Path integral theory of anharmonic crystals

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J. Phys. C : Solid State Phys., Vol. 6 , 1973. Printed in Great Britain. @ 1973

Path integral theory of anharmonic crystals

V SAMATHIYAKANIT? and H R GLYDEt Physics Department, Faculty of Science, Chulalongkorn University, Bangkok 5, Thailand

Received 21 August 1972, in revised form 9 January 1973

Abstract. The path integral method of quantum mechanics introduced by Feynman is used to derive the recently developed selfconsistent theory of anharmonic lattice dynamics. The method allows a sinple and rigorous derivation of the theory to all orders. The simplicity is obtained from integral representation of the partition function while the rigour is obtained from use of a general extremum principle rather than reliance on a variational principle. The first and second orders are derived explicitly including a derivation of the phonon frequencies via the method in each order.

Close comparisons are made with the previous derivations by Choquard, Horner and Werthamer and the Green function expansion, which underline the differences in method and result.

1. Introduction

Until the middle sixties, all treatments of the anharmonic character of atomic vibrations in solids had used standard perturbation theory. In this method, the interatomic potential 4 is expanded in a power series in the atomic displacements, q5 = 4,, + bz + & + $4 . . , The quadratic term & is then treated exactly in a harmonic approximation and the lowest order anharmonic terms & and qj4 are retained as perturbation corrections. This work has been reviewed by Cowley (1963,1968) and more recently for insulators by Glyde and Klein (1971).

In highly anharmonic crystals, such as the rare gas crystals, the vibrational amplitudes are sufficiently large that the above power series converges too slowly to be useful. As a result, a selfconsistent (sc) theory of lattice dynamics has been developed in which all the anharmonic terms in the power series are retained. Alternate expansions or variational methods are then used to derive expressions for the thermodynamic properties and phonon frequencies. The variational methods (Hooton 1958, Boccara and Sarma 1965, Koehler 1966, Gillis et al 1968) provide the simplest derivation of the lowest order, selfconsistent harmonic (SCH) theory. However, these variational treatments are strictly restricted to lowest order since there is no guarantee that the higher order approximations are an upper bound to the exact energy or free energy.

Choquard (1965, 1967) derived the higher order theory using a diagram cumulant expansion of the partition function and a resummation procedure. Horner (1967) used the equation of motion method coupled with an extremum principle for the free energy

t Associate Member of the International Centre for Theoretical Physics (ICTP), Trieste, Italy. $ On leave from Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada.

1166

Path integral theory of anharmonic crystals 1167

in the presence of external forces to obtain higher orders. Similarly, for the T = OK limit, Koehler (1968) employed an expansion of the above power series in Hermite polynomials to obtain higher orders. Introducing a time dependent, model harmonic action, Werthamer (1969, 1970) employed a variational principle to obtain an approxi- mate second order theory. Finally, the higher order selfconsistent theory can be derived using the Green function and free energy expansion common to standard perturbation treatment (Glyde 1971, Glyde and Klein 1971).

In the present paper we present a simple and rigorous derivation of the selfconsistent theory of lattice dynamics employing the Feynman path integral method. Following the spirit of the Feynman method, the derivation opens by introducing a trial harmonic action, So. The exact partition function Z is then expanded in cumulants about the corresponding trial harmonic partition function Z,. Successive orders of the self- consistent theory are then obtained by keeping successive cumulants and requiring that their contribution to the crystal dynamics vanishes. Taking advantage of the gaussian averaging appearing in the Feynman integral, the SCH theory can be obtained as easily as in the variational methods and higher orders can be obtained rigorously with little additional effort. The method provides a description of both the vibrational dynamics and thermodynamics.

The paper can also be regarded as an application of the Feynman path integral method to a new area. Since the original formulation (Feynman 1948), this method has been applied most successfully to several problems such as quantum electrodynamics (Feynman 1950), liquid helium (Feynman 1953), polarons (Feynman 1955), and disordered systems (Edwards and Gulyaev 1964). A generalization of the Feynman path integral method to the functional integral method has been applied successfully to the problems such as superconductivity (Muhlschlegel 1962), the Ising model (Muhlschlegel and Zittartz 1963), the Anderson model (Wang et a1 1969) and the Hubbard model (Cyrot 1970). Since this method has not yet been used to discuss nuclear vibration, a short derivation of the path integral representation of the partition function suitable for lattice dynamics in crystals is presented in Appendix 1.

