Low-temperature properties of ferromagnetic spin chains in a magnetic field

The thermodynamic properties of ferromagnetic spin chains have been analyzed with a variety of microscopic methods over the years: Bethe ansatz, spin-wave theory, Schwinger-boson mean-field theory, Green functions, and renormalization group methods. Surprisingly, in all these different studies, the manifestation of the spin-wave interaction in the low-temperature series for the thermodynamic quantities, in the presence of a finite magnetic field, has been largely neglected. In the present work, we address this problem by following a different path, based on the systematic effective Lagrangian method. We evaluate the partition function up to two-loop order and derive the low-temperature expansion of the energy density, entropy density, heat capacity, magnetization, and susceptibility in the presence of a weak external magnetic field. Remarkably, the spin-wave interaction only manifests itself beyond two-loop order. In particular, there is no term of order $T^2$ in the low-temperature series of the free energy density. This is the analog of Dyson's statement that there is no term of order $T^4$ in the low-temperature series of the free energy density in the case of three-dimensional ideal ferromagnets. The range of validity of our series is critically examined in view of the nonperturbatively generated energy gap. We also compare our results with the condensed matter literature and point out that there are some misleading statements.

Figure 1

Feynman graphs related to the low-temperature expansion of the partition function for the ferromagnetic spin chain up to order $p^5$. 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 filled circle. Note that ferromagnetic loops are suppressed by one power of momentum in one spatial dimension, $d_s$=1.

Figure 2

Range of validity of the effective low-temperature expansion of the ferromagnetic spin chain. The allowed parameter regime corresponds to the area above the curve and is restricted to very low temperatures.

Figure 3

Range of validity of the effective low-temperature expansion of the two-dimensional ideal ferromagnet. The allowed parameter regime corresponds to the area above the curve.

Figure 4

Ferromagnetic spin chain: the Taylor expansion for the free energy density in the parameter $\sigma$, Eq. (25), only makes sense at very low temperatures compared to the scale $J$.

Figure 5

Two-dimensional ideal ferromagnet: the Taylor expansion for the free energy density in the parameter $\sigma$, Eq. (III.8) of Ref. 20, makes sense up to temperatures which are about one fifth of the scale $J$.