j fivez, b de raedt and h de raedt- low-temperature phonon dynamics of a classical compressible...

8/3/2019 J Fivez, B De Raedt and H De Raedt- Low-temperature phonon dynamics of a classical compressible Heisenberg chain

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J . Phys. C :Solid State Phy s., 14 (1981) 2923-2933. Printed in Gr ea t Britain

Low-temperature phonon dynamics of a classicalcompressible Heisenberg chainJ Fivez t$ , B D e RaedtO and H D e Raedt$$ Departem ent N atuurkunde, Universitaire Instelling Antwerpe n, B-2610 Wilrijk,Belgium0 Institut fur Festkorperforscbung der Kernforschungsanlage Julich, Postfach 1913,D-5170 Julich, W GermanyReceived 23 De cem ber 1980Abstract. Alo w-te mp eratu re calculationoftheinfluenceof theanharmonicity of the phononson the dynam ic structure factor is made for a compressible Heisenberg system. The anh ar-monicity is caused by the coupling of phonons and spins. The memory function of thedisplacement relaxation function is calculated to lowest order in the temperature. Onlytwo-magnon processes are found to contribute to the memory function in this limit. Thedisplacement relaxation function turns ou t to be very simple. T he collision processes canlead to a second resonance. In two limits, the results are compared with the results of acontinued fraction approximation. T he qualitative agreement is good.

1. IntroductionIn a previous paper (Fivez et a1 1980) we presented a detailed study of the phonondynamics in a compressible Heisenberg chain (CHC).As a starting point we took thecontinued fraction expansion of the displacement relaxation function. A four-poleapproximation was then used. H er e wc take u p the subject again from a different pointof view. W e use a met ho d, originally due to Reite r and Sjolander (R eiter and Sjola nde r1980, Reiter 1980), that yields exact results in the low-temperature limit. T he app roac his based o n a low -tem pera ture expansion of th e mem ory function, retaining only thelowest-ord er non-vanishing term . W e use a slightly different but equivalent formu lationof their approach .W e find it interesting to study the ph ono ns in the C H C again within this theor y for thefollowing reasons. First, the truncation approximation for the continued fraction thatwas used is based on th e assumption th at the mem ory function contains no slow processes(D e Raed t and D e Rae dt 1977). A t low temperatures however the memory functionturns ou t to have a very slow and oscillatory time dep end enc e and the approximationbecomes qu estionable. Therefore i t seems worthwhile to com pare the results of thefour-pole approx imation with the results of a low-tem peratue the ory (LTT). Second, thepres ent study shed s light on fea tures of th e L ~ Tha t were not present in the case of th erigid Heisen berg chain (RHC) Re iter et a1 1980). Th e memory function in th e LTT haspoles an d a cut-off. Th e crucial point is that for the RHC the spinwave peak is alwayssituate d below poles and cut-off. For the ph ono n resonance of the CHC this is not alwaystrue. We find a displacement relaxation function of strikingly simple functional formt Research assistant of the Belgian National Fund for Scientific Research (NF W O).0022-3719/81/212923+ 11$01.50 @ 1981 T he Institute of Physics 2923


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2924 J Fivez , B De Raedt and H De Raedtthat has a peculiar pa rame ter de penden ce. Thirdly, the theory not only de mon strateswhat kind of interaction processes are responsible for th e phon on relaxation to lowestorder in the temperature, but also how they cause the relaxation. In this way themechanism of the non-linear processes becom es mo re transp are nt. Last but not least,a t reatmen t of the dynamics of a coupled system tha t yields all frequency m om en ts exactup to first or de r in the te m pe ra tur e is interesting in its own right.T he plan of the p ape r is as follows. In 4 2 we define the mo del. Th e basic theo ry isthen given. The harmonic approximation of the model is discussed. In 3 we give theresults for th e mem ory fun ction. In 5 4 the results are discussed and we co mp are the mwith o ur previous wo rk. T he conclusions are summarised in 5.

