The Faddeev equations for the Heisenberg ferromagnet

Physica

f%A (1977) 93-110

@ North-Holland

Publishing

Co.

THE FADDEEV EQUATIONS FOR THE HEISENRERG FERROMAGNET J.E.

VAN

HIMBERGEN*

Institute for Theoretical Physics, University of Utrecht, Utrecht, The Netherlands

Received

21 June 1976

The Faddeev equations for the three-magnon T-matrix of the nearest neighbour interactions are derived for the cubic lattice extreme case of spin i is considered and the kinematical restriction, site is possible, has been taken into account rigorously. Hence the for the study of bound state as well as scattering state properties. homogeneous Faddeev equations in one dimension is given.

Heisenberg ferromagnet with in arbitrary dimensions. The that only one spin deviation per T-matrix is unitary and suited The analytic solution of the

1. Introduction Some time ago the Faddeev equations for the three-magnon T-matrix of a Heisenberg ferromagnet of spins one half were derived by Majumdar in his pioneering work’). In this derivation not the Heisenberg hamiltonian itself but the well-known Dyson hamiltonian2) was taken as a starting point. The latter describes the spin system by means of ideal spin waves of bosonic nature. It was shown later ‘) that these Faddeev equations also exhibit spurious solutions. This is a consequence of the fact, that, without a special indefinite metric on ideal spin wave space2), the Dyson hamiltonian is not hermitean and does not preserve the kinematical restriction that exists for the real spins. For instance, calculation of the three-magnon bound state energy spectrum in one and two dimensions, from these equations, has shown that, apart from physical bound states, also unphysical bound state solutions occur3.4). Nevertheless, the advantage of Majumdar’s approach is that it leads to relatively simple Faddeev equations, and three-magnon bound state energies can be determined from them, since unphysical bound states can easily be recognized as such4”). However, the information, that can be obtained in this way, is restricted to the spectrum, since the T-matrix itself displays the spurious features inherent * Present address: bridge, Massachusetts

Massachusetts 02139, USA.

Institute

of Technology,

93

Department

of Chemistry,

Cam-

94

J.E.

VAN

HIMBERGEN

in the hamiltonian. The subject of this paper will be the derivation of rigorous Faddeev equations for the three-magnon T-matrix. For this purpose either the Dyson hamiltonian has to be considered in connection with a special indefinite metric, which serves to project out such unphysical features as mentioned above, or the Heisenberg hamiltonian itself has to be taken as the starting point. Although the latter approach will be adopted here, the former will also be discussed briefly in order to allow for a direct comparison with the earlier work’.3). It should be noted that the concept of ideal spin wave space, introduced by Dyson, has far more power than is needed for the present purpose, for which an analysis starting directly from the Heisenberg hamiltonian will serve equally well. The main complication of the analysis will arise from the fact that the kinematical restriction on the number of spin deviations possible at each lattice site has to be taken into account. This gives rise to a three-magnon interaction, which poses some extra problems in the derivation of the Faddeev equations. The results, however, are only slightly more complicated than those obtained earlier for the Dyson hamiltonian. They may be used as a basis for the calculations of both bound state and scattering state properties of three magnons. In the subsequent analysis we shall consider the Heisenberg hamiltonian for spins one half with nearest neighbour interaction on a cubic lattice in arbitrary dimensions. In section 2 we shall investigate the hamiltonian in the subspace of three-spin deviation states, and discuss the way in which the kinematical restriction is taken into account. In section 3 the Faddeev equations will be derived in reciprocal lattice space. Many details of the derivation are given in the appendix. Finally, in section 4, the Faddeev equations will be discussed for the case of a linear chain and the analytic solution of the homogeneous Faddeev equations will be given and compared with earlier results’).

2. The hamiltonian We consider the hamiltonian for a d-dimensional (d = 1, 2, 3) simple cubic Heisenberg ferromagnet of spins 4 with isotropic nearest neighbour exchange Y, =

c

-;.lc I

si . si,A

(J > 0).

(2.1)

A

Here i runs over all lattice sites, A over the nearest neighbours of a given spin. The lattice spacing is taken to be unity, periodic boundary conditions are imposed, with a periodicity cube of sides L that contains N = Ld spins. The operator n = -;N + Ei Si for the total number of spin deviations from the ground state IO) (n = 0) defines a good quantum number. For the derivation of the Faddeev equations we need to know the matrix elements of X, in the subspace of three-spin deviation states (n = 3). For the case S = $ under

FADDEEV

EQUATIONS

FOR HEISENBERG

FERROMAGNET

95

consideration, S :,S :,S :,lO) = 1m,m2m3) is a three-spin deviation state if m, # m2 # m3 # ml, and equals zero if at least two of the mi are equal. We could, nevertheless, extend the space of real three-spin-deviation states by formally defining physically non-existing states Im,m,m,) with at least two of the m, equal, such that all states are orthogonal

(2.2) and form a complete

set

IXpCP,j indicates the sum over all permutations of 1, 2 and 3. To ensure that this extension has no effect whatsoever on the dynamical features of the spin system, the matrix element (m~m~m~~Xslm,m,m,) has to be zero, if either Im,m2m3) or (m;m;mil or both belong to the extension. This can easily be arranged, and using the commutation rules for the spin operators one readily finds how Xs acts on an arbitrary state Im,m2m,) of the extended subspace

- 5 2 c c1 - 8m,,+A.ol,, - 8n,,+A,m,l)lmp, + Am,,m,,)l, A CP(Pi)

(2.4)

z cpCp,jis a sum over cyclic permutations

of 1, 2 and 3. The advantage of the above procedure is that multiple lattice sums can be performed independently, each summation index running over all lattice sites, since contributions of terms with equal indices are excluded by construction of the hamiltonian. By Fourier transformation we may now define three-magnon states Ik,k:W=&

c exp (i c k, ” mlvm3

m,,) )m,mzm3).

(2.5)

We shall often use the symbolic notation k to indicate the momenta k,, k?, k, of the three magnons. The states Ik) form a complete set of orthogonal states within the subspace of three-magnon states with normalization (2.6) and completeness t

c klk+,

relation

Ik,WJ (k,W = 1.

(2.7)

J.E. VAN

96

HIMBERGEN

The possible values of each ki are determined by the periodic boundary conditions that we have imposed. It is a straightforward matter to calculate the matrix-elements (k )Zslk’) given by

Using

(2.4) we find

(kISYTlk’>= i t3.J -J( cos k,, + cos k,, + cos kj, )I(k Ik’) n-l

spy

+;

2i i.j=I

-hU’,, c

P:,)

n=l

(2.8b)

K:,, pi,,)1 d

+6p,p,

SK;,K,.[~~~ pi,,(cos $K;, - cos

(2.8a)

3 2

n=I i=l

g;‘(k)f;‘(k’).

(2.8c)

It is convenient to indicate coordinate axes by numbers n = I,. . . , d. If a momentum is labeled by two indices, the first one invariably indicates the magnon(s) involved, the second one the component. For instance kin is the nth component of the momentum of the magnon labeled i. Furthermore we have defined P=k,+kz+k,;

K, = kz + k,, p, = f(kz - k,);

K,=k,+k,,p,=;(k,-k,);

(2.9)

K3=k,+kz,p3=f(k,-kJ,

E (I’,,, K :,, p :, ) = 3 - cos (P,, - K :,,) - 2 cos K :, /2 cos p ;,, fl:‘ct’,=$$

/-I

(2.10)

[2 cos(P,,-K’,,,)-cosK:,-11,

3

g(,)(k) = c cos (I’, -K,,); ,=I

(2.1 la)

$ [2 sin (P, -K’,,,) I I

f(n?)(k’) = 3

- sin K’,,,],

3 g!?(k) = c sin (P, - Kj,); ,=I fF’(k’)=$$[l-COS(P,~ /-I As one sees from

x, =

Ye,,+

2 ,=I

(2.1 lb)

-K’,,,)],

g:;“(k)=

(2.8), XX has the following

vi + v”‘.

I.

(2.1 lc)

structure (2.12)

FADDEEV

EQUATIONS

FOR HEISENBERG

FERROMAGNET

97

The matrix elements of the diagonal part 2, are given by (2.8a), which describes three free magnons of momenta ki and energies .I El=, (1 - cos kin). The remaining part describes their interaction; there are two-magnon interactions Vi in which the magnon labeled i remains a spectator at the interaction of the other two. Their matrix elements are given by (2.8b). Finally there is a three-magnon interaction V”‘, in which all three magnons interact and, accordingly, only total momentum is conserved, as one sees from (2.8~). It is important to note the separability of all interactions with respect to their dependence on the momenta of incoming and outgoing magnons. This property turns out to be crucial for the solution of the Faddeev equations for the three-magnon T-matrix. Finally, we make the following observation. Disregarding the term involving E(P,, K :,,, p :, ) in (2.8b), and (2.8c), we are left with the interaction of three magnons as it occurs in the matrix elements (k\Xnlk’) of the Dyson hamiltonian2). Indeed we could have resorted to Dyson’s ideal spin wave space, transforming %‘, by means of a Dyson-Maleev transformation) into ZXn.But then spurious features, not present in the spin system, are also observed, since the kinematical restriction on the number of spin deviations per site is not fulfilled within the ideal spin wave space. It could be restored, however, by projecting out the non-physical part of the ideal spin wave space. For the three-magnon subspace, in the case s = f, this can be achieved by considering P%?J’ instead of %n, where the projection operator is given by P = 1- f 2

(a :>‘(a,)’ + 4 c (a ;)‘(ai)‘.

