Excitation Spectra of Disordered Dimer Magnets Near Quantum Criticality

For coupled-dimer magnets with quenched disorder, we introduce a generalization of the bond-operator method, appropriate to describe both singlet and magnetically ordered phases. This allows for a numerical calculation of the magnetic excitations at all energies across the phase diagram, including the strongly inhomogeneous Griffiths regime near quantum criticality. We apply the method to the bilayer Heisenberg model with bond randomness and characterize both the broadening of excitations and the transfer of spectral weight induced by disorder. Inside the antiferromagnetic phase this model features the remarkable combination of sharp magnetic Bragg peaks and broad magnons, the latter arising from the tendency to localization of low-energy excitations.

Figure 1

Numerical results for the disordered bilayer Heisenberg model, with a concentration of $p$ weak interlayer couplings with $J_2=J_1/2$. (a) Phase diagram, showing the staggered magnetization $M_s$ as function of $K/K_c$ for different levels of disorder $p$, with $K_c=J_1/4$ denoting the critical coupling of the clean system in the linearized bond-operator approach. A regime with weak and strongly inhomogeneous order emerges for small $p$ and $K\lesssim K_c$. (b) $M_s$ as function of doping level $p$ for different $K$ values. [(c)–(h)] Transverse dynamic susceptibility ${\chi }^{\prime \prime }(\overrightarrow{q},\omega )$ along a path in the 2D Brillouin zone ($q_z=\pi$) for different combinations of $K/K_c$ and $p$. The green dashed lines indicate the dispersion of the clean system with the respective $K$.

Figure 2

Momentum-integrated transverse susceptibility ${\chi }^{\prime \prime }(\omega )$, for the bilayer model with random weak interlayer couplings as in Fig. 1. Disorder tends to close the spin gap by the transfer of spectral weight to low energies.

Figure 3

Bilayer Heisenberg magnet with $K/K_c=0.92$ and a single disorder realization of $p=0.02$ weak intradimer bonds with $J_2=J_1/2$, as in Fig. 1g, with $L=64$. (a) Spatial distribution of $J$ values, indicating the location of the weak bonds. (b) Corresponding spatial distribution of the local magnetization $M_i$. [(c)–(e)] Wave function amplitudes of selected eigenmodes of ${\mathcal{H}}_2$. (f) Disorder-averaged probability distribution of local (staggered) magnetization values for parameters as in (a)–(e).

Figure 4

(a) Static structure factor $S(\overrightarrow{q})$ and (b) susceptibility ${\chi }^{\prime \prime }(\overrightarrow{q},\omega )$ at different $\omega$ values, for $K/K_c=0.92$ and $p=0.02$ weak intradimer bonds for $L=128$. The inset in (a) shows the finite-size scaling of $[S(\overrightarrