- Email: [email protected]
Integral representation for the spin 12 Ising ferromagnet
Solid State Communications, Vol. 24, p. 343, 1977.
Pergamon Press.
Printed in Great Britain
INTEGRAL REPRESENTATION FOR THE SPiN 1/2 ISING FERROMAGNET B. Frank and 0. Mitran Department of Physics, Concordia University, Sir George Williams Campus, 1455 de Maisonneuve Blvd. W., Montreal, Quebec H3G 1M8, Canada (Received 20 June 1977 by R. Barrie) A simpler derivation is given for a recently discovered integral representation for the spin 1/2 Ising ferromagnet. A NEW integral representation for spin 1/2 systems with Ising bond interactions has recently been found [1], using spectral representations for the relevant time correlation functions. This representation allows, inter alia, the determination of the transition temperature to within 2% for the cubic lattices. In this note, it will be shown how to obtain this representation using a much simpler method. Let us consider the Hamiltonian H
=
—
~
~
—~
~
[i.e. all the operators in (2) commutej is now exploited by making use of the integral formula [41 1air Sm (ax) ir sinh (bx) = 2b tanh (3) with the identifications b = ir/(3 and a = 0, + h~.The result is, as obtained in reference [11, =
__L
f
dt exp (ilz~t)(exp (i0~t))cosech (irt/i3).
(1) The thermal average term in the integrand can now be (4)
lJf~Jg
in which the exchange integral J~which couples the sites f and g is unrestricted (except that J~ 0) and h~ represents the value of the non-uniform applied magnetic field at site f. The formula (S~)= ~
written [1] in terms of semi-invariant averages [5, 6J, with the O~as the relevant statistical variables. Equations for higher-order correlation functions can be generated from the above equation by differentiating it with respect to the hg [7] The translationally invariant case can be obtained by setting h~= h (2) (independent ofg) in (4) and in all other equations
(13 = l/kT)
where O,
.
derived from it, while appropriate operations can be performed on the same equations to treat the case where the bond interactions are random.
~ J~gS~, g
is most easily derived using the method of Suzuki [2]. Equation (2) is the generalization to the non-uniform field case of a well-known [3] result. The fact that the Ising model is a classical system
Acknowledgements The authors wish to thank the National Research Council of Canada for financial support (Grant No. A3 151). —
REFERENCES 1.
FRANK B. &MITRAN O.,J.Phys. C1O, 2641 (1977).
2.
SUZUKI M.,Physics Letters 19, 267 (1965).
3.
MATTIS D.C., The Theory ofMagnetism, p. 272. Harper and Row, New York (1965).
4.
GRADSHTEYN, I.S. & RYZHIK I.W., Table ofIntegrals Series and Products, p. 503, No. 3.981. Academic Press, London (1965).
5.
KIRKWOOD J.G., J. Chem. Phys. 6,70(1938).
6.
HORWITZ G. & CALLEN H.,Phys. Rev. 124, 1757 (1961).
7.
LEE K.C. & BARRIE R., Can. J. Phys. 47,769(1969).
343
Pergamon Press.
Printed in Great Britain
INTEGRAL REPRESENTATION FOR THE SPiN 1/2 ISING FERROMAGNET B. Frank and 0. Mitran Department of Physics, Concordia University, Sir George Williams Campus, 1455 de Maisonneuve Blvd. W., Montreal, Quebec H3G 1M8, Canada (Received 20 June 1977 by R. Barrie) A simpler derivation is given for a recently discovered integral representation for the spin 1/2 Ising ferromagnet. A NEW integral representation for spin 1/2 systems with Ising bond interactions has recently been found [1], using spectral representations for the relevant time correlation functions. This representation allows, inter alia, the determination of the transition temperature to within 2% for the cubic lattices. In this note, it will be shown how to obtain this representation using a much simpler method. Let us consider the Hamiltonian H
=
—
~
~
—~
~
[i.e. all the operators in (2) commutej is now exploited by making use of the integral formula [41 1air Sm (ax) ir sinh (bx) = 2b tanh (3) with the identifications b = ir/(3 and a = 0, + h~.The result is, as obtained in reference [11, =
__L
f
dt exp (ilz~t)(exp (i0~t))cosech (irt/i3).
(1) The thermal average term in the integrand can now be (4)
lJf~Jg
in which the exchange integral J~which couples the sites f and g is unrestricted (except that J~ 0) and h~ represents the value of the non-uniform applied magnetic field at site f. The formula (S~)= ~
written [1] in terms of semi-invariant averages [5, 6J, with the O~as the relevant statistical variables. Equations for higher-order correlation functions can be generated from the above equation by differentiating it with respect to the hg [7] The translationally invariant case can be obtained by setting h~= h (2) (independent ofg) in (4) and in all other equations
(13 = l/kT)
where O,
.
derived from it, while appropriate operations can be performed on the same equations to treat the case where the bond interactions are random.
~ J~gS~, g
is most easily derived using the method of Suzuki [2]. Equation (2) is the generalization to the non-uniform field case of a well-known [3] result. The fact that the Ising model is a classical system
Acknowledgements The authors wish to thank the National Research Council of Canada for financial support (Grant No. A3 151). —
REFERENCES 1.
FRANK B. &MITRAN O.,J.Phys. C1O, 2641 (1977).
2.
SUZUKI M.,Physics Letters 19, 267 (1965).
3.
MATTIS D.C., The Theory ofMagnetism, p. 272. Harper and Row, New York (1965).
4.
GRADSHTEYN, I.S. & RYZHIK I.W., Table ofIntegrals Series and Products, p. 503, No. 3.981. Academic Press, London (1965).
5.
KIRKWOOD J.G., J. Chem. Phys. 6,70(1938).
6.
HORWITZ G. & CALLEN H.,Phys. Rev. 124, 1757 (1961).
7.
LEE K.C. & BARRIE R., Can. J. Phys. 47,769(1969).
343