(2.13)

Here, at and ai are the boson operators that create and annihilate quasi-spin deviations in the ideal spin wave space. If one calculates the matrix elements (k jPXDPIk'), one obtains the full expression (2.8). Hence this approach is completely equivalent to that given above, the second term of the two-magnon interaction and the three-magnon interaction being due to the projection of non-physical three-spin deviation states. The approach of this section has been followed earlier by Boyd and Callaway’) in their calculation of the two-magnon scattering amplitude.

3. The Faddeev equations Now we want to derive the Faddeev equations for three magnons whose interaction is given by (2.8bc). The Lippmann-Schwinger equation for the three-magnon T-matrix reads 4

T(E) = V + VG,(E)T(E)

4

= C Vi + C ViG”(E)T(E), i=l i=l

G,,(E) = [E - X0]-‘; the three-magnon

interaction

(3.1)

is denoted by V4. Following

98

J.E.

Faddeev’) we write Lippmann-Schwinger (1 - V,G,)T”’ where

VAN

HIMBERGEN

T = C:=, T”‘. The partial equations

T-matrices

obey

the following

= Vi + V,G,( T”’ + T(k) + Tc4’),

(i, j, k) runs

over

the cyclic

permutations

of (1, 2, 3), and

(1 - V4Go)T’4’ = V4 + V 4G 0(T”’ + T”’ + T”‘). We define

the matrices

T,=V,+ViG,T,

T, by (i=l,2,3,4).

(3.2)

Here T, involves the three-magnon interaction, but is defined by (3.2) completely similar to the familiar two-magnon T-matrices T, (i = 1, 2, 3). We then obtain the Faddeev equations for the partial T-matrices T”’ = T, + T,G,( T’” + Ttk’ + Tc4’)

(i, j, k) = CP( 1,2, 3),

T’4’ = T, + T,G,( T”’ + Tf2’ + T”‘).

(3.3) (3.4)

Usually one deals with pairwise interactions only, here we have an additional three-particle interaction. It can be taken along with the pairwise interactions in the simple way described above. In order to derive equations for the threemagnon T-matrix elements from (3.3) and (3.4), we first have to solve the matrix elements of 7) between three-magnon states from (3.2). First we consider the two-magnon T-matrices involving two-magnon interactions. It will be convenient to use states lPKjpj)j with j(PKjPj

IP’KlPj), =

~P.P,~K,,K;(~~,,~; + SpI,mp;)y

; c IPKipi)i j (PKjpj 1= 1. P“,p,

(3.5) (3.6)

For instance IPK,p,), is the state symmetric with respect to the magnons labeled 2 and 3, which have total momentum K, and relative momentum p,. Owing to the separable structure of the two-magnon interaction, it can readily be shown that the solutions of eqs. (3.2) are given by

j(PKjPjITjIk’)=

6p,p’g hi(pj)$,(Kj, E; k’).

The form factors have been arranged into a 2d-dimensional vector l), and the functions $J, are obtained as the solution (cos PII, 1,...,COSP,,, set of algebraic equations $I; = 2 Ai,,c, “,=I

(i=1,...,2d).

(3.7) i = of a

(3.8)

The details of the derivation, together with the definitions of A and 6, are given in the appendix. Quite similarly the solution of eq. (3.2) for the matrix elements (klT,Jk’) can be obtained. The detailed derivation is again given in the

FADDEEV

appendix.

EQUATIONS

The solution

(kIT,(k’)

= a,,,

FOR HEISENBERG

FERROMAGNET

99

is

5 gi(k)ri(k’; ,=I

E).

(3.9)

Here gi are the form factors of the three-magnon interaction defined by (2.1 l), arranged into a 3d-dimensional vector S = (gy’, gy’, gy’, . . . , gy’, g(d2),ge’). Ti is again the solution of a set of algebraic equations 3d T, = c a,,(P, E)fi(k’)

(i = 1,. . . ,3d).

(3.10)

J:I

The matrix (Y is defined in the appendix. With the solutions by (3.7) and (3.9), we can proceed to derive equations for T-matrix elements. Here we shall only state the result, while is given in the appendix. For the matrices T”’ (i = 1, 2, 3) we solution ,(PK,p,IT”‘jk’)

[cos p,,cp;‘(Ki)

= i

of eqs. (3.2) given the three-magnon the full derivation find the following

+ &‘(Ki)]

n-l

?d

=

c h,(~i)cp,(Ki;

(3.11)

E, P>t

q=l

where

the functions

(pq are solutions

of the set of integral

equations

u,(K;E,P)=IL,(K,E;k’)+CT,,(K;E,P)T,(k’) r=, +77

+g+,

1

dK”(Y&,(K”,

K ; P, E)

(2Tr)d I

+ L&(K”,

K-nP, E))cp,,(K”;

E, P>

(q = 1, . . . ,2d).

