Symplectic $N$ and time reversal in frustrated magnetism

Identifying the time-reversal symmetry of spins as a symplectic symmetry, we develop a large $N$ approximation for quantum magnetism that embraces both antiferromagnetism and ferromagnetism. In $\text{SU}\left(N\right)$, where $N>2$, not all spins invert under time reversal, so we have introduced a large $N$ treatment that builds interactions exclusively out of the symplectic subgroup $\left[\text{SP}\left(N\right)\right]$ of time-reversing spins, a more stringent condition than the symplectic symmetry of previous $\text{SP}\left(N\right)$ large $N$ treatments. As a result, we obtain a mean-field theory that incorporates the energy cost of frustrated bonds. When applied to the frustrated square lattice, the ferromagnetic bonds restore the frustration dependence of the critical spin in the Néel phase and recover the correct frustration dependence of the finite temperature Ising transition.

Figure 1

(Color online) The time-reversal operator $\theta$ and rotation operators $U$ acting on a spin must commute. (a) depicts $\vec{S}\stackrel{\theta U}{\to }-R\vec{S}$. (b) $\vec{S}\stackrel{U\theta }{\to }R\left(-\vec{S}\right)$, where $R$ is the rotation performed by $U$. These two are equivalent for all $U$ in SU(2), and $\text{SP}\left(N\right)$, but not in $\text{SU}\left(N\right)$.

Figure 2

(Color online) The toy picture of $\text{SU}\left(N\right)$ spins, where the symplectic, time-reversing components are represented along the $\widehat{y}$ axis (in blue) and the antisymplectic, non-time-reversing components are along the $\widehat{x}$ axis (in red). The full $\text{SU}\left(N\right)$ spin obtained by adding its symplectic and antisymplectic components (in purple). (a) depicts the ferromagnetic state. (b) depicts the antiferromagnetic state, where we obtain the antiferromagnet by time reversing every other spin. While the symplectic components are antiparallel, the antisymplectic components are still aligned, causing the total spins to be orthogonal at neighboring sites.

Figure 3

$\text{SU}\left(N\right)$ spins consist of two components: symplectic directions that reverse under time-reversal and antisymplectic directions that are invariant under time reversal, which prevent $\text{SU}\left(N\right)$ spins from forming two-particle singlets. However, if the spins are projected into the symplectic plane, these components can form two-particle singlets, which are well defined as long as the antisymplectic components are noninteracting.

Figure 4

(Color online) Integrating out the fluctuations. (a) $\Delta$ is integrated along real axes $x$ and $y$, and its saddle point is a minimum at some ${\Delta }_0$. (b) $h$ is integrated along the imaginary axes $u$ and $v$, with a maximum, real saddle point $h_0$.

Figure 5

(Color online) The $J_1\text{−}{J}_2$ model. (a) depicts antiferromagnetic order as described by antiferromagnetic valence bonds (blue arrows) and ferromagnetic bonds (red, dashed line), and the spin order $\vec{Q}=\left(\pi ,\pi \right)$. (b) depicts the collinear order, $\vec{Q}=\left(0,\pi \right)$, where there are two different antiferromagnetic valence bonds (blue and green arrows) and ferromagnetic bonds (red, dashed lines).

Figure 6

(Color online) We compare the critical spin $S_c={\left(\frac{n_b}{N}\right)}_c$ below which there is no long-range order in the ground state calculated within $\text{SP}\left(N\right)$ (bold red line), symplectic $N$ (blue and green lines), and spin-wave theory (Ref. 39) (thin black line). For small $J_2/J_1$, the spin configurations are staggered, while for large $J_2/J_1$, the ground state breaks lattice symmetry to develop collinear order as shown in the figure. $\text{SP}\left(N\right)$ (bold red line) tends to overstabilize the long-range-ordered phases most dramatically on the one sublattice side, where the critical spin is independent of the strength $J_2$ of the frustrating diagonal bonds (Ref. 5). Symplectic $N$ restores the frustration-induced fluctuations by treating both ferromagnetic and antiferromagnetic bonds on equal footing, which corrects this overstabilization. The physical spin $S=1/2$ is indicated by a horizontal dashed line.

Figure 7

(Color online) Finite temperature phase diagram for $S=1/2$. $T_d$ [Eq. (79)] and $T_N$ [Eq. (A2)] are the transitions into short-range two sublattice and Néel antiferromagnetic orders, respectively. The Ising transition $T_c$ is shown for both symplectic $N$ (blue) and $\text{SP}\left(N\right)$ (red). The Ising order is long range even though the underlying antiferromagnetic order is not. The dashed (green) line indicates a first-order transition from Ising order to short-range antiferromagnetic order. Just as we saw by examining $S_c$, $\text{SP}\left(N\right)$ overstabilizes the Ising order. Insets show the appropriate valence bond order.

Figure 8

(Color online) (a) The antiferromagnetic triangular plaquette has three possible bond orderings: (left) just a single $\Delta$ connecting two of the spins and leaving the third completely free; (middle) $\Delta$’s (blue) and $h$’s (red, dashed) segregated; and (right) the uniform state, which is the ground state of symplectic $N$. (b) The tetrahedral plaquette. When all sites are assumed to be equivalent, there are three different types of bonds—as shown in black, gray, and thin black lines.

Figure 9

(Color online) Thin lines indicate the second-order transitions from short-range to long-range Néel and collinear orders. For points within the region of collinear long-range order, the ground-state energies of the two possible orders were compared. Where the collinear order is lower, a circular blue dot is placed on the phase diagram; when Néel order is lower, the dot is green and square. The black dot indicates the classical second-order phase transition, which was calculated analytically. The physical spin $S=1/2$ is indicated by the dashed line.