linear theory

3LINEAR RESPONSE THEORY

This chapter is devoted to a concise presentation of linear response the-ory, which provides a general framework for analysing the dynamicalproperties of a condensed-matter system close to thermal equilibrium.The dynamical processes may either be spontaneous uctuations, ordue to external perturbations, and these two kinds of phenomena areinterrelated. Accounts of linear response theory may be found in manybooks, for example, des Cloizeaux (1968), Marshall and Lovesey (1971),and Lovesey (1986), but because of its importance in our treatment ofmagnetic excitations in rare earth systems and their detection by inelas-tic neutron scattering, the theory is presented below in adequate detailto form a basis for our later discussion.

We begin by considering the dynamical or generalized susceptibility,which determines the response of the system to a perturbation whichvaries in space and time. The KramersKronig relation between thereal and imaginary parts of this susceptibility is deduced. We derivethe Kubo formula for the response function and, through its connectionto the dynamic correlation function, which determines the results of ascattering experiment, the uctuationdissipation theorem, which relatesthe spontaneous uctuations of the system to its response to an externalperturbation. The energy absorption by the perturbed system is deducedfrom the susceptibility. The Green function is dened and its equation ofmotion established. The theory is illustrated through its application tothe simple Heisenberg ferromagnet. We nally consider the calculationof the susceptibility in the random-phase approximation, which is themethod generally used for the quantitative description of the magneticexcitations in the rare earth metals in this book.

3.1 The generalized susceptibility

A response function for a macroscopic system relates the change of anensemble-averaged physical observable B(t) to an external force f(t).For example, B(t) could be the angular momentum of an ion, or the mag-netization, and f(t) a time-dependent applied magnetic eld. As indi-cated by its name, the applicability of linear response theory is restrictedto the regime where B(t) changes linearly with the force. Hence wesuppose that f(t) is suciently weak to ensure that the response is lin-ear. We further assume that the system is in thermal equilibrium before

3.1 THE GENERALIZED SUSCEPTIBILITY 135

the external force is applied.When the system is in thermal equilibrium, it is characterized by

the density operator

0 =1

ZeH0 ; Z = Tr eH0 , (3.1.1)

where H0 is the (eective) Hamiltonian, Z is the (grand) partition func-tion, and = 1/kBT . Since we are only interested in the linear part ofthe response, we may assume that the weak external disturbance f(t)gives rise to a linear time-dependent perturbation in the total Hamilto-nian H:

H1 = A f(t) ; H = H0 +H1, (3.1.2)where A is a constant operator, as for example

i Jzi, associated with

the Zeeman term when f(t) = gBHz(t) (the circumex over A or Bindicates that these quantities are quantum mechanical operators). Asa consequence of this perturbation, the density operator (t) becomestime-dependent, and so also does the ensemble average of the operatorB:

B(t) = Tr{(t) B}. (3.1.3)The linear relation between this quantity and the external force has theform

B(t) B = t

BA(t t) f(t)dt, (3.1.4)

where B = B(t = ) = Tr{0 B}; here f(t) is assumed to vanishfor t . This equation expresses the condition that the dierentialchange of B(t) is proportional to the external disturbance f(t) andthe duration of the perturbation t, and further that disturbances atdierent times act independently of each other. The latter conditionimplies that the response function BA may only depend on the timedierence tt. In (3.1.4), the response is independent of any future per-turbations. This causal behaviour may be incorporated in the responsefunction by the requirement

BA(t t) = 0 for t > t, (3.1.5)

in which case the integration in eqn (3.1.4) can be extended from t to+.

Because BA depends only on the time dierence, eqn (3.1.4) takesa simple form if we introduce the Fourier transform

f() =

f(t) eitdt, (3.1.6a)

136 3. LINEAR RESPONSE THEORY

and the reciprocal relation

f(t) =12

f() eitd. (3.1.6b)

In order to take advantage of the causality condition (3.1.5), we shallconsider the Laplace transform of BA(t) (the usual s is replaced byiz):

BA(z) = 0

BA(t) eiztdt. (3.1.7a)

z = z1 + iz2 is a complex variable and, if0|BA(t)|etdt is assumed

to be nite in the limit 0+, the converse relation is

BA(t) =12

+i+i

BA(z) eiztdz ; > 0. (3.1.7b)

When BA(t) satises the above condition and eqn (3.1.5), it can readilybe shown that BA(z) is an analytic function in the upper part of thecomplex z-plane (z2 > 0).

In order to ensure that the evolution of the system is uniquely de-termined by 0 = () and f(t), it is necessary that the externalperturbation be turned on in a smooth, adiabatic way. This may beaccomplished by replacing f(t) in (4) by f(t) et

, > 0. This force

vanishes in the limit t , and any unwanted secondary eects maybe removed by taking the limit 0+. Then, with the denition of thegeneralized Fourier transform

B() = lim0+

(B(t) B

)eit etdt, (3.1.8)

eqn (3.1.4) is transformed into

B() = BA() f(), (3.1.9a)

where BA() is the boundary value of the analytic function BA(z) onthe real axis:

BA() = lim0+

BA(z = + i). (3.1.9b)

BA() is called the frequency-dependent or generalized susceptibilityand is the Fourier transform, as dened by (3.1.8), of the response func-tion BA(t).

The mathematical restrictions (3.1.5) and (3.1.7) on BA(t) havethe direct physical signicance that the system is respectively causaland stable against a small perturbation. The two conditions ensure that

3.2 RESPONSE FUNCTIONS 137

BA(z) has no poles in the upper half-plane. If this were not the case,the response B(t) to a small disturbance would diverge exponentiallyas a function of time.

The absence of poles in BA(z), when z2 is positive, leads to a rela-tion between the real and imaginary part of BA(), called the KramersKronig dispersion relation. If BA(z) has no poles within the contourC, then it may be expressed in terms of the Cauchy integral along C bythe identity

BA(z) =1

2i

C

BA(z)

z z dz.

The contour C is chosen to be the half-circle, in the upper half-plane,centred at the origin and bounded below by the line parallel to the z1-axis through z2 = , and z is a point lying within this contour. SinceBA(t) is a bounded function in the domain

> 0, then BA(z) must

go to zero as |z| , whenever z2 > 0. This implies that the partof the contour integral along the half-circle must vanish when its radiusgoes to innity, and hence

BA(z) = lim0+

12i

+i+i

BA( + i)

+ i z d( + i).

