Z. Phys. B - Condensed Matter 60, 57-60 (1985) Condensed Zeitschrift Matter far Physik B

�9 Springer-Verlag 1985

Ising Spin System with a Rectangular Distribution of Internal Energy Parameters

A. Freudenhammer

Universit~t-GH-Duisburg, FB 10 - Theoretische FestkSrperphysik, Federal Republic of Germany

Received March 21, 1985

The model of a spin system with internal degrees of freedom is solved for a rectangular distribution of the internal energy parameters. For simplicity, the interaction between the spin clusters is taken as the infinite range interaction. We obtain second and first order phase transitions with a tricritical point or a critical end point and an isolated critical point.

I. Introduction

The first example of an Ising System with internal degrees of freedom [ ! ] was examined in order to show that the very weak coupled additional spins in a lattice cause typical spin glass properties in the susceptibility and specific heat. A second example in which both spins of a cluster take part in the in- teraction with the surrounding spins [2] already showed parts of the phase diagram presented in this paper for instance a critical end point and an isolat- ed critical point. The simplest subsystem of spins which possesses internal energy levels is a two spin cluster that may be taken as Ising spins for sim- plicity. Therefore we define the total spin S~ = sil + s~2 in order to build a subsystem. This spin variable can take the values S~ = _+ 1/2 _+ 1/2 = 4-_ 1, 0, 0. This model deviates considerably from that introduced by Blume [3], Capel [4] and Griffith [5] because of the double degeneracy of both energy levels in the term (S~)< Systems with random fields also can be in- terpreted as those with internal energy levels as in- vestigated by Aharony, Galam etal. and Morgen- stern et al. [6]. As the interaction between the clusters we took the infinite range interaction, which is iiquivalent to the mean field approximation. As a consequence the mean values of a quantity may be calculated using the mean values of the distribution. This result comes out if one takes the macroscopic limit in the case of infinite range interaction.

A larger part of the phase diagram for the model with the two 5-peak distribution of the internal in- teraction parameters is given in [7]. There we have obtained two phase transitions of the first order and one of the second order. As shown for the con- tinuous distribution, isolated critical points as well as tricritical points and critical end points occur for the two 6-peak distribution. In the next Sect. II the model is given with the Gibbs free energy G. The expansion for low temperatures and the Landau ex- pansion of G is given in Sect. III. The phase diagram is discussed in Sect. IV and a conclusion is drawn in Sect. V.

II. The Model

A spin system with infinite range interaction Jo > 0 is considered. The hamiltonian is given by

H= -~n ~ SIS1- JiS~-h S~, (1) i ~ j i = 1 i = I

where S i = _+ 1, 0, 0 and h is the external field. The parameters of the internal energies Ji determine the internal ground state of -J iS~. For ferromagnet- ic Ji (Ji >0) we have tSi] = 1 and for antiferromagnet- ic (J~<0) we obtain a noninteracting ground state S~ =0. For simplicity we choose the distribution of Ji to be rectangular, that is

ji ~1/(J1-J2) for Jl>__J~>J2 and J~>J2 = ~0 otherwise. (2)

58 A. Freudenhammer: Rectangular Distr ibution of Internal Energy Parameters

Normalizat ion of all parameters to J0 yields x 1 =J1/Jo, x2=J2/Jo, H=h/J o and k B �9 r / Jo~ T. Performing the saddle point method on the partition function first with the discrete Hamil tonian yields the Gibbs free energy G per subsystem

n / x,'r r e + H \ Gin=m2/2 - T ~, xiln [1 + e / c o s h ~ ) . (3)

i=1

Taking into account the distribution of (2) leads to the integral

m + H \ G/n=m2/2 T ~ldxln l + e ~ / r c o s h ~ - - ) . (4)

X 1 - - X 2 x2

The stationary points of the Gibbs free energy can be evaluated by the derivative of G

, T G (m) = m - - - tanh ((m + H)/T)

0.