In $2.1, the action and model action for the crystal are introduced. In $2.2 the general condition for choosing the selfconsistent force constants is derived and in $ 2.3 and $ 2.5 the lowest and second order selfconsistent partition functions and atomic force constants are derived. The phonon frequencies in each order are obtained in $2.4 and $ 2.6. The results are discussed and compared with other presentations in 0 3.

2. The selfconsistent theory

2.1. lhe model action and partition function

We consider a crystal of N identical atoms containing one atom per unit cell. Assuming a unique potential V(rl, r 2 , . . . , rN) depending upon the nuclear coordinates vi only, the hamiltonian for the nuclear motion is

where u,(l) = r,(l) - R,(l) denotes the displacement of atom I along the c( axis from its lattice point R(1).

1168 V Sarnathiyakanit and H R Glyde

In the Feynman formulation (see Appendix l), the partition function is given by

Z = dNri @(ri(z)) exp S s j where @(ri(z)) denotes that the integration is over all possible paths and the action S = - j EH(z) dz corresponding to (1) is

To describe the crystal by a harmonic approximation, we introduce the model action

Here the model harmonic force constants @ (U', z - 7') are explicitly time dependent to include the possibility of phonons with a finite lifetime. On writing exp S = exp So x exp (S - So) and introducing the model harmonic partition function,

:p

Z , = dNri 9' ( r i (z) ) exp So ( 5 ) s s we may write (2) as

Z = Z , (exp (S - So)) ( 6 ) where

( 0 ) = Z ; 1 d"ri j @'(ri(z)) exp S,O. (7)

On expanding (exp (S - So)) in cumulants,

z = Z , exp [(S - s,) + +{((s - sol2) - ( S - So)') + . . .]. (8)

From (3) and (4)

where summation over repeated indices is implied.

2.2. The selfconsistent force constants

To obtain the general condition for choosing the selfconsistent force constants, we introduce the free energy function W = In Z. From (6) this may be written as a sum of harmonic and anharmonic parts

W = In Z = WO({@}) + AW({@}, ( B } ) . (10)

Regarding M and V ( [ r i } ) as given, WO (exp(S - So)) is a function of @ and the correlation function

In 2, is a function of @ only and A W = In

B,&r ; z - 7 ' ) = (U& 7) up(l', 2')). (11)

Path integral theory of anharmonic crystals 1169

Dropping subscripts and time variables, variation of 0 and B gives

From (5) 6 Wo/6@ = - B and 6 Wo/6B = 0 so that

On introducing the function R = W + BQ,, which amounts to a Legendre transforma- tion, (13) gives

6(W +BO) = 6R = 0 6 B + {gg) 6 0 + (1%) 6 B )

so that

If we had truncated (8) at the harmonic term (AW not included), then (15) tells us that the harmonic force constants are given by (6R/6B),. At this stage, however, O is simply an arbitrary parameter and there is no condition for choosing 0. We now choose Q, by including the anharmonic contribution to the free energy A W and requring that the contribution from it to the force constants vanishes, that is from (15) that

pg)* = 0 .

This will mean that 0 is chosen so that the anharmonic contribution to the force constants is already included in the effective harmonic approximation.

The condition (16) is exactly equivalent to choosing the selfconsistent field by requiring that W is an extremum with respect to B. That is, if (16) holds then

This equivalence also follows directly from (12) since (6AW/6@) = B and this last relation and (12) shows that W is not really a function of Q, at all. The O dependence was artificially introduced via the model So--an artefact useful in seeking a model harmonic approximation. (17) is a classical analogue of the extremum condition discussed by Dominicis and Martin (1964). For the present purpose (15) is useful since it displays that (6AW/6B) corresponds to the contribution to the dynamics arising from the free energy contribution AW.

The force constants for any order can now be derived by retaining AW up to that order in (8) and using (16).