2. Model and basic theoryThe compressible spin system we consider is described by the Hamiltonian (Fivez et a11980 and references qu ote d therein)

H = N p + H s + H S p (2 . la)(2.16)

(2. c )

(2 . I d )The spins are located on a harmonic lattice with an exchange interaction dependinglinearly o n th e atom-atom sep ara tion . W e consider classical spins of unit lengt h.The quantity we want to calculate is the equilibrium displacement correlationfunction

r xdt exp(izt) C ( q , ) ; z = w + i E , (2.2a)

whereC(q9 t>

( A B ) = ( (A- (4)( B - @)I),

( X q ( t ) , xq ) = ( x q> xp(-iLt) X J .He re we use the notat ion

an d the Liouville o pe ra tor is given byLA = [ H , A ]Following M ori (1965) we can write

(2 .2b )


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Classical compressible Heisenberg chain 2925where (w2), is the second frequency m omen t given by

(Lx,,xy)(xq>q)(U2 ) ,=

which can be calculated rigorously by mea ns of the transfer op era tor metho d (Fivez eta1 1980).In th e low -tem pera ture limit we have(w2), = 2 ( a / m )(1- cosq) (1- T ) ( 2 . 7 ~ )

(2.7b)( 2 . 7 ~ )

Z,(z) is th e mem ory function defined by2,(2) = - (Lx , , Lx,)-'QL2xq,2 - QLQ)- 'QL% J , (2 .8a)

where Q is th e projection o pe ra to r which projects o nt o the non-secular variables and isdefined by

In time space the m emory function readsq t ) = -(Lx,, X q ) - I M q ( t )

(2 .8b)

(2 .9a)M q ( t )= (QL 2xq , xp(- iQLQt) QL2x,). (2 .9b)

In principle the exact calculation of 2, is prohibitively complicated. However, asT + 0 2, vanishes proportional to T because the phonons decouple from the spinfluctuations and consequently the system becomes harmonic (Fivez et a1 198 0). If wecontent ourselves with a displacement c orrelation function th at has frequency m om ent scorrect up to first orde r in T , t follows tha t we can limit ourselves to t he calculation ofthe lowest-order term of equ ation (2 .9 ) . Re iter and Sjolander (1980) showed that inorde r to calculate this lowest-order term we can replace Q L Q in equ ations (2.8) and(2.9) by the normal time evolution ope rator L . M ore specifically, we can writeM q ( t )= ( Q L 2 x q , xp(-iLt)QL2x,) + O ( T 3 ) . (2.10)

Th eir argu me nt is formulated in time space. Th e sam e result can also be obta ined directlyin frequency spac e. DefiningA,(z) = - ( L x q , L x q ) - ' ( Q L 2 x q ,Z -L) - 'QL2Xq) ,

( z 2- w2),) C,(z) = ( z 2- (w2)q )A&) + zA,(z) X q ( 2 ) .(2.11)

(2.12)on e can derive the exact relation

Now both A, and 2, vanish proportional t o T and hence& ( z ) =A,(z) + 0 (T2 ) . (2.13)

Strictly speak ing, equa tion (2.13) only holds when the low est-order term of Aq( z )s nottoo large. How ever, if A&) has singularities at som e frequencies we can not claim tohave an exact result. T he result (2.13) is however exact to the extent tha t it reproduces


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2926 J Fiuez, B De Raedt and H De Raedtall the frequ ency mom ents of 2 , exact to lowest non-vanishing or der in T ,The advantageof equation (2.10) is tha t we can use the harmonic approximation for the evaluation ofthe low est-ord er term . This harmo nic approximation is based o n the Holstein-Primakofftransfo rm ation. A justification of the use of this transfor ma tion has been given by Rei ter(1980). Fo r th e ferromag net we w rite ( 2 . 1 4 ~ ~ )

(2.14b)s: = 1 - 4s;s:S = Sf t Sy

and after Fourier transform ation we findH = + E X 1 - cos q ) A x ~ A x - ~B 2 w(q ) SiS? , , (2.15a)

q 2m 4 9(2.15b)

4 q ) = 2 4 1 + r) (1 - cosq) , ( 2 . 1 5 ~ )and where the ground sta te energy has been om itted. Consequen tly,

S, i ( t ) = exp(Tiw(q)t) S: .(2.16)(2.17)

Fo r the antiferrom agnet we writes; = (- 1)' (1 - Bs;s;), (2.18)

combined with equation (2.14b). Again we find the H amiltonian (2.15a) but now wehave(2.19)4 2 i J / ( 1 + I4 1 + COS^).

