Nematic, vector-multipole, and plateau-liquid states in the classical $O\left(3\right)$ pyrochlore antiferromagnet with biquadratic interactions in applied magnetic field

The classical bilinear-biquadratic nearest-neighbor Heisenberg antiferromagnet on the pyrochlore lattice does not exhibit conventional Néel-type magnetic order at any temperature or magnetic field. Instead spin correlations decay algebraically over length scales $r\lesssim {\xi }_c\sim 1/\sqrt{T}$, behavior characteristic of a Coulomb phase arising from a strong local constraint. Despite this, its thermodynamic properties remain largely unchanged if Néel order is restored by the addition of a degeneracy-lifting perturbation, e.g., further neighbor interactions. Here we show how these apparent contradictions can be resolved by a proper understanding of way in which long-range Néel order emerges out of well-formed local correlations and identify nematic and vector-multipole orders hidden in the different Coulomb phases of the model. So far as experiment is concerned, our results suggest that where long-range interactions are unimportant, the magnetic properties of Cr spinels which exhibit half-magnetization plateaux may be largely independent of the type of magnetic order present.

Figure 1

(Color online) 16-site cubic unit cell of the pyrochlore lattice—a network of corner-sharing tetrahedra. Exchange interactions $J_1$ are associated with the first neighbor bonds ${⟨ij⟩}_1$, and $J_3$ with the two (inequivalent) types of third-neighbor bond ${⟨ij⟩}_3$.

Figure 2

(Color online) (a) Magnetic phase diagram of the classical pyrochlore antiferromagnet with biquadratic interactions $b=0.1$, and additional third-neighbor interactions $J_3=-0.05$, as determined by classical Monte Carlo simulation (Refs. 16, 17). The form of four-sublattice Néel order is illustrated together with the irreducible representation (irrep) of the tetrahedral symmetry group ${\mathcal{T}}_d$ to which it belongs. For these parameters, the model provides a good description of the half-magnetization plateau seen in ${\text{CdCr}}_2{\text{O}}_4$ (Refs. 3, 4, 5, 6, 7, 8). (b) Equivalent magnetic phase diagram in the absence of any longer-range interactions. Canted Néel states are replaced by phases with multipolar order while the collinear half-magnetization plateau state gives way to a collinear spin liquid. Crosses denote the crossover at temperature $T^{\ast }$ from paramagnet to the plateau-liquid state as determined by a peak in the heat capacity. In both (a) and (b) circles with solid (red) lines denote first-order phase boundaries while those with dashed (blue) lines denote second-order ones. Both $T$ and $h$ are measured in units of $J_1$.

Figure 3

(Color online) (a) Gound-state phase diagram of a single classical tetrahedron as a function of magnetic field $h$ and dimensionless coupling constant $b$, taken from Ref. 2. Solid (red) lines denote first and dashed (blue) lines second-order transitions. Spin configurations and relevant irreducible representations (irreps) are shown in each case. (b) Symmetries of the ${\mathsf{A}}_1$, $\mathsf{E}$, and ${\mathsf{T}}_2$ irreps of the tetrahedral group ${\mathcal{T}}_d$ used to classify different states. Solid (red) lines have negative weight and hollow (blue) lines have positive weight. Thin (black) lines have zero weight. See also Eq. (7).

Figure 4

(Color online) (a) Illustration of collinear $\mathsf{E}$-symmetry nematic state at $h=0$, showing looplike coordination of parallel spins associated with the ice manifold. (b) Four-sublattice long range order with $\mathsf{E}$ symmetry induced by FM $J_3$. (c) Canted nematic state with partial magnetization under applied field.

Figure 5

(Color online) Spin collinearity $P$ [Eq. (19)] at $h=0$ for $b=0.1$, showing the onset of nematic order at $T_Q\approx 0.13$. $P$ is measured for the farthest spin pair along the $⟨110⟩$ chains in the pyrochlore lattice in each system size ranging from $L=4$ to $L=16$, and averaged over the $⟨110⟩$ chains running in different directions.

