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Sixth frequency moment of two-spin correlation functions in a linear chain of spins at high temperature
Volume 49A, number 1
PHYSICS LETTERS
12 August 1974
SIXTH FREQUENCY MOMENT OF TWO-SPIN CORRELATION FUNCTIONS IN A LINEAR CHAIN OF SPINS AT HIGH TEMPERATURE M. PLAINDOUX Laboratoire de Chimie Organique Physique. Départernent de Recherche Fondarnentale, Centre d’Etudes Nucléaires de Grenoble, B.P. 85, Centre de Tn, F. 3804] Grenoble Cedex, France Received 11 July 1974 We have computed for an arbitrary spin the sixth frequency moments of the two-spin correlation function for an anisotropic Heisenberg Hamiltonian in a linear chain of spins at high temperature, and we compare our results with those previously known.
The knowledge of the first frequency moments (W2’~)rL,of the frequency-wave-vector-dependent correlation function (S7(t) S7(O))k,~(ct = x,y, z) is important because one may describe both the short time and the long time behaviour of the two-spin time correlation function F~°’(t) = (S7(t) S7(Op. This latter behaviour is particularly interesting for linear chains of spins where diffusion processes take on a great importance. This problem has been investigated many times in the literature. Nevertheless, for the sixth frequency moments there remains some discrepancies between different calculations performed in the case of a linear chain of spins at infinite temperature. The calculation of McFadden and Tahir-Kheli [1] for an isotropic Heisenberg Hamiltonian is not in agreement with two other computations: i) The spin 1/2 calculation of Morita et al. [2]. ii) The infinite spin result of Tomita and Mashiyama [3]. In order to clarify this point and to improve the description of the correlation function, we have computed these moments for the more general case of an anisotropic Heisenberg Hamiltonian with interaction limited to the nearest neighbor and for an arbitrary spin: =
2
E [J
11S~S~÷1 +J1(S~S~~1 +S~S~~1)J.
(1)
The normalised sixth frequency moment is explicitly given by exp {—ik(r1—r1)} tr ~ [1C,[~C,[~C,S~]]] [JC,[~JC, [~IC,S7]] J}.
(2)
After performing computation of the third2order commutators and evaluating the different traces, we get: 1011 X2+47)+2.J~J~(l2228 X2—ll99X---27) (~6)zz 4~XJ~(l—cosk){4J~(6792X —
+J~(9696X2 2168X + 111)— 35OXJ~[2J~(52X—3)+J~(26X+1)] cosk+ 14000 X2J~cos2k} —
=
=
j~-X{14J~ (3092 x2—486x+22) +J~J~(29392 X2
—
(3)
3236X— 153)
+5J~J~(7832X2—1 l36X1-57)1-50J~(2l6X2—30X+1)—2J
2—2l88X+5 1) 11.J1[2J~(21O6lX +2J’~(7824X2—592X+9)+25J~J~(1962X2—192X+5)] cosk +7OOJ~X[3J~(4X—l)+2J~Jf(37X+2) +
6J~(14X—1)]cos2k
—
28O0OJ~J~X2 cos3k}
(4)
Volume 49A, number I
PHYSICS LETTERS
12 August 1974
where X=4S(S +1). For the XY model (J 11 = 0 in eq. (3))of spin S = 1/2, one gets for the longitudinal moment: 6)~ = l28OJ~sin6k/2,
(5)
(w
which is the value that one obtains by treating exactly this model [4] Ifwe extract from (w6)~~ the k independent term we obtain for S = 1/2: .
=
4(100J~+ 55J~J~ + 8J~J~).