(3.12)

The quantities 4 and ? are given by (3.8) and (3. lo), respectively. The matrix r and both kernels X and _Y are defined in the appendix. For the matrix elements of T14’ the following result is derived Id (kIT’4’(W

=

with the functions

.T,,,(k’, E,P)

by

c

g,(k)Y,-,,,(k’;

T,,, obeying

= T,(k’)+;

(3.13)

E)

the equation

I

$fj

;,

a,,(P,

E)L,(K;

E>P)cp,(K;

EP)

-77

(m =1,...,3d), where

(p is the solution

(3.14) of eq. (3.12).

100

J.E.

VAN

HIMBERGEN

The results (3.11) up to (3.14) have the same status as those obtained by Majumdar in his pioneering work’). There, however, the Dyson hamiltonian is taken as a starting point for the derivation of Faddeev’s equations, which consequently exhibit spurious features not present in the real spin system3). Here we have shown how the kinematical restriction on the spins can be incorporated rigorously, so as to obtain the Faddeev equations for the three-magnon T-matrix of the Heisenberg hamiltonian in arbitrary dimensions. Therefore these results provide a suitable platform for the investigation of three-magnon scattering phenomena as well as bound state properties. The complications, arising from the interaction between the spins that is due to the kinematical restriction, are relatively mild. The contribution from threemagnon interactions is eliminated from eq. (3.12) at the expense of a more complicated kernel (the term involving 9 originates from the three-magnon interactions). That this elimination should be possible is already clear from (3.3) and (3.4).

4. The linear chain In this section we shall investigate the Faddeev equations (3.12) and (3.14) for the linear chain, and give the analytic solution for the three-magnon T-matrix when E = E, = t(l - cos P), the three-magnon bound state energy found long ago by Bethe’). In one dimension the integrations (A.6) can be performed, and therefore closed expressions can be obtained for the quantities (A.7), (A.8), (A.18) and (A.22)~(A.24), since all these depend on the functions defined by (A.6) in a simple manner. The matrix elements aii defined by (A.12) and (A.13) are an exception and have to be calculated numerically even in one dimension, because they involve integrals, containing the functions (A.6), that cannot be evaluated otherwise. Apart from that, all quantities mentioned above, which in turn determine the kernels of the integral equations (3.12) and (3.14), are easily calculated as indicated in the appendix. For the linear chain there are two coupled integral equations (3.12). The homogeneous Faddeev equations, from which the bound state energy eigenvalues follow, have been studied numerically. Using gaussian quadrature with 16 mesh points, the integrals are turned into discrete sums, yielding a (32 x 32) kernel. If for certain values of energy and total momentum, this kernel exhibits an eigenvalue of one, a bound state energy has been found. In this way Bethe’s bound state energy at EB = $1 - cos P) has been recovered together with the corresponding eigenfunction at the mesh point values. Also, as expected, this is found to be the only bound state, in contrast to the case of Faddeev’s equations for the Dyson hamiltonian’). If one substitutes the bound state eigenfunction into eq. (3.14), to solve for Y,,,, one finds numerically that Y,,, = 0 for all m at E = E”. The bound state solution of eqs. (3.12) can also be found analytically in the following manner. Near the bound state pole (E = EB) the three-magnon

FADDEEV

T-matrix

EQUATIONS

FOR HEISENBERG

FERROMAGNET

101

is given by (4.1)

Hence, apart from a normalization (LIWW)

= #]

factor, (4.2)

v($Bh

Since the bound state wave function is known, the three-magnon T-matrix can be found by means of (4.2). From Bethe’s work we record for the bound state wave function

with, in the limit of N +x, a mlm2m3

UN

=e

e

the amplitudes

i(P-2u)m2+iu(m,+m3)

given by

e-o(m3-m,),

(4.4)

where we have substituted k; = u + iv, ki = P - 2u, k j = u - iv for the momenta of the individual magnons. The bound state parameters u and v are given in terms of the total momentum P by 2 = 3 cotg ;P,

(4Sa)

u = arctan [2z/(z2 + 3)],

(4.5b)

e m2U= (1 + z’)/(9 + 22).

(4Sc)

In our approach the bound state is written as a superposition belonging to the extended subspace of three-spin deviation states, IrL)

=

z

of states

(4.6)

~m,m*m311711~2~3).