2.3. f i e first order or selfconsistent harmonic approximation

In first order, we retain only exp (S - So) in (8) so that

AW1 = ( S - So)

1170 V Samathiyakanit and H R Glyde

On expanding V ( r l , . . . , rN) in the usual Taylor series of the displacements u,(l) about the lattice points R(I)

<v({ r i ( T 1 ) ) ) ) = ( ~ X P {uz(l> ~ 1 ) Vci(4)) V ( { R i } )

= exp {i&W’, z1 - zl) V,(4 vp(U) V { R l } ) . (19)

Here V,(l) 3 i?/?u,(l) and V ( { R i } ) is the potential with each atom fixed at its lattice point. The second equality in (19) is obtained by expanding (exp {u,(l, z) V,(l)}) in cumulants and using the fact that, for gaussian averaging such as appears in (7), only the second cumulant is nonzero (Kubo 1962).

Using the expression (19) for (V({Y, ( T , ) } ) ) we have

The condition (16) thus gives via (18) and (20)

or

@ @ ( I / ’ , z - 7 ‘ ) = 6(z - z’) (V#)v/j(r) v({ri(z)})). (21) Equation (21) is the usual expression for the SCH force constants which are explicitly time independent.

2.4. ?he SCHphonon frequencies

The SCH phonon frequencies can be obtained from (21) following the usual harmonic theory of Born and Huang (1954). However, to follow the path integral formulation and to prepare for higher orders, we obtain the equation of motion for U,(/, z) by variation of the model action So. This leads to

To solve this, we introduce the usual transformation to the normal oc.iordinates Qn(q, z)

The equation of m o t h then becomes

Q j.( q, z) = j: @( qR, z - z’) Qn( q, z’) dz’ (24,

where

@(qA, z - z’) = E , ( q A ) ~ exp [iq . (R(1) - R(I’)}] @,&I/’. T - z’) E a ( q A ) . (25)

The term in large parentheses in (25) is a slight generalization of the dynamic matrix D of Born and Huang (1954) and the eigenvectors E, (qA) are always chosen to diagonalize D.

) 1 ab ( N M 11’

Path integral theory of anharmonic crystals 1171

Assuming an oscillatory time dependence for Qn(q, z) in (24) and introducing (21), the SCH frequency for phonon with wavevector q and branch A is

w i n = E(&) 2‘ (exp [iq . { N O ) - R(I)}] - 1) <v(O) VU) v ( (~4) ) ~ ( q 4 . (26) [’ 1 1 On Fourier transforming V ( { R i + q}), the expectation value in (26) can be expressed

as

(V(0) V ( 4 v ( { r i } ) > = V(0) V ( 0 S. - . Sd{k} (exp {ik(l) - 4O)) V({k))

= V(0) V(I) f. . , [d{k) exp { -qk(l). B(11’). k(l‘)} V({k}) (27)

which, on transforming V({k}) back to V ( ( r } ) gives

where on using (23)

Commencing, say, with an assumed value of B, equations (26), (28) and (29) are then iterated to obtain the SCH frequencies (Koehler 1966, Gillis et a1 1968).

2.5. 7he second order theory

In second order, we retain the first two cumulants in (8) so that

AW = Awl + AWz = (S - So) + % { ( ( S - So)’) - ( S - So)’} . (30) In using these conditions to obtain 0, it is convenient to break down the expectation values appearing in the product ( ( S - So)’) into averages over single functions. This can be readily done using the properties of the averaging in (7). That is, since (7) is a gaussian average, a pairing theorem holds and as a result only the second term in a cumulant expansion of the displacements u,(l) is nonzero. We may also take advantage of the fact that the u,(l) are classical numbers. The first two terms of ( ( S - So)*) are broken down in Appendix 2 as examples.

Breaking ((S - So)z) in this way and making the cancellations in ( ( S - So)’) - ( S - So)’ gives

( S - s o > + it<@ - So)’> - ( S - S0)’I

= - job d71[(v({ri(T1)})> -3 d~’@(Il’, 712) B(ll’9 71211

1172 V Samathiyakanit and H R Glyde

B B P B x ('(0 ' (1 ' ) V ( { r i ( z l ) ) ) ) @(pk, 234) + $ C 5 S { dT1 d ~ , dz3 dt,

X B ( ~ P , 713) B(l 'k T24) @ ( l P , 212) @(l'k t,,)]. 0 0 0 0

(31) Here, we have deleted the component indices on u,(l) etc and abreviated V,(l) to V(/). The prime is introduced on V'CQ to denote that it operates on V({vi(z)}).

