Low-temperature properties of two-dimensional ideal ferromagnets

The manifestation of the spin-wave interaction in the low-temperature series of the partition function has been investigated extensively over more than seven decades in the case of the three-dimensional ferromagnet. Surprisingly, the same problem regarding ferromagnets in two spatial dimensions, to the best of our knowledge, has not been addressed in a systematic way so far. In the present paper the low-temperature properties of two-dimensional ideal ferromagnets are analyzed within the model-independent method of effective Lagrangians. The low-temperature expansion of the partition function is evaluated up to two-loop order and the general structure of this series is discussed, including the effect of a weak external magnetic field. Our results apply to two-dimensional ideal ferromagnets which exhibit a spontaneously broken spin rotation symmetry $\mathrm{O}\left(3\right)\to \mathrm{O}\left(2\right)$ and are defined on a square, honeycomb, triangular, or kagome lattice. Remarkably, the spin-wave interaction only sets in at three-loop order. In particular, there is no interaction term of order $T^3$ in the low-temperature series for the free energy density. This is the analog of the statement that, in the case of three-dimensional ferromagnets, there is no interaction term of order $T^4$ in the free energy density. We also provide a careful discussion of the implications of the Mermin-Wagner theorem in the present context and thereby put our low-temperature expansions on safe grounds.

Figure 1

Feynman graphs related to the low-temperature expansion of the partition function for a two-dimensional ferromagnet up to order $p^8$. The numbers attached to the vertices refer to the piece of the effective Lagrangian they come from. Vertices associated with the leading term ${\mathcal{L}}_{\mathrm{eff}}^2$ are denoted by a dot. Note that ferromagnetic loops are suppressed by two powers of momentum in $d_s$ $=$ 2.

Figure 2

Feynman graphs occurring in the low-energy expansion of the two-point function for a two-dimensional ferromagnet up to order $p^6$.

Figure 3

Feynman graphs related to the low-temperature expansion of the partition function for a three-dimensional ferromagnet up to order $p^{11}$. The numbers attached to the vertices refer to the piece of the effective Lagrangian they come from. Vertices associated with the leading term ${\mathcal{L}}_{\mathrm{eff}}^2$ are denoted by a dot. Note that ferromagnetic loops are suppressed by three powers of momentum in $d_s$ $=$ 3.