(6)
This value is in agreement with the results of Morita et a]. [2]. For infinite spin in the isotropic case (J=J 1 =J1 in eq. (3)) one gets: 6(1—cosk)(438—325cosk+lOOcos2k) (7) (w)k,w =~J which is in agreement with the calculation of Tomita and Mashiyama [3]. On the other hand, for the isotropic Heisenberg Hamiltonian (J = J 1~= J1 in eq. (3)) we have: 6)kW ~4~XJ6(1 cos k){1752X2— 246X+7 —50X(26X—l) cos k+400X2 cos2 k}. (8) —
(w
This expression does not agree with the results of McFadden and Tahir-Kheli [1]. Martin and Tahir-Kheli [5,6] described the two-spin time dependent correlation function Fr(t) by giving phenomenologically a gaussian shape to the diffusivity function D°°(k,w). This procedure uses only the second and fourth moments. With the sixth moment we are now able to improve this description, using for D~(k,w)a Cram-Charlier expansion [81.The diffusion constant D = urn (k 0, w 0)D~(k,b.,) has been evaluated in this approximation for the isotropic case. Our values are significantly larger than the values obtained by using the simple Gaussian approximation. For S = 1/2 D is increased by 19%, and by 30% for S> 1/2. For infinite spin we get a diffusion constant of 1.62 J(4X)L’2 this value is drawing nearer to the value 1.33 ~ = 2.30 ±0.2 obtained recently by Lurie, Huber and Blume [7] by computer simulation of classical Heisenberg chain. The non Gaussian behaviour of the diffusivity will be discussed in a future paper. -+
~
We wish to thank Prof. A. Rassat, Dr. J.P. Boucher, Dr. C. Anderson Evans and Dr. M. Nechtschein for many valuable comments.
References [1]
D.G. McFadden and R.A. Tahix-Kheli, Phys. Rev. Bl (1970) 3671. [2] T. Morita, K. Kobayashi, S. Katsura and Y. Abe, Phys. Lett. 32A (1970) 367. [31 I-I. Tomita and H. Mashiyama, Prog. Theor. Phys. 48 (1972) 1133. [4] S. Katsura, 1. Horiguchi and M. Suzuki, Physica 46 (1970) 67. [5] H.S. Bennett and P.C. Martin, Phys. Rev. 138 (1965) A608. [61 D.G. McFadden and R.A. Tahir, Phys. Rev. Bi (1970) 3649. [7] N.A. Lurie, D.L. Huber and M. Blume, Phys. Rev. B9 (1974) 2171. 181 M.F. Collins and W. Marshall, Proc. Phys Soc. 92 (1967) 390.
2
PHYSICS LETTERS
12 August 1974
SIXTH FREQUENCY MOMENT OF TWO-SPIN CORRELATION FUNCTIONS IN A LINEAR CHAIN OF SPINS AT HIGH TEMPERATURE M. PLAINDOUX Laboratoire de Chimie Organique Physique. Départernent de Recherche Fondarnentale, Centre d’Etudes Nucléaires de Grenoble, B.P. 85, Centre de Tn, F. 3804] Grenoble Cedex, France Received 11 July 1974 We have computed for an arbitrary spin the sixth frequency moments of the two-spin correlation function for an anisotropic Heisenberg Hamiltonian in a linear chain of spins at high temperature, and we compare our results with those previously known.
The knowledge of the first frequency moments (W2’~)rL,of the frequency-wave-vector-dependent correlation function (S7(t) S7(O))k,~(ct = x,y, z) is important because one may describe both the short time and the long time behaviour of the two-spin time correlation function F~°’(t) = (S7(t) S7(Op. This latter behaviour is particularly interesting for linear chains of spins where diffusion processes take on a great importance. This problem has been investigated many times in the literature. Nevertheless, for the sixth frequency moments there remains some discrepancies between different calculations performed in the case of a linear chain of spins at infinite temperature. The calculation of McFadden and Tahir-Kheli [1] for an isotropic Heisenberg Hamiltonian is not in agreement with two other computations: i) The spin 1/2 calculation of Morita et al. [2]. ii) The infinite spin result of Tomita and Mashiyama [3]. In order to clarify this point and to improve the description of the correlation function, we have computed these moments for the more general case of an anisotropic Heisenberg Hamiltonian with interaction limited to the nearest neighbor and for an arbitrary spin: =
2
E [J
11S~S~÷1 +J1(S~S~~1 +S~S~~1)J.
(1)
The normalised sixth frequency moment is explicitly given by exp {—ik(r1—r1)} tr ~ [1C,[~C,[~C,S~]]] [JC,[~JC, [~IC,S7]] J}.