The expression is exact and can be differentiated with respect to B(1l'. z - z') to obtain CD(ll', T - T') to second order. In all previous treatments, however, the @ in the second cumulant is approximated by (21). the first order form. This approximation is introduced implicitly in the perturbation nature of the previous cumulant (Choquard 1967, Horner 1967) and Green function expansion methods (Glyde 1971) and explicitly in the variational treatment due to Werthamer (1969, 1970). The approximation has the distinct advantage that the last two terms in (31) will then cancel with the second term in the expansion of

(exp {B(!l'. 213) V(I) V'(/')} - 1) = B(lP, T ~ ~ ) V(l) V'(!') + +(B(Ll', z l 3) V(l) V'(lr))2 + . . . . Making this approximation, and recognizing that (V(1) V({r i ( z ) } ) ) 5 0, (31) reduces to

- COB dzI[<V({ri(Z1)})) - 3 d 7 2 ~ ( l r > z12)~(1~' , 71211 + +i

x (V(Ir,(z I)>)) < V({4(T3)})) (32) where

Differentiating (32) with respect to B(11', T - z') and noting that from the definition (33)

From (35) we see that the second cumulant contribution has two distinct types of terms. On expansion of Os, these are the series of terms (V,) (V,) + (V,) (V,) + . . . and ( V 3 ) (V,) + (V,) (V,) + . . . . In view of the delta function 6 ( ~ - 7') the second set make a purely real contribution to CD. The full expression for CD, obtained without introducing @(U, Z,T& = 6 ( ~ , - zb) (V(1) V(1') V((ri(s)})) into the second cumulant, is written out in Appendix 2. It differs only in that the cancellation which removes the derivative of V({ri(z)}) below the cubic cannot be made. If the scheme is useful, that is if the higher order approximations to CD do not differ markedly from the first, in 4 2.3, then the approximation leading from (31) to (32) should not be a restrictive one.

Path integral theory of anharmonic crystals 1173

2.6. %e second order phonon frequencies

To obtain the phonon frequencies to second order we again use the equation of motion (22) and the transformation (23) leading to equation (24). Since in second order CD depends on the complex time T, we now introduce an oscillatory time dependence for QL(q, z) of

Q L ( q , TI = Q L ( q , O ) ~ X P ( - i ~ n ~ ) (36)

in which on is in general complex. Substituting (36) in (24) gives

(-io,)’ = lop d(z - T‘) @(qA, z - z’) exp {iw,(z - T’)} = CD(qA, iwJ. (37)

The frequency and lifetime for phonon with wavevector q and branch.2 is then given by continuing io, to the real axis following the prescription io, L= R + ie (see Abrikosov et al 1963) and taking the real and imaginary parts of CD(qA, SZ + ie).

Using the definition of @(qA, icon) in terms of (35), the expression (37) is exactly the same as that obtained following the Green function expansion method (see Appendix B of Glyde 1971). In the Green function method (D(q2, iw,) is the total self energy to second order and the phonon frequency is given by the position of the pole in the corresponding Green function. The present result can be expressed in a similar form by expanding B(ll‘,z - z’) in terms of a response function.

In a Peierls-like ordering of anharmonic terms, the lowest order term in the second order part of (35) is the (V,)( V,) term. On introducing the transformation ua(l, z) = Ta,qj, QL(q, z) defined by equation (23) and

(40) Apart from the work of Goldman et al (1970), this is the only term so far included in calculations of the phonon frequency spectrum. It is generally included as a perturba- tion to the first order SCH frequencies (eg Koehler 1969) although Horner (1972) has included it in an iteration scheme with the first order term.

F7

1174 V Samathiyakanit and H R Glyde

3. Discussion

In the present derivation of the selfconsistent theory of lattice dynamics, we expand the partition function Z in cumulants of ( S - S o ) where So is the model harmonic action. The contribution to the force constants (self energy) (D arising from the cumulants is obtained by differentiating them with respect to B (see equation 15). The CD is then chosen so that this contribution vanishes-which corresponds to choosing CD in S o such that the contributions from S and So are equal.