It follows tha t now

where(2.20a)

(2.20b)Equ ation (2.16) remains unchange d. O ur general scheme to calculate equation (2.10)is the n as follows. U se the exact Liouville op era tor L to calculate L'x,. Then expandL2x, in the normal coordinates Axq ,S and Si using th e Holstein-Primakoff tran sfo r-mation and retain only linear and qu adratic terms. Th e Q operator then erases the termslinear in Axq .The n use th e harm onic time evolution and calculate the expectation valuesusing the form ulas (2.16) and (2 .15~ )r ( 2 . 1 9 ) .T o close this section , we briefly discuss anothe r pro cedu re. Following M ori (1965),we can equally well write instead of equ atio n (2.5)

( z - A:/[,? + T q ( z ) ]( q ,2 ) = C ( q , = 0) z - (2.21)


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Classical compres sible Heisenberg chain 2927

(2.22)can be calculated exactly (Fivez et al1980). T q( z) s now given by

T,(z) - (Aq ,AJ1(QL3xq , 2 - QLQ) - 'QL3xJ , ( 2 . 2 3 ~ )where

A , = L'x, - ( o ~ ) ~q , (2.23b)and Q is defined by

As A i cc T in the low -temperature limit we only need the T = 0 result for Tq(z) n ord erto get the sam e kind of approximation for C(q,z ) as befo re. In Tq(z),owe ver, we maynot replace QLQ by L to lowest ord er in T . This difficulty can be circum vented in thefollowing w ay.Definingyq = - (A, ,Aq)- ' (QL3x,, ( Z - L ) - ' QL3xq), (2.24)

on e has the exact relation

For T = 0 this becomes(2.25)

(2.26)where yq may again be calculated in the harmonic approx imation. Th e same scheme asfor the calculation of equatio n (2.10) can then be use d.We will show that both procedures lead to the same result for the correlationfunctions, show ing the consistency of the theo ry.3. The memory functionLe t us first focus on equ ation (2.10 ). Consider th e ferrom agn et. Following ou r generalsche me , we find

This yields, using eq ua tion s (2.14-17)42 2 q T 2m2 2 j 2 ( 1 + y)'(QL2xq, -'L'QL2x) = -sin -

1x -N k exp[i(w(k - qi2) - o ( k + qi2))tI


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2928 J F iuez , B D e R a ed t and H D e R a ed tUsing the form ula

1lim -2 cos(a sin k t ) = Jo(at) ,N + ~ Nwhere Jo s the B essel function of ord er zero , on e findsc,(t)= -A $ Jo(a t ) ,

(3.3)

(3.4)and consequently

(3.5),(z) = iA$/ (a2- z2)ll2.Here a is defined by

a ( q ) = 4J( 1+ y ) I sin q/2 1, (3.6)Azis defined in eq uatio n (2.22) and use has been m ade of the low-temp erature result

2am$ = - ( l - c o ~ q ) r T + O ( T ~ ) . (3.7)

Fo r the antiferromagnet we must use equations (2.18-20) but the result turns out to bethe same, when J and y are replaced by their absolute value. Remark that X 4 ( t ) isessentially t he self-correlation function of the spin energy density at zero tem pera ture(D e Rae dt and D e R aedt 1980). This is so because in the Ham iltonian (2.1) the displace-me nts are coupled with the spin energy.Now the energy relaxation at T = 0 is independent of the sign of the spin-spininteraction (De Raedt and De Raedt 1980). However, for f inite T the spin energyrelaxation function is different for ferro- and antiferromagnets (D e R aed t et a1 1981).Con seque ntly, the indepen dence of the p hono n function of the sign of J will not persistat higher temperatu res.Now consider equation (2.21). The n we have to calculate (2.26). Ta ke again theferro m agn et. Following the general scheme we find

x [cos(k+ q/2) - cos@ - q/2)]. (3 8)Making use of equ ation s (2.14-17), equ ation (2.24) and the integral representation ofthe Bessel functions o ne finds

iz 2 (3.9)q(4 = ( a 2 - 2)l /2 + 2 .Com bining equ ation s (2.21), (2.26) and (3.9) we finally obtain

(3.10)

Though the calculation uses equa tion s (2.18-20), again the same results (3.9) and (3.10)are recovered for the antiferrom agnet. Comparing e quation (3.10) with (2.5) combinedwith (3.5 ), we see th at the app roac hes (2.5) and (2.21) lead to exactly the same result.It is possible t o pro ve generally the equivalence of both approaches within the scheme


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Classical compressible Heisenberg chain 2929for the calculat ion of the m emo ry functions. This proves that the ha rmon ic treatme ntleads to consistent results.

