Thermodynamics of the pyrochlore Heisenberg ferromagnet with arbitrary spin $S$

We use the rotation-invariant Green's function method (RGM) and the high-temperature expansion (HTE) to study the thermodynamic properties of the spin-$S$ Heisenberg ferromagnet on the pyrochlore lattice. We examine the excitation spectra as well as various thermodynamic quantities, such as the order parameter (magnetization), the uniform static susceptibility, the correlation length, the spin-spin correlations, and the specific heat, as well as the static and dynamic structure factors. We discuss the influence of the spin quantum number $S$ on the temperature dependence of these quantities. We compare our results for the pyrochlore ferromagnet with the corresponding ones for the simple-cubic lattice both having the same coordination number $z=6$. We find a significant suppression of magnetic ordering for the pyrochlore lattice due to its geometry with corner-sharing tetrahedra.

Figure 1

The pyrochlore lattice can be visualized as a structure which consists of alternating kagome and triangular planar layers. The kagome (triangular) planes are colored in green (blue). The four-site unit cell is marked with the red bonds.

Figure 2

Dispersion of the excitation energies ${\omega }_{\mathbf{q}\gamma }$ [Eq. (9), $J=-1]$ at zero temperature $T=0$ (upper panel) and in the infinite-temperature limit $T\to \infty$ (lower panel). Note that ${\omega }_{\mathbf{q}\gamma }/S$ is independent of $S$ at $T=0$, whereas ${\omega }_{\mathbf{q}\gamma }/\sqrt{S(S+1)}$ is independent of $S$ at $T\to \infty$. The points $\mathrm{\Gamma }$, X, W, and K in the first Brillouin zone of a face-centered-cubic Bravais lattice are given by $\mathrm{\Gamma }=(0,0,0),\mathrm{X}=(0,2\pi ,0),\mathrm{W}=(\pi ,2\pi ,0),\mathrm{K}=(3\pi /2,3\pi /2,0)$, see, e.g., Ref. [50].

Figure 3

Main panel: normalized spin stiffness $\rho /\left(S\right|J\left|\right)$ (dashed) and normalized excitation velocity $v/\left(S\right|J\left|\right)$ (solid) as a function of the normalized temperature $T/T_c$. We report results for different spin values $S=1/2,1,3/2,3$ for the pyrochlore-lattice case (thin lines) and for $S=1/2$ for the simple-cubic case (thick lines). Inset: spin stiffness $\rho /\rho \left(0\right)$ versus $T/\left|J\right|$ for the $S=1/2$ pyrochlore (dashed thin red line) and simple-cubic (dashed thick black line) lattices.

Figure 4

RGM data for the normalized magnetization $M/S$ of the ferromagnet on the simple-cubic lattice $\left(S=1/2\right)$ (thick solid black line) and the pyrochlore lattice $\left(S=1/2,1,3/2,3\right)$ (thin solid lines) as a function of the normalized temperature $T/T_c$. Dashed curves correspond to the inverse uniform susceptibility $1/{\chi }_{\mathbf{0}}$ above $T_c$. Note that the thin dashed curves for $S>1/2$ almost coincide.

Figure 5

Inverse uniform susceptibility $1/{\chi }_{\mathbf{0}}$ of the ferromagnet on the pyrochlore lattice obtained by the RGM (thin solid lines) and by the HTE approach (Padé [5,6]—thin dashed lines) as a function of the normalized temperature $T/\left(S\right(S+1\left)\right)$ for several spin quantum numbers $S$. We also show the RGM results for the simple-cubic-lattice case with $S=1/2$ (thick black line). Note that the energy scale is set by $J=-1$.

Figure 6

Normalized Curie temperatures $T_c/\left(S(S+1)\right)$ of the ferromagnet on the simple-cubic lattice and the pyrochlore lattice within the RGM approach and the HTE approach (up to the eleventh order) as a function of the inverse spin quantum number $1/S$. The HTE data labeled by “pyro,HTE,[5,5]” are taken from Ref. [14].

Figure 7

Main panel: normalized correlation functions $\langle {\widehat{\mathbf{S}}}_{\mathbf{0}}·{\widehat{\mathbf{S}}}_{\mathbf{R}}\rangle /S^2$ (nearest neighbors—solid; next-nearest neighbors—dashed) as a function of the normalized temperature $T/T_c$ for the spin-$S$ pyrochlore ferromagnet for several spin quantum numbers $S$ (thin lines). We also show the results for the $S=1/2$ simple-cubic ferromagnet (thick lines). Inset: $\langle {\widehat{\mathbf{S}}}_{\mathbf{0}}·{\widehat{\mathbf{S}}}_{\mathbf{R}}\rangle /S^2$ versus $T/\left|J\right|$ for the $S=1/2$ case.

Figure 8

Specific heat of the ferromagnet on the pyrochlore lattice within the RGM (thin solid lines) and the HTE approach (Padé [5,6]—thin dashed lines) as a function of the normalized temperature $T/\left(S\right(S+1\left)\right)$ for several values of the spin quantum numbers $S$. We also show the RGM results for the $S=1/2$ simple-cubic ferromagnet (thick solid line).

Figure 9

Two top rows: magnetic structure factor $S_{\mathbf{q}}/\left(S(S+1)\right)$ of the $S=1/2$ ferromagnet on the pyrochlore lattice within the RGM approach at $T=1.3T_c$ (upper row) and $T=2T_c$ (middle row) in the Bragg plane $q_z=0$ (left panels) and in the Bragg plane $q_x=q_y$ (right panels). Bottom row: magnetic structure factor $S_{\mathbf{q}}/\left(S(S+1)\right)$ of the $S=1/2$ ferromagnet on the pyrochlore lattice within the HTE approach (ninth order) at $T=2T_c$ in the Bragg plane $q_z=0$ (left panel) and in the Bragg plane $q_x=q_y$ (right panel).

Figure 10

Magnetic structure factor $S_{\mathbf{q}}/\left(S(S+1)\right)$ of the pyrochlore ferromagnet along the line $q_x=q_y=q_z$ for two temperatures: $1.3T_c$ (red) and $2T_c$ (blue). RGM results are shown by solid lines, whereas HTE results are shown by dashed lines. Thin lines correspond to the $S=1/2$ case, thick lines correspond to the $S=3$ case.

Figure 11

Dynamic structure factor $S_{\mathbf{q}}^{zz}\left(\omega \right)$ of the $S=1/2$ pyrochlore ferromagnet along the line $q_x=q_y=q_z$ for $T=0.0425$ (top) and $T=0.425$ (bottom). We set $\epsilon =0.1$. The white lines correspond to the excitation energies ${\omega }_{\gamma \mathbf{q}}$ (9).

Figure 12

Dynamic structure factor $S_{\mathbf{q}}^{zz}\left(\omega \right)$ of the $S=1/2$ pyrochlore ferromagnet as a function of the reduced momentum $t=2-D_{\mathbf{q}}$ with $\mathbf{q}=(q,q,q)$ for $T=0.0425$. We set $\epsilon =0.1$. The white lines correspond to the excitation energies ${\omega }_{\gamma \mathbf{q}}$ (9).