A quantum field theory of diffusion near the liquid-glass transition

ELSEVIER

Physica A 245 (1997) 543-559

A quantum field theory of diffusion near the liquid-glass transition Toyoyuki Kitamura* Nagasaki Institute of Applied Science, Nagasaki 851-01, Japan

Received 29 January 1997; revised 29 April 1997

Abstract

Randomly distributed atoms are considered in the harmonic potentials with random eigenfrequencies. The hopping of atoms makes energy dispersion curves of an atom. The correlation functions of density fluctuations in the intraband transition yield the freezing point, where the hopping is prohibited. The entropy due to the density fluctuations in the intraband transition yields the hopping matrix which shows the Vogel-Fulcher law. The diffusion is determined by the randomness of the eigenfrequencies and the hopping matrices. The diffusion coefficient is proportional to the first or third power of the configurationally averaged hopping matrix at high or low temperatures, respectively, and obeys the Vogel-Fulcher law. The randomness of eigenfrequencies and hopping matrices also yields the relaxation time of atoms, which obscures the liquid-glass transition. PACS: 64.70.Pf; 66.10.-x Keywords: Quantum field theory; Liquid-glass transition; Density fluctuations; Hopping; Diffusion

I. Introduction

We have established a quantum field theory of the structure of phonons in solids and liquids using the two-band model in the harmonic potential approximation [ 1-7]: an atom feels a harmonic potential made up by the surrounding atoms. We consider two levels: the ground state and the first excited states. When an atom drops to the ground state from the first excited state, a surrounding atom in the ground state is excited to the first excited state through the interaction and vice versa. Thus, the particle-hole pair spreads over the matter. We call this collective excitation a phonon. In amorphous solids and liquids, the distribution of atoms is represented by a radial distribution function. This fact yields the roton-type minimum of phonons near the reciprocal atomic distance. * E-mail:

kitamura @elc.nias.ac.jp.

0378-4371/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved PII S0378-4371 (97)00329-4

544

T. Kitamura/ Physica A 245 (1997) 543-559

The energy spectrum of atoms has bands due to the hopping of atoms; there is a gap. The gap ensures the rigidity. When a phonon frequency merges in the continuum of the particle-hole excitations, the phonon has a finite lifetime. So far, we have dealt with the particle-hole pairs in the interband transition, but we have not dealt with the particle-hole pairs in the intraband transition. The density fluctuations consist of the particle-hole pairs in the interband and intraband transitions, which hereafter we call off-diagonal density fluctuations and diagonal density fluctuations, respectively. The correlation functions of the off-diagonal density fluctuations yield the structure of phonons, while the correlation functions of the diagonal density fluctuations relate to the density response function to the external field and yield the phase transition. As the temperaatre decreases, finally the density response function diverges, where atoms cannot hop; atoms freeze. This is the liquid-glass transition since we are not concerned with the first-order transition. In actual liquids, frequencies of atoms are determined by the poles of the atom Green's functions; a frequency consists of energy dispersion curve and its imaginary part. The former essentially comes from the hopping of atoms; the band width of the energy dispersion is essentially equal to the magnitude of the hopping matrix. As the temperature decreases toward the liquid-glass transition, this band width becomes narrower. The critical temperature for the freezing of atoms is determined by the divergence of the density response function considering these energy dispersion curves. On the other hand, the latter comes from both random eigenfrequencies of atoms and random hopping matrices, which yields the relaxation time of atoms. Since the randomness remains above and below the liquid-glass transition, the randomness obscures the liquid-glass transition. The hopping relates to the random configuration of atoms which reflects the entropy due to diagonal density fluctuations. Thus, we can estimate the hopping matrix from the entropy due to the diagonal density fluctuations. The entropy is derived by a thermodynamical function due to the diagonal density fluctuations. The temperature dependence of the configurationally averaged hopping matrix shows the Vogel-Fulcher law. On the other hand, the hopping matrix connects with the self-diffusion. We calculate the diffusion by considering random eigenfrequencies of atoms and random hopping matrices. The diffusion coefficient is proportional to the first or third power of the configurationally averaged hopping matrix at high or low temperatures, respectively. Thus, the diffusion coefficient shows the Vogel-Fulcher law. The purpose of the present paper is as follows: (i) we calculate the correlation functions of diagonal density fluctuations in the random-phase approximation at low temperatures and show the critical temperature for the freezing of atoms. (ii) We calculate the entropy due to the diagonal density fluctuations and estimate the temperature dependence of the hopping matrix which shows the Vogel-Fulcher law. (iii) We calculate the diffusion coefficient due to random eigenfrequencies of atoms in the harmonic potentials and random hopping matrices, and show the Vogel-Fulcher law. Dynamical models of the liquid-glass transition have been proposed by Leutheusser [8] and Bengtzetius et al. [9] in simple liquids and glasses taking into account the mode coupling theory [10,11]: at low temperatures, the nonlinear parts in the kinetic