99m3 all

Let us consider the Schrbdinger equation (m,m2m31Xsl$> = E,(m,m,m,~~) = Because Xs, given by (2.4), has been constructed such that E~am,m~m,. (mlm,m,lXs~+) = 0 if at least two of the mi are equal, we see that amlmlml= 0 if equal indices occur. The amplitudes with all indices different, are of course given by (4.4), when these indices are arranged in order of increasing magnitude. It is straightforward to calculate X&I) with the aid of (2.4). For the terms representing the interaction we find VI+)

=

-35

+

[

C

a,, ml+,mzlml

ml

+

1 m2>

m1m2 m22mI+l

c mlm2 m2
am2m,m,+l Im2mlml+l)+

2 ml*? m2>ml

aml~lm,mzlm,-l

m,m2>+

102

J.E.

+ 3J

+

+

VAN

HlMBERGEN

a m, m,+l mzlm, + 1 ml + 1 m2)

m,m,+lm,+l)+ C am,m,ml+~l

c

2

%2m,-lm, Im, m, - 1 m, - 1)

a,,~lm,mzlml-lm,-lm~)

1.

(4.7)

J

mim2

The contributions to VI+,) appear to originate only from the two-magnon interaction (that is the part of (2.4) involving terms with one S-function). Furthermore, the contributions to 5YS($), that involve amplitudes with at least two equal indices, all cancel each other out, as they should, of course, in order that (m,m,m,~X,~~) = 0 if some m, are equal. Nevertheless, we consider the effect of the three-magnon interaction (terms of (2.4) containing two &functions) on I$) separately. One finds Vc3’1+) = 6Jc

ammmlm m m) - 3Jc

-3Jc

2 m

m

c a,,,+dl m d

+6JC

2 m

%,m+Alm

a,,,,,,,,lm m m + A)

A

mm+Am+A)-3J~~a,,,,,+,~mmm) m A 111 m

+

A).

(4.8)

A

The first two terms cancel against contributions to Y&l+) of amplitudes with equal indices, and, what is more important, the others cancel against such contributions in the two-magnon interaction. Therefore, remembering relation (4.2), the effect of V’” upon the three-magnon T-matrix is the following. Its presence modifies the contribution of the two-magnon interaction to the total T-matrix, in that it has eliminated from (4.7) the two-magnon contributions that are equal to the last three terms of (4.8), but of opposite sign. On the other hand there is no direct contribution to the three-magnon T-matrix from V”‘, since all terms in (4.7) derive from the two-magnon interaction. This is precisely the situation reflected by the Faddeev equations. The kernel of (3.12) contains a contribution .Z’ from the three-magnon interaction, indicating the contribution of its effect on the functions cp, and (p?, that determine two-magnon interactions to the T-matrix. With the solution of (3.12) subthat there stituted into (3.14), one finds Y,,, = 0 for all m if E = E,, indicating is no direct contribution from Vc3’ to the three-magnon T-matrix at E = E,.

FADDEEV

EQUATIONS

FOR HEISENBERG

FERROMAGNET

103

It is easy to calculate (kl VI+) f rom (4.7), using (4.4). The indices already been arranged in order of increasing magnitude. Since we consider limit of N +m, one of the sums may be converted into a sum over Im, - m,], which gives rise to an infinite geometric series. The remaining provides conservation of total momentum. Thus we find, according to

(kITIk’) = ~P,Pi

[ COSPi

i-_l

+

e-“[cos

(P - Ki)

- iKi) - cos (P - &)I (2e-"[em"cashcosu(u- cos (U - P + Ki)

+ cos (P - 2K,) - e-”

COS

U -

em”COS (U - Ki)]

cash 2, - cos (U - P + KJ

have the p = sum (4.2)

II3 (4.9)

where u and 2, are given by (4.5). This solution has the structure of (3.1 l), the functions cp, and (p2 accompanying the form factors cos pi and 1 respectively, are the solutions of the integral equations (3.12); they perfectly agree with the numerical eigenvector. It should be stressed that, although the analytic solution for the bound state has only been found in one dimension, the result that Y,,, = 0 for all m at every energy eigenvalue, is true in all dimensions and not only for the bound states but for all eigenstates of the hamiltonian. The argument is entirely similar to that given above. Again, the T-matrix is seen to exhibit the particular structure, observed earlier for three particles interacting through S-function potentials”“‘), and for three magnons described by Dyson’s hamiltonian. In contrast to the latter case, there is now only one pole in expression (4.9), corresponding to the one physical bound state, since the unphysical bound state at E = 7/8 [ref.3)] has disappeared because of the rigorous treatment of the kinematical restriction. Finally, it should be mentioned that in connection with our earlier work3), Majumdar has shown an elegant way of directly solving the homogeneous Faddeev equation for the Dyson hamiltonian12). It depends, however, on an educated guess about the denominator of the solution, which, it seems, is not obvious for the physical bound state. Moreover, although it has been successfully applied to unphysical solutions, his analysis has not yet proved possible for the physical bound state. That is why in this paper we have again chosen our previous approach of finding the T-matrix using relation (4.2).

