THEOREMS FOR SERIES EXPANSIONS FOR THE GENERALIZED HEISENBERG MODEL

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THEOREMS FOR SERIES EXPANSIONS FOR THE GENERALIZED HEISENBERG MODEL

G. Paul, H. Stanley

To cite this version:

G. Paul, H. Stanley. THEOREMS FOR SERIES EXPANSIONS FOR THE GENERAL- IZED HEISENBERG MODEL. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-350-C1-351.

�10.1051/jphyscol:19711119�. �jpa-00213935�

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JOURNAL DE PHYSIQUE Colloque C 1, supplkment au no 2-3, Tome 32, Fe'vrier-Mars 1971, page C 1 - 350

THEOREMS FOR SERIES EXPANSIONS FOR THE GENERALIZED HEISENBERG MODEL

G. PAUL (") and H. E. STANLEY (**)

Department of Physics and Center for Materials Science and Engineering Massachusetts Institute of Technology, Cambridge, Massachusetts

R6sum6. - Dans ce travail nous demontrons deux identites extrdmement utiles dans le developpement en series a hautes tempkatures, de I'Hamiltonien isotrope #Heisenberg pour des spins de dimensions quelconques. Une identite relie les valeurs associ6es aux diagrammes pour la fonction de correlation spin-spin aux diagrammes de I'knergie libre.

La seconde identit6 relie les valeurs associkes a diffkrents diagrammes de 1'6nergie libre, qui diffkre, dans le langage de la theorie des graphes, par l'insertion d'un vortex de valence deux.

Abstract. - In this paper we prove two identities which are extremely useful in developing high temperature series expansions for the isotropic Heisenberg Hamiltonian of arbitrary spin dimensionality. One identity relates the values associated with diagrams for the spin-spin correlation function to the values associated with diagrams for the free energy. The second identity relates the values associated with different diagrams for the free energy which differ, in the language of graph theory, by the insertion of a vertex of valence two.

which has been discussed extensively earlier [I 1, [2], where the spins sID' are D-dimensional unit vectors.

Here the sum is over all pairs of sites of the lattice and all JLj are in general different. For the moment it will be useful to maintain this generality ; in any final results we will set all J i j equal to a constant, J, for nearest neighbour sites and equal to zero otherwise.

Two quantities of interest are the free energy F defi- ned by

- PF = In Tr e-flx (2)

Consider the Hamil tonian dn ~r S, .s, ^e-Px

x(D) ⁼- ^J s j D ) . s y ) (1) %(f, g ) = a/?" - re-^^

and the spin-spin correIation function TCf, g ) defined bv

p = o (8)

where p = ilk, T . For any function of the spin varia- bles, O(S,, S2 ... ^S,),the trace operation is defined by

i i

Tr O(S1, S2 ... S,), =

1 ^{dB, dB2}^...^dB,^{O(S,, S,}^...^S,)

-

- -- (4)

The integrals are over surfaces of hyperspheres in D- dimensional space and we have dropped the supers- cript (D).

Where a series expansion is valid we can write

with An and a, given by Taylor's theorem as

(*) NSF Predoctoral FeIlow.

(**) Supported by NSF Grant No. GP-15428.

The basic data needed in a series expansion analysis are the values of An and a, up to a given order.

The first step in the proof of our theorems is t o write (7) formally as

J"

{Ji j ) = 0

where the second equality follows from an applica- tion of the multinomial theorem and where C' ^signi-

'"1

fies a sum over all values of the set { n ) fo"; which nij = n. We have set J i j = J and limited the sum

i i

over i and j to nearest neighbour sites. Proceeding simi- larly for orn we can now write (7) and (8) as

an(f, g) = C' ^@f, ^{ⁿ1

{n) (1 1)

with

We can associate a diagram with every A { n ) or xfg ( n } by drawing n,, lines between vertices 1 and 2, n,, lines between vertices I and 3, and so on. The function A { n } and orfg { ⁿ) were introduced earlier

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711119

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THEOREMS FOR SERIES EXPANSIONS FOR THE GENERALIZED HEISENBERG MODEL C 1 - ³⁵¹

by Stanley and Kaplan [3] by means of these diagrams and were defined in terms of a recursion relation.

It is advantageous to work in terms of these functions because A { n } and olfg { n ) are identically zero for all disconnected and tree diagrams. Thus large classes of diagrams do not contribute and need not be consi- dered in (10) and (11). We have used the representation in (12) and (13) here rather than the recursion relation representation because it will thus be easier to prove the theorems which follow.

Proof: From (12)

Taking one derivative with respect to Jf, and compar- ing with (13) Theorem I follows immediately.

THEOREM 11. - For a diagram W , which, in the language of graph theory 141, is obtained from a diagram 6 of nth order by the insertion of a vertex of valence two (Fig. 1)

(0) (b)

FIG. 1. - Example of two diagrams related by the insertion of a vertex of valence two. Diagram (b) is ohtained from diagram (a) by the insertion of a vertex of valence two between

vertices 1 and 2.

Without loss of generality we label the vertices so that the ifnserted vertex is between vertices 1 and 2:

The number of lines between 1 and 2 in 9 is denoted by n,,. Labeling the inserted vertex with a zero we can write (15) more explicitly as

A(nlo = 1, no, = 1, n12 - 1 ...) =

= - J (n + 1) n12 4 0 , 0, nl, ...) . (16) D

Proof : By definition 5"' 1

n(a1) = - ⁽ⁿ+ ¹⁾!

n 1 2 - 1

jY+' (nlz - 1) ! nl, ! ...

x (&) ^...In Tr e-@"

Taking the derivatives a/aJl0 and d/aJo2 explicitly we have

~ n 1 +

n(6') = - ⁽ⁿ+ ¹⁾^! _X

pR-l (n12 - 1) ! ...

'12-' Tr ( s G s ~ ) ( s o . ~ ~ ) e - ~ ~

"'[ re-^"^

- ^{(Tr S,}^.Soe-aK) (Tr So. S, e-BX) (Tr e-PX)2

Because there are no further derivatives involving lines which end a t vertex 0 we can set all

and perform the trace integration over dSo. Using the facts that

where a and p denote cartesian components of the spin vector, we have

The second theorem is particularly useful. To cal- culate expansion coefficients for the free energy to tenth order for loose packed lattices 298 topologically different diagrams are required. However, only the values associated with 15 basic diagrams must be calculated directly, the rest being obtained by applica- tion of Theorem 11. A similar theorem has been proved for the cluster series method [5].,

References

[I] STANLEY (H. E.), Phys. Rev. Letters, 1968, 20, 589. [4] ESSAM (J. W.) and FISHER (M. E.), Rev. Mod. Phys., [2] STANLEY (H. E.) and LEE (M. H.). Int. J. Quantum 1970, 42, 272.

Chem. 4S, (in press).

[3] STANLEY (H. E.) and KAPLAN (T. A.), Phys. Rev. 15] (G. S.)y Phys. ¹⁹⁶⁷⁹ ¹⁵⁵⁹ 478.

Letters, 1966, 16, 981.