4. DiscussionT he mem ory function given by equation ( 3 . 5 )diverges fo r z = a (equation ( 3 . 6 ) ) .Thisis a conse que nce of a divergence in the two-magnon density of states

This is easily unders tood as follows. Following e qua tion ( 3 . 2 ) , wo processes contribu teto t he mem ory function. In the f irst process a magnon of wavevector k - qi2 is abso rbedand a m agnon with wavevector k + qi2 is em itted . Ano the r possibility is the decay of amagnon k + qi2 into a magnon k - qi2. T h e possible processes are sk etch ed in figure 1.At t he k vector for which n diverges th ere a re infinitely many possible processes w ith thesame energy gain or loss Q- and consequently the relaxation is very large at thisf requency L2-.Th e qua ntity of inte rest is the dynamic struc ture factor C q ( w ) ,which is minus theimaginary part of C ( q ,2) an d is given by

Phonon absorpt ion

Phonon radiat ionFigure 1.Th e interaction processes that are responsible for the phonon dam ping. o enotesthe magnon frequency. Q , q respectively de note the energy and wavevector gain or loss inthe process. There a re absorption and radiation processes.


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2930 J Fivez , B D e Raedt and H De Raedt

where C, has be en normalised such that1Im,(w) do) = 1.Jd -- m

2; and are the real and imaginary part of Zq(w + i ~ )E-+ 0). From equat ion (3.5)c ; ( w ) = 0; O < o < a ,C; (w ) = -A$/(w2 - a2)2; CO > a ,X i ( o ) = A;i(a - w); 0 < w < U ,q w > 0; w 2 .

( 4 . 5a )( 4 . 5 b )( 4 . 5 d )(4 .5c )

Rem ark that we ch ose the sign of the square root such thatZI2 0 , (4 .6a)2 6 0 for w > a. (4.6b)

R ela tio n ( 4 . 6 ~ )s the damp ing condition and equ ation ( 4 . 5 b ) follows from it throughth e Kramers-Kronig relations. Following equ atio ns ( 4 . 5 )we must distinguish two case s,Iw( a . For IwI d a we have to combine equations (4.3) and ( 4 . 5 2 , c ) . ForI w (> a equat ions ( 4 . 3 ) and (4.5b, d ) giveC , (W ) = n(wZ), , IC S(w - mi)/W J > Q (4 .7a)

wh ere we definedf(w) = w2- (d),wx;(o), ( 4 . 7b )

and wi are th e zeros of f ( o ) .Defining

measuring th e freque ncy in units of a and the structure factor in units of l i a we find

Equ ation (4.9) dep ends only on two parameters , f and B . Neither B nor Fdepends onthe wavevector q . He nc e the scaled structure factor (4.9) is indep end ent of q . This veryimp ortant result is du e to th e coincidence that the cut-off frequency a ( q ) of th e collisionprocesses has th e same q-dep endence as the phon on dispersion. E qua tion (4.9) must beinterpreted as follows. Phonon processes of wavevector q with a frequ ency lower tha na(q ) ar e dam pe d by two-magnon processes. All processes with a freque ncy higher thana ( q ) are un dam ped bec ause the frequ ency does not fit the energy gain or loss of an ypossible two-magnon process of figure 1. Indeed, a ( q ) is the m aximum of Q - defined inequation (4 .2 ) .


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Classical compressible Heisenberg chain 293 1In the discussion of equ atio n (4 .9 ) , the par t of th e spectrum below o = 1 causes nodifficulties. Th e higher part how ever, requ ires a knowledge of the zerosoff( w ) quat ion( 4 . 7 b ) ,which are the solutions of a third-o rder equatio n. Let us distinguish betweenthr ee different regimes depending on the value of B.

4.1.B > 1H er e th e ph on on frequ ency is sufficiently lower than th e highest frequency of a collisionprocess. H er e the s tructu re facto r (4.9) essentially becomes a pair of L oren tz lines at thefrequencies

U = k 1IB (4.10)with a half width a t half maximum given by

T2B(B2 - ) 2 A, = (4.11)

Figure 2. The dynamic structure factor in the two different regimes: B = 2 (-) an dB = 213 (-- -). Th e scaling factor a (q ) is given by equa tion ( 3 . 6 ) .Th e scaled structure factoris indepe ndent of q . Fo r B = 213 the d elta function has a weight of approximately 80 % of thetotal area. F = 0.1.

Th ere is an additional pea k of very low weight at w = 1.T he part of the spectrum forw > 1 s of negligible weight. Figu re 2 shows a typical plot of t he s tructu re facto r. Thedepen dence o n T is not show n. Its only effect is to change the width of the Lo rentz linesand th e weight of the small peak at o = 1.