Acknowledgement The author discussions.

is greatly

indebted

to Professor

J.A.

Tjon

for

stimulating

104

J.E. VAN HIMBERGEN

Appendix 1. The two-magnon For definiteness T, are similar.

T-matrices

in the three-magnon

we shall solve eq. (3.2) for T,, since the solutions Using (3.6) we find from (3.2)

+i

,(P~,P,IV,IP”K’~P’~),

2

,(P”K;p’;IT,Ik’)

with (2.10), we have

X

[cosp,“(cos;K;”

= 6,,, From (A.l),

5

n-l

- cos pi,) - &(I’,, K;,, p;,)]

[cosp,.v,(K,;k’)+w,(K,;

k’)l.

(A.3

the last equality the definitions of v,, and w, are obvious. It is clear from after substitution of (A.2), that the solution has the form

,O’K,p,lT,lk’)=

a~,,,f:

n:I

(cos ;K,,

k’)l.

[cos~,.rlr,(K,,E;k')+x,(K,,E;

Having substituted (A.3) into (A.l), equations for C/I”and xn

one obtains

- cos p’l,) cos p’l,.

E - E(P, K,, P’,‘)

the following

MK,,

XJK,,

the eigenvalues

of I-I, are denoted

E(P,K,,p,)&

(3-

COS (P,

-

set of algebraic

(A.4a)

E; k’)

cjl,JK,, E; k’) = w,.

p; E-EU=,K,,p;)

(A.3)

E; k’)

,y”z(K,, E; k’) = un,

Here

(A.1)

E -E(k”)

P”Krpi

(2.8b),

for T2 and

= dPK,~,lV,lk’)

I(PK,P,IT,IW

From

subspace

(A.4b)

as E(P, Ki, p,), with

K, ~ - 2

COS

K,, /2 COS

Pin )

,I = I

=

J i (3 - cos k,, - cos k?, -cosk,,)=E(k). n-1

(A.4c)

FADDEEV

EQUATIONS

FOR HEISENBERG

Now we take the limit

N + m, then

Furthermore

the following

we define

quantities: (ASa)

t,=:d-tE,J-1~cos(P,-K,,), n=l pli =

COS

(ASb)

(i = 1,. . . , d),

fK,i

105

FERROMAGNET

57

Do(t,, K,) =

dp, L 7TdI 0

I

Do(t,, K,) = i rrd

1

LiPlnlcos Lb’

tl-

m=,

dCoS ‘In

dp,

0

(A.6a)

tl-

c

P,m

(A.6b)

(n=l,...,d),

COSPlm

m=l T

&,(t,,

K,)

=

4 rr

I

CosdPIn ‘OS‘In

dp,

0

t,

-

c

(n,n’=l,...,

d),

(A.6c)

COSPI,

Ph

WI=,

Eqs.

(A.4) may now be written

as follows

d

c [h,,, + P,nDAt,, K,) - D,,Af,r K,)I$,W,,

E ; k’)

,,‘=I

d +

2

II’=,

[P&At,,

K,) - D,(t,, K,Nxn,(K,,E; k’j = u,

(n = 1, . . . , d), (A.7a)

,$, [P,nDnn,(f,,

K,) - ;(3 - cos (P, - K,,))DJr,,

K,)I4,(K,,

E; k’)

d

+

c L&n,+ P,nDn(t,,K,)-$3

--OS

(P, -

K,,)Po(t,,

Kdl

“‘=I

x x”f(K,, E; k’) = w,

(n=l,...,d).

(A.7b)

We may write the solution of these equations in the compact form of eq. (3.8). We define a 2d -dimensional vector 4 = (+,, x,, . . . , Gd,Xd) for the solution of (A.7), fi = (v,, w,, . . . , ud, wd) for its inhomogeneous term and K = (cos p ,,, 1I..., 1) for the form factors of expression (A.3). Then one has as cos P Id,

J.E. VAN

106

solution

HIMBERGEN

of eqs. (A.7)

$, (K,, E ; k’) = 2 Aijvi, ,=I

64.8)

the (2d x 2d) matrix A being the inverse (A.7), properly arranged. Furthermore the two-magnon T-matrix 2d ,(PK,P,IT,Ik’)

=

~P,P,

of the matrix elements

of coefficients

(A.3) are given

of eq.

by

2d

C

C

i=l j=1

hAijui

=

6,~

2

hib,)+i(K,,E;

k’).