This procedure is similar in spirit to Choquard’s. He first makes a cumulant expansion. For each successive order he next constructs a ring diagram which is formed from each successive cumulant by breaking displacement correlations or “links’. This breaking of links is identical to differentiating the cumulant with respect to B(ZZ’, z - 7’) (see Choquard 1967, p 97) and represents the procedure for going from the free energy to the dynamics. A model dynamical matrix (see equation 19) rather than a force constant is introduced (Choquard 1967, p 86) and this model is determined so that successive pertur- bation ring diagram corrections to it vanish. In calculating subsequent perturbation corrections, the dynamical matrix of the previous order is used. Thus the approximation equivalent to introducing the previous order force constants such as in going from (31) to (32) is always made.

Horner (1967) employs functional derivatives of the free energy to generate the Green functions and the equation of motion method. External forces (first one-point forces, then pair forces and so on) then are successively introduced and incorporated into a new, enlarged free energy via a Legendre transformation. By coupling the equation of motion with derivatives of the enlarged free energy, successive orders of the self energy are obtained by varying this free energy until the equilibrium contribution from the external forces vanishes. The extremum principle used in this variation is exactly the one discussed by Dominicus and Martin (1964) and is, in principle, the same as the one used here. However, Horner’s derivation differs in the introduction of external forces and in the use of the equation of motion to introduce the self energy.

The advantage of the present method is simplicity, the explicit exposure of the time dependence of each term entering @(U’, z - 7’) and the straightforward generalization to an arbitrary rather than two-body potential.

The concept of a model action combined with a variational treatment was employed first by Feynman (1955) to treat polarons. This procedure was introduced into lattice dynamics by Werthamer (1969) and we have made use of many ideas set out by him. Werthamer, however, preferred to retain an operator expression for the partition func- tion and use operator calculus rather than an integral expression and classical numbers.

Werthamer then obtains his model force constants CD by minimizing the free energy of the crystal when successive order cumulant corrections are retained. This procedure, however, is strictly correct for the lowest order cumulant only since there is no guaran- tee that the model free energy is an upper bound beyond first order. In fact, to obtain the second order, Werthamer minimized the second order cumulant independently by differentiating it with respect to the force constants and setting the result to zero (using the general argument this correction should be small). This procedure turns out to be algebraically equivalent to setting the correction to zero and differentiating with respect to B as done here. Hence his result does not actually depend on the second order approxi- mation being an upper bound to the real free energy. In his treatment only the first term in the brackets of (35) is included since it is the dominant one in second order. Algebraically, the second term is omitted since Werthamer uses the difference coordin-

Path integral theory of anharmonic crystals 1175

ates U&). In this system two types of correlations enter Di,(. - z') = (uijuij) and D,, = ( y j u k L ) and both must be differentiated to obtain both terms.

In 3 2.6 we noted that the expression for the second order force constants is exactly the same as that obtained by a Green function expansion. The Green function method has the advantage that the result is expressed in precisely the form needed for comparison with a scattering experiment. Here, however, the advantage is that 2 and @ are obtained simultaneously.

Finally, we note that since the integral method is done in configuration space and uses first quantization, the method may offer a new way to combine a treatment of short range correlations and lattice dynamics simultaneously. A consistent combination of these two features which is computationally realizable remains an important outstanding problem in solid helium (Koehler and Werthamer 1971, Glyde and Khana 1972). Ex- change can also be included here by including a permutation of particles as discussed in Appendix 1. In view of the recent anomalous neutron scattering results for 4He (Osgood et a1 1972) exchange may be important although Horner (1972) has provided an explana- tion of the anomalous behaviour in terms of nuclear vibration only.

Acknowledgments

HRG acknowledges the kind hospitality of Chulalongkorn University where this work was done. VS gratefully acknowledges the support of UNESCO and IAEA via the International Centre for Theoretical Physics (ICTP), Trieste. Valuable discussions with Dr N R Werthamer and Dr M L Klein are also gratefully acknowledged.

