High-temperature series expansion study

1. Effect of easy-axis single-ion anisotropy on phase transitions of Heisenberg antiferromagnetic films Kok-Kwei Pan (潘國貴) Physics Group, Center of General Education Chang Gung University ( 長庚大學 ) No. 259, Wen-Hua 1st Road Kwei-San, Tao-Yuan Taiwan High-temperature series expansion study

2. Outline • Motivation Magnetic thin films ( quasi-two-dimensional systems) offer unique opportunities for studying finite-size scaling effects on the critical behavior. Single-ion anisotropy plays a major role in determining the magnetic behavior of the system . • High-temperature series expansion • Results • Effect of easy-axis single-ion anisotropy on thickness-dependent Néel temperature TN(n) • The thickness dependence of Néel temperature TN(n) for n layers cubic lattice films • Conclusions

3. Hamiltonian ofHeisenberg Antiferromagnet (HAF) with single-ion anisotropy

4. 2D D D Spin-1 Phase diagram (3D HAF with single-ion anisotropy ) Quantum Paramagnetic phase -Dc/J D/J Easy-plane 0 Easy-axis Planar Antiferromagnetic phase Ising Antiferromagnetic phase K. K. Pan, Phys. Rev. B 79 , 134414 (2009). K. K. Pan, Phys. Lett. A 374 , 3225 (2010). Heisenberg antiferromagnet Neel order with quantum fluctuations

5. High-temperature series study

6. Cubic lattice film n-layers ∞ ∞

7. Free energy and staggered susceptibility due to the quantum and thermal fluctuation correlations Free energyof n=2, 3, 4, 5 and 6 interacting layers films Staggered susceptibility of n interacting layersfilms K. K. Pan, Phys. Rev. B 71, 134524 (2005); Phys. Rev. B 64, 224401 (2001); Phys. Rev. B 59, 1168 (1999). two-rooted connected diagrams

9. Analysis of the Series and Results Néel temperature and critical exponent n layers films • Three-dimensional ( 3D) bulk Néel temperature and critical exponent ( 3D) • Pseudocritical temperature of the n-layer and critical exponent ( 2D) Ratio method Ratio method D-log Pade approximant

10. Spin-1 Spin-3/2 ( 3D) ( 3D) ( 2D) ( 2D) ( 3D) ( 3D) ( 2D) ( 2D)

11. Effect of easy-axis single-ion anisotropy on thickness-dependent Néel temperature( S=3/2)

12. The thickness dependence of Néel temperature TN(n) for n layers cubic lattice films 3D Ising 3D Ising

13. 3D Ising 3D Ising

14. Conclusions • The thickness dependence of Néel temperature TN(n) for n layers cubic lattice films with easy-axis anisotropy is described by a finite-size scaling relation with a shift exponent of • The obtained shift exponents for spin-1and spin-3/2 of the sc and bcc lattices with easy-axis anisotropy show good agreement with the finite-size scaling prediction for the 3D Ising universality class and the general universality principles of the spin independent shift exponents.

16. The thickness dependence of the Neel temperature $T_{N}$ for the $n-$layers cubic lattice films is described by a finite-size scaling relation with a shift exponent of $\lambda \simeq 1.1 \pm 0.2$. Although the obtained shift exponents for spin-$1$ and spin-$\frac{3}{2}$ of the sc and bcc lattices are not accurate compared with the expected value, they show good agreement with the finite-size scaling prediction for the 3D Heisenberg universality class and the general universality principles of the spin independent shift exponents.