Introducing z = + i and applying Diracs formula:

lim0+

1 i = P

1 + i(

),

in taking the limit 0+, we nally obtain the KramersKronig rela-tion (P denotes the principal part of the integral):

BA() =1iP

BA()

d, (3.1.10)

which relates the real and imaginary components of ().

3.2 Response functions

In this section, we shall deduce an expression for the response functionBA(t), in terms of the operators B and A and the unperturbed Hamil-tonian H0. In the preceding section, we assumed implicitly the use ofthe Schrodinger picture. If instead we adopt the Heisenberg picture,the wave functions are independent of time, while the operators becometime-dependent. In the Heisenberg picture, the operators are

B(t) = eiHt/h B eiHt/h, (3.2.1)


corresponding to the equation of motion

d

dtB(t) =

i

h[H , B(t) ] (3.2.2)

(assuming that B does not depend explicitly on time). Because thewave functions are independent of time, in the Heisenberg picture, thecorresponding density operator H must also be. Hence we may write(3.1.3)

B(t) = Tr{(t) B} = Tr{H B(t)}. (3.2.3)Introducing (3.2.1) into this expression, and recalling that the trace isinvariant under a cyclic permutation of the operators within it, we obtain

(t) = eiHt/h H eiHt/h,

ord

dt(t) = i

h[H , (t) ]. (3.2.4)

The equation of motion derived for the density operator, in the Schro-dinger picture, is similar to the Heisenberg equation of motion above,except for the change of sign in front of the commutator.

The density operator may be written as the sum of two terms:

(t) = 0 + 1(t) with [H0 , 0] = 0, (3.2.5)

where 0 is the density operator (3.1.1) of the thermal-equilibrium statewhich, by denition, must commute with H0, and the additional contri-bution due to f(t) is assumed to vanish at t . In order to derive1(t) to leading order in f(t), we shall rst consider the following densityoperator, in the interaction picture,

I(t) eiH0t/h (t) eiH0t/h, (3.2.6)

for which

d

dtI(t) = e

iH0t/h{ ih[H0 , (t) ] +

d

dt(t)

}eiH0t/h

= iheiH0t/h [H1 , (t) ] eiH0t/h.

Because H1 is linear in f(t), we may replace (t) by 0 in calculatingthe linear response, giving

d

dtI(t)

i

h

[eiH0t/hH1 eiH0t/h , 0

]=

i

h[ A0(t) , 0 ]f(t),


using (3.2.5) and dening

A0(t) = eiH0t/h A eiH0t/h.

According to (3.2.6), taking into account the boundary condition, thetime-dependent density operator is

(t) = eiH0t/h( t

d

dtI(t

)dt + 0)eiH0t/h

= 0 +i

h

t

[ A0(t t) , 0 ] f(t)dt,

(3.2.7)

to rst order in the external perturbations. This determines the timedependence of, for example, B as

B(t) B = Tr{ ((t) 0) B}=

i

hTr{ t

[ A0(t

t) , 0 ] B f(t)dt}

and, utilizing the invariance of the trace under cyclic permutations, weobtain, to leading order,

B(t) B = ih

t

Tr{0 [ B , A0(t

t) ]} f(t)dt=

i

h

t

[ B0(t) , A0(t) ] 0 f(t)dt.(3.2.8)

A comparison of this result with the denition (3.1.4) of the responsefunction then gives

BA(t t) = ih(t t) [ B(t) , A(t) ] , (3.2.9)

where the unit step function, (t) = 0 or 1 when t < 0 or t > 0 respec-tively, is introduced in order to ensure that BA satises the causalityprinciple (3.1.5). In this nal result, and below, we suppress the index 0,but we stress that both the variations with time and the ensemble aver-age are thermal-equilibrium values determined byH0, and are unaectedby the external disturbances. This expression in terms of microscopicquantities, is called the Kubo formula for the response function (Kubo1957, 1966).

The expression (3.2.9) is the starting point for introducing a numberof useful functions:

KBA(t) =i

h [B(t) , A ] = i

h [B , A(t) ] (3.2.10)


is also called a response function. A is a shorthand notation for A(t = 0).The inverse response function KAB(t), which determines A(t) causedby the perturbation H1 = f(t)B, is

KAB(t) =i

h [A(t) , B ] = KBA(t),

and KBA(t) can be expressed in terms of the corresponding causal re-sponse functions as

KBA(t) =

{BA(t) for t > 0

AB(t) for t < 0.

The susceptibility is divided into two terms, the reactive part

BA(z) = AB(z) 12

{BA(z) + AB(z)

}, (3.2.11a)

and the absorptive part

BA(z) = AB(z) 12i{BA(z) AB(z)

}, (3.2.11b)

so thatBA(z) =

BA(z) + i

BA(z) (3.2.11c)

and, according to the KramersKronig relation (3.1.10),

BA() =1P

BA()

d ; BA() =

1P

BA()

d.

(3.2.11d)In these equations, AB() is the boundary value obtained by takingz = + i, i.e. as lim0+ AB(z = + i), corresponding to thecondition that AB(z), like AB(z), is analytic in the upper half-plane. The appropriate Laplace transform of KBA(t) with this propertyis

KBA(z) =

KBA(t) ei(z1t+iz2|t|)dt

= 0

BA(t) eiztdt

0

AB(t) eiztdt.

HenceKBA(z) = 2i

BA(z). (3.2.12)

Next we introduce the dynamic correlation function, sometimes re-ferred to as the scattering function. It is dened as follows:

SBA(t) B(t) A BA = B A(t) BA, (3.2.13)


and is related to the response function introduced earlier by

KBA(t) =i

h

{SBA(t) SAB(t)

}. (3.2.14)

The dierent response functions obey a number of symmetry rela-tions, due to the invariance of the trace under a cyclic permutation ofthe operators. To derive the rst, we recall that the Hermitian conjugateof an operator is dened by

(< |B | >) = < |B | > .