Appendix 1

In this Appendix, we give a formal derivation of the partition function appearing in equation 2. We assume that the potential is symmetric in the ri coordinates. Then the partition function which corresponds to the hamiltonian fi is

(A. 1.1)

where P = l/k,Tand E, are the eigenvalues of H. Equation A.l.l may be put into the form of a trace

Z = dNri( rilexp ( -/?A)@ I ri) s (A.1.2)

where ri denotes a set of coordinates rl, rz, . . . r, and dNri denotes the integrations over them. The state lri) stands for the product of N single particle states Irl)lr,). . . IrN). 8 denotes the projection operator which projects out the many-particle state into sym- metrical and antisymmetrical states depending on the statistics of the particles. Formally the projection operator can be represented as

where p is the parity of the permutation.

1176 V Samathiyakanit and H R Glyde

The standard method of transforming equation A.1.2 in the path integral represen- tation is to make use of the identity

with I factors on the right hand side, where I = / l ie is a positive integer. By inserting complete sets of many particle states between the brackets, equation A.1.2 becomes

Z = j d N r i j d”r,(l) . . .j dNriG). . .I dNri(l - l ) ( r i l ( l - &)@Iri(l))

. . . (ri(j)1(1 - ~ f i ) @ l r ~ ( j + 1)) . . . (v , (I - 1)1(1 - cA)@Iri) (A.1.3)

where j goes from 1 to I - 1. Considering any of the brackets in (A.1.3), one can show that

- eV(r1, r 2 , . . . 2 Y N ) ] f 0 (e2), 1 Substituting this expression back to equation A.1.3, and upon neglecting O(E’), we finally obtain

(A.1.4)

where

and

9 ( r i ( z ) ) = (y)3’2 dri(z). O < Z < P

This expression was first derived by Feynman (1953) to treat liquid helium. The partition function written in this form is exact and could be applied to any many-particle system.

In a crystal, the partition function can be simplified by recognizing that atoms in the crystal very seldom move beyond their own unit cell. It is therefore justified to neglect all permutation contribrutions in equation A.1.4 except the identity permutation. This approximation of course neglects all contributions due to atomic diffusion in solids. The justification for this can be understood from the fact that the jump frequency involved in the diffusion process is much smaller (- 10- ’)than the vibrational frequency of the lattice. Hence we briefly write

(A.1.5)

Path integral theory of anharmonic crystals

where

1177

(A.1.6)

Appendix 2

To evaluate the second cumulant, [((S - So)') - ( S - we consider first

where z12 = z1 - z2. The first term, obtained on multiplying the two potential expressions, is

Hence the first term of (A.2.1) is

The second or cross term in (A.2.1) is

1178 V Samathiyakanit and H R Glyde

On introducing the auxiliary parameter La(l, t ) and using the identity,

(A.2.4)

Here the symbol l A z ( l , t ) = l implies that after the differentiation, we must set Aa(l, t ) = 1 to obtain the final result. On expanding the average in a cumulant series and using the fact that, for the gaussian averaging, only the second cumulant is nonzero (A.2.4) becomes

Carrying out the differentiations and setting all the auxiliary parameters & ( I , t) = ,I&’, t’) = 1, (A.2.4) becomes

(A.2.5)

which aside from a difference in dummy index is identical to the second term (A.2.5). Using the pairing theorem, the last term of A.2.1 is

Here again the last two terms of (A.2.7) are identical aside from a difference in dummy index.

Path integral theory of anharmonic crystals 1179

Hence collecting terms we have

f8 fB

On introducing (18), (A.2.9) and (A.2.81 equation (31) is obtained. On differentiating (31) with respect to B(ll’, z - 7’) and using (16) gives the exact second order force con- stants as,

W ’ , z - 7’) = 6(z - z‘) (V(OV(1’) V({ri(z)})> - [(exp {B(ll’, z - z’)v(I)v’(r)) - I)

x (V(O v({ri(z)>)> <V(U v({ rh‘)})) + 6(z - 7’) job dz3

x ( ~ X P {W’, 7 - ~ d v ( I ) V ’ ( r ) ) - 1) (VU) W ) v({ri(z)})> (V({r)i(t’))))

- 2 dz4B(ik, z - z4) (V(i) V(j) V({ri(z)))) @(l‘k, - z4)

dz3 dz4B(ik, z3 - z4) Q(ip, z - z3) @(l’i, z’-z4)]. Jo Jo x Q(pk,z3 - z4) + (A.2.10)

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1180 V Samathiyakanit and H R Glyde

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