T. Kitamura I Physica A 245 (1997) 543-559

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equation for the density correlation functions play the role of the cages for atoms. At the liquid-glass transition, the cages trap atoms. Our theory approaches the liquid-glass transition from the direction opposite to theirs; our theory starts with the cage model taking into account the hopping of atoms, while theirs starts with the free atom model taking into account the mode coupling theory. In Section 2, we establish the model Hamiltonian. The unperturbed Hamiltonian consists of the configurationally averaged eigenfrequency of atoms and hopping matrix. The model Hamiltonian is presented in two forms in terms of localized operators and extended operators. In Section 3, we calculate the correlation functions of diagonal density fluctuations. The liquid-glass transition is determined by the divergence of the correlation functions. In Section 4, we derive the entropy due to diagonal density fluctuations and relate the hopping matrix to the entropy. The hopping matrix shows the Vogel-Fulcher law. In Section 5, we derive the diffusion coefficient from the configurational average of two Green's functions due to random eigenfrequencies of atoms and random hopping matrices making use of the Ward-Takahashi identity. In Section 6, we make the ladder approximation to the vertex parts. We show that the diffusion coefficient obeys the Vogel-Fulcher law. Section 7 is devoted to some concluding remarks.

2. The model Hamiltonian

We consider the following simple liquid: an atom is well-localized in an isotropic harmonic potential at a site Rm, temporarily. We are concerned with two levels of the harmonic potential; the ground state and the first excited states [ 1-7]. The eigenfrequency and the eigenfunction for the ground state are denoted as hC3m0 and ~ m o ( X R m ) = WmO(Xl - Rml )Wmo(X2 - Rm2 )WmO(X3 - Rrn3 ),

respectively. The eigenfrequencies

and eigenfunctions for the first excited states are denoted a s ~COmiand ~mi(X - Rm) = Wml(Xi- Rmi)WmO(Xj- Rmj)WmO(Xk --Rmk) for i = 1,2,3, where xi,xj and xk are three different Cartesian coordinates to each other. There is a relation between the ground state and the first excited state wave functions for the ith Cartesian coordinate: Wml (Xi ) = --2(m ~7iWmO(Xi ),

h ~m = ~/ 2M~om '

(2.1/

where ~,n is the mean width of the zero-point motion, M is the mass of an atom and (D m is an eigenfrequency at the site gm; h~rn0 = 3 ~ ( D m and ~OJmi = 5~0) m. Here the wave functions of the second excited states are written a s Wm2ii(xi) = -~2{WmO(Xi)2 (m~TiWml(Xi)}. Since we are not concerned with the states more than the first excited states, we obtain WmO(Xi ) = 2(m ~TiWml (Xi ) .

(2.2)

Eqs. (2.1) and (2.2) shows that the operator V' plays the role of changing the states like the spin-rotational operator in magnetism.

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In actual amorphous solids and liquids, there are a lot of vacancies. A vacancy raises the potential of its surrounding atoms in the direction of the vacancy. If there is a vacancy around the atom at Rm, the potential of the atom is raised in the direction of the vacancy and the atom can hop of the vacancy. In order to see this situation more explicitly, we start with the following Hamiltonian:

H =

/

1/

d3xOt(x)h~°(-i~7)~k(x) + ~

d3xd3yn(x)V(x - y)n(y),

(2.3)

where ~ is an annihilation field operator of an atom, he°(-i~7) is the energy operator of a free atom, n(x) is a density operator of the atom, n(x) = ~9t(x)~b(x) and V is the interaction potential between atoms. If we introduce an annihilation operator of the atom in the harmonic potential at the site Rm, bmu, we can write ~l(X) = Z W m p ( X

--

Rm)bm#, # = 0, 1,2,3.

