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Spin glass transition in an Ising spin system with transverse field
0038-1098/86 $3.00 + .00 Pergamon Journals Ltd.
Solid State Communications, Vol. 58, No. 9, pp. 629-630, 1986. Printed in Great Britain.
SPIN GLASS TRANSITION IN AN ISING SPIN SYSTEM WITH TRANSVERSE FIELD K.D. Usadel Theoretische Physik, Universit~it Duisburg, 4100 Duisburg 1, West Germany
(Received 29 January 1986 by C. W. McCombie) The effect of a transverse field A on the spin glass freezing temperature for an Ising spin glass model is studied. The freezing temperature is calculated for all values of A and it is shown that for A greater than twice the exchange interaction no spin glass transition takes place.
IN A RECENT PAPER Ishii and Yamamoto [1] studied the effect of a transverse field on the spin glass freezing temperature of the conventional SherringtonKirkpatrick [2] model of an Ising spin glass. The decrease of the transition temperature Tc with increasing strength A of the transverse field was calculated close to A = 0 and it was argued that the system always undergoes a spin glass transition irrespective of the magnitude of A. We have reexamined this problem since we were sceptical with respect to this statement. Indeed it is possible to argue that in a very large transverse field spins will align to this field, the exchange in the spin components perpendicular to the field will be unimportant and thus no phase transition will occur. In this paper we present detailed calculations of To(A) for all values of A. Our results agree with the results obtained by Ishii and Yamamoto [1] close to T¢(0), the region their calculation is restricted to. In addition we present a numerical calculation of To(A) for all values of A and we will show that To(A)= 0 for A > 2 J . A transverse field A applied to an Ising spin system brings about quantum effects by causing spin-flips. For these quantum spin glasses a well established theory is available which was obtained by Bray and Moore [3] and independently by Sommers [4] and Sommers and Usadel [5]. Meanwhile this theory has been applied with success to spin glass systems with uniaxial anisotropy [6, 7]. In this paper we will show how this theory can be used for treating an Ising system in a transverse field. The Hamiltonian for N interacting Ising spins in a transverse field A reads
HIS =
- - i~: ,j
Ji'iOizOJz q-
A "2
~i" (ai+ + oi-),
and a variance of J2/N. On the other hand a theory of quantum spin glasses usually starts with an Hamiltonian which is isotropic in the exchange, Hex = -- Z JijSi'Si,
(2)
i,i
where Si denotes the spin operator of the i'th spin. Thus local anisotropic terms have to be added to Hex in order that those spin eigenstates are projected out which do not appear in equation (1). This is achieved with the following ansatz
H = -- • JijSi" S j - Z D@]z -- l) i,)
+
A
j
(3)
Z (s?++ s?_), $
where magnetic moments with spin 1 are considered. The z-components m of these magnetic moments are 0 and -+ 1. The states m = 0 are projected out for D -~ + oo and the operators S~ connect the states m - - - - 1 and + 1 as is required for the Ising limit. Thus, for D + + 0% H goes over into His. Following closely the paper by Sommers and Usadel [5] a functional for the averaged free energy for a Hamiltonian of the type presented in equation (3) in the so-called static approximation can be derived. This functional which is given by 1
f(ux, u
,uz) =
+
+
l" d3r expl (-- r2/2) Tr eg(r), (4) + In j (21r)S/2
(1) with
where a/ are the Pauli matrices for the j'th spin and oi+_ denotes the usual spin raising and lowering operators. The exchange interactions J# in equation (1) are quenched and independently distributed with zero mean 629
h(r) = flD(S2z -- 1 ) + ~ @ + 2 + S2) + ~
+
+
Iz
z,
xS x
(5)
630
SPIN GLASS TRANSITION IN AN ISING SPIN SYSTEM '
I
Vol. 58, No. 9
The transition temperature can be calculated analytically for A ~ 0 by expanding equation (6) or the resulting equation for/a(T, A) to first order in A2. Using equation (7) one obtains the following result
'
For the coefficient a appearing in equation (8) we obtain 1 o~ = e -In _~ dx ex2n -- 1/2 = 0, 225 . . . ,
(9)
0
0
J
0
I
L
I
05
1
Fig. 1. Phase diagram for the Ising spin glass in a transverse field. T/J = t reduced temperature, A/J = 8 reduced field strength. The dashed lines are the results of an expansion around t = 1 and t = 0, respectively, see equations (8) and (10). depends on variational parameters /ax, /lr, ~Uz. Selfconsistency requires that f is stationary with respect to small variations of these parameters. The trace appearing in equation (4) is the sum of the three eigenvalues of the operator exp (h(r)). These eigenvalues can be obtained analytically and after a longer but straightforward calculation we end up with the following expression for the free energy in the limit D = oo 1
f(u~, uy, uz) = - ~ (u~ + + In
which agrees numerically with the result of Ishii and Yamamoto [1]. It is also possible to study the low temperature limit /3--*oo analytically by expanding equation (6) in terms of (2~lazZ2)/(~A) 2 with the result that no transition is obtained for A > 2J. For 8 = A/J smaller than 2 but very close to 2 the reduced transition temperature te = Tc(A)/J is given by tc = ~ ( 2 - 8 ) ( 8
Acknowledgement - The author would like to thank K. Bien for the numerical calculation of the Tc(A)curve.