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2932 J Fivez , B De Raedt and H D e R aed t4 .2 . B < 1Now th e unda mp ed p art of th e spectrum is dom inan t. It consists of a pair of delta pea ksat

(4.12)In ad dition , the re ar e resonances of low weight at w = +1.T he relative weight of theseresonances is given by

( 4 . 1 3 ~ )= X / ( l - ) ,where x is defined a s

x = F(1- B 2 / 2 ) / ( 1 B2)32, (4.13b)Although this is relatively u nim porta nt it is detectab le because t he d om inan t pea ks arevery narrow and hence the second resonances are not lost in the tails of the mainresonanc e. Figure 2 also depicts the structure factor in this regime. Again th e Td epe nd -ence is om itte d. It merely affects the weight of the second resonanc e.4 .3 . B = 1In this case th e pho non frequency is close to the cut-off frequency and the whole w eightof th e spe ctrum is now concentrated in a narrow region arou nd the cut-off. Fo r B z 1the lines are no longer Lorentzian but have an asymmetric shape and the undampedpart of th e spectrum is no longer negligible. Fo r B = 1 the separation between the cut-off and the delta function is propo rtiona l to T23.W e now com pare ou r results with those fou nd previously using a continued fractionapproach (Fivez et a1 1980). Fo r this aim, co nsider the results of th e continued frac tionmethod for T = 0. A t low tem pera tures narrow lines are observed and ther efore it issufficient to examine the sh ap e near th e peak positions. T hen th e memory function maybe evaluated a t w = (w2)l*.For B % 1we find

2 = 0, ( 4 . 1 4 ~ )

(4.14b)Comp aring this with eq uatio ns (4.5a, c) we see th at th e only difference is that Eis&times larg er. Consequently, the continued fraction gives a Lorentzian line at the sa meposition, ( w ~ ) ~ ~nshifted, but with a width fi imes as large. For B 6 1 the resultreads

2 = -artiB, ( 4 . 1 5 ~ )2l= 0. (4.15b)Comp aring with equ ation s (4.5b , d ) ,we se e that t he results a re identical. In addition tothe line at the phonon frequency the continued fraction produces a central peak forB 4 . This peak is to be co mp ared with the pea k at w = 16 B in the present theo ry. Ofcou rse, the four-pole approximation necessarily smears out such peaks leading to theobserved central pe ak. F or B = 1no agreement is found between the continued fractionand the p resen t results.


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Classical compressible Heisenberg chain 29335. ConclusionsT he phon on relaxation function of a compressible Heisenberg chain at low temp eratu reswas studied . Fo r this purpose th e lowest-order contribution to the mem ory function wascalculated using a prescription t ha t was shown to be co nsistent. Th e results are exact tothe exte nt that all frequency m oments of the correlation function are given rigorouslyup to o rder T . Scaling the frequency and intensity, the structure factor was found to beinde pen den t of th e wavevector. For some para me ter values a second resonance besidesthe usual pho non peak was observed . This is du e to a divergency in the two-magnondensity of sta tes .O ur mo del is confined to nearest-neighbour interactions. Th e proced ure to calculatethe mem ory function is equally well applicable if further-neighbour interactions areincluded. Obviously the dispersions and hence the divergencies in the two-magnondensity of sta tes would ch ange, bu t qualitatively we d o not expect much difference. Apractical difficulty would be the exact calculation of the second frequency moment( C O ~ ) ~p to first order in the tem perature, because the transfer op erator m ethod can onlybe applied to the nearest-neighbour case.Le t us also discuss what will hap pen in two o r three dimensions. In o ne dimensionthe mem ory function ha s inverse sq uare ro ot divergencies, because it is essentially givenby the two-magnon density of state s. In two dimensions we expect at most logarithmicdivergencies. In three dimensions the mem ory function remains finite but it has a sha rpstruc ture with discontinuous derivatives.

ReferencesDe Raedt B and De Raedt H 1980Phys. Reu. B 21 4108De Raedt H and De Raedt B 1977 Phys. Reu. B 15 5379De Raedt H , Fivez J and De Raedt B 1981Phys. Reu. B in pressFivez J , De Raedt H and D e Raedt B 1980Phys. Reu. B 21 5330Mori H 1965 Progr. Theor. Phys. 34 399Reiter G 1980 Phys. Reu. B 21 5356Reiter G and S jolander A 1980J. Phys. C: Solid State Phys. 13 3027

j fivez, b de raedt and h de raedt- low-temperature phonon dynamics of a classical compressible...

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