(A.9)

i=l

i (PKjpj 1Tj Ik’) (j = 1, 2, 3) are given by (A.9) with

In general the matrix elements K,, p, replaced by K,, pi. 2. The matrix T, With the three-magnon (3.2) for T4 reads (kjT,lk’)

given

by (2.8~) together

with (2.11), eq.

= 6p.p, i [c g:‘(k)fj:‘(k’) n=I i=l

+i

c 2 bd,?k)f~‘W’) k”

Thus

interaction

the structure

+“I T,lk’)

E _

E(k,,)]

(A.10)

i=,

of the solution

is

(kIT,lk’) = cS~,~.i i g’,“(k)t:‘(k’). n=1 i=l Upon substitution equations:

(A.ll)

of (A.1 1) into (A.10) one again

obtains

an algebraic

set of

f(i)(k”)gti)(kf!) ”

E - E;;“)

I

@(k’) = f”‘(k’). n

(A.12)

Denoting the various quantities as vectors, now 3d -dimensional since there are three different form factors to each component, with ? = (t :I’, t y’, t I)‘, . . . , t :“, t y’, t T’) for the solution of (A.12), f = (f:“, . . , f’,“) for its inhomogeneous term and 2 = (g:“, . . . , gy’) for the form factors, we can write the solution as follows 7, = 2

(A.13)

ais,

and the T-matrix

elements 3d

(klT,lk')

=

6,p

C

(A.ll)

as 3d

3d C

,=I j=,

gi"iif

=

6.p'

C

,=I

gi(k)T,(k';

El.

(A. 14)

FADDEEV

EQUATIONS

FOR HEISENBERG

107

FERROMAGNET

The (3d x 3d) matrix (Y is the inverse of the matrix of coefficients (A.12). Its matrix elements involve certain integrals over the functions in (A.6). 3. Derivation of the Faddeev Let us first consider

equations

TC4’. For its matrix

(klT”‘lk’)=(klT,/k’)+&

elements

we have

i(&fii

~

(3.4)

IT”‘lk’)

E - E(k”)

Here we have used (2.7) and (3.6) in order that we encounter analysis only expressions involving the quantities defined Upon substitution of (3.9), we find 6,,,

from

cc b,~b,dklT4lk”) i=, k” #+ESi

X (k”\PK@i)i

(kIT’4’lk’)=

of eqs. defined

gi(k)Ti(k’;

in our subsequent by (3.7) and (3.9).

E)

i=l

i=l

k" I%,#,

I=I

3d

x

mz, g,

(k)T,,, (k”) ‘(‘;~~;),,;‘).

We now anticipate a result to be derived soon, i(PKipi IT”‘lk’) (i = 1, 2, 3) has the structure i’:PKip,(T”‘Ik’)=

~ h,(pi)cp,(K,, 4’1

namely

that the solution

k’, E).

We only denote the dependence of various quantities on such variables important in each stage of the analysis. With (A.15) we have

for

(A.15) as are

(k IT’4’jk’)

(A.16) Here we have performed four of the sums in the above expression, which are easy because of the 6 -functions. For each value of I, we take & to be &,P,Ki.,pi. Also T,,,(k”) is expressed in terms of PI’, K;’ and p;’ only. It turns out that these expressions are identical for all 1, moreover they are symmetric under p’; + -p; since only cos p;‘, occurs. This gives a factor of six and turns f”(k”) into a

108

J.E.

VAN

HIMBERGEN

function of P, ~~ and pi. In this way one arrives that the structure of the solution is (klT’4’)k’)=6,.,,

at (A.16). From

(A.16) we see

2 g,,,(k)Y,,,(k’,E) WI=,

(A.17)

and defining (A.18) the following

result

T,,,(k’, E,

for F,,, is derived

P> = qn(k’)+&~

5 K

from $

(A.16)

a,,(P,

E)I,,(K;

E, P)qy(K)

n=lq=I

,3d).

(m=l,... From

definitions

(A. 19)

(2.11) one finds

45 f(,l,(K,, pi) = 9 [2

COS

(P,

-

K,“)

-

COS

Kin + 4

COS

iKin

COS

Pin

-2 cos (P, - ;K,, ) cos p,,, - 31 fjl?)(Ki, pi) = $

[2 sin (P, - Kin ) - sin Ki, + 4 sin ;K,, cos pin

-2 sin (P, - iKin) f’,3’(K,, pi) = $

cos

p,,],

[3 - cos (P, - Ki,) - 2 cos fK,

cos p,,].