If we assume that a certain set of state vectors | > constitutes a diag-onal representation, i.e. H0| >= E| >, then it is straightforward toshow that

B(t) A = A(t) B,leading to the symmetry relations

KBA(t) = KBA(t)

andBA(z) = BA(z). (3.2.15)

Another important relation is derived as follows:

B(t) A = 1ZTr{eH0 eiH0t/h B eiH0t/h A

}= 1

ZTr{eiH0(t+ih)/h B eiH0(t+ih)/h eH0 A

}= 1

ZTr{eH0 A B(t + ih)

}= A B(t + ih),

implying thatSBA(t) = SAB(t ih). (3.2.16)

In any realistic system which, rather than being isolated, is in con-tact with a thermal bath at temperature T , the correlation functionSBA(t) vanishes in the limits t , corresponding to the conditionB(t = ) A = BA. If we further assume that SBA(t) is an an-alytic function in the interval |t2| of the complex t-plane, then theFourier transform of (3.2.16) is

SBA() = eh SAB(), (3.2.17)

which is usually referred to as being the condition of detailed balance.Combining this condition with the expressions (3.2.12) and (3.2.14), we


get the following important relation between the correlation functionand the susceptibility:

SBA() = 2h1

1 eh BA(), (3.2.18)

which is called the uctuationdissipation theorem. This relation ex-presses explicitly the close connection between the spontaneous uctu-ations in the system, as described by the correlation function, and theresponse of the system to external perturbations, as determined by thesusceptibility.

The calculations above do not depend on the starting assumptionthat B (or A) is a physical observable, i.e. that B should be equal toB. This has the advantage that, if the Kubo formula (3.2.9) is taken to

be the starting point instead of eqn (3.1.4), the formalism applies moregenerally.

3.3 Energy absorption and the Green function

In this section, we rst present a calculation of the energy transferredto the system by the external perturbation H1 = A f(t) in (3.1.2),incidentally justifying the names of the two susceptibility componentsin (3.2.11). The energy absorption can be expressed in terms of AA()and, without loss of generality, A may here be assumed to be a Hermitianoperator, so that A = A

. In this case, f(t) is real, and considering a

harmonic variation

f(t) = f0 cos (0t) =12f0

(ei0t + ei0t

)with f0 = f0,

then

f() = f0{(0)+(+0)}, as

ei(0)tdt = 2(0),

and we have

A(t) A = 12f0{AA(0) ei0t + AA(0) ei0t

}.

The introduction of A = B = A

in (3.2.15), and in the denition(3.2.11), yields

AA() = AA() =

AA()

AA() = AA() = AA(),

(3.3.1)

3.3 ENERGY ABSORPTION AND THE GREEN FUNCTION 143

and these symmetry relations allow us to write

A(t) A = f0 {AA(0) cos (0t) + AA(0) sin (0t)} .

The part of the response which is in phase with the external force is pro-portional to AA(0), which is therefore called the reactive component.The rate of energy absorption due to the eld is

Q =d

dtH = H/t = A(t) f/t,

which shows that the mean dissipation rate is determined by the out-of-phase response proportional to AA():

Q = 12f20 0

AA(0) (3.3.2)

and AA() is therefore called the absorptive part of the susceptibility.If the eigenvalues E and the corresponding eigenstates | > for

the Hamiltonian H(= H0) are known, it is possible to derive an explicitexpression for BA(). According to the denition (3.2.10),

KBA(t) =i

h

1ZTr{

eH [ eiHt/h B eiHt/h , A ]}

=

i

h

1Z

eE{eiEt/h < |B | > eiE t/h < |A | >

< |A | > eiE t/h < |B | > eiEt/h }.Interchanging and in the last term, and introducing the populationfactor

n =1

ZeE ; Z =

eE , (3.3.3a)

we get

KBA(t) =i

h

< |B | >< |A | > (n n) ei(EE)t/h,(3.3.3b)

and hence

BA() = lim0+

0

KBA(t) ei(w+i)tdt

= lim0+

< |B | >< |A | >E E h ih (n n

),(3.3.4a)


or equivalently

AB() = lim0+

AB( + i)

= lim0+

< |A | >< |B | >E E + h ih (n n

).(3.3.4b)

An interchange of and shows this expression to be the same as(3.3.4a), with replaced by . The application of Diracs formula thenyields the absorptive part of the susceptibility (3.2.11b) as

BA() =

< |B | >< |A | > (nn) (h (E E)

)(3.3.5)

(equal to KBA()/2i in accordance with (3.2.12)), whereas the reactivepart (3.2.11a) is

BA() =E =E

< |B | >< |A | >E E h (n n

) + BA(el) 0,

(3.3.6a)where

0 lim0+

i

+ i=

{1 if = 00 if = 0,

and the elastic term BA(el), which only contributes in the static limit = 0, is

BA(el) = {E=E

< |B | >< |A | > n BA

}. (3.3.6b)

We remark that BA() and BA() are often referred to respectively as

the real and the imaginary part of BA(). This terminology is not validin general, but only if the matrix-element products are real, as they areif, for instance, B = A

. The presence of the elastic term in the reactive

response requires some additional consideration. There are no elasticcontributions to KBA(t), nor hence to BA(), because n n 0if E = E . Nevertheless, the appearance of an extra contribution at = 0, not obtainable directly from KBA(t), is possible because theenergy denominator in (3.3.4) vanishes in the limit | + i| 0, whenE = E . In order to derive this contribution, we consider the equal-time correlation function

SBA(t = 0) = (B B)(A A)=

< |B | >< |A | > n BA (3.3.7a)


which, according to the uctuationdissipation theorem (3.2.18), shouldbe

SBA(t = 0) =12

SBA() d =1

11 eh

BA()d(h).

(3.3.7b)Introducing (3.3.5), the integration is straightforward, except in a nar-row interval around = 0, and we obtain

SBA(t = 0) =E =E

< |B | >< |A | > n + lim

0+

BA()

d

after replacing 1 eh with h in the limit 0. A comparison ofthis expression for SBA(t = 0) with (3.3.7a) shows that the last integralhas a denite value:

lim0+

BA()

d =E=E

< |B | >< |A | > n BA.

(3.3.8)The use of the KramersKronig relation (3.1.10), in the form of (3.2.11d),for calculating BA(0) then gives rise to the extra contribution

BA(el) = lim0+

1

BA()

d (3.3.9)

to the reactive susceptibility at zero frequency, as anticipated in (3.3.6b).The zero-frequency result, BA(0) =

BA(0), as given by (3.3.6), is the

same as the conventional isothermal susceptibility (2.1.18) for the mag-netic moments, where the elastic and inelastic contributions are respec-tively the Curie and the Van Vleck terms. This elastic contribution isdiscussed in more detail by, for instance, Suzuki (1971).