REFERENCES 1. 2.
expl (-- z2/2)
3.
(/~A)2) 1/2.
(6) 4.
Stationarity of f with respect to variations of the parameters #x,/~y,/az immediately leads to/~x =/ay = 0. From the resulting equation for/az =~u(T, A) the transition temperature can be obtained as that temperature for which
2U J
- - =
1.
(10)
The full Tc(A)-curve has been obtained numerically and the result is shown in Fig. I. This curve corresponds to the so-called de AlmeidaThouless [8] line for an Ising spin glass in a longitudinal magnetic field.
u2 + u~)
x 2 cosh (21~lazZ2 +
- 1) 2.
(7)
5. 6. 7. 8.
H. Ishii & T. Yamamoto, J. Phys. C.: Solid State Phys. 18, 6225 (1985). D. Sherrington & S. Kirkpatrick, Phys. Rev. Lett. 32, 1972 (1975). A.J. Bray & M.A. Moore, J. Phys. C.: Solid State Phys. 13L, 655 (1980). H.J. Sommers, J. Magn. Magn. Mater. 22, 267 (1981). H.J. Sommers & K.D. Usadel, Z. Phys. B47, 63 (1982). K.D. Usadel, K. Bien & H.J. Sommers, Phys. Rev. B27, 6957 (1983). G. Brieskorn & K.D. Usadel, to appear in J. Phys. C. (1986). J.R.L. de Almeida & D.J. Thouless, J. Phys. A l l , 983 (1978).
Solid State Communications, Vol. 58, No. 9, pp. 629-630, 1986. Printed in Great Britain.
SPIN GLASS TRANSITION IN AN ISING SPIN SYSTEM WITH TRANSVERSE FIELD K.D. Usadel Theoretische Physik, Universit~it Duisburg, 4100 Duisburg 1, West Germany
(Received 29 January 1986 by C. W. McCombie) The effect of a transverse field A on the spin glass freezing temperature for an Ising spin glass model is studied. The freezing temperature is calculated for all values of A and it is shown that for A greater than twice the exchange interaction no spin glass transition takes place.
IN A RECENT PAPER Ishii and Yamamoto [1] studied the effect of a transverse field on the spin glass freezing temperature of the conventional SherringtonKirkpatrick [2] model of an Ising spin glass. The decrease of the transition temperature Tc with increasing strength A of the transverse field was calculated close to A = 0 and it was argued that the system always undergoes a spin glass transition irrespective of the magnitude of A. We have reexamined this problem since we were sceptical with respect to this statement. Indeed it is possible to argue that in a very large transverse field spins will align to this field, the exchange in the spin components perpendicular to the field will be unimportant and thus no phase transition will occur. In this paper we present detailed calculations of To(A) for all values of A. Our results agree with the results obtained by Ishii and Yamamoto [1] close to T¢(0), the region their calculation is restricted to. In addition we present a numerical calculation of To(A) for all values of A and we will show that To(A)= 0 for A > 2 J . A transverse field A applied to an Ising spin system brings about quantum effects by causing spin-flips. For these quantum spin glasses a well established theory is available which was obtained by Bray and Moore [3] and independently by Sommers [4] and Sommers and Usadel [5]. Meanwhile this theory has been applied with success to spin glass systems with uniaxial anisotropy [6, 7]. In this paper we will show how this theory can be used for treating an Ising system in a transverse field. The Hamiltonian for N interacting Ising spins in a transverse field A reads
HIS =
- - i~: ,j
Ji'iOizOJz q-
A "2
~i" (ai+ + oi-),
and a variance of J2/N. On the other hand a theory of quantum spin glasses usually starts with an Hamiltonian which is isotropic in the exchange, Hex = -- Z JijSi'Si,
(2)
i,i
where Si denotes the spin operator of the i'th spin. Thus local anisotropic terms have to be added to Hex in order that those spin eigenstates are projected out which do not appear in equation (1). This is achieved with the following ansatz
