NUMERICAL QUANTUM TRANSFER-MATRIX RESULTS FOR A SPIN CHAIN CORRESPONDING TO CHAB

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NUMERICAL QUANTUM TRANSFER-MATRIX

RESULTS FOR A SPIN CHAIN CORRESPONDING

TO CHAB

Th. Delica, R. Gerling, H. Leschke

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C8, Suppl6ment au no 12, Tome 49, d6cembre 1988

NUMERICAL QUANTUM TRANSFER-MATRIX RESULTS FOR A SPIN CHAIN

CORRESPONDING TO CHAB

Th. Delica,

R. W.

H.

Institut fur Theoretische Physik der Universitat, 8520 Erlangen, F. R. G .

Abstract. - A variant of the quantum transfer-matrix method is used to obtain numerically accurate results for the specific heat of an easy-plane spin chain with an in-plane field, corresponding to the ferromagnetic compound CHAB. They are contrasted with predictions of the classical sine-Gordon approximation for the chain and with experimental data.

The interest in easy-plane spin chains with in- plane magnetic fields as described, e.g., by the Hamil- tonian N-1 ,E 1.6 -7 H = -J

x

[SSST+~

+

s:s,Y+,+

-

results mainly from two facts:

first, from the discovery of quite a lot of mag- netic compounds exhibiting quasi one-dimensional be- haviour above their ordering temperatures

T,,

second, from the observation [I, 21 that, in the limit of a continuum of classical spins and of large anisotropy y, the model (1) is equivalent t o the La- grangian of a sine-Gordon field.

One of the best studied compounds in this context is the ferromagnet ( C B H ~ ~ N H ~ ) CuBr3 also called CHAB

[3, 41. It is believed that, above its T, = 1.5 K, CHAB may be described by (1) with the parameters [5]

Fig. 1. - Specific heat C versus 1/N, the inverse total spin number. The temperature T is 5 K , the field B is 5 kG.

the different parameters (2)) in comparison with [a], because it is expected that deviations of CHAB from ideal model behaviour become less important when the field is increased 151.

Figures 2-5 display our results for the excess spe- cific heat AC = C (B) - C (0). They were obtained by a numerical differentiation of the internal energy

s = 112, J / l c ~ = 126 K, y = 0.044, g = 2 ₍₂₎

which slightly differ from previous values.

Recently, numerical results for CHAB, showing dis- tinct differences t o the previous ones [6, 71, have been published by one of the present authors 181. These re- sults were obtained by a variant [9] of the quantum transfer-matrix method which we beIieve to be supe- rior to other numerical procedures for two reasons:

first, better convergence in the Trotter number m has been established by computing the internal energy from the nearest-neighbour spin-correlation functions in the real-space decomposition [9, 101,

second, for a total spin number N = 150 an ex- _{B (kG1} trapolation t o N = oo seems to be unnecessary (see,

e.g., Fig. 1 for the model (I), (2)). _{Fig. 2.}

- Field dependence of the excess specific heat A C = In the present study we have applied the Same _C (B) -C (0)

.

method to the model (1) for higher fields B (and with for the temperature in units of K.

'present address: Center for Simulational Physics, University of Georgia, Athens, GA 30602, U.S.A.

C8 - 1586 JOURNAL DE PHYSIQUE which, in its turn, was extrapolated to m = co from

the results for m = 7 and m = 8 and N = 150. From figure 3 one observes, in comparison with the results for lower fields [8], worse agreement with the univer- sal behaviour predicted by the classical sine-Gordon picture. Figure 4 shows that the linear variation of the peak height AC,, with temperature T breaks down for higher T . The data points of figure 5 for

Fig. 3. - Reduced excess specific heat A ~ / f i versus t h e reduced temperature T/&. Values for the field &re indicated.

Fig. 4. - Height AC,, of the maximum in the excess specific heat as a function of T. Numerical results from figure 2 are given by circles, experimental results [4] by

triangles.

the peak position are well fitted by Bpeak = b~~ with b = 0.25 ~ G / K ~ , while the experimental value [4] is b = 0.21 ~ G / K ~ , and the (original) sine-Gordon pic- ture [ll] predicts b = 0.20 ~ G / K ~ .

It will be intersting to compare the present results with additional experimental data for higher fields.

Fig. 5. - Magnetic field Bpeak a t which the excess specific heat versus B is a maximum as a function of T2. Numeri- cal results from figure 2 are given by circles, experimental results [4] by triangles.

Acknowledgments

This work was supported in part by the Deutsche Forschungsgemeinschaft. We want t o thank K. Kopinga for encouragement and helpful discussions. We acknowledge the hospitality of the Center for Sim- ulational Physics a t the University of Georgia where part of this research was carried out. The final com- putations were performed on the CYBER 205 of the HFK Karlsruhe.

[I] Mikeska, H. J., J. Phys. C 11 (1978) L29; 13

(1980) 2913.

[2] Leung, K. M., Hone, D. W., Mills, D. L., Risebor- ough, P. S. and Trullinger, S. E., Phys. Rev. B

21 (1980) 4017.

[3] Kopinga, K., Tinus, A. M. C. and de Jonge, W. J. M., Phys. Rev.

B

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[5] Kopinga, K. (private communication).

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Wysin, G. and Bishop, A. R., Phys. Rev. B 34

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