energy is obtained from the relation hH � �Ni ¼ FCPA þ ��1S as

hH � �Ni ¼1

�

Xn

XL

i!nFLðnÞ �1

4ðU0 � 2U1 þ JÞ

Xm

h ~nnLð�Þ2i þ ð2U1 � JÞh ~nnlð�Þ2i �X�

~JJ� h�2�i �2

� ~JJ�

� �" #

þ ��1

X1�¼1

X1n¼1

nDðnÞ�

1þX1�¼1

ðD� � 1Þ

* +: ð92Þ

The first term at the r.h.s. of eq. (92) is the coherentcontribution of the kinetic energy, the second term corre-sponds to the double counting correction in the Hartree–Fock energy. The last one is the dynamical correction to theenergy.

The sum rule n0 ¼P

LhnnLi determines the chemicalpotential for a given valence electron number n0. The CPAequation (73) and effective potential (71) with eqs. (72),(55), and (64) form the self-consistent equations to deter-mine the dynamical coherent potential f�Lði!nÞg. Afterhaving solved the self-consistent equations, we can calculatethe magnetic moments and charge from eqs. (77), (78), and(80), the square of local charge and spin fluctuations fromeqs. (83) and (84), as well as the other thermodynamicquantities [see eqs. (70), (87), and (92)].

5. Numerical Calculations: Fe and Ni

The simplest approximation to the dynamical CPA is toneglect the dynamical potential Edynð�Þ in the self-consistentequations. This is called the static approximation and may bejustified in the high temperature limit. The next approxima-tion is to add the dynamical potential Edynð�Þ by takinginto account the higher-order terms D

ðnÞ(n 1) in a series

expansion (64). We have taken into account the terms up tothe second order (n 2) in eq. (64). We call this level ofapproximation the second-order dynamical CPA. Within theapproximation, we have performed numerical calculationsfor Fe and Ni in order to examine the quantitative aspects ofthe theory and the dynamical effects on their electronic andmagnetic properties.

We obtained the intraorbital Coulomb interaction U0,interorbital Coulomb interaction U1, and exchange inter-action energy parameter J from the parameters U and J

in the LDA + U via the relations: U0 ¼ U þ 8J=5, U1 ¼U � 2J=5, and J ¼ J. (Note that U0 ¼ U1 þ 2J.) Weadopted in the present calculations the LDA+U valuesused by Anisimov et al.;27) U ¼ 0:1691 Ry and J ¼ 0:0662

Ry for Fe, and U ¼ 0:2205 Ry and J ¼ 0:0662 Ry forNi. These values yield U0 ¼ 0:2749 Ry, U1 ¼ 0:1426 Ry,J ¼ 0:0662 Ry for Fe, and U0 ¼ 0:3263 Ry, U1 ¼ 0:1940

Ry, J ¼ 0:0662 Ry for Ni, respectively.In the numerical calculations, we adopted an approximate

expression of the coherent Green function36)

FLðnÞ ¼Z

�Lð�Þ d�i!n � ���Lði!nÞ

: ð93Þ

The expression takes into account the effect of hybridizationbetween different l blocks in the nonmagnetic state via thelocal densities of states �Lð�Þ, but neglects that in the spinpolarized state. Moreover, we adopted a decoupling approx-

imation13) to the thermal average of the impurity Greenfunction in the dynamical CPA equation (73).

hGLðn; �z; �2?Þi

¼Xq¼�

1

21þ q

h�ziffiffiffiffiffiffiffiffih�2z i

p !

GLðn; qffiffiffiffiffiffiffiffih�2z i

q; h�2?iÞ: ð94Þ

The approximation is correct up to the second moment andreasonably describe the thermal spin fluctuations. We havesolved the dynamical CPA equation for the bcc Fe using theexpressions (93) and (94). The densities of states (DOS) for3d, 4s, and 4p states were calculated by using von Barth–Hedin LDA potential. The total DOS and the d DOSfor eg and t2g electrons are shown in Fig. 1. Single-particleexcitation spectra have been calculated by using the Padenumerical analytic contribution.

Figure 2 shows the calculated d DOS of paramagnetic Feat T=TC ¼ 1:19 as the single-particle excitation spectra. TheDOS in the static approximation is broadened as comparedwith the LDA result in the nonmagnetic state becauseof the strong thermal spin fluctuations. The dynamicalcharge and spin fluctuations produce a satellite peak around! ¼ �0:5 Ry (¼ �6:8 eV), and suppress the band broad-ening by about 22% as compared with the static one. Theexistence of the satellite peak is consistent with the previousresults of the ground-state calculations37) as well as those atfinite temperatures.29) The d band width in the presentcalculations, though it is strongly reduced as compared withthe static one, is comparable to that of the LDA calculations,

0

5

10

15

20

25

30

35

40

45

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

DO

S (

stat

e/ R

y at

om s

pin)

Energy (Ry)

EgT2gTotal

Fig. 1. Densities of states (DOS) calculated by the LDA and TB-LMTO

method. Dashed curve: local DOS for eg electrons, dotted curve: local

DOS for t2g electrons, solid curve: total DOS consisting of 4s, 4p, and 3d

orbitals.

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-9

while the XPS experiments suggest the 10% reduction ofthe width as compared with the LDA results, and no dip at! ¼ �0:1 Ry.5) These inconsistencies may be attributed toan overestimate of the local exchange splitting above TC.

Below the Curie temperature, the up and down DOS aresplit as shown in Fig. 3. In the up-spin band, the satellitepeak at ! ¼ �0:45 Ry remains, and the quasiparticle bandsat ! � �0:2 Ry shifts to the Fermi level as comparedwith those in the static approximation, showing the bandnarrowing. The satellite peak for the down-spin banddisappears because of a large value of j Im �LðzÞj in thisenergy region. These behaviors are consistent with recentQMC calculations without transverse spin fluctuations.29)

It is not easy to calculate the DOS at low temperatures inthe QMC calculations. The present approach allows us toinvestigate the DOS even at low temperatures. Figure 4shows the DOS at T=TC ¼ 0:3. The DOS in the staticapproximation approaches to the Hartree–Fock one withdecreasing temperature, but are still broadened at thistemperature by thermal spin fluctuations. Dynamical termssuppress the thermal spin fluctuations and develops thequasiparticle states, so that sharp peaks of eg electrons

appear at ! ¼ �0:15 Ry in the DOS. The present calcu-lations reduce to the second-order perturbation theory atT ¼ 0, so that the DOS in Fig. 4 is close to those obtained atthe zero temperature by Drchal et al.38)

The effective potential determines the behavior ofmagnetic moments. Figure 5 shows the potential for Febelow TC. It has double minima along z axis, and mono-tonically increases with increasing �? ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2x þ �2y

p. [Note

that the effective potential is spherical on the xy plane:Eeffð�z; �?Þ.] The dynamical contribution Edynð�Þ to theeffective potential is given in Fig. 6. The dynamical partshows a ‘butterfly’ structure; it increases along the z-axiswith increasing the amplitude j�j, while it decreases on thexy plane. This implies that the dynamical effects reduce thelongitudinal amplitude of spin fluctuations, and enhance thetransverse spin fluctuations. In fact, we find 6% reduction offfiffiffiffiffiffiffiffih�2z i

pand 6% enhancement of

ffiffiffiffiffiffiffiffiffih�2?i

pat T=TC ¼ 1:19.

Magnetic properties of Fe are summarized in Fig. 7. Bothstatic and dynamical calculations yield the Curie–Weisssusceptibility. Calculated effective Bohr magneton numbersare 3.1 �B in the static approximation and 3.0 �B in thedynamical calculations, respectively, being in good agree-ment with the experimental value 3.2 �B.39) CalculatedCurie temperature is 2020 K (2070 K) in the second-orderdynamical calculations (the static approximation). They are

0

10

20

30

40

50

60

70

80

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

DO

S (

stat

es/ R

y at

om)

Energy (Ry)

T/Tc=1.19

Dyn. StaticLDA

Fig. 2. Single particle excitation spectra (DOS) for d electrons in the

paramagnetic Fe. Results for the LDA, the static approximation, and the

second-order dynamical CPA are shown by the dotted curve, the dashed

curve, and the solid curve, respectively.

0

5

10

15

20

25

30

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

DO

S (

stat

es/ R

y at

om s

pin)

Energy (Ry)

T/Tc=0.7

Fig. 3. Up and down d DOS in the ferromagnetic Fe at T=TC ¼ 0:7.

Results for the static approximation are shown by the dotted curves.

0

5

10

15

20

25

30

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

DO

S (

stat

es/ R

y at

om s

pin)

Energy (Ry)

T/Tc=0.3

Fig. 4. Up and down d DOS in the ferromagnetic Fe at T=TC ¼ 0:3.

T/Tc=0.50-4-2

02

4ξz

-4-2

02

4

ξx

-0.06-0.05-0.04-0.03-0.02-0.01

00.01

Eeff(ξ)

Fig. 5. Effective potential in the ferromagnetic Fe at T=TC ¼ 0:5 on the

�x–�z plane.

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-10

much smaller than the Hartree–Fock value 12200 K, butstill twice as large as the experimental value (1040 K).40)

The present results are comparable to the QMC result ofcalculations without transverse spin fluctuations (1900 K).29)

The reduction of TC due to dynamical corrections is 50 K,which is rather small. Dynamical effects in general reducethe magnetic energy, but also reduce the magnetic entropy ofthe static approximation. Both effects are competitive toeach other, resulting in the reduction of TC by 50 K in thecase of Fe.

The magnetization increases with decreasing temperature,and reach the Hartree–Fock value 2.61 �B at T ¼ 0 K in thestatic approximation. The latter is overestimated as com-pared with the experimental value 2.216 �B.41) The second-order dynamical CPA calculations yield M ¼ 2:59 �B

(extrapolated value); the calculations hardly reduce theground-state magnetization as seen in Fig. 7. One has to takeinto account the higher-order electron–electron scattering

effects as found in the low-density approximation16) toreduce the magnetization. The amplitude of local magneticmoment was calculated by means of eq. (84). The results areplotted in the same figure. Because of the strong Coulombinteraction, it hardly changes with increasing temperature.The dynamical fluctuations enhance the amplitude

ffiffiffiffiffiffiffiffiffiffihmm2i

pby

1%, and reduce the d charge fluctuationsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihð�nndÞ2i

pby 5% at

T=TC ¼ 1:2.We have also calculated the electronic and magnetic

properties of the fcc Ni at finite temperatures. Figure 8shows the DOS in the paramagnetic state. In the staticapproximation, the details of the structure are smeared bythermal spin fluctuations and the d band width is broadenedby about 0.1 Ry. The dynamical effects suppress the thermalspin fluctuations and develop the quasiparticle states.Reduction of the quasiparticle band width is 17% ascompared with that of the static approximation. Furthermorewe find the satellite peak at ! ¼ �0:45 Ry. These resultsexplain well the XPS data8) as shown in Fig. 8.

Below TC, the peak of the down-spin band is on the Fermilevel, as shown in Fig. 9. On the other hand, the top of the

T/Tc=0.50-4-2

02

4ξz

-4-2

02

4

ξx

-0.01

-0.005

0

0.005

0.01

Edyn(ξ)

Fig. 6. Dynamical contribution to effective potential in the ferromagnetic

Fe at T=TC ¼ 0:5 on the �x–�z plane.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 500 1000 1500 2000 2500 3000

M (μ

B)

T (K)

Fe

Mχ−1

⟨m2⟩1/2

Fig. 7. Calculated magnetization (M), inverse susceptibility (��1), and

amplitude of local moment (hm2i1=2) as a function of temperature (T) in

Fe. The dynamical results are shown by the solid curves. Results in the

static approximation are shown by dotted curves. Magnetization

calculated by the DMFT without transverse spin fluctuations29) is also

shown by open squares. Experimental data of magnetization42) are shown

by +. Note that the absolute values of the DMFT magnetization are not

given in ref. 29. Thus they are plotted here by assuming that the

extrapolated value at T ¼ 0 agrees with the experimental one.

0

10

20

30

40

50

60

-0.6 -0.4 -0.2 0 0.2 0.4

DO

S (

stat

es/ R

y at

om)

Energy (Ry)

T/Tc=1.4

Dyn. StaticExpt.

Fig. 8. Calculated DOS in the paramagnetic Ni. Solid curve: second-order

dynamical CPA, dashed curve: static approximation, dotted curve: XPS

data.8)

0

5

10

15

20

25

30

35

40

-0.6 -0.4 -0.2 0 0.2 0.4

DO

S (

stat

es/ R

y at

om)

Energy (Ry)

T/Tc=0.79

Dyn. dσStatic dσDyn. total

Fig. 9. Calculated DOS in the ferromagnetic Ni. Solid curves: spin-

polarized d DOS in the second-order dynamical CPA, dashed curves:

spin-polarized d DOS in the static approximation, dotted curve: total DOS

in the second-order dynamical CPA.

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-11

up-spin d band is away from the Fermi level, so that the peakis weakened due to larger damping of the quasiparticlestates. The satellite peak for the down-spin band disappearsdue to strong incoherent scatterings around ! ¼ �0:35 Ry,while the satellite peak for the up-spin band is enhanced at! ¼ �0:45 Ry.

The effective potential for Ni shows a single minimumstructure as shown in Fig. 10. The minimum position shiftsto the origin with increasing temperature. This should becontrasted to the case of Fe, in which the effective potentialhas a double minimum structure even above TC as shown inFig. 5, and the paramagnetic state is realized by changingthe energy difference between the two minima. Thedynamical potential Edynð�Þ in Ni has a butterfly structureas in the case of Fe, but it is highly asymmetric along the z

axis in the ferromagnetic state so that considerable reductionof the magnetization due to dynamical corrections occurs.We find the reduction of

ffiffiffiffiffiffiffiffih�2z i

pby 5.0%, and the enhance-

ment offfiffiffiffiffiffiffiffiffih�2?i

pby 1.5% at T=TC ¼ 1:3.

The magnetic moment and the inverse susceptibilitycalculated from the effective potential are presented inFig. 11 as a function of temperature. The susceptibility

follows the Curie–Weiss law. Both the static and dynamicalcalculations yield the effective Bohr magneton number1.2 �B, which should be compared with the experimentalvalue 1.6 �B.43) Calculated Curie temperature in Ni is1260 K (1420 K) in the second-order dynamical (static)calculations. These values are much smaller than theHartree–Fock value 4950 K, but are still twice as large asthe experimental value 630 K.40)

The magnetization increases with decreasing temperaturebelow TC. Extrapolated value at T ¼ 0 is 0.67 (0.71) �B inthe second-order dynamical (static) calculations. Thesevalues are considerably larger than the experimental one(0.62 �B).41) The amplitude of Ni local moment slightlyincreases with increasing temperature and hardly showsanomaly at TC. The second order dynamical corrections tothe amplitude of local moment and the local chargefluctuations are small; the enhancement of

ffiffiffiffiffiffiffiffiffiffihmm2i

pis only

1.8% and the reduction of the d charge fluctuationsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihð�nndÞ2i

pis 5.9% at T=TC ¼ 1:3.

6. Summary

We have developed the dynamical CPA on the basis of theLDA+TB-LMTO Hamiltonian towards realistic calculationsof the itinerant electron system. The theory is a directextension of the single-site theory developed by Cyrot,Hubbard, Hasegawa, and Kakehashi, to the degenerate bandcase. It is based on the functional integral method whichtransforms the interacting electron system into an independ-ent electron system with time-dependent random charge andexchange potentials. Using the method, we have taken intoaccount the spin fluctuations as well as charge fluctuations inthe degenerate band system. We have then introduced aneffective medium �Lði!nÞ, and derived the self-consistentdynamical CPA equation for the medium, making use of asingle-site approximation.

We adopted the harmonic approximation (HA) to treat thefunctional integrals in the dynamical CPA. The HAdescribes the dynamical effects from the weak- to thestrong-Coulomb interaction regime. The approximationallows us to obtain analytical expressions of the physicalquantities, and takes into account the dynamical correctionssuccessively starting from a high-temperature approximation(i.e., the static approximation). We can calculate theexcitation spectra as well as the thermodynamic quantitieseven at low temperatures using the HA because we obtainedtheir analytic expressions.

We have investigated the dynamical effects in Fe and Niwithin the second-order dynamical CPA, and have shownthat the second-order dynamical corrections much improvethe single-particle excitation spectra in these systems. Thestatic approximation broadens the DOS due to thermal spinfluctuations at finite temperatures. The dynamical effectssuppress the thermal spin fluctuations and create thequasiparticle states with narrow band width near the Fermilevel. Furthermore, the correlations create the satellite peakat 6 eV below the Fermi level in both Fe and Ni. The XPSdata in the paramagnetic Ni is well explained by the presenttheory.

We verified that the dynamical CPA yields the Curie–Weiss susceptibilities. Calculated effective Bohr magnetonnumbers, 3.0 �B for Fe and 1.2 �B for Ni, explain the

T/Tc=0.47-3 -2

-10 1

23ξz

-3-2

-10

12

3

ξx

-0.015

-0.01

-0.005

0

0.005

Eeff(ξ)

Fig. 10. Effective potential in the ferromagnetic Ni at T=TC ¼ 0:47 on the

�x–�z plane.

0

0.5

1

1.5

2

2.5

0 500 1000 1500 2000

M (μ

B)

T

Ni

Mχ−1

⟨m2⟩1/2

Fig. 11. Magnetization, inverse susceptibility, and amplitude of local

moment as a function of temperature in Ni. The dynamical results are

shown by the solid curves, while the results in the static calculation are

shown by dotted curves.

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-12

experimental data quantitatively or semiquantitatively. Cal-culated Curie temperatures, 2020 K for Fe and 1260 K forNi, are however overestimated by a factor of two. Extrapo-lated values of the ground state magnetization, 2.59 �B forFe and 0.67 �B for Ni, are also overestimated considerablyas compared with the experimental ones (2.22 �B for Fe and0.62 �B for Ni).

We found that the static approximation provides us with agood starting point to calculate finite-temperature magneticproperties of Fe and Ni, but the dynamical calculations to gobeyond the static approximation have been limited to thesecond-order dynamical CPA in the present work. Over-estimate of the ground-state magnetization and the Curietemperature should be reduced by taking into account thehigher-order dynamical fluctuations. Further improvementsof the dynamical CPA theory are left for future investiga-tions.

Acknowledgment

We would like to express our sincere thanks to Dr. OveJepsen for sending us the Stuttgart TB-LMTO program andfruitful advice on how to install the program on ourcomputer.

Appendix A: Expression of DDðnÞf��gð�; k;mÞ

We calculate in this appendix the coefficientsDDðnÞf�gð�; k;mÞ in the n-th order expansion of the determinantD�ðk;mÞ with respect to the dynamical potential v�ð�;mÞ.

Let us rewrite D�ðk;mÞ defined by eq. (57) as follows bymaking use of the Laplace expansion.

D ¼ jað0ÞjD20D20 þX�

ðað0ÞÞ��� þ D10D10; ðA:1Þ

�11 ¼ �ðD20 � D24ÞðD20 � D24Þ � D23D23; ðA:2Þ�12 ¼ �D23ðD20 � D21Þ � ðD20 � D24ÞD22; ðA:3Þ�21 ¼ �D22ðD20 � D24Þ � ðD20 � D21ÞD23; ðA:4Þ�22 ¼ �ðD20 � D21ÞðD20 � D21Þ � D22D22: ðA:5Þ

In the above equations, we have omitted the suffixes �, k,m for simplicity, and jað0Þj denotes the determinant of the2� 2 matrix akð�;mÞ. fDn�g at the r.h.s. of eqs. (A·1)–(A·5)are defined by

Dn� ¼

bðn�1Þ� 1 0

aðnÞ� 1 1

aðnþ1Þ 1 1

. ..

0

; ðA:6Þ

Dn� ¼

bðn�1Þ� aðnÞ� 0

1 1 aðnþ1Þ

1 1 aðnþ2Þ

. ..

0

: ðA:7Þ

Here aðnÞ (aðnÞ) stands for an�þkð�;mÞ (a�n�þkð�;mÞ). aðnÞ� , bðnÞ� ,aðnÞ� , and bðnÞ� are defined by aðnÞ0 ¼ aðnÞ, bðnÞ0 ¼ 1, aðnÞ0 ¼ að�nÞ,bðnÞ0 ¼ 1, and for � ¼ 1{4,

aðnÞ1 ¼ aðnÞ2 ¼0 aðnÞ12

0 aðnÞ22

!; aðnÞ3 ¼ aðnÞ4 ¼

0 aðnÞ11

0 aðnÞ21

!; ðA:8Þ

bðnÞ1 ¼aðnÞ11 0

aðnÞ21 1

!; bðnÞ2 ¼

aðnÞ12 0

aðnÞ22 1

!; bðnÞ3 ¼

aðnÞ11 1

aðnÞ21 0

!; bðnÞ4 ¼

aðnÞ12 1

aðnÞ22 0

!; ðA:9Þ

aðnÞ1 ¼ aðnÞ2 ¼0 0

að�nÞ21 að�nÞ22

!; aðnÞ3 ¼ aðnÞ4 ¼

0 0

að�nÞ11 að�nÞ12

!; ðA:10Þ

bðnÞ1 ¼

að�nÞ11 að�nÞ12

0 1

!; b

ðnÞ2 ¼

að�nÞ21 að�nÞ22

0 1

!; b

ðnÞ3 ¼

að�nÞ11 að�nÞ12

1 0

!; b

ðnÞ4 ¼

að�nÞ21 að�nÞ22

1 0

!: ðA:11Þ

It should be noted that eq. (A·1) is calculated from a set ðD10;D20;D21;D22;D23;D24Þ. Thus we define DðnÞ bytDðnÞ ¼ ðDn0;Dnþ1 0;Dnþ1 1;Dnþ1 2;Dnþ1 3;Dnþ1 4Þ. By making use of the Laplace expansion, we can derive a recursionrelation as follows.