1 exp (x 1/T) cosh ((m + H)/T) �9 I n ( 5 )

1 + exp (xz/T) cosh ((m + H)/T)"

The width is 0 . = x l - x 2. Calculating the low tem- perature behaviour will show that, beside the re- moval of the second order transition, the magnetic field causes only a linear shift of the transition lines of the first order. Therefore it will be set to zero where not important. The second order phase transition is determined by the change of the sign of the second derivative of G at r e = H = 0 :

G" (m = 0) = 1 - 1_ in (1 + exp (x l/T)) = 0. (6) a 1 + exp (x2/T)

Inverting (6) yields the functional form

x~ = T. in ((exp (0.) - 1)/(1 - exp (1 - l/T))). (7)

This transition line of the second order will be shown to be destabilized by a first order transition in certain parameter regions. The fourth derivative will be needed for calculating the tricritical points of the transition. GW(0) is ex- pressed along the line given by (7):

0. T 2. G TM (m = 0) = 2 0. - 3 (1/( 1 + exp ( - x 1/T))

- 1/(1 + exp ( - xz/T))). (8)

And finally the derivative of G to the sixth order will serve as a condition for destabilization of the tricritical line

aT4Gva(m = 0)= - 160.+ 15(1/(1 + exp ( - x l / T ) )

- 1/(1 + exp (-xz/T)))

+ 15 (1/(1 + exp ( - xl/T)) 2

-- 1/(1 + exp (-xz/T)):) . (9)

III. Low Temperature and Landau Expansion

The numerical calculation of the phase transition lines for low temperatures as well as certain tran- sition points are well supported by analytical ex- pressions. For low temperatures the Gibbs free energy G can be easily expanded in certain parameter regions:

Region: x~>0 , x2 < - i

G(m) has only one minimum at

m o = (x i + H - T in 2)/(0. - 1) (10)

with the condition x 1 + H > 0 and x 2 + H < - 1. The external magnetic field causes a linear shift of the behaviour as expected. The unusual linear decrease of re(T) is shown in Fig. 4 for H = 0 . The phase transition to the para- magnetic state is of second order.

Region: x z < x 1 < 0

G(m) has two minima. One at m 0 = 0 (the para- magnetic state) and at m o= 1 the ferromagnetic or- dered state. A first order transition occurs at

T 1 =(2(x l + H ) + 1-0-)/(2 ln2) (11)

for - l < x 2 + H < x ~+H<O. This transition terminates in a tricritical point (T~) or in an isolated critical point (Tk). The Landau expansion has to be taken up to the eighth power because a critical end point may occur

[2, 83.

G(m)= 2 m z + u m 4 + v m 6+win s. (12)

The tricritical points are determined by r - -u = 0 pro- vided v > 0 therefore (7) and (8) are needed. The coefficient v has to be examined to see whether v > 0 is fulfilled. It emerges that there is a point (a) of instability, i.e.v, changes sign at (a) in Fig. 1. The multicritical point a is obtained by variing 0. on the line of tricritical points, i.e. r = u = 0 and v = v(0 . ) .

The stability conditions G'(mo)=0, G"(mo)>O and G(mo)<0 have to be fulfilled

mg=3(-v)/(4w); G"(mo)~(-v)m~; (13)

G(mo)~ -v4/w 3.

Therefore a critical end point will always occur in this system on the destabilized tricritical line if v < 0 and w > 0 is satisfied.

A. Freudenhammer : Rectangular Distribution of Internal Energy Parameters 59

T

0.6

0.4

02

0 .0 V -0.5

0.001

P

TI

-0.3 -0.1

0 ~0 0 ~,95

0~5 0 ~7 0 498

& '• 0._% o.o o.1 Fig. 1. Phase diagram of the Ising model with a rectangular distribution (xa<x~<x 0 of the internal interaction parameters. The transition temperature is plotted as a function of the upper internal interaction parameter x 1. The numbers at the curves are the half of the width of the distribution (a/2). The ferromagnetic region is separated by the line T t- T 2 are the second order tran- sition lines and T~ the first order transition lines. The line of tricritical points becomes unstable and splits up at the point (a) with the coordinates (xa=5.4673 �9 10 2, Ta=0,22873 ' a/2,=0.3892) into a line of critical end points (CE) and isolated critical points (Tk). The point x 1 =0, T = 0 is one of high singularity - all transition lines with c~/2 > 0.5, terminate at this point

The destabilization of the tricritical line does not always need the condition of changing the sign of v. In a model with more than two minima of the Gibbs free energy a different destabilization mecha- nism will be shown [9].