Figure 6

(Color online) (a) Nematic order parameter $Q=\sqrt{Q^{\left(2\right)}}$ [Eq. (18)], showing the onset of nematic order at $T_Q\approx 0.13$; (b) heat capacity; (c) absence of long-range four-sublattice order ${\lambda }_{\mathsf{E}}^{\mathsf{global}}$ is accompanied by well-formed local correlations ${\lambda }_{\mathsf{E}}^{\mathsf{local}}$; and (d) the associated local susceptibility shows a sharp jump at $T_Q$, where spins in tetrahedra with preformed local order gain energy by selecting long range collinearity. All data are for $h=0$, $b=0.1$, and for system size ranging from $L=4$ to $L=16$.

Figure 7

(Color online) (a) Spin correlations $D$ [Eq. (5)] at $h=0$ for $b=0.6$, for $r\le L$ measured along $⟨110⟩$ chains in the pyrochlore lattice and averaged over different chain directions as in Fig. 5. $r$ is in units of the distance between nearest-neighbor spins. (b) Characteristic $1/r^3$ power-law decay of spin correlations in the low-temperature nematic phase for $T<T_Q\sim 0.5$ associated with the ice manifold of 2:2 states, plotted on a log-log scale. (c) Exponential decay of spin correlations in the high-temperature paramagnetic phase for $T>T_Q\sim 0.5$, plotted on a log-linear scale. In (b) and (c), gray lines are guides for the eye, $\sim 1/r^3$ and $\sim \text{exp}\left(-r\right)$, respectively. In all figures the data are for temperatures ranging from $T=0.30$ to $T=0.84$ and the system size $L=16$.

Figure 8

(Color online) Spin correlations spanning the nematic state and high-temperature paramagnet plotted as a function of the rescaled distance $r\sqrt{T}$ on (a) log-log scale and (b) log-linear scales. The data are identical to those plotted in Figs. 7b, 7c; gray lines are guides for the eye, showing $\sim 1/r^3$ and $\sim \text{exp}\left(-r\right)$, respectively. Temperatures should be compared with the nematic ordering temperature $T_Q\approx 0.5$.

Figure 9

(Color online) (a) Nematic order parameter $Q^{\perp ,2}=\left|{\mathbf{Q}}^{\perp ,2}\right|$ [Eq. (22)], signaling the nematic long-range order in finite magnetic field and (b) heat capacity [Eq. (31)]. Data are at $h=1.2$ and $h=2.4$ for $b=0.1$, and for system size ranging from $L=4$ to $L=16$.

Figure 10

(Color online) Dependence of the magnetization $m$ [Eq. (6)] on magnetic field $h$ for $b=0.1$ and $J_3=0$, showing the existence of the magnetization plateau in the absence of long-range Néel order. Symbols show the result of Monte Carlo simulations for temperatures ranging from $T=0.04$ to $T=0.24$ and the system size $L=8$. The dashed line is the result obtained by minimizing the energy for $T=0$.

Figure 11

(Color online) Half-magnetization plateau states ($uuud$ states) on a pyrochlore lattice with exactly three up and one down spins per tetrahedron. (a) Schematic picture of an $uuud$ state with no long-range order, associated with the dimer manifold; (b) $uuud$ state with long-range four-sublattice order with ${\mathsf{T}}_2$ symmetry induced by FM $J_3$, as considered in Refs. 2, 17; and (c) 16-sublattice order. See also Fig. 15.

Figure 12

(Color online) Absence of long-range magnetic order in the plateau-liquid state for $b=0.6$, $J_3=0$, and $h=4$. (a) The reduced spin correlation function $D\left(r\right)$, defined by Eq. (5), measured along the $⟨110⟩$ chains, as in Fig. 7a. (b) Characteristic $1/r^3$ power-law decay of spin correlations at low temperatures $T<T^{\ast }\sim 0.6$, associated with the dimer manifold of $uuud$ states, plotted on a log-log scale. (c) Exponential decay of spin correlations at high temperatures $T>T^{\ast }\sim 0.6$, plotted on a log-linear scale. In (b) and (c), gray lines show guides for the eye, $\sim 1/r^3$ and $\sim \text{exp}\left(-r\right)$, respectively. In all figures the data are for temperatures ranging from $T=0.30$ to $T=0.84$ and the system size $L=16$.