The results (3.3.46) show that, if the eigenstates of the Hamil-tonian are discrete and the matrix-elements of the operators B and Abetween these states are well-dened, the poles of BA(z) all lie on thereal axis. This has the consequence that the absorptive part BA()(3.3.5) becomes a sum of -functions, which are only non-zero when his equal to the excitation energies E E. In such a system, no spon-taneous transitions occur. In a real macroscopic system, the distributionof states is continuous, and only the ground state may be considered as awell-dened discrete state. At non-zero temperatures, the parameters ofthe system are subject to uctuations in space and time. The introduc-tion of a non-zero probability for a spontaneous transition between thelevels and can be included in a phenomenological way by replac-ing the energy dierence E E in (3.3.4) by (E E) i(),


where the parameters, including the energy dierence, usually dependon . According to the general stability and causality requirements,the poles of BA(z) at z = z = (E E) i must lie in thelower half-plane, implying that has to be positive (or zero). In thecase where |E E| , the -dependence of these parametersis unimportant, and the -function in (3.3.5) is eectively replaced by aLorentzian:

BA()

< |B | >< |A | >(E E h)2 + 2

(n n)

+h0

(h)2 + 20BA(el),

(3.3.10)

with a linewidth, or more precisely FWHM (full width at half maximum),of 2. In (3.3.10), we have added the quasi-elastic response due to apole at z = i0, which replaces the one at z = 0. The correspondingreactive part of the susceptibility is

BA()

< |B | >< |A | >(E E h)2 + 2

(E E h)(n n)

+20

(h)2 + 20BA(el). (3.3.11)

The non-zero linewidth corresponds to an exponential decay of the oscil-lations in the time dependence of, for instance, the correlation function:

SBA(t) eizt/h = ei(EE)t/h et/h.

The absorption observed in a resonance experiment is proportionalto AA(). A peak in the absorption spectrum is interpreted as an ele-mentary or quasi-particle excitation, or as a normal mode of the dynamicvariable A, with a lifetime = h/. A pole at z = i0 is said torepresent a diusive mode. Such a pole is of particular importance forthose transport coecients determined by the low-frequency or hydro-dynamic properties of the system. Kubo (1957, 1966) gives a detaileddiscussion of this subject. As we shall see later, the dierential scatter-ing cross-section of, for example, neutrons in the Born-approximation isproportional to a correlation function, and hence to (). This impliesthat the presence of elementary excitations in the system leads to peaksin the intensity of scattered neutrons as a function of the energy transfer.Finally, the dynamic correlation-functions are related directly to variousthermodynamic second-derivatives, such as the compressibility and the


magnetic susceptibility, and thereby indirectly to the corresponding rst-derivatives, like the specic heat and the magnetization. Consequently,most physical properties of a macroscopic system near equilibrium maybe described in terms of the correlation functions.

As a supplement to the response function BA(t t), we now in-troduce the Green function, dened as

GBA(t t) B(t) ; A(t) i

h(t t) [ B(t) , A(t) ] = BA(t t).

(3.3.12)

This Green function is often referred to as the double-time or the retardedGreen function (Zubarev 1960), and it is simply our previous responsefunction, but with the opposite sign. Introducing the Laplace transformGBA(z) according to (3.1.7), we nd, as before, that the correspondingFourier transform is

GBA() B ; A = lim0+

GBA(z = + i)

= lim0+

(0)

GBA(t) ei(+i)tdt = BA().(3.3.13)

We note that, if A and B are dimensionless operators, then GBA() orBA() have the dimensions of inverse energy.

If t = 0, the derivative of the Green function with respect to t is

d

dtGBA(t) = i

h

((t) [ B(t) , A ] + (t) [ dB(t)/dt , A ]

)= i

h

((t) [ B , A ] i

h(t) [ [ B(t) , H ] , A ]

).

A Fourier transformation of this expression then leads to the equationof motion for the Green function:

hB ; A [ B , H ] ; A = [ B , A ] . (3.3.14a)

The sux indicates the Fourier transforms (3.3.13), and h is short-hand for h( + i) with 0+. In many applications, A and B arethe same (Hermitian) operator, in which case the r.h.s. of (3.3.14a) van-ishes and one may proceed to the second derivative. With the conditionthat [ [ [ A(t) , H ] , H ] , A ] is [ [ A(t) , H ] , [ A , H ] ] , the equationof motion for the Green function [ A , H ] ; A leads to

(h)2A ; A + [ A , H ] ; [ A , H ] = [ [ A ,H ] , A ] . (3.3.14b)


The pair of equations (3.3.14) will be the starting point for our applica-tion of linear response theory.

According to the denition (3.2.10) of KBA(t), and eqn (3.2.12),

KBA() = 2iBA() = 2iGBA().We may write

i

BA() eitd = i

h [ B(t) , A ] (3.3.15)

and, setting t = 0, we obtain the following sum rule:

h

BA()d = [ B , A ] , (3.3.16)

which may be compared with the value obtained for the equal-time corre-lation function B ABA, (3.3.7). The Green function in (3.3.14a)must satisfy this sum rule, and we note that the thermal averages in(3.3.14a) and (3.3.16) are the same. Equation (3.3.16) is only the rstof a whole series of sum rules.

The nth time-derivative of B(t) may be written

dn

dtnB(t) =

(i

h

)nLnB(t) with LB(t) [H , B(t) ].

Taking the nth derivative on both sides of eqn (3.3.15), we get

i

(i)nBA() eitd =(

i

h

)n+1 [LnB(t) , A ] .

Next we introduce the normalized spectral weight function

FBA() =1

BA(0)1

BA()

, where

FBA()d = 1.