Therefore the quantities I,, of (A.18) are linear combinations of D,, D,, D,,., defined by (A.6), with arguments t, and K,, the coefficients being simple goniometric functions of the momenta P and K,. Next we turn to the solution of eqs. (3.3). For definiteness, we only consider T”‘. From (3.3) we have

,O’K,p,lT”‘lk’)=,O’K,p,l’Clk’)+i~d-‘Kd~,lW Ir’

+f i c ,(PK,p,IT,IP”K’;p’+ i -2 P”K’;p;

Or upon

substitution

,(PK,p,IT”‘(k’)

of expression = 6,~

E_E(k,,)

(P”K’;p’;IT”‘lk’) ’ ’ ’ E - E(P”, K’;, p’:)’

(A.9)

5 h,,(p,)[lli,(K,, In=,

(k”(Tc4’jk’)

k’)

(k”]T’4’lk’) + +t c 6 p,,.,&,, (K, ; k”) E - E(k”) L”

FADDEEV

EQUATIONS

FOR HEISENBERG

109

FERROMAGNET

(A.20) Defining v’(” = (u$‘,, w\“, . . . , vy’, wf’), and its components the terms with i = 1 only, we have from (A.@ $;,(K,;

K;“,, pfl(i,) = 2 A,iv;,,(K,; j=l

by taking

K!;“),p;‘(i)).

in (A.2)

(A.21)

KY”’ and p’(“’ are the values of the total and relative momentum of magnons and 3, given that magnon i has momentum ki, as may be seen from ,(PK,p,IT,IP”K~pp!:)i

= ,(PK,p,(T,IP”K;~“p;‘“), = a,,,,, 2 h,(p,)q:‘(K,; m=,

They are P_;KI;_p;,

2

K’I”‘, p’fi)).

given by K;‘2’=P-iK;+p;, P;I”‘=-;P+$K;-$;.

~“(~)=iP-iK;--ip!j and Kyc3)= From (A.20) now follows the general structure of the solution, it is given by (A.15), which justifies its earlier use. Substituting (A.15) into (A.20) and using (A.17), one finds q+,,(K,)=$,(K,;k’)+i +;

Let us define T,,(K,;

P,P” ;

,rWJ]

K’;“‘, p’fi)) 2d 2 h,(p’i’)cp,(K’:) iz2 K:ps;E - E(P, K’,‘,p’,‘) q=,

i

$;‘(K,;

c

the following E, P) = t 2

yr(k’,

El

(WI = 1,. . . ,2d).

quantities

6p,p,, hn ‘,“” ;;;;,;k”).

(A.22)

k”

Using (A.2) (A.8) and (2.1 l), it is easily seen that these quantities are again linear combinations of the functions defined by (A.6) with simple goniometric expressions as coefficients xm, (K,~, K,; p, E) ~ N c P,

(L!i’(K,; K’?‘, P’?‘)~,(P’~)~ E - E(P, K’;, p’;)

These expressions are identical for i = 2, 3, hence just gives a factor of 2. With (A.21) the summation 6 K,,KI(l) at one’s disposal. Finally

-r?,,(K, K,; P, E) -+ 2 r,,,,(K,; E, P)w,,(P,

(A.23)

C;=? in the above equation is trivial, since one has a

E)In,(K;

E, P).

(A.24)

r.n

If we now substitute (A.19) to eliminate !Yr from the equations for qrn above, one arrives at the result of eqs. (3.12). The matrix _!??defined by (A.24) enters as result of elimination of Yr.

References Majumdar. Phys. Rev. !41 : ,“it:, : Phys. Rev. 102 ( I’;‘>\,: I’ See also: S.T. Dembinski, PI:\, ‘,:~.~i.$j: i ‘c.: 41 :: J.E. van Himbergen and J.A tl,,: I’;:. : 5. J.E.vanHimbergenandJ.A. i ,.,, 1 V. :‘. .I’ 1.G. Gochev. Phys. Stat. SW ‘,“. t !‘: :.z 21I S. Maleev, Sov. Phys. JET! R.G. Boyd and J. Callaway I+;!. ,. : ) .;: :’ L. Faddeev. Sov. Phys. JE.4 Jo ?: i 2: 1 I’~/ H.A. Bethe, Z. Physik (1931i ‘:b ‘~1 ,, .I, ’ L.R. Dodd, J. Math. Phys. 3: ‘2, < C.K. Majumdar, J. Math, Pit , 1’ ;,: ;c C.K. Majumdar, J. Math. Pi .‘. i ’

I) C.K.

2) F.J. Dyson, 3) 4) 5) 6) 7)

8) 9) 10) 11)

I?)

i’;‘?.?)3 1. IL,‘,,, :I.:“:.:: ., ~ir,-sicag2A(1976)389.

‘i I

i