(2.4)

m#

The Heisenberg equation for bm~ is given by

h~bm~ : Z

I d3xwm~(X- Rm)[I~e°(-i~7) +

~)R"(X)]Wnv(X --

Rn)bnv , (2.5)

nv

¢J~R°(X) =

R.}

d3yV(x - y)n(y),

(2.6)

where we have used O(z) = em/hOe-m/~ for any operator O at finite temperatures and {R,} means summing up the surrounding sites of the atom at R~ in n(y). The term q'R° means a potential made up by the surrounding atoms of the site R,. We assume that the atom at the site Rn feels a harmonic potential:

qbR,(X) ~-- ~M°o2n(x -- Rn) 2 ,

(2.7)

where M is the mass of an atom and ~Omis the eigenfrequency of the harmonic potential at the site Rn. Thus, we obtain {h~°(-iV ") + ~R,(x)}~nv(x - Rn) = hcbnvJnv(X - R~),

(2.8)

The magnitude of potentials ~ . distributes randomly, so does that of the eigenfrequencies. At this stage, we can take the following model Hamiltonian in terms of localized operators:

H = Z ~m~btmubmu + Z Jmunvbtmpbnv' m#

(2.9)

m#nv

where the hopping matrix is given by

ffmpnv : l d3xff~m#(X - Rm){h~°(-iW) + fbg°(x)}~nv(X -- Rn) . d

(2.10)

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Note that the model Hamiltonian is nonlinear through the term n(y) in q~R, in Eqs. (2.9) and (2.10). Now, we first establish the unperturbed Hamiltonian by taking the configurational average of the model Hamiltonian Eq. (2.9):

Ho = (H>c : y ~ hCoubtmubmu+ Z Ju(Rm - Rn)btm,,bna , I.*m

(2.11)

pmn

o5u = (aSm,}~,

(2.12)

Jv(Rm - Rn)

(2.13)

= (Jm,un#)c ,

where (...) = Tr e -/~(/4-dv) • • •/Tr e -#(H-uN) and the suffix c means the configurational average. The hopping between different states is neglected because of simplicity; this effect only modifies energy spectra. Next we establish the altemative model Hamiltonian in terms of extended operators. If we introduce the following extended operators:

1

bm~-- x/~ ~_e'PRmaut~,

/~=0,1,2,3,

(2.14)

P

we obtain

14o = Z hc~t'a~ea~t"

(2.15)

#P

~

= hoS~ + j ~ ( p ) ,

(2.16)

1

Jt,(p) - ~ ~ eil"R'Ju(R,,).

(2.17)

m

The field operator is written down as

Ill(x) = ~

1

~___eit)'Rm~mu(X-- Rm)a~ .

(2.18)

I~prn

From Eq. (2.18), we obtain

1

n ( x ) = ~ - - ~ e -`¢'R" mq

[~u

1,v2#(x-Rm)P~#q-~m~7i'W2o(x-Rm)P}q i

+...

1

,

(2.19) where

:

4a..-q,

(2.20)

P

P*iq= Z(a~aop - , + a*o,a,,-,) .

(2.21)

P

ptd.¢ and p~q are diagonal and off-diagonal density fluctuation operators, respectively. Substituting Eq. (2.19) into the interaction Hamiltonian of Eq. (2.3)

T. Kitamura/Physica A 245 (1997) 543-559

548

yields

HI = ~

1

Z

1

Vi~a(q)P[,PJ' + ~-~ Z

ijq

' Va

t

,v(q)PaucPa~¢'

(2.22)

Ilvq

v, d(q) - V,ojo.

V dv - v..,w,

1

V~#',vv'(q) = - ~ Z e -iq'(R'~-R") ×

(2.23)

/

d3xd3yWml~(x - Rm)wm~'(x - em)

m~n × V(x

-

y)

.vCv -

-

R,),

(2.24)

where there do not appear the cross terms of the diagonal and off-diagonal density operators, because of the symmetry of the wave functions. In this paper the suffixes i,j mean 1, 2 and 3, and #, v mean 0, 1, 2 and 3. Here it should be noted that since the unperturbed Hamiltonian H0 in Eq. (2.9) involves the term fd3xd3yn(x)V(xy)In(y))c, the interaction Hamiltonian HI excludes the term. Thus, the prime on ~ in Eq. (2.22) means that HI excludes one loop diagrams. The correlation functions of the off-diagonal density fluctuations, Piq yield phonons. The operators Piq correspond to the spin-flip operator S+ + S_ in magnetism. The correlation function of the spin-flip operator yields a magnon. In the present case, there are three components corresponding to the Cartesian coordinates. Thus, there are three modes: one longitudinal and two transverse modes. The operator limq-~0pa~q corresponds to the number density of atoms at /z state. The difference of number density between the excited state and the ground state corresponds to the spin operator Sz. The correlation function of the spin operator Sz determines the critical tempetature in magnetism. In the present case, the correlation function of the sum of the diagonal density operators limq__+0~ u Pa,q determines the critical temperature for the freezing of atoms. Here, we consider the sum of the diagonal density operators opposite to the case for Sz, but the sum or difference of the diagonal density operators does not matter, since the contractions of the operators in correlation functions are carried out only between the same operators so that the signs of + are always squared. Next we consider the wave functions in the number density operator (2.19). The wave functions E m eiq'RmW2m#(x - - R m ) of the diagonal density fluctuations represent the fluctuations of the number density of the /z state atoms. The wave functions ~-~meiq'Rm~m~7iW20(X-Rm) of the off-diagonal density fluctuations correspond to the current density fluctuations of the ground-state atoms. Thus, the correlations of Pa/,q and Piq correspond to those of the density and current fluctuations in the classical theory [12,13], respectively. There is no conservation law between the density and the current in our theory, but they conserve the number density by themselves. Thus, the correlation functions of Pa/~ and Piq yield sound and phonon modes, respectively. In the previous papers [ 1-7], we have presented the correlation functions of off-diagonal density fluctuations. Here we are concerned with the correlation functions of diagonal density fluctuations.