(2)
After performing computation of the third2order commutators and evaluating the different traces, we get: 1011 X2+47)+2.J~J~(l2228 X2—ll99X---27) (~6)zz 4~XJ~(l—cosk){4J~(6792X —
+J~(9696X2 2168X + 111)— 35OXJ~[2J~(52X—3)+J~(26X+1)] cosk+ 14000 X2J~cos2k} —
=
=
j~-X{14J~ (3092 x2—486x+22) +J~J~(29392 X2
—
(3)
3236X— 153)
+5J~J~(7832X2—1 l36X1-57)1-50J~(2l6X2—30X+1)—2J
2—2l88X+5 1) 11.J1[2J~(21O6lX +2J’~(7824X2—592X+9)+25J~J~(1962X2—192X+5)] cosk +7OOJ~X[3J~(4X—l)+2J~Jf(37X+2) +
6J~(14X—1)]cos2k
—
28O0OJ~J~X2 cos3k}
(4)
Volume 49A, number I
PHYSICS LETTERS
12 August 1974
where X=4S(S +1). For the XY model (J 11 = 0 in eq. (3))of spin S = 1/2, one gets for the longitudinal moment: 6)~ = l28OJ~sin6k/2,
(5)
(w
which is the value that one obtains by treating exactly this model [4] Ifwe extract from (w6)~~ the k independent term we obtain for S = 1/2: .
=
4(100J~+ 55J~J~ + 8J~J~).
(6)
This value is in agreement with the results of Morita et a]. [2]. For infinite spin in the isotropic case (J=J 1 =J1 in eq. (3)) one gets: 6(1—cosk)(438—325cosk+lOOcos2k) (7) (w)k,w =~J which is in agreement with the calculation of Tomita and Mashiyama [3]. On the other hand, for the isotropic Heisenberg Hamiltonian (J = J 1~= J1 in eq. (3)) we have: 6)kW ~4~XJ6(1 cos k){1752X2— 246X+7 —50X(26X—l) cos k+400X2 cos2 k}. (8) —
(w
This expression does not agree with the results of McFadden and Tahir-Kheli [1]. Martin and Tahir-Kheli [5,6] described the two-spin time dependent correlation function Fr(t) by giving phenomenologically a gaussian shape to the diffusivity function D°°(k,w). This procedure uses only the second and fourth moments. With the sixth moment we are now able to improve this description, using for D~(k,w)a Cram-Charlier expansion [81.The diffusion constant D = urn (k 0, w 0)D~(k,b.,) has been evaluated in this approximation for the isotropic case. Our values are significantly larger than the values obtained by using the simple Gaussian approximation. For S = 1/2 D is increased by 19%, and by 30% for S> 1/2. For infinite spin we get a diffusion constant of 1.62 J(4X)L’2 this value is drawing nearer to the value 1.33 ~ = 2.30 ±0.2 obtained recently by Lurie, Huber and Blume [7] by computer simulation of classical Heisenberg chain. The non Gaussian behaviour of the diffusivity will be discussed in a future paper. -+
~
We wish to thank Prof. A. Rassat, Dr. J.P. Boucher, Dr. C. Anderson Evans and Dr. M. Nechtschein for many valuable comments.
References [1]
D.G. McFadden and R.A. Tahix-Kheli, Phys. Rev. Bl (1970) 3671. [2] T. Morita, K. Kobayashi, S. Katsura and Y. Abe, Phys. Lett. 32A (1970) 367. [31 I-I. Tomita and H. Mashiyama, Prog. Theor. Phys. 48 (1972) 1133. [4] S. Katsura, 1. Horiguchi and M. Suzuki, Physica 46 (1970) 67. [5] H.S. Bennett and P.C. Martin, Phys. Rev. 138 (1965) A608. [61 D.G. McFadden and R.A. Tahir, Phys. Rev. Bi (1970) 3649. [7] N.A. Lurie, D.L. Huber and M. Blume, Phys. Rev. B9 (1974) 2171. 181 M.F. Collins and W. Marshall, Proc. Phys Soc. 92 (1967) 390.
2