(3.3.17a)The normalization of FBA() is a simple consequence of the KramersKronig relation (3.2.11d). The nth order moment of , with respect tothe spectral weight function FBA(), is then dened as

nBA =

nFBA()d, (3.3.17b)

which allows the relation between the nth derivatives at t = 0 to bewritten

BA(0) (h)n+1BA = (1)n [LnB , A ] . (3.3.18a)

3.4 LINEAR RESPONSE OF THE HEISENBERG FERROMAGNET 149

These are the sum rules relating the spectral frequency-moments withthe thermal expectation-values of operators obtainable from B, A, andH. If B = A = A, then (3.3.1) shows thatFBA() is even in , and allthe odd moments vanish. In this case, the even moments are

AA(0) (h)2nAA = [L2n1A , A ] . (3.3.18b)

3.4 Linear response of the Heisenberg ferromagnet

In this section, we shall illustrate the use of linear response theory byapplying it to the case of the three-dimensional Heisenberg ferromagnet,with the Hamiltonian

H = 12

i=j

J (ij)Si Sj , (3.4.1)

where Si is the spin on the ith ion, placed in a Bravais lattice at theposition Ri. The spatial Fourier transform of the exchange coupling,with the condition J (ii) 0, is

J (q) = 1N

ij

J (ij) eiq(RiRj) =j

J (ij) eiq(RiRj), (3.4.2a)

and conversely

J (ij) = 1N

q

J (q) eiq(RiRj) = VN(2)3

J (q) eiq(RiRj)dq,

(3.4.2b)depending on whether q, dened within the primitive Brillouin zone, isconsidered to be a discrete or a continuous variable (we shall normallyassume it to be discrete). N is the total number of spins, V is thevolume, and the inversion symmetry of the Bravais lattice implies thatJ (q) = J (q) = J (q). The maximum value of J (q) is assumed tobe J (q = 0), in which case the equilibrium state at zero temperature,i.e. the ground state, is the ferromagnet:

Si = S z at T = 0, (3.4.3)

where z is a unit vector along the z-axis, which is established as thedirection of magnetization by an innitesimal magnetic eld. This resultis exact, but as soon as the temperature is increased above zero, it isnecessary to make a number of approximations. As a rst step, we


introduce the thermal expectation-values Si = S in the Hamiltonianwhich, after a simple rearrangement of terms, can be written

H =

i

Hi 12

i=j

J (ij)(Si S) (Sj S), (3.4.4a)

withHi = Szi J (0)Sz+ 12J (0)Sz2, (3.4.4b)

and S = Sz z. In the mean-eld approximation, discussed in the pre-vious chapter, the dynamic correlation between spins on dierent sites isneglected. This means that the second term in (3.4.4a) is disregarded,reducing the original many-spin Hamiltonian to a sum of N indepen-dent single-spin Hamiltonians (3.4.4b). In this approximation, Sz isdetermined by the self-consistent equation

Sz =+S

M=SM eMJ (0)S

z/ +SM=S

eMJ (0)Sz (3.4.5a)

(the last term in (3.4.4b) does not inuence the thermal average) which,in the limit of low temperatures, is

Sz S eSJ (0). (3.4.5b)

In order to incorporate the inuence of two-site correlations, toleading order, we consider the Green function

G(ii, t) = S+i (t) ; Si . (3.4.6)

According to (3.3.14a), the variation in time of G(ii, t) depends onthe operator

[S+i , H ] = 12

j

J (ij) (2S+i Szj + 2Szi S+j ) .The introduction of this commutator in the equation of motion (3.3.14a)leads to a relation between the original Green function and a new, moreelaborate Green function. Through its equation of motion, this newfunction may be expressed in terms of yet another. The power of theexchange coupling in the Green functions which are generated in thisway is raised by one in each step, and this procedure leads to an in-nite hierarchy of coupled functions. An approximate solution may beobtained by utilizing the condition that the expectation value of Szi isclose to its saturation value at low temperatures. Thus, in this limit,


Szi must be nearly independent of time, i.e. Szi Sz. In this random-

phase approximation (RPA) the commutator reduces to

[S+i , H ] j

J (ij)Sz (S+j S+i ) ,and the equations of motion lead to the following linear set of equations:

hG(ii, ) +

j

J (ij)Sz {G(ji, )G(ii, )}= [S+i , Si ] = 2Sz ii .

(3.4.7)

The innite set of RPA equations is diagonal in reciprocal space. Intro-ducing the Fourier transform

G(q, ) =i

G(ii, ) eiq(RiRi ), (3.4.8)

we obtain

hG(q, ) + Sz {J (q)G(q, ) J (0)G(q, )} = 2Sz,or

G(q, ) = lim0+

2Szh + ih Eq , (3.4.9)

where the dispersion relation is

Eq = Sz {J (0) J (q)} . (3.4.10)

Introducing the susceptibility +(q, ) = G(q, ), we obtain

+(q, ) =2Sz

Eq h + i 2Sz (h Eq). (3.4.11a)

Dening +(q, ) analogously to +(q, ), but with S+ and S in-

terchanged, we obtain similarly, or by the use of the symmetry relation(3.2.15),

+(q, ) =2Sz

Eq + h i 2Sz (h + Eq), (3.4.11b)

so that the absorptive susceptibility is

+(q, ) = +(q,) = 2 Sz (h Eq). (3.4.11c)


The above susceptibilities do not correspond directly to physical observ-ables but, for instance, xx(q, ) (where S

+ and S are both replacedby Sx) does. It is straightforward to see (by symmetry or by directverication) that ++(q, ) = (q, ) 0, and hence

xx(q, ) = yy(q, ) =1

4

{+(q, ) + +(q, )

}.

The presence of two-site correlations inuences the thermal averageSz. A determination of the correction to the MF result (3.4.5b) forSz, leading to a self-consistent RPA result for the transverse suscepti-bility, requires a relation between Sz and the susceptibility functionsdeduced above. The spin commutator-relation, [S+i , S

i ] = 2S

z ii ,turns out to be satised identically, and thus leads to no additionalconditions. Instead we consider the Wortis expansion

Szi = S 12S

Si S+i

18S2(S 12 )

(Si )2(S+i )

2 (3.4.12)

for which the matrix elements between the p lowest single-spin (or MF)levels are correct, where p 2S+1 is the number of terms in the expan-sion. Using (3.4.11), we nd from the uctuationdissipation theorem(3.2.18):

Si S+i = 1Nq

S+(q, t = 0)

= 1N

q

1

11 eh

+(q, )d(h) = 2Sz,

(3.4.13a)with

= 1N

q

nq ; nq =1

eEq 1 , (3.4.13b)

where nq is the population factor for bosons of energy Eq. If S = 12 ,then Sz is determined by the two rst terms of (3.4.12), and

Sz = S Sz/S,or

Sz = S2/(S + ) 12 + 22 In general one may use a HartreeFock decoupling, (Si )2(S+i )2 2(Si S+i )2, of the higher-order terms in (3.4.13) in order to show that

Sz = S + (2S + 1)2S+1 S 1N

q

nq, (3.4.14)


where the kinematic correction, of the order 2S+1, due to the limitednumber of single-spin states, which is neglected in this expression, isunimportant when S 1. Utilizing the HartreeFock decoupling oncemore to write Szi Szj (i=j) Sz2 S2 2Sz, we nd the internalenergy to be

U = H = 12NJ (0)S2 +q

Eq nq

= 12NJ (0)S(S + 1) +q

Eq(nq + 12 ).(3.4.15)

The second form, expressing the eect of the zero-point motion, is de-rived using J (ii) = 1N

q J (q) 0.