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T. Kitamura/ Physica A 245 (1997) 543-559

3. The correlation functions of diagonal density fluctuations and the liquid-glass transition

In order to consider the density-response function to the external potential, we introduce the following diagonal density Green's functions: F~,,q(Zl

- "c2) =-- - - ~ ( T ~ p d l u l ( Z l ) P ~ d u , q ( Z 2 ) ) c

=--

Z

e-iV"(~'-~2)Fw~,(q)

. (3.1)

v° Here we consider the interaction Hamiltonian (2.22). If we make the random phase approximation in Eq. (3.1), we obtain 1

1

d

Fm,,(q) = ..~P~(q)fuu' + ~ Z

,

Pu(q)(V~v(q))cFv~ ( q ) ,

(3.2)

V

where 1

Pu(q) = 7:R--~---~ ~ Z

Gu(p)Gt'(P - q ) '

(3.3)

,

(3.4)

p

Gu(p) -- .

1

10.)n -- F,~p

and qz signs in Eq. (3.3) correspond to bosons and fermions, respectively. The explicit expression for P~(q) is given by

1 P~(q) = - N Z

P

f(eut, ) - f(eup_q) . . . . . .

iVn -- (Elip -- et~P--q) '

(3.5)

where f is a boson or fermion distribution function. The configurationally averaged cue is given by

e~t, -- ff9~ + 1 f d3Rg(R)eip.Rj~(R) '

(3.6)

where 9(R) is a pair-distribution function. The configurationally averaged diagonal potential function is given by (V~v(q)) c =

V; (Rm) =- /

d3Rg(R)e-iq.R

V ~d ( R ) ,

(3.7)

d3xd3 y(ff~2~(x)V( x - Y ) ~ v ( Y - Rm))c .

(3.8)

For simplicity, we consider that all the potential functions due to diagonal density fluctuations are the same:

V(q) =_ (Vffv(q)) c .

(3.9)

Then we obtain

F(q) =

O/tOP(q) 1 - (1/h)V(q)P(q) '

(3.10)

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550

where

F(q) = Z F t , v(q),

(3.11)

~v

(3.12)

P(q) = #

At high temperatures, P(q) becomes the density correlation function of the classical 0

o

ideal gas: P(q)-- - ~ 1 ~-~ f(el,)--f(ee_q) iv _(~o_~o ), where e~0 is a free-atom energy spectrum. This n

p

p_q

case has been discussed by Schneider et al. [14,15]. However, the freezing of atoms in cages occurs at the lower-temperature regime. We confine ourselves to the low-temperature regime near the freezing point: h~ofl >> 1 and IJu]/~ << 1. In this regime, we can put V(q) TM (Voo(q))c. Here we put 9(R) TM p f ( R - a) in Eq. (3.6), where a is an atomic distance and then we have evp ~ oSv + j

~

41ta2

sin~__, a~~ =~ ~

pJv(a),

~ - pa.

(3.13)

The function sin~/~ behaves as a decreasing oscillatory function and it has the maximum at ~ -- 0 and the minimum at k near ~ = 3rc/2. Since Ji > 0 and J0 < 0, there is a band gap between ei and e0 at ~ = K. Under the conditions h~/~ >> 1 and /~l)~l << 1, we can restrict ourselves to the lowest band and obtain

1

P(q) ~- Po(q)

TM

-~ Z

f'(___~0 c ~)/~Jo ' / _ . _( _sin _ _ _ ~ - - sinc~/_______~) iv, + Jo(sincd/~' - sin~/~) '

(3.14)

P

where ~' = ~p_q = alp - ql and f'(ch0) is the derivative of f(oS0) with respect to /~O5o; f¢(o5o) = f(aSo){1 4- f(¢3o)}. In order to obtain the freezing point To, we continue analytically as iv, --+ qo + i3 in P(q). The potential function is obtained:

V(q) ~- 47ta2pVodo(a)sin~ . o~

(3.15 )

Since limqo-+oPo(q) ~= -f'(~o)t~ and V(q) has the negative minimum value a t / £ , the freezing point/~o is determined by f'(cOo)/~o limqo_+O{1 _ hl V(K)P(I£,qo)} = 1 + - -V ( / £ ) h

= 0.

