Field-Tuned Order by Disorder in Frustrated Ising Magnets with Antiferromagnetic Interactions

We demonstrate the appearance of thermal order by disorder in Ising pyrochlores with staggered antiferromagnetic order frustrated by an applied magnetic field. We use a mean-field cluster variational method, a low-temperature expansion, and Monte Carlo simulations to characterize the order-by-disorder transition. By direct evaluation of the density of states, we quantitatively show how a symmetry-broken state is selected by thermal excitations. We discuss the relevance of our results to experiments in 2D and 3D samples and evaluate how anomalous finite-size effects could be exploited to detect this phenomenon experimentally in two-dimensional artificial systems, or in antiferromagnetic all-in–all-out pyrochlores like ${\mathrm{Nd}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ or ${\mathrm{Nd}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$, for the first time.

Figure 1

(a) Conventional unit cell of the pyrochlore lattice ($L=1$) with 4 tetahedra pointing up (colored) and 4 pointing down (uncolored). The arrows represent the spin direction at each site. The color of the spheres marks the sign of the charges and the size is proportional to their modulus. (b) Planar projected configuration: a tetrahedron becomes a square with crossings, in a checkerboard lattice. The field $\mathbf{B}\parallel [110]$ couples to two spins on each tetrahedron ($\alpha$ chains, yellow arrows), and is orthogonal to the other two ($\beta$ chains, blue arrows). The configuration shown is a zero-temperature ground state for large fields. (c) The energy of a singly charged ($S$), doubly charged ($D$), and neutral tetrahedron ($N$) as functions of the magnetic energy normalized by the exchange energy scale; only the energies corresponding to the $S$ and $N$ configurations favored by the field are shown. (d) Linear size dependence of the residual entropy $S_{\text{res}}$.

Figure 2

The order parameters (a) ${\rho }_S^s$ and (b) ${\rho }_S^d$ and (c) the specific heat as functions of temperature. The symbols show MC data for the 3D system, and the dashed lines (with the same color code) show the results of the low-$T$ expansion for the 2D projection, both with similar number of spins $N$. The solid (blue) lines are the CVM outcome ($N\to \infty$).

Figure 3

(a) Density of states $\delta$ as a function of excitation energy $\mathrm{\Delta }E$ and total staggered charge density ${\rho }_S$, for a system of size $L=3$ and for $\mu B/|J|=7.562$. Each colored circle corresponds to a single energy of the system. The two symmetric peaks are centered at $|{\rho }_S^{\text{peak}}|\approx 1.2$. Inset (b) compares $|{\rho }_S|$ calculated from the WL method and the MC simulation with the values directly obtained from the maxima of $\delta$. Inset (c), copied from Ref. [31], represents the ground-state manifold (black curve) in parameter space and the selection of an ordered state (cross) by thermal excitations (green region).