IV. The Phase Diagram

The phase diagram shown in Fig. 1 is calculated numerically, taking into account the results of the previous section. The numbers shown in Fig. 1 are half of the width of the distribution from the in- ternal energy parameters. The ordering is ferromag- netic because of the positive infinite range interac- tion J0 >0 between the subsystems. The line of the tricritical points T t starts from the paramagnetic phase P, where o-= 0 and terminates at the destabilization point (a) separating the lines of first order transition T1 and the lines of second order transition T 2. The tricritical point splits up into the line of critical end points (CE) and the line of isolated critical points T K at the point (a). These lines terminate in the point xl = T = 0 , which is a highly singular point because in that point the ferromagnetic ground state of the subsystem with x~ near zero has a ground state which is degenerated four times. The transition lines with a half width or/2 larger than 0.5 are completely of the second order type. For strong enough antiferromagnetic internal interac-

Fig. 2. The region of the critical end points (CE) and of the isolated critical points (Tk) of Fig. 1 is enlarged for the parameters ~r/2 = 0.40 to or/2 = 0.50

1.0

0.5

0.0 . . . . ' . . . . ' ,

0.00 005 0.10 0.15 Fig. 3. The magnetization as a function of the temperature is shown in the vicinity of a critical end point (~/2=0.47, x 1 =0.06). There is a first order discontinuity followed by a second order transition. At the critical end point there is a discontinuity from below and a singularity from above

tions (x 1 >0, x 2 < - 1 , 0 ) the ordering at zero temper- ature may not be complete. According to (10) we obtain a magnetization of Xl/(a-1)=0.5 at zero temperature with a linear decrease for increasing temperature (x 1 =0.1, a = 1.2, Fig. 4). The region of critical end points and of isolated critical points is depicted in Fig. 2. The first order transition line T 1 with a/2=0.47 meets the second order transition line T 2 at the critical end point (full circle) and terminates at the isolated critical point (open circle). The phase tran- sition at the critical end point depends on the ap- proach at this point; the magnetization has a dis- continuity for an approach from low temperatures, whereas the approach from above gives singular be- haviour for instance for the susceptibility.

60 A. Freudenhammer: Rectangular Distribution of Internal Energy Parameters

ili ........... /it 0.00 0.05 0.a0 r 0.15

Fig. 4. For a half width a/2>=0.5 of the distribution of internal interaction parameters no first order phase transition occurs. The magnetization for the parameters (x 1 =0.1, a/2=0.6) shows a lin- ear behaviour in a large region. The large decrease of the magne- tization is due to the large antiferromagnetic internal interaction present for the chosen values x 1 =0.1 and x 2 = -1.1

The ferromagnetic ordering is completely suppressed for larger ant iferromagnetic internal interaction pa- rameters. Figure 3 shows the magnet izat ion as a function of the temperature for parameters x t = 0 . 0 6 and a/2 = 0.47 in the vicinity of this critical end point where the described behaviour can be seen. An external magnet ic field causes a shift of the tran- sition lines of the first order and the second order phase transit ion disappears. The dependence of the isolated critical points is discussed in [9].

V. Conclusion

The ordinary Ising model with infinite range in- teract ion possesses only a ferromagnetic phase tran-

sition. The addit ion of antiferromagnetic internal in- teract ion enlarges the number of phase transitions as has been shown for a distribution of two &peaks [7, 9]. These addit ional phase transitions of the first order disappear in the case of a cont inuous distribu- t ion of the internal interaction parameters. But, as in the case of two 6-peaks, the second order transit ion is suppressed by increasing the antiferromagnetic in- ternal distr ibution; further increasing of the antifer- romagnet ic part leads to a tricritical point and a transit ion line of the first order so that at the end the whole ordering can be damaged be a sufficiently large antiferromagnetic internal interaction. The line of tricritical points becomes unstable at the parameter a /2=0.3892, x r = 5 . 4 6 7 3 . 1 0 -2 and T =0.22873. It splits off into a line of critical end points and a line of isolated critical points.

References

1. Freudenhammer, A.: Z. Phys. B - Condensed Matter 40, 111 (1980)

2. Freudenhammer, A.: J. Magn. Magn. Mater. 28, 219 (1982) 3. Blume, M.: Phys. Rev. 141, 517 (1966) 4. Capel, H.W.: Physica 32, 966 (1966) 5. Griffith, R.B.: Physica 33, 689 (1966) 6. Aharony, A.: Phys. Rev. B18, 3318 (1978)

Galam, S., Birman, J.L.: Phys. Rev. B28, 5322 (1983) Morgenstern, I., Binder, K., Hornreich, R.M.: Phys. Rev. B23, 287 (1981)

7. Freudenhammer, A.: Proceedings of the PMM in Jadwisin Poland, (September, 1984). Suppl. Acta Phys. Polon. (in press)

8. Ziman, T.A.L., Amit, D.J., Grinstein, G., Jayaprakash, C.: Phys. Rev. B25, 319 (1982)

9. Freudenhammer, A.: (to be published)

A. Freudenhammer Fachbereich 10 Theoretische Festk~Srperphysik Universit~it - Gesamthochschule Duisburg Bismarckstrasse 90 D-4100 Duisburg 1 Federal Republic of Germany