Degeneracy and strong fluctuation-induced first-order phase transition in the dipolar pyrochlore antiferromagnet

We show that a continuous set of degenerate critical soft modes strongly enhances the first-order character of a fluctuation-induced first-order transition in the pyrochlore dipolar Heisenberg antiferromagnet. Such a degeneracy seems essential to explain the strong first-order transition recently observed in ${\mathrm{Gd}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. We present some evidence from Monte Carlo simulations and a perturbative renormalization group expansion.

Figure 1

The $A$ state $(\mathbf{q}=\mathbf{0})$ of the pyrochlore lattice stabilized at low temperatures by the dipole-dipole interactions (Refs. 3, 5, 8, 10). Since the ordering is at $\mathbf{q}=0$, all tetrahedra have the same spin configuration as the one shown. In the figure, all the spins lie onto the $\left(xy\right)$ plane and form pairs of antiparallel spins that are parallel to the opposite edge of the tetrahedron they belong to. There are equivalent $\left(xz\right)$ and $\left(yz\right)$ states. The magnetic order is therefore characterized by a $n=3$-component order parameter, $\psi$ with $\left(xy\right)$ corresponding to $\psi =(1,0,0)$ at $T=0$.

Figure 2

(Color online). Order parameter squared $(\times N)$ for the $A$ phase, shown in Fig. 1, and the $B$ phase, for $J_2=J_3=0$ as function of temperature $T$ in a Monte Carlo simulation of the dipolar pyrochlore Heisenberg antiferromagnet. We see a large jump in the order parameter for the largest system size $L=4$ signaling the onset of a strong first-order transition. The number of spins is given by $N=16L^3$.

Figure 3

(Color online). Binder ratio for the $A$ phase for different system sizes $L$ for $J_2=J_3=0$.

Figure 4

The lowest part of the spectrum of $J\left(\mathbf{q}\right)$ (from Ref. 10). One sees that the degeneracy of the critical soft modes is lifted by second and third neighbor couplings. A positive (resp. negative) $J_2=J_3$ favors $\mathbf{q}=\pi$ (resp. $\mathbf{q}=0$).

Figure 5

(Color online). Order parameters squared $(\times N)$ for $J_2=J_3=-0.061J$. A transition occurs to the $A$ phase $(\mathbf{q}=0)$ which is more gradual than for the degenerate case with $J_2=J_3=0$ shown in Fig. 2.

Figure 6

(Color online). The Binder ratio for the $A$ and $B$ phases for different system sizes $L$ for $J_2=J_3=-0.061J$.

Figure 7

(Color online). Order parameters obtained by Monte Carlo simulation for $J_2=J_3=0.01J$. The transition gives rise to the $B$-like state $(\mathbf{q}=\pi )$ and is more gradual than for the case of $J_2=J_3=0$ shown in Fig. 2.

Figure 8

(Color online). Binder ratio for the $A$ and $B$ phases for different system sizes $L$ for $J_2=J_3=0.01J$.

Figure 9

Lines of minimum energy in reciprocal space, given by the four equivalent (1,1,1) directions (the cube is drawn for convenience). The (lack of) dispersion along the (1,1,1) lines is shown in Fig. 4.