Since )-]pf(~5o) ~ N, f'(oSo) has the value f(OSo) ----2 for bosons or f'(~o) fermions. I f we take the Boltzmann distribution function as f , ft(~o) TM 1.

(3.16) TM

0 for

4. The entropy due to diagonal density fluctuations and the Vogel-Fulcher law The hopping matrix element Jl, should be determined by the distribution of atoms self-consistently. The hopping probability relates to the entropy due to the diagonal

T. Kitamura/Physica A 245 (1997) 543-559

551

density fluctuations. Here to obtain the hopping probability, we calculate the entropy due to diagonal density fluctuations. To do so, we introduce the Hamiltonian with a parameter 2: H(2) = Ho + 2HI.

(4.1)

The thermodynamical function f2;. for the above Hamiltonian is given by ..O;. = - ~ln{Tr e -I~(H°-pJv+;J4')}

(4.2)

o

Here we calculate the thermodynamical function due to diagonal density fluctuations denoted by f2a~. From Eq. (4.1), we obtain 0Qa;~

--

~

1 (2H,~),~= ~ Z

(4.3)

V(q)(Ptdm pam)z'

/tq

where (-.-);. is given by the equation replaced HI by ~-/I in (.-")c. In the random-phase approximation, putting 2 ~ 1, we obtain the thermodynamical function due to the diagonal density fluctuations 12a: •a=

Zln

1-~V(q)P(q)

(4.4)

,

qVn

where we have neglected the 2-dependence of P(q) because of the condition ~lJ~l << 1, when we integrate Eq. (4.3) with respect to 2. The dominant contribution of the diagonal density fluctuations to the thermodynamical function comes from Vn = 0 in P(q) under the condition fllJ, I << 1 and the regions q ~ K. If we denote the number of q-states near/£ by No, we obtain ~2a- ~ l n

1+

.

(4.5)

The entropy due to the diagonal density fluctuations is given by Sd ~ -

v,l,

~

To

2

- + In T - To ~

To } '

(4.6)

The thermodynamical meaning is as follows [16]: if we consider vacancies as a body in an external medium in a nonequilibrium state and the entropy Sa - -Rmin/T as the total entropy of the body together with medium, Rmin is the minimum work which must be done from the external object when the total system comes to an equilibrium state. On the other hand, the exponential of the entropy Sa is proportional to the hopping probability. Thus, we obtain IJu] oc e &/uk8 ~- e -&/r-r°,

Eo -- NoTo 2N

Eq. (4.7) is nothing but the Vogel-Fulcher law.

(4.7)

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552

5. Diffusion Diffusion is calculated from the configurational average of two Green's functions due to the random scattering [17,18]. At this stage, since we take into account random eigenfrequencies of atoms and random hopping matrices more explicitly, we start with Eq. (2.9) in terms of localized operators. Hereafter, since we confine ourselves to the lowest band, we abbreviate the band index. In order to introduce the random scattering, we take the difference of Eq. (2.9) from Eq. (2.11) as the interaction Hamiltonian,

HI = ~ ~(ff3m og)btmbm+ ~ ~{Jmn J(Rm -

-

-

m

-

-

gn)}btmbn .

(5.1)

mR

Note that since the term Jmn involves operators n(y) in ~R~,//i is essentially nonlinear. But at low temperatures an atom stays at the same site longer. Thus, we make the approximation that the term .Inn is a random c-number. Using Eq. (2.14), we rewrite Eq. (5.1) as HI = ~ ~ pp'

tt(dgm -- ~ ) e -'(p-p )'Rm

+ ~ ~{Jmn -- J ( R m - R n ) } e-t~'Rm+ip''Rn mn

atpap' .

(5.2)

Elemental scattering processes, U,o due to random eigenfrequencies of atoms and Uj due to random hopping matrices, are shown in Fig. 1 and the corresponding terms are given by

U'°= ~I ~--~((~2m)c _

(32) ,

(5.3)

m

uj=~1 f

d3Re(R)e_,q.R{(jZm,(R))c _ j2(R)}

(5.4)

where Jmn(R) = "]mOnOlR=Rm--R," Here the crosses on the diagrams mean summing up the scattering processes in terms of random positions of atoms.

p+

p'+

p

p

p+

p'+

T p-

p

Fig. 1. Elemental scattering processes, Uo~ due to the random eigenfrequencies of the harmonic potentials and Uj due to the random hopping matrices.