Figure 13

(Color online) Spin correlations spanning the plateau liquid and high-temperature paramagnetic phases, plotted as a function of the rescaled distance $r\sqrt{T}$ on (a) log-log scale and (b) log-linear scales. The data are identical to those plotted in Figs. 12b, 12c; gray lines are guides for the eye, showing $\sim 1/r^3$ and $\sim \text{exp}\left(-r\right)$, respectively. Temperatures should be compared with the crossover scale $T^{\ast }\approx 0.6$.

Figure 14

(Color online) Temperature dependence of (a) collinearity [Eq. (19), cf. Figure 5], (b) heat capacity [Eq. (31)], (c) the related measure of local correlation ${\lambda }_{{\mathsf{T}}_2}^{\mathsf{local}}$ defined by Eq. (10), and the global order parameter ${\lambda }_{{\mathsf{T}}_2}^{\mathsf{global}}$ defined by Eq. (8), and (d) the associated local susceptibility. Simulations were performed for $h=4$, $b=0.1$, in clusters with $L=4$ to $L=16$.

Figure 15

(Color online) (a) Probability distribution of the flippable hexagons within the dimer manifold. (b) The fluctuation entropy per site $s_{\mathsf{f}}$ [Eq. (40)] as a function of the number of “flippable” hexagons. The lower bound $s_{\mathsf{f}}=-4.19874$ is set by the four-sublattice state shown in Fig. 11b, which has no flippable hexagons. The upper bound $s_{\mathsf{f}}=-4.19806$ is set by the 16-sublattice state with the maximum number of flippable hexagons [see Fig. 11c]. The blue dots represent a sample of $10000$ random configurations. (c) Probability distribution of the fluctuation entropy per site $s_{\mathsf{f}}$ within the dimer manifold. All results are for a cluster of $N=1024$ sites with $b=0.6$.

Figure 16

(Color online) (a) DOS for eigenvalues of $\mathcal{M}$ for four-sublattice $uuud$ state in thermodynamic limit, showing finite gap $\Delta$ [Eq. (42)] and flat band at $\epsilon =16b$. Integrated DOS (b) of four-sublattice $uuud$ state (points), (c) of a typical disordered $uuud$ state (points), and (d) averaged within disordered $uuud$ states (dashed line). In all cases $J_3=0$, $h=4$, and $b=0.6$, and a cluster size is $N=1024$. The integrated DOS corresponding to (a) is shown in shading on (b)–(d) for comparison.

Figure 17

(Color online) Magnetic field dependence of (a) primary order parameter $Q^{\perp ,1}=\left|{\mathbf{Q}}^{\perp ,1}\right|$ [Eq. (23)] and (b) secondary order parameter $Q^{\perp ,2}=\left|{\mathbf{Q}}^{\perp ,2}\right|$ [Eq. (22)] in the vector-multipole phase for $b=0.1$. The continuous transition from the plateau-liquid state into the vector-multipole phase at lower magnetic fields, and the direct transition from the vector-multipole phase into the (saturated) paramagnet at higher fields are clearly visible. Dashed lines show behavior at $T=0$ in a single-tetrahedron theory. Points show results of MC simulations for the system size $N=8$ from $T=0.02$ to $T=0.08$.

Figure 18

(Color online) Temperature dependence of (a) the primary order parameter $Q^{\perp ,1}=\left|{\mathbf{Q}}^{\perp ,1}\right|$ defined by Eq. (23), showing the onset of the vector-multipole order at $T=T_V\approx 0.042$, (b) heat capacity Eq. (31), (c) the related measure of local correlation ${\lambda }_{{\mathsf{T}}_2}^{\mathsf{local}}$ defined by Eq. (10), and the global order parameter ${\lambda }_{{\mathsf{T}}_2}^{\mathsf{global}}$ defined by Eq. (8), and (d) the associated local susceptibility Eq. (11). Simulations were performed for $h=6$, $b=0.1$, in clusters with $L=4$ to $L=16$.

