Goldstone modes in the emergent gauge fields of a frustrated magnet

We consider magnon excitations in the spin-glass phase of geometrically frustrated antiferromagnets with weak exchange disorder, focusing on the nearest-neighbor pyrochlore-lattice Heisenberg model at large spin. The low-energy degrees of freedom in this system are represented by three copies of a $\mathrm{U}\left(1\right)$ emergent gauge field, related by global spin-rotation symmetry. We show that the Goldstone modes associated with spin-glass order are excitations of these gauge fields, and that the standard theory of Goldstone modes in Heisenberg spin glasses (due to Halperin and Saslow) must be modified in this setting.

Figure 1

Integrated densities $N\left(\lambda \right)$ of Hessian eigenvalues $\lambda$ [(a) and (b)] and $D\left(\omega \right)$ of magnon frequencies $\omega$ [(c) and (d)], with dependence on system size $L$ and disorder strength $\mathrm{\Delta }/J$. (a) and (c): for $2^{-10}\le \mathrm{\Delta }/J\le 2^{-6}$ at $L=3$; (b) and (d): for $3\le L\le 7$ at $\mathrm{\Delta }/J=2^{-6}$. Dashed line in (b) represents $N\left(\lambda \right)\propto {(\lambda /\mathrm{\Delta })}^{3/2}$; dashed line in (d) represents $N\left(\omega \right)\propto {(\omega /\mathrm{\Delta })}^3$. Insets in (a) and (c) show full range of $N\left(\lambda \right)$ and $D\left(\omega \right)$.

Figure 2

Characterization of low-lying Hessian and dynamical eigenvectors. (a) Scaling collapse of angular correlator for lowest nontrivial Hessian eigenvectors, with $3\le L\le 7$ and $\mathrm{\Delta }/J=2^{-6}$. (b) Equivalent for dynamical eigenvectors. (c) Smooth magnetization density in dynamical modes vs. $\omega /\mathrm{\Delta }$ at $L=3$ for $2^{-10}\le \mathrm{\Delta }/J\le 2^{-6}$. Inset: correlator of magnetizations of neighboring tetrahedra in Hessian eigenvectors. See main text for discussion. Fluctuations evident in (a) and (b) for smaller $L$ are predominantly a finite-size effect, rather than because of sample-to-sample fluctuations.