T. KitamuraI Physica A 245 (1997) 543-559

553

Now we derive the Bethe-Salpeter equation. First, we introduce the atom Green's functions:

Gpp,(~ - z') =-- -(T~ap(~)atp,(~')) = ~ Ze-iC°"(~-~')Gpp'(iOn).

(5.5)

O)n

We derive the Bethe-Salpeter equation for configurationally averaged two Green's functions:

~alJp,(ivn,q;ion)=(G,+q/2,1/+q/2(i°n+i~)Gp-q/:,t,'-q/:(i°n-i~))c

" (5.6)

Using the vertex function, we can write Eq. (5.6) as

~)pp' (iVn,q; ion ) ~ G(p+ )G(p'- )(~pp' +G(p +)G( p - ) F ( p +, p-; p'+, p'- )G(p '+ )G(p'- ),

(5.7)

where G ( p ) is configurational average of the Green's function (5.5):

(5.8)

(Gpp,(iOn))c = Gp(iOn)~pp, = G(p)~pp,

iVn and the abbreviations p + = (/7 + ~, ion 4- T ) have been used. Note that pl = ~ , ion). The vertex function is written down as

F(p +, p - ; p'+, p'- ) = U(p +, p - ; p'+, p'- ) 1

~z~-~ZU(p+,p-;p+,p[)G(p+)G(p[)F(p+,p[;p'+,p'-

),

(5.9)

Pl

and illustrated in Fig. 2. The term U is the irreducible vertex part. Substituting Eq. (5.9) into Eq. (5.7), we obtain the Bethe-Salpeter equation: (5.10)

49pt,,(ivn,q, ion)= G(p+)G(p - ) [6pp, + Z UPt~'49P'I/ ' p~ where we abbreviate

(5.11)

Upp, =- U(p+, p-; p'+, p'- ) . p+

p'+

p+

p'+

p+

p+ p+ _1

p

p

p

p

p

p" +

_1

p? p ?

p

Fig. 2. The vertex function F for two Green's functions due to the random scattering processes, Uj. The vertex function U is the in'educible vertex part.

U,o and

T. KitamuralPhysicaA 245 (1997) 543-559

554 I f we rewrite

AG(p,q) G(p +) - G(p-) _ , G(p+)G(p -) = G _ l ( p _ ) - G - l ( p + ) G - l ( p - ) - G-l(p+)

(5.12)

and introduce the self-energy parts of the Green's function as

G-l(p) ----io9n - ~p - S(p), after we analytically continue iogn ~ o9 and ivn -+ v

(5.13) +

i6, we obtain

{ - v + ~7ep'q + AS(p,q)}4'p= AG(p,q) [a + Z

Upp,4'p,l

(5.14)

where we consider the regime: q ~ 0, v ~ 0 and we put

AG(p,q) = G+(p +) - G - ( p - ) ,

(5.15)

A,~(p,q) = N+(p+) - S - ( p - ) ,

(5.16)

4" = Z

(5.17)

4'~/ '

pr

Here + signs on Green's functions and the self-energy parts means advanced and retarded ones. Using the Ward-Takahashi identity [17]

A~(p,q) = Z Ut'j"AG(p"q)

(5.18)

p'

in Eq. (5.14) and summing up both sides of Eq. (5.14) in p, we obtain

- v4' + q. • = Z AG(p,q),

(5.19)

P

where 4' =- Z

4"'

(5.20)

Ue,4'p,

(5.21)

P

- Z P

which have the characters of a density-relaxation function and of a current-relaxation function, respectively. In order to obtain dynamical equation for 4' and • in a closed form, we multiply ~7e~ • q on both sides of Eq. (5.14) and sum up with respect to p. Since the effective interaction between atoms due to Eqs. (5.3) and (5.4) are independent of p, the self-energy parts of Green's function are independent of p. To consider the equation within the accuracy of the order of q2, we develop AG(p,q) up to the order of q:

AG(p, q) = 2i Im G+(p)+ { G + Z ( p ) + G - Z ( p ) } ~ ~7~p • q .

(5.22)

7". KitamuralPhysica A 245 (1997) 543-559

555

Thus, we obtain

- v q . ¢b +

Z(~78p)2q~p --t-2ilmX+ q • P

=Z

W~p. qAG(p) + Z ( V ~ p " q)Up,,c~e,

p

pp~

+~1 E(~78p " q)Z { G+2(p) + G-2(p)} P

1

+~ Z(~7~p . q)Z{G+2(p) + G-2(p)}Ut, t/4& .

