Some physical properties of disorder Blume-Emery-Griffiths model S.L. Yan, H.P Dong Department of Physics Suzhou University CCAST

Outlines 1. Introduction 2. Theory 3. Results & discussions 4. Summary

1. Introduction (1) Ising model and its development Ising model ( 1925 ) Transverse Ising model ( Gennes, SSC, 1 (1963) 132 ) Blume-Capel model ( PR 141 (1966) 517; Physica 32 (1966) 966 ) Blume-Emery-Griffiths model ( PRA 4 (1971) 1071 ) ………..

1. Introduction Multi component fluid liquid crystal mixtures, Magnetic materials (PRB 71 (2005) ) Microemulsion Semiconducting alloy systems Ternary mixtures (PRL 93 (2004) ) (2) Applications of the BEG model

1. Introduction (3) Methods of research the BEG model Mean field method Real space renormalization group theory Effective field theory (EFT) Cluster variational method in pair approximation Monte Carlo simulations ……… In this report, we give the phase diagrams and magnetic properties of the disorder BEG model

2. Theory The Hamiltonian for the spin-1 BEG model can be defined as: The effective Hamiltonian is and K is the exchange and biquadratic interaction between the nearest-neighbor pairs. D i is a crystal field parameter. h is a uniform magnetic field. K>0 or K<0. (1) (2)

2. Theory By intruducing the ratio of biquadratic and exchange interaction, defined to be. whereand. (3) (4) (5) (6)

2. Theory Forcase, it is a single lattice problem. Within the EFT, by using differential operator technique and the van der Waerden identities, the average magnetization and the quadrupolar moment can be formulated by: (7) (8)

2. Theory and where (9) (10) (11) (12)

2. Theory Since order parameter equations deal with multi-spin correlations, we cannot treat exactly all the spin-spin correlations. Therefore, a cutting approximation procedure shall be adopted: for Then equations may be rewritten to be (13) (14)

2. Theory where z is the coordination number. By solving above- mention equations, the expression can be obtained: According to Laudau theory, the second-order phase transition equation is determined by: (15) (16) (17)

2. Theory where and are given by: (18) (19)

2. Theory The first-order phase transition equation is determined by and The tricritical point at which the second-order phase transition changes to the first-order one must satisfy and

2. Theory For case, it is necessary to adopt two sublattices. We can discuss staggered quadrupolar (SQ) phase and bicritical point of the BEG model. The Hamiltonian is (1’) (2’) (3’) (4’) (5’)

2. Theory where and The SQ phase satisfiesand conditions. This is condition of the two-cycle fixed points. In addition, the paramagnetic phase corresponds and conditions. (6’) (7’)

2. Theory Thus, one has: Obviously, the paramagnetic phase satisfies the single fixed point condition. In order to obtain the SQ and the paramagnetic phase boundary, the single fixed point must bifurcate into two-cycle fixed points. The limit of stability for bifurcation of the fixed point is as follows: (8’) (9’)

2. Theory The solution of the SQ-P phase boundary is given by equations (8’) and (9’). The F-P second-order phase boundary is still given by equations (16) and (17). The intersection of the SQ-P line and the F-P one is a BCP. The coordinate of the BCP is solution of the set of coupled equations (8’)-(9’) and (16)-(17). For simplification, a simple cubic lattice is selected as the three-dimensional version

3. Results & Discussions (1) Critical behaviors of the BEG model for Ferromgnetic region; Reentrant phenomenon; Finite critical temperature; Double TCPs problem

3. Results & Discussions Shrink of Double TCPs

3. Results & Discussions Fig. (a) apparent reentrant phenomenon, Fig.(b) weak reentrant phenomenon The TCP is depressed monotonically with decreasing crystal field concentration.

3. Results & Discussions The bond percolation threshold is for The bond percolation threshold has many different values for There exist two double bond percolation threshold in Fig. 4(a)

3. Results & Discussions (2) Staggered quadrupolar phase and bicritical point SQ phase, SQ-P boundary, F-P boundary bicritical point different effect of two dilution factors PRB 69 (2004)

3. Results & Discussions Influence of different parameters on the SQ phase and the BCP

(3) Magnetic properties of the BEG model for 3. Results & Discussions Influence of positive and negative ratio on Initial magnetizations

3. Results & Discussions Influence of ratio and bond dilution on Initial magnetizations

3. Results & Discussions Influence of positive and negative ratio on susceptibility

3. Results & Discussions Influence of ratio and bond dilution on susceptibility

3. Results & Discussions The temperature dependence of the magnetization under different parameters

3. Results & Discussions (PRB 71 (2005) )Our result

4. Summary Phase diagrams for positive ratio are compared with those for negative one. There exist double TCPs. The first-order phase transition is enlarged with increasing of ratio at a fixed random crystal field concentration, while the first-order phase transition will shrink due to a fixed bond dilution concentration. Bond percolation threshold is always p= for and has many different values. The reentrant phenomenon is shown. The system has shown the SQ phase, the ferromagnetic phase and the BCP. A large negative ratio and two different dilution factors magnify the range of the SQ phase and reduce that of in or plane. These parameters can assist the reentrant behavior of the SQ-P lines and suppress that of the F-P lines. The influence of bond dilution on the BCP is dissimilar to that of anisotropy dilution. The initial magnetization curves and susceptibility curves exhibit an irregular behavior in the region of low temperature when the crystal field takes a smaller value. The peak of the susceptibility curve has a explicit decline. The magnetization curves transform from ferromagnetism to paramagnetism at high temperatures. The magnetization curves have a remarkable fluctuation process for a negative ratio and show a discontinuity due to larger crystal field and a positive ratio.

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