Figure 19

(Color online) Temperature dependence of (a) heat capacity defined by Eq. (31), (b) and (c) the related measure of local correlation ${\lambda }_{{\mathsf{T}}_2}^{\mathsf{local}}$ and its susceptibility defined by Eqs. (10, 11), and (d) and (e) the global order parameter ${\lambda }_{{\mathsf{T}}_2}^{\mathsf{global}}$ and its susceptibility defined by Eqs. (8, 9) for $b=0.6$, $h=4$, and a range of values of system size. The results are for $J_3=-0.06$, showing a single first-order transition into the ordered phase at $T_N=0.70\left(1\right)$ (indicated by the vertical dashed line).

Figure 20

(Color online) The same plots as Fig. 19 but for $J_3=-0.02$, showing both a crossover into the plateau liquid for $T^{\ast }\approx 0.6$ (bold gray line) and a transition into the ordered phase at $T_N=0.46\left(1\right)$ (dashed line).

Figure 21

(Color online) Temperature dependence of the order parameter ${\lambda }_{{\mathsf{T}}_2}^{\mathsf{global}}$ for four-sublattice $uuud$ order [as defined by Eq. (8)], for a range of values of $J_3<0$. The transition temperature becomes smaller as $\left|J_3\right|\to 0$. At the same time the transition becomes more strongly first order. All results are for $h=4$ and $b=0.6$, in a cluster with $L=8$. The lines are guides for the eye.

Figure 22

(Color online) Phase diagram for the classical Heisenberg antiferromagnet on a pyrochlore lattice in applied magnetic field $h=4$ with additional biquadratic interactions $b=0.6$ [Eq. (1)]. The transition temperature $T_N$ associated with the gapped, ordered, half-magnetization plateau state vanishes as the strength of ferromagnetic third-neighbor interactions $J_3\to 0$, as determined by Monte Carlo simulation. A state exhibiting a half-magnetization plateau but no long-range magnetic order exists above $T_N$ up to a crossover temperature $T^{\ast }$. Estimates of the crossover temperature are taken from peaks in the local susceptibility and heat capacity.

Figure 23

(Color online) Temperature dependence of (a) nematic order parameter [Eq. (18)], (b) heat capacity [Eq. (31)], [(c) and (d)] the related measure of local correlation ${\lambda }_{\mathsf{E}}^{\mathsf{local}}$ and its susceptibility [Eqs. (10, 11)], and [(e) and (f)] the global order parameter ${\lambda }_{\mathsf{E}}^{\mathsf{global}}$ and its susceptibility [Eqs. (8, 9)] for $b=0.6$, $h=0$, and a range of values of system size. The results are for $J_3=-0.02$, showing both a first-order transition into the nematic phase at $T_Q=0.518\left(6\right)$ (indicated by the vertical dotted line) and a transition into the ordered phase at $T_N=0.42\left(2\right)$ (dashed line).

Figure 24

(Color online) Phase diagram for the classical Heisenberg antiferromagnet on a pyrochlore lattice, Eq. (1), in applied magnetic field $h=0$, with additional biquadratic interactions $b=0.6$. The transition temperature $T_N$ associated with the $\mathsf{E}$-symmetry long-range order vanishes as the strength of ferromagnetic third-neighbor interactions $J_3\to 0$, as determined by Monte Carlo simulation. A phase exhibiting nematic order exists above $T_N$ up to a $T_Q\sim b$.

Figure 25

(Color online) (a) Temperature dependence of the primary order parameter $Q^{\perp ,1}$ in the vector-multipole phase. In the limit $L\to \infty$ these results extrapolate to a mean fieldlike behavior $Q^{\perp ,1}\sim \sqrt{T_V-T}$ with $T_V\simeq 0.091$ (solid black points and gray line). (b) The secondary order parameter $Q^{\perp ,2}$ also takes on a finite value for $T<T_V$ but grows more slowly at the transition. (c) Heat capacity, showing no measurable singularity at $T=T_V$. All data are for $h=5$, $b=0.1$, and $J_3=0$ with system sizes ranging from $L=4$ to $L=16$.