H = -- • JijSi" S j - Z D@]z -- l) i,)
+
A
j
(3)
Z (s?++ s?_), $
where magnetic moments with spin 1 are considered. The z-components m of these magnetic moments are 0 and -+ 1. The states m = 0 are projected out for D -~ + oo and the operators S~ connect the states m - - - - 1 and + 1 as is required for the Ising limit. Thus, for D + + 0% H goes over into His. Following closely the paper by Sommers and Usadel [5] a functional for the averaged free energy for a Hamiltonian of the type presented in equation (3) in the so-called static approximation can be derived. This functional which is given by 1
f(ux, u
,uz) =
+
+
l" d3r expl (-- r2/2) Tr eg(r), (4) + In j (21r)S/2
(1) with
where a/ are the Pauli matrices for the j'th spin and oi+_ denotes the usual spin raising and lowering operators. The exchange interactions J# in equation (1) are quenched and independently distributed with zero mean 629
h(r) = flD(S2z -- 1 ) + ~ @ + 2 + S2) + ~
+
+
Iz
z,
xS x
(5)
630
SPIN GLASS TRANSITION IN AN ISING SPIN SYSTEM '
I
Vol. 58, No. 9
The transition temperature can be calculated analytically for A ~ 0 by expanding equation (6) or the resulting equation for/a(T, A) to first order in A2. Using equation (7) one obtains the following result
'
For the coefficient a appearing in equation (8) we obtain 1 o~ = e -In _~ dx ex2n -- 1/2 = 0, 225 . . . ,
(9)
0
0
J
0
I
L
I
05
1
Fig. 1. Phase diagram for the Ising spin glass in a transverse field. T/J = t reduced temperature, A/J = 8 reduced field strength. The dashed lines are the results of an expansion around t = 1 and t = 0, respectively, see equations (8) and (10). depends on variational parameters /ax, /lr, ~Uz. Selfconsistency requires that f is stationary with respect to small variations of these parameters. The trace appearing in equation (4) is the sum of the three eigenvalues of the operator exp (h(r)). These eigenvalues can be obtained analytically and after a longer but straightforward calculation we end up with the following expression for the free energy in the limit D = oo 1
f(u~, uy, uz) = - ~ (u~ + + In
which agrees numerically with the result of Ishii and Yamamoto [1]. It is also possible to study the low temperature limit /3--*oo analytically by expanding equation (6) in terms of (2~lazZ2)/(~A) 2 with the result that no transition is obtained for A > 2J. For 8 = A/J smaller than 2 but very close to 2 the reduced transition temperature te = Tc(A)/J is given by tc = ~ ( 2 - 8 ) ( 8
Acknowledgement - The author would like to thank K. Bien for the numerical calculation of the Tc(A)curve.
REFERENCES 1. 2.
expl (-- z2/2)
3.
(/~A)2) 1/2.
(6) 4.
Stationarity of f with respect to variations of the parameters #x,/~y,/az immediately leads to/~x =/ay = 0. From the resulting equation for/az =~u(T, A) the transition temperature can be obtained as that temperature for which
2U J
- - =
1.
(10)
The full Tc(A)-curve has been obtained numerically and the result is shown in Fig. I. This curve corresponds to the so-called de AlmeidaThouless [8] line for an Ising spin glass in a longitudinal magnetic field.
u2 + u~)
x 2 cosh (21~lazZ2 +
- 1) 2.
(7)
5. 6. 7. 8.
H. Ishii & T. Yamamoto, J. Phys. C.: Solid State Phys. 18, 6225 (1985). D. Sherrington & S. Kirkpatrick, Phys. Rev. Lett. 32, 1972 (1975). A.J. Bray & M.A. Moore, J. Phys. C.: Solid State Phys. 13L, 655 (1980). H.J. Sommers, J. Magn. Magn. Mater. 22, 267 (1981). H.J. Sommers & K.D. Usadel, Z. Phys. B47, 63 (1982). K.D. Usadel, K. Bien & H.J. Sommers, Phys. Rev. B27, 6957 (1983). G. Brieskorn & K.D. Usadel, to appear in J. Phys. C. (1986). J.R.L. de Almeida & D.J. Thouless, J. Phys. A l l , 983 (1978).