Disorder from order among anisotropic next-nearest-neighbor Ising spin chains in ${\mathrm{SrHo}}_2{\mathrm{O}}_4$

We describe why Ising spin chains with competing interactions in ${\mathrm{SrHo}}_2{\mathrm{O}}_4$ segregate into ordered and disordered ensembles at low temperatures $\left(T\right)$. Using elastic neutron scattering, magnetization, and specific heat measurements, the two distinct spin chains are inferred to have Néel $\left(\uparrow \downarrow \uparrow \downarrow \right)$ and double-Néel $\left(\uparrow \uparrow \downarrow \downarrow \right)$ ground states, respectively. Below $T_{\mathrm{N}}=0.68\left(2\right)\mathrm{K}$, the Néel chains develop three-dimensional long range order (LRO), which arrests further thermal equilibration of the double-Néel chains so they remain in a disordered incommensurate state for $T$ below $T_{\mathrm{S}}=0.52\left(2\right)\mathrm{K}$. ${\mathrm{SrHo}}_2{\mathrm{O}}_4$ distills an important feature of incommensurate low dimensional magnetism: kinetically trapped topological defects in a $\text{quasi}-d-\text{dimensional}$ spin system can preclude order in $d+1$ dimensions.

Figure 1

(a) Crystallographic unit cell of ${\mathrm{SrHo}}_2{\mathrm{O}}_4$. Sr atoms were omitted for clarity. Red and blue spheres show two distinct Ho sites, and the corresponding arrows show the Ising spin directions. (b) Magnetic lattice formed by Ho and a schematic representation of the spin structure determined by neutron diffraction.

Figure 2

Magnetic susceptibility of ${\mathrm{SrHo}}_2{\mathrm{O}}_4$ along three axes measured under 200 Oe. Dashed lines show fits to $J_1\text{−}{J}_2$ Ising models. The inset shows the inverse magnetic susceptibility up to 300 K.

Figure 3

(a) Specific heat of ${\mathrm{SrHo}}_2{\mathrm{O}}_4$ as a function of $T$. Filled symbols show the experimental total specific heat; the magenta long-dashed line shows the calculated specific heat due to a nuclear Schottky anomaly; open symbols show the magnetic specific heat obtained by subtracting the nuclear Schottky anomaly from the total specific heat. (b) Magnetic specific heat over $T$ versus $T$. (c) shows the entropy versus $T$. The dashed line shows the entropy of an Ising doublet.

Figure 4

$T$-dependent elastic magnetic neutron scattering indicating quasi-one-dimensional short range order in ${\mathrm{SrHo}}_2{\mathrm{O}}_4$. (a) and (c) show measurements in the $\left(HK0\right)$ reciprocal lattice plane, while (b) and (d) are from the $\left(0KL\right)$ plane. Measurements at 50 K were subtracted to eliminate nuclear scattering. (e) and (f) show $J_1\text{−}{J}_2$ model calculations at 1.4 K with exchange constants determined from Fig. 2.

Figure 5

Thermal evolution of interchain correlations for red and blue sites probed by neutrons. (a) shows $(0,K,0)$ scans probing the correlations along $\mathbf{b}$ between red chains at several temperatures. (b) and (c), respectively, show $(0,K,\frac{1}{2})$ and $(H,0,\frac{1}{2})$ scans that probe correlations along $\mathbf{b}$ and $\mathbf{a}$ for blue chains. Horizontal bars represent instrumental resolution. Sharp peaks at $K\sim 5.5$ in (b), at $H\sim 4.6$ and $H\sim 5.4$ in (c) are the results of Bragg scattering from the copper sample mount.

Figure 6

Interchain correlations for blue sites in ${\mathrm{SrHo}}_2{\mathrm{O}}_4$ probed by neutrons. Filled and empty circles are from $H$ and $K$ scans at $\left(00\frac{1}{2}\right)$ measured at 0.28 K. The solid lines show fits to the data. The inset shows a sharp component of the $H$ scan in detail; the horizontal bar is the instrumental resolution. Data from different sample orientations are normalized to the peak count rate.

Figure 7

Calculated single ion magnetic susceptibility for the (a) red Ho1 site and (b) blue Ho2 site in ${\mathrm{SrHo}}_2{\mathrm{O}}_4$ along three axes based on CEF level schemes calculated according to point charge approximation.

Figure 8

Magnetic structure refinement for ${\mathrm{SrHo}}_2{\mathrm{O}}_4$ based on $L$-integrated peak intensities measured for $T=0.3$ K. (a) and (b) show the refinement results for red and blue sites, respectively. Long-dashed lines represent the $y=x$ line.

Figure 9

Spin correlations versus $T$ in ${\mathrm{SrHo}}_2{\mathrm{O}}_4$ probed by neutrons. (a)–(e) show results from $L$ scans at $\left(030\right)$ (red) and $\left(00\frac{1}{2}\right)$ (blue) that probe correlations along red and blue chains, respectively. (a) and (b) show integrated intensities and peak widths for $\left(030\right)$. The green dashed line in (b) shows the instrumental resolution. (c) shows integrated intensities for $\left(00\frac{1}{2}\right)$. (d) shows the peak shift from $\left(00\frac{1}{2}\right)$. The inset shows the peak shift along ${\mathbf{c}}^{*}$ from $\left(0K\frac{1}{2}\right)$. The dashed line shows the predicted shift based on the $J_1\text{−}{J}_2$ model. (e) shows the inverse correlation length ${\kappa }_c$ derived as the half width at half maximum of resolution convoluted Lorentzian fits. (f) shows ${\kappa }_b$ extracted from $K$ scans at $\left(00\frac{1}{2}\right)$. Black dashed lines indicate $T_{\mathrm{N}}$ and $T_{\mathrm{S}}$. Blue dashed lines in (d) and (e) are ANNNI model calculations based only on the exchange constants obtained from the data in Fig. 2.

Figure 10

Ground state degeneracy and domain walls of ANNNI chains in ${\mathrm{SrHo}}_2{\mathrm{O}}_4$. (a) and (b) show the twofold degenerate ground states for red chains and the corresponding domain wall. (c) illustrates that a $R1$ to $R{1}^{*}$ domain wall is identical to a $R{1}^{*}$ to $R1$ domain wall (up to time reversal). (d) and (e) show the more complicated situation for blue chains where the ground state is fourfold degenerate and the domain walls are chiral. (f) illustrates the chiral character of the domain walls in blue chains: a $B1$ to $B2$ domain wall is different from a $B2$ to $B1$ domain wall. Refer to Sec. 5 for a detailed description. Dashed rectangles in (c) and (f) encircle spins whose bond energies are affected by transition to a different ground state, and thus represent the domain walls.