Thermodynamics of a two-dimensional Heisenberg ferromagnet with dipolar interaction

Thermodynamics of quantum and classical two-dimensional (2D) Heisenberg models with long-range dipole–dipole interaction has been investigated using various forms of self-consistent spin-wave theory (SSWT). It has been found that SSWT gives a much lower transition temperature $T_c$ than the free-magnon (spin-wave) theory. For the classical spin, the $T_c$ from SSWT lies within 9% of the Monte Carlo value, making SSWT the best approximation among those considered. It is proven that the random phase approximation vertex corrections to SSWT are rather small. The results depend strongly on the value of the spin, emphasizing the importance of using the quantum and not the classical 2D Heisenberg model even for large spins such as $S=7∕2$.

Figure 1

(Color online) The free-magnon (SW) transition temperature $T_c$ vs dipolar interaction $J_d$. The symbols are numerical results, while the curves are the asymptotical formulas (31, 34).

Figure 2

(Color online) SSWT relative magnetization $\overline{s}$ and short-range order parameter $\gamma$ vs temperature for $S=1∕2$ and $j_d={10}^{-3}$. For comparison, SW magnetization for $j_d={10}^{-3}$, and $\gamma$ from SSWT for $j_d=0$ (Mermin–Wagner situation) are also shown.

Figure 3

(Color online) The transition temperature $T_c$ vs dipolar interaction $J_d$ from different approaches for $S=1∕2$. The symbols are numerical results and the curves are the asymptotical formulae (31),(62).

Figure 4

(Color online) The transition temperature $T_c$ vs dipolar interaction $J_d$ from different approaches for classical spin. The symbols are numerical results and the curve is the free-magnon asymptotical formula (34). The Monte Carlo result for $J_d∕J=0.1$ is taken from Ref. 30.

Figure 5

(Color online) The $\gamma {\overline{s}}^2$-SSWT transition temperature $T_c$ as a function of dipolar interaction $J_d$ for different values of spin. The symbols are numerical results, while the curves are the asymptotical formulas (62),(63).

Figure 6

The diagrams corresponding to the RPA renormalization of (a) magnon–magnon vertex and (b) magnon Green’s function.