DðnÞ ¼ ðc0 þ cðnÞ1 þ cðnÞ2 ÞDðnþ2Þ: ðA:12Þ

Here ðc0Þij ¼ �i1�j1 þ �i2�j1, and

cðnÞ1 ¼

�aðnÞ11 � aðnÞ22 �aðnþ1Þ11 � aðnþ1Þ

22 aðnþ1Þ22 �aðnþ1Þ

21 �aðnþ1Þ12 aðnþ1Þ

11

0 �aðnþ1Þ11 � aðnþ1Þ

22 aðnþ1Þ22 �aðnþ1Þ

21 �aðnþ1Þ12 aðnþ1Þ

11

aðnÞ11 0 0 0 0 0

aðnÞ12 0 0 0 0 0

aðnÞ21 0 0 0 0 0

aðnÞ22 0 0 0 0 0

0BBBBBBBBBB@

1CCCCCCCCCCA; ðA:13Þ

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

094706-13

cðnÞ2 ¼

jaðnÞj cðn;nþ1Þ12 �cðn;nþ1Þ

11221221 cðn;nþ1Þ22212122 cðn;nþ1Þ

11121211 �cðn;nþ1Þ22112112

0 jaðnþ1Þj 0 0 0 0

0 �cðn;nþ1Þ11222112 aðnÞ11 a

ðnþ1Þ22 aðnÞ21 a

ðnþ1Þ22 �aðnÞ11 a

ðnþ1Þ12 �aðnÞ21 a

ðnþ1Þ12

0 cðn;nþ1Þ22121222 aðnÞ12 a

ðnþ1Þ22 aðnÞ22 a

ðnþ1Þ22 �aðnÞ12 a

ðnþ1Þ12 �aðnÞ22 a

ðnþ1Þ12

0 cðn;nþ1Þ11212111 �aðnÞ11 a

ðnþ1Þ21 �aðnÞ21 a

ðnþ1Þ21 aðnÞ11 a

ðnþ1Þ11 aðnÞ21 a

ðnþ1Þ11

0 �cðn;nþ1Þ22111221 �a

ðnÞ12 aðnþ1Þ21 �aðnÞ22 a

ðnþ1Þ21 aðnÞ12 a

ðnþ1Þ11 aðnÞ22 a

ðnþ1Þ11

0BBBBBBBBBB@

1CCCCCCCCCCA: ðA:14Þ

Here cðn;nþ1Þ����0�0 0�0 ¼ aðnÞ��a

ðnþ1Þ� � aðnÞ�0�0a

ðnþ1Þ 0�0 and cðn;nþ1Þ

12 ¼ cðn;nþ1Þ11221221 þ cðn;nþ1Þ

22112112 þ jaðnþ1Þj. Note that c0, cðnÞ1 , and cðnÞ2 matrices areof the zeroth order, the first order, and the second order with respect to the dynamical potential v, respectively.

Using the relation c20 ¼ c0 and D10 ¼ 1, we obtain the relation.

DðmÞ ¼ E2 þX1n¼0

cn0ðDðmþ2nÞ � c0D

ðmþ2nþ2ÞÞ: ðA:15Þ

Here tE2 ¼ ð1; 1; 0; 0; 0; 0Þ.Substituting eq. (A·12) into eq. (A·15), and using the recursion relation successively, we reach the expansion of Dð1Þ with

respect to the dynamical potential v.