The thermodynamic properties of the Heisenberg ferromagnet aredetermined by (3.4.10), (3.4.14), and (3.4.15), which are all valid at lowtemperatures. In a cubic crystal, the energy dispersion Eq is isotropicand proportional to q2 in the long wavelength limit, and (3.4.14) thenpredicts that the magnetization Sz decreases from its saturation valueas T 3/2. The specic heat is also found to be proportional to T 3/2. Thethermodynamic quantities have a very dierent temperature dependencefrom the exponential behaviour (3.4.5b) found in the MF approxima-tion. This is due to the presence of elementary excitations, which areeasily excited thermally in the long wavelength limit, since Eq 0when q 0 in the RPA. These normal modes, which are describedas spin waves, behave in most aspects (disregarding the kinematic ef-fects) as non-conserved Bose-particles, and they are therefore also calledmagnons.

We shall not present a detailed discussion of the low-temperatureproperties of the Heisenberg ferromagnet. Further details may be foundin, for instance, Marshall and Lovesey (1971), and a quite completetreatment is given by Tahir-Kheli (1976). The RPA model is correct atT = 0 where Sz = S, but as soon as the temperature is increased, themagnons start to interact with each other, giving rise to nite lifetimes,and the temperature dependence of the excitation energies is modied(or renormalized). The temperature dependence of Eq = Eq(T ) is re-sponsible for the leading order dynamic corrections to Sz and to theheat capacity. A more accurate calculation, which we will present inSection 5.2, adds an extra term to the dispersion:

Eq = Sz {J (0) J (q)}+ 1N

k

{J (k) J (k+ q)}nk, (3.4.16)

from which the heat capacity of this non-interacting Bose-gas can bedetermined as

C = U/T =q

Eq dnq/dT. (3.4.17)


We note that there are corrections to U , given by (3.4.15), of secondorder in . The low-temperature properties, as determined by (3.4.14),(3.4.16), and (3.4.17), agree with the systematic expansion performed byDyson (1956), including the leading-order dynamical correction of fourthpower in T (in the cubic case), except for a minor kinematic correctionwhich is negligible for S 1.

3.5 The random-phase approximation

Earlier in this chapter, we have demonstrated that many experimentallyobservable properties of solids can be expressed in terms of two-particlecorrelation functions. Hence it is of great importance to be able to cal-culate these, or the related Green functions, for realistic systems. Weshall therefore consider the determination of the generalized susceptibil-ity for rare earth magnets, using the random-phase approximation whichwas introduced in the last section, and conclude the chapter by apply-ing this theory to the simple Heisenberg model, in which the single-ionanisotropy is neglected.

3.5.1 The generalized susceptibility in the RPAThe starting point for the calculation of the generalized susceptibilityis the (eective) Hamiltonian for the angular momenta which, as usual,we write as a sum of single- and two-ion terms:

H =

i

HJ(Ji) 12i=j

J (ij)Ji Jj . (3.5.1)

For our present purposes, it is only necessary to specify the two-ionpart and, for simplicity, we consider only the Heisenberg interaction. Asin Section 2.2, we introduce the thermal expectation values Ji in theHamiltonian, which may then be written

H =

i

HMF(i) 12i=j

J (ij) (Ji Ji) (Jj Jj), (3.5.2)

where

HMF(i) = HJ(Ji)(Ji 12 Ji

) j

J (ij)Jj. (3.5.3)

From the mean-eld Hamiltonians HMF(i), we may calculate Ji asbefore. The Hamiltonian (3.5.3) also determines the dynamic suscepti-bility of the ith ion, in the form of a Cartesian tensor oi (), accordingto eqns (3.3.46), with A and B set equal to the angular-momentumcomponents Ji. We wish to calculate the linear response Ji(t) of

3.5 THE RANDOM-PHASE APPROXIMATION 155

the system to a small perturbative eld hj(t) = gBHj(t) (the Zeemanterm due to a stationary eld is taken as included in HJ(Ji) ). From(3.5.2), we may extract all terms depending on Ji and collect them inan eective Hamiltonian Hi , which determines the time-dependence ofJi. Transformed to the Heisenberg picture, this Hamiltonian is

Hi(t) = HMF(i, t)(Ji(t) Ji

) (j

J (ij)(Jj(t) Jj) + hi(t)).

(3.5.4)We note that a given site i appears twice in the second term of (3.5.2),and that the additional term Ji hi has no consequences in the limitwhen hi goes to zero. The dierences Jj(t) Jj(t) uctuate in a vir-tually uncorrelated manner from ion to ion, and their contribution tothe sum in (3.5.4) is therefore small. Thus, to a good approximation,these uctuations may be neglected, corresponding to replacing Jj(t)in (3.5.4) by Jj(t) (when j = i). This is just the random-phase ap-proximation (RPA), introduced in the previous section, and so calledon account of the assumption that Jj(t) Jj(t) may be described interms of a random phase-factor. It is clearly best justied when theuctuations are small, i.e. at low temperatures, and when many sitescontribute to the sum, i.e. in three-dimensional systems with long-rangeinteractions. The latter condition reects the fact that an increase in thenumber of (nearest) neighbours improves the resemblance of the sum in(3.5.4) to an ensemble average. If we introduce the RPA in eqn (3.5.4),the only dynamical variable which remains is Ji(t), and the Hamiltonianbecomes equivalent to HMF(i), except that the probing eld hi(t) is re-placed by an eective eld hei (t). With Ji() dened as the Fouriertransform of Ji(t) Ji, then, according to eqn (3.1.9),

Ji() = oi ()hei (),where the eective eld is

hei () = hi() +j

J (ij)Jj(). (3.5.5)

This may be compared with the response determined by the two-ionsusceptibility functions of the system, dened such that

Ji() =

j

(ij, )hj(). (3.5.6)

The two ways of writing the response should coincide for all hj(), whichimplies that, within the RPA,

(ij, ) = oi ()(ij +

jJ (ij)(jj, )

). (3.5.7)


This self-consistent equation may be solved under various conditions.For convenience, we shall consider here only the uniform case of a ferro-or paramagnet, where HMF(i) is the same for all the ions, i.e. Ji = Jand oi () =

o(), in which case we get the nal result

(q, ) ={1 o()J (q)}1 o(). (3.5.8)

Here 1 is the unit matrix, and we have used the Fourier transform (3.4.2)of J (ij)

J (q) =

j

J (ij) eiq(RiRj). (3.5.9)

In the RPA, the eects of the surrounding ions are accounted forby a time-dependent molecular eld, which self-consistently enhancesthe response of the isolated ions. The above results are derived from akind of hybrid MF-RPA theory, as the single-ion susceptibility oi () isstill determined in terms of the MF expectation values. A self-consistentRPA theory might be more accurate but, as we shall see, gives rise to fur-ther problems. At high temperatures (or close to a phase transition), thedescription of the dynamical behaviour obtained in the RPA is incom-plete, because the thermal uctuations introduce damping eects whichare not included. However, the static properties may still be describedfairly accurately by the above theory, because the MF approximation iscorrect to leading order in = 1/kBT .

The RPA, which determines the excitation spectrum of the many-body system to leading order in the two-ion interactions, is simple toderive and is of general utility. Historically, its applicability was ap-preciated only gradually, in parallel with the experimental study of avariety of systems, and results corresponding to eqn (3.5.8) were pre-sented independently several times in the literature in the early 1970s(Fulde and Perschel 1971, 1972; Haley and Erdos 1972; Purwins et al.1973; Holden and Buyers 1974). The approach to this problem in thelast three references is very similar, and we will now present it, followingmost closely the account given by Bak (1974).

We start by considering the MF Hamiltonian dened by (3.5.3). Thebasis in which HMF(i) is diagonal is denoted |i > ; = 0, 1, . . . , 2J ,and we assume that HMF(i) is the same for all the ions:

HMF(i)|i > = E |i >, (3.5.10)

with E independent of the site index i . The eigenvalue equation denesthe standard-basis operators

a(i) = |i >< i |, (3.5.11)


in terms of which HMF(i) =

Ea(i). Dening the matrix-elements

M = < i |Ji Ji|i >, (3.5.12)

we may writeJi Ji =

M a(i),

and hence

H =

i

E a(i) 12

ij

J (ij)M M a(i) a(j).

(3.5.13)We have expressedH in terms of the standard-basis operators, as we nowwish to consider the Green functions G,rs(ii, ) = a(i) ; ars (i).According to (3.3.14), their equations of motion are

h G,rs(ii, ) [ a(i) , H ] ; ars(i) = [ a(i) , ars(i) ] .(3.5.14)

The MF basis is orthonormal, and the commutators are

[ a(i) , ars(i) ] = ii{ras(i) sar(i)},

so we obtain

{h (E E)}G,rs(ii, )+j

J (ij)

{a(i)M a(i)M} M a(j) ; ars(i)

= iir as(i) s ar(i). (3.5.15)

In order to solve these equations, we make an RPA decoupling of thehigher-order Green functions:

a(i) a(j) ; ars(i)i=j a(i)a(j) ; ars(i) + a(j)a(i) ; ars(i).

(3.5.16)

This equation is correct in the limit where two-ion correlation eectscan be neglected, i.e. when the ensemble averages are determined by theMF Hamiltonian. The decoupling is equivalent to the approximationmade above, when Jj(t) in (3.5.4) was replaced by Jj(t). The thermalexpectation value of a single-ion quantity a(i) is independent of i,and to leading order it is determined by the MF Hamiltonian:

a a0 = 1Z Tr{eH(MF) a

}= n , (3.5.17)


and correspondingly J in (3.5.12) is assumed to take the MF value J0.Here Z is the partition function of the MF Hamiltonian, and thus n isthe population factor of the th MF level. With the two approximations(3.5.16) and (3.5.17), and the condition that

Ma(j)0 =

Jj Jj00 = 0 by denition, (3.5.15) is reduced to a closed set ofequations by a Fourier transformation:

{h (E E)}G,rs(q, )+

J (q)(n n)M M G,rs(q, ) = (n n) rs.

(3.5.18)We now show that these equations lead to the same result (3.5.8) asfound before. The susceptibility, expressed in terms of the Green func-tions, is

(q, ) = ,rs

MMrsG,rs(q, ). (3.5.19)

MMrs is the dyadic vector-product, with the ()-component givenby (MMrs) = (M)(Mrs) . Further, from eqns (3.3.46), theMF susceptibility is

o() =

E =E

MME E h (n n) +

E=E

MM n 0.

(3.5.20)Multiplying (3.5.18) by MMrs/(E E h), and summing over(, rs), we get (for = 0)

(q, ) o()J (q)(q, ) = o(), (3.5.21)

in accordance with (3.5.8). Special care must be taken in the case ofdegeneracy, E = E , due to the resulting singular behaviour of (3.5.18)around = 0. For = 0, G,rs(q, ) vanishes identically if E = E ,whereas G,rs(q, = 0) may be non-zero. The correct result, in thezero frequency limit, can be found by putting E E = in (3.5.18),so that n n = n(1 e) n. Dividing (3.5.18) by , andtaking the limit 0, we obtain in the degenerate case E = E:

G,rs(q, 0)

J (q)nM M G,rs(q, 0) = n r s.

(3.5.22)Since (q, ) does not depend on the specic choice of state-vectors inthe degenerate case, (3.5.22) must also apply for a single level, i.e. when = . It then follows that (3.5.18), when supplemented with (3.5.22),


ensures that (3.5.21) is also valid at = 0, as (3.5.22) accounts forthe elastic contributions due to o(), proportional to 0. This zero-frequency modication of the equations of motion was derived in thiscontext in a slightly dierent way by Lines (1974a).

Although eqns (3.5.18) and (3.5.22) only lead to the result (3.5.8),derived previously in a simpler manner, the equations of motion clarifymore precisely the approximations made, and they contain more infor-mation. They allow us to keep track in detail of the dierent transitionsbetween the MF levels, which may be an advantage when performing ac-tual calculations. Furthermore, the set of Green functions G,rs(q, )is complete, and hence any magnetic single- or two-ion response functionmay be expressed as a linear combination of these functions.

In the derivation of the RPA result, we utilized two approximateequations, (3.5.16) and (3.5.17). The two approximations are consistent,as both equations are correct if two-ion correlation eects are negligible.However, the RPA Green functions contain implicitly two-ion correla-tions and, according to (3.3.7), we have in the linear response theory:

a(i) ars(j) a(i)ars(j) =1

N

q

eiq(RiRj)1

11 eh G

,rs(q, )d(h),

(3.5.23)where, by the denition (3.2.11b),

G,rs(q, ) =12i

lim0+

{G,rs(q, + i)Grs,(q, + i)

}.

Equation (3.5.23), with i = j, might be expected to give a better esti-mate of the single-ion average a than that aorded by the MF ap-proximation used in (3.5.17). If this were indeed the case, the accuracy ofthe theory could be improved by using this equation, in a self-consistentfashion, instead of (3.5.17), and this improvement would maintain mostof the simplicity and general utility of the RPA theory. Unfortunately,such an improvement seems to occur only for the Heisenberg ferromagnetdiscussed previously, and the nearly-saturated anisotropic ferromagnet,which we will consider later. Equation (3.5.23) allows dierent choicesof the Green functions G,rs(q, ) for calculating a, and the resultsin general depend on this choice. Furthermore, (3.5.23) may lead tonon-zero values for a(i) ars(i), when = r, despite the fact that< i |ri >= 0 by denition. The two-ion correlation eects which areneglected by the RPA decoupling in (3.5.18) might be as important,when using eqn (3.5.23) with i = j, as those eects which are accountedfor by the RPA. Nevertheless, it might be possible that certain choices


of the Green functions, or a linear combination of them, would lead toan accurate determination of a (the most natural choice would be touse G0,0(q, ) ). However, a stringent justication of a specic choicewould require an analysis of the errors introduced by the RPA decou-pling. We conclude that a reliable improvement of the theory can onlybe obtained by a more accurate treatment of the higher-order Greenfunctions than that provided by the RPA. General programs for ac-complishing this have been developed, but they have only been carriedthrough in the simplest cases, and we reserve the discussion of theseanalyses to subsequent sections, where a number of specic systems areconsidered.

3.5.2 MF-RPA theory of the Heisenberg ferromagnetWe conclude this chapter by applying the RPA to the Heisenberg model,thereby demonstrating the relation between (3.5.8) and the results pre-sented in the previous section. In order to do this, we must calculate

o(). The eigenstates of the MF Hamiltonian (3.4.4b) are |Sz = M > ,with M = S,S + 1, , S, and we neglect the constant contributionto the eigenvalues

EM = MJ (0)Sz0 = M with = J (0)Sz0,

denoting the MF expectation-value (3.4.5a) of Sz by Sz0. Accordingto (3.3.4a), we then have (only terms with = M + 1 and = Mcontribute):

o+() =S1M=S

< M + 1 |S+ |M >< M |S |M + 1 >EM EM+1 h (nM+1 nM )

= 1Z

S1S

S(S + 1) M(M + 1) h

(e(M+1) eM

)

= 1 h

1

Z

( SS+1

{S(S + 1) (M 1)M

}eM

S1S

{S(S + 1) M(M + 1)

}eM

)

= 1 h

1

Z

SS

2MeM =2Sz0 h ,

as all the sums may be taken as extending from S to S. Similarly o+() =

o+(), whereas o++() = o() = 0, from which we


obtain

oxx() = oyy() =

1

4

{ o+() +

o+()

}=

Sz02 (h)2 , (3.5.24a)

and

oxy() = oyx() = i4{ o+() o+()

}=

ihSz02 (h)2 . (3.5.24b)

We note here that oxy() and oxy

(), obtained by replacing by + i and letting 0+, are both purely imaginary. Of the remainingcomponents in o(), only ozz() is non-zero, and it comprises only anelastic contribution

ozz() = (Sz)20, with (Sz)2 (Sz)20 Sz20. (3.5.25)

Because oz() = 0, the RPA equation (3.5.8) factorizes into a 2 2(xy)-matrix equation and a scalar equation for the zz-component. In-verting the (xy)-part of the matrix {1 o()J (q)}, we nd

xx(q, ) = oxx() | o()|J (q)

1 { oxx() + oyy()}J (q) + | o()|J 2(q),

where the determinant is

| o()| = oxx() oyy() oxy() oyx() =Sz20

2 (h)2 .

By a straightforward manipulation, this leads to

xx(q, ) =E0qSz0

(E0q)2 (h)2, (3.5.26a)

with

E0q = Sz0J (q) = Sz0{J (0) J (q)}. (3.5.26b)

The same result is obtained for yy(q, ). We note that (3.5.26a) shouldbe interpreted as

xx(q, ) =12 Sz0 lim0+

(1

E0q h ih+

1E0q + h + ih

).

This result is nearly the same as that deduced before, eqns (3.4.1011), except that the RPA expectation-value Sz is replaced by its MF


value Sz0, reecting the lack of self-consistency in this analysis. As asupplement to the previous results, we nd that

zz(q, ) = ozz()

1 ozz()J (q)=

(Sz)2

1 (Sz)2 J (q) 0, (3.5.27a)

and the corresponding correlation function is

Szz(q, ) = 2h(Sz)2

1 (Sz)2 J (q)(h). (3.5.27b)

The zz-response vanishes in the zero-temperature limit and, in this ap-proximation, it is completely elastic, since (Sz)2 is assumed indepen-dent of time. However, this assumption is violated by the dynamiccorrelation-eects due to the spin waves. For instance, the (n = 1)-sum-rule (3.3.18b) indicates that the second moment (h)2zz is non-zero,when q = 0 and T > 0, which is not consistent with a spectral functionproportional to (h).

Although this procedure leads to a less accurate analysis of theHeisenberg ferromagnet than that applied previously, it has the advan-tage that it is easily generalized, particularly by numerical methods, tomodels with single-ion anisotropy, i.e. where HJ(Ji) in (3.5.1) is non-zero. The simplicity of the RPA result (3.5.8), or of the more generalexpression (3.5.7), furthermore makes it suitable for application to com-plex systems. As argued above, its validity is limited to low tempera-tures in systems with relatively large coordination numbers. However,these limitations are frequently of less importance than the possibility ofmaking quantitative predictions of reasonable accuracy under realisticcircumstances. Its utility and eectiveness will be amply demonstratedin subsequent chapters.

linear theory

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