(5.23)

pp+

We denote the three terms on the left hand side of Eq. (5.23) as L1, L2 and L3 in order, and the four terms on the right-hand side as R 1, R2, R3 and R4, respectively. We need equations for ~b and • in a closed form, but Eq. (5.23) has the term ~bp. So we should construct ~bp with 4) and ~b. In Eq. (5.10) the dominant p-dependence of ~bp comes from the factor G+(p+)G-(p-), since Upp, does not depend on p and p'. We develop the factor up to the order of q:

AG(p) + {AG(p)} 2 1 V~p .q. G+(p+)G-(P-) ~- -2-i-I-m- -Z- T 2iImZ + 2

(5.24)

Thus, we put

c~, ~ AG(p) [~b(°) + ½(Vsp . q)AG(p)c~ 0)] .

(5.25)

Summing up both the sides of Eq. (5.25) with respect to p, we obtain ~b = -2inp(~o)q~ (°) ,

(5.26)

where we have put

Z

AG(p) = 2i Z

P

Im G+(p) - -2i7zp(o9),

(5.27)

P

and p(~o) is the density of states for atoms. Multiplying Vep • q on both the sides of Eq. (5.25) and summing up in p, we obtain q. ~ = -Koq2tp O) ,

(5.28)

K o = - ~ 1 Z ( V s p . #)2{AG(p)} 2

(5.29)

where

P

and ~ is the unit vector of q. Since the integration value of G+(p)G-(p) in p dominates that of {G+(p)} 2 and { G - ( p ) } 2, we can put

{AG(p)} 2 ~-- _2G+(p)G-(p) -

2 Im G+(p) imZ+

(5.30)

T. KitamuralPhysica A 245 (1997) 543-559

556

Using Eq. (5.30), we obtain ^ 2 Im

Ko ~ Y~A~7~p • q)

~

G+(p)

.

(5.31)

p

From Eqs. (5.25), (5.26) and (5.28), we obtain 149_1 49p ~- A G ( p ) [ 27~ip(o9)

2-k-~oqZ ( V e , . q ) A G ( p ) q

(5.32)

. ¢I,] .

Making use of Eq. (5.32), we obtain L2 ~- - Z ( ~ 7 ~

(5.33)

. q)2 AG(P)-2ni~tog).,.,

p

49

R4 ~ -~1 Z ( g T e p . q)2 [{G+(p)}2 + { G - ( p ) } 2 ] 2 i Im Z + 2xip(o~) '

(5.34)

P

where in L2 we have considered 49 ~,, O(q) and in deriving Eq. (5.34), we have used the Ward-Takahashi identity, Eq. (5.18). Using the above equations, we obtain Im S + 2

L2 - R4 = - K o ~ q

49.

(5.35)

In our model, since U~, is isotropic, R 1,R2 = 0. L 1 is a higher-order term and R3 is a source term. Combining the above results, Eq. (5.23) yields q . • = - i D ( co )q2 49 ,

(5.36)

where

K0

D ( o g ) - 2~p(o9) "

(5.37)

Substituting Eq. (5.36) into Eq. (5.19), we obtain

2xp(¢o) 49 -- - i v + D(~o)q 2 "

(5.38)

Thus D ( ~ ) is the diffusion coefficient.

6. Calculation of the diffusion coefficient and the Vogel-Fulcher law

In order to obtain the diffusion coefficient, we calculate the self-energy part of atom Green's function. The Ward-Takahashi identity guarantees that the irreducible vertex part of two Green's functions and the self-energy part are determined self-consistently. Here we make the ladder approximation to the vertex part for two Green's functions: Upp, ~ U~o + U j ,

(6.1)

T. KitamuralPhysica A 245 (1997) 543-559

i"""a+(p)""i

557

a+(p)

Fig. 3. The self-energy part corresponding to the ladder approximation to the vertex function is shown. The irreducible vertex function for the ladder approximation is shown in Fig. 1.

which are illustrated in Fig. 1. This corresponds to the following renormalized selfenergy part: S+(og) ~- (Uo~ + U j ) E

G + ( p ) = (U~o + U j ) E

p

1 - S+(og) , o9 + i6 - ep

(6.2)

P

which are illustrated in Fig. 3. The self-energy part should be solved self-consistently in Eq. (6.2), but near the liquid-glass transition, Uj is small enough proportional to j2 and we can expect U~, is also small. Thus, we put G + ( p ) ~-

1

(6.3)

og + i6 - ep

in Eq. (6.2). Then we obtain ImX + ~= -(Uo, + Uj)np(og),

(6.4)

Im G+ ( p ) ~ n6(o9 - et,) .

(6.5)

The diffusion constant D(og) is given by 1 D(og) = (U~ + Ua)np2(og) Z ( V ' e P " #)2(~(o9 __

ep).

(6.6)

p

We put Z""

V 3 f dSd~p... 6(o9 - gP)de,,,/dp 1 ' 6(o9 - gp) -- (2n)

(6.7)

p

where V is a volume of the system, S is the equi-energy surface of co = gp and dp on the right-hand side of the equation is an infinitesimal element of the width between the equi-energy surfaces. Combining Eqs. (6.6) and (6.7), we obtain D(og)-

2n

3V(U~o + U j )

(~7eP)---~3~o=~, p2

"

(6.8)

Here ~7ee is proportional to J , Uj to j 2 and Uo) is nearly constant above and below the liquid-glass transition. We obtain

T. Kitamura/Physica A 245 (1997) 543-559

558

(i) Uo~ > Uj: D(~) oc j3 .

(6.9)

(ii) U~o < Uj: D(co) cx J .

(6.10)

In both cases, since J is given by Eq. (4.7), D(og) obeys the Vogel-Fulcher law.

7. Concluding remarks We have investigated the correlation functions of diagonal density fluctuations which correspond to the density response function to the external potential. The potential which an atom feels is determined by the density of the surrounding atoms Eqs. (2.6) and (2.7). Thus, Eq. (3.16) determines the freezing point, where the hopping is prohibited. In determining the freezing point in Eq. (3.16), we have neglected the self-energy parts and the vertex corrections, which originate from the scattering processes, Uo~ due to random eigenfrequencies of atoms and Uj due to random hopping matrices. The self-energy parts relate to the relaxation time of atoms z(~) at a frequency co; 1/2z(09) = Im Z+(og) in Eq. (6.4). The freezing temperature To is accurate within the order of floh/z(~o) << 1. The relaxation time obscures the freezing point. Uj is proportional to the second power of the configurationally averaged hopping matrix, Uj (x j2. The configurationally averaged hopping matrix obeys the VogelFulcher law, so does Uj. Thus, as the temperature become lower towards the freezing point, Uj becomes smaller according to the Vogel-Fulcher law. the quantity Uj is zero below the freezing points. On the other hand, the quantity U~o is almost constant above and below the freezing point. This fact reflects the diffusion coefficient, Eqs. (6.9) and (6.10). The structure of phonons and sound is determined by the correlation functions of off-diagonal and diagonal density fluctuations, respectively. The lifetime of the modes appear, when the frequencies of the modes merge into the continuum of the particlehole excitations in the interband and intraband transition, respectively [4,6]. The lifetime of the modes also comes from the relaxation of atoms. The width of the first peak of the pair-distribution function 9(R) is expected to originate from the relaxation of atoms. In this paper we have taken 9(R) ~- pf(R - a). This approximation means that we have considered the two discrete levels in the harmonic potential. The freezing point has been determined by Eq. (3.16), which has been obtained in the mean field approximation. Since the freezing point exists apparently, the mean-field approximation is reasonable to estimate the liquid-glass transition. If we include the nonlinear terms in the interaction Hamiltonian [ 19-24], we can deal with more realistic potentials of atoms depending on temperatures.

72 KitamuralPhysica A 245 (1997) 543-559

559

Acknowledgements I w o u l d like to thank Professor K y o z i Kawasaki and Professor S h o z o Takeno for valuable discussions.

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E. Leutheusser, J. Phys. C19 (1982) 2801,2827. J. Bosse, W. Gftze, M. Liicke, Phys. Rev. A 17 (1978) 434. T. Schneider, R. Brout, H. Thomas, J. Feder, Phys. Rev. Lett. 25 (1970) 1423. T. Schneider, Phys. Rev. A 3 (1971) 2145. L.D. Landau, E.M. Lifshitz, Statistical Physics, Pergamon Press, London 1958. D. Vollhandt, P. W61fle, Phys. Rev. B 22 (1980) 4666. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena, Academic Press, London, 1995. [19] T. Kitamura, S. Takeno, Phys. Lett. A 172 (1992) 186. [20] T. Kitamura, Phys. Lett. A 186 (1994) 351. [21] T. Kitamura, S. Takeno, Phys. Lett. A 190 (1994) 327. [22] T. Kitamura, Physica A 213 (1995) 525, 539. [23] T. Kitamura, Physica A 214 (1995) 295. [24] T. Kitamura, Phys. Lett. A 203 (1995) 395.