Dð1Þ ¼ E2 þX1n¼1

Xnkn=2

X1lk¼0

Xlklk�1¼0

� � �Xl2l1¼0

Xi1þ���þik¼n

cl10 c2l1þ1i1� � � clk�lk�1

0 c2lkþ2k�1ik

E2; ðA:16Þ

where i1; . . . ; ik take a value 1 or 2.In the same way, we obtain the expansion of D

ð1Þas

Dð1Þ ¼ E2 þ

X1n¼1

Xnkn=2

X1lk¼0

Xlklk�1¼0

� � �Xl2l1¼0

Xi1þ���þik¼n

cl10 c2l1þ1i1� � � clk�lk�1

0 c2lkþ2k�1ik

E2: ðA:17Þ

Here cðnÞ1 and cðnÞ2 are defined by c1 and c2 in which faðnÞg have been replaced by faðnÞg.Substituting eqs. (A·16) and (A·17) into eq. (A·1), we obtain the expansion of D with respect to dynamical potentials.

DðnÞ� ðk;mÞ ¼X1n¼0

X�11����nn

v�1ð�;mÞv1

ð��;mÞ � � � v�nð�;mÞvn ð��;mÞDDðnÞf�gð�; k;mÞ: ðA:18Þ

Note that �n and n take 0, x, y, and z.The first few terms of DDðnÞf�gð�; k;mÞ are expressed as follows.

DDð0Þð�; k;mÞ ¼ 1; ðA:19Þ

DDð1Þ� ð�; k;mÞ ¼ �X1

n¼�1

X

aa�ð�;m; n�þ kÞ; ðA:20Þ

DDð2Þ��0 0 ð�; k;mÞ ¼1

2DDð1Þ� ð�; k;mÞDD

ð1Þ�0 0 ð�; k;mÞ �

1

2

X1n¼�1

X

aa�ð�;m; n�þ kÞ

! X

aa�0 0 ð�;m; n�þ kÞ

!

þX1

n¼�1

"aa�ð�;m; n�þ kÞ""aa�0 0 ð�;m; n�þ kÞ## � aa�ð�;m; n�þ kÞ#"aa�0 0 ð�;m; n�þ kÞ"#

�X0

aa�ð�;m; n�þ kÞ0 aa�0 0 ð�;m; n�þ kÞ0

#: ðA:21Þ

Here aa�ð�;m; nÞ is defined by

aa�ð�;m; nÞ ¼ ð1þ O1x þ O2y þ O3zÞ �hhðm; n� �; nÞ� �

�: ðA:22Þ

O1, O2, O3, and �hh in eq. (A·22) are 4� 4 matrices defined by

O1 ¼x 0

0 �y

� �; O2 ¼

0 �� þ �z� þ ��z 0

� �; O3 ¼

0 �ðx þ yÞ��ðx þ yÞ 0

� �; ðA:23Þ

�hhðm; n� �; nÞ ¼

e0 þ ex þ ey þ ez aðþÞx � ibð�Þx aðþÞy � ibð�Þy aðþÞz � ibð�Þz

aðþÞx þ ibð�Þx e0 þ ex � ey � ez bðþÞz � iað�Þz bðþÞy þ iað�Þy

aðþÞy þ ibð�Þy bðþÞz þ iað�Þz e0 � ex þ ey � ez bðþÞx � iað�Þx

aðþÞz þ ibð�Þz bðþÞy � iað�Þy bðþÞx þ iað�Þx e0 � ex � ey þ ez

0BBBB@

1CCCCA: ðA:24Þ

Here � ¼ ð1þ iÞ=2, and

094706-14

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

e�¼ g�Lðn� �Þg�LðnÞ ð� ¼ 0; x; y; zÞ; ðA:25Þ

að�Þ� ¼ g�Lðn� �Þg0LðnÞ � g0

Lðn� �Þg�LðnÞ ð� ¼ x; y; zÞ; ðA:26Þ

bð�Þ� ¼ g�Lðn� �Þg

LðnÞ � g

Lðn� �Þg

�LðnÞ ð� ¼ x; y; zÞ: ðA:27Þ

Note that ð�; �; Þ in eq. (A·27) denotes a cyclic change of ðx; y; zÞ. The static Green functions g�LðnÞ (� ¼ 0; x; y; z) are definedby ~ggL0 ðnÞ [see eq. (51)] as

~ggL0 ðnÞ ¼ g0LðnÞ�0 þ

Xx;y;z�

g�LðnÞð�Þ0 : ðA:28Þ

Appendix B: Calculation of the Gaussian Average of Dynamical Potentials

We calculate here the Gaussian average of the n-th order products of dynamical potentials.

Y2lþ1

m¼1

YnðmÞk¼1

ðv�kðmÞð�;mÞvkðmÞð��;mÞÞ

" #¼Z Yxyz

�

�2lþ1 detB�

ð2 Þ2lþ1

Y2lþ1

m¼1

d2�m�ð�Þ

" #�2lþ1 detA

ð2 Þ2lþ1

Y2lþ1

m¼1

d2�mð�Þ

" #

�Y2lþ1

m¼1

YnðmÞk¼1

v�kðmÞð�;mÞvkðmÞð��;mÞ

!" #

� exp ��

2��ð�ÞA�ð�Þ þ

X�

���ð�ÞB���ð�Þ

!" #: ðB:1Þ

Here integers fnðmÞg satisfy the constraintP

m nðmÞ ¼ n. ��ð�ÞA�ð�Þ stands forP

mm0 ��mð�ÞAmm0�m0 ð�Þ. v�ð�;mÞ are given by

eqs. (61) and (62).The average is calculated from a generating function Iðs; tÞ as follows.

Y2lþ1

m¼1

YnðmÞk¼1

ðv�kðmÞð�;mÞvkðmÞð��;mÞÞ

" #¼

Y2lþ1

m¼1

@2nðmÞ

@sm�1ðmÞ@tm1ðmÞ � � � @sm�nðmÞðmÞ@tmnðmÞðmÞ

" #Iðs ¼ 0; t ¼ 0Þ: ðB:2Þ

Here

Iðs; tÞ ¼Z Yxyz

�

�2lþ1 detB�

ð2 Þ2lþ1

Y2lþ1

m¼1

d2�m�ð�Þ

" #�2lþ1 detA

ð2 Þ2lþ1

Y2lþ1

m¼1

d2�mð�Þ

" #

� exp ��

2��ð�ÞA�ð�Þ þ

X�

���ð�ÞB���ð�Þ

!þX2lþ1

m¼1

X4

�¼0

sm�v�ð�;mÞ þ tm�v�ð��;mÞð Þ

" #: ðB:3Þ

The latter is obtained as follows.

Iðs; tÞ ¼ exp1

2�

Xmm0�

sm�C�mm0 tm0�

" #: ðB:4Þ

Here the Coulomb interactions C�mm0 are defined by eq. (66).By differentiating Iðs; tÞ with respect to sm� (tm), we have a new factor ð2�Þ�1

Pn C

�mntn� (ð2�Þ�1

Pn snC

nm). When we

take the 2n-th derivative of Iðs; tÞ with respect to the variable ðsm�1; tm1

; . . . ; sm�n ; tmnÞ, we have a 2n-th order polynomialtimes Iðs; tÞ. When we put sm� ¼ 0 and tm� ¼ 0 in the derivative, we have Iðs ¼ 0; t ¼ 0Þ ¼ 1, and only the zeroth order termsof the polynomial remain. The latters were created by taking a derivative of the factor ð2�Þ�1

Pn sn�C

�nm or

ð2�Þ�1P

n0 Cmn0 tn0 with respect to the variable conjugate to smi or tm0�i . A created constant ð1=2�ÞC�imm0��ij may be

indicated by a contraction sm�i tm0j . Then the zeroth order terms, and therefore the Gaussian average (B·1) should be given bythe sum over all possible products of contractions.

Y2lþ1

m¼1

YnðmÞk¼1

ðv�kðmÞð�;mÞvkðmÞð��;mÞÞ

" #¼

1

ð2�ÞnX

P

Y2lþ1

m¼1

YnðmÞk¼1

C�kðmÞmmp��kðmÞkp

ðmpÞ

" #: ðB:5Þ

Here the permutation P is taken with respect to the n elements fðk;mÞj k ¼ 1; . . . ; nðmÞ; m ¼ 1; . . . ; 2lþ 1g;Pfðk;mÞg ¼ fðkp;mpÞg. Application of the formula (B·5) to eq. (63) yields eq. (65) in §4:

DðnÞ� ¼

1

ð2�ÞnX

Pkm

lðk;mÞ¼n

Xf�jðk;mÞg

XP

Y2lþ1

m¼1

Y��1

k¼0

Ylðk;mÞj¼1

C�jmmp

!DDðlðk;mÞÞf��p�1 gð�; k;mÞ

" #: ðB:6Þ

094706-15

J. Phys. Soc. Jpn., Vol. 77, No. 9 Y. KAKEHASHI et al.

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9) See for example, V. L. Moruzzi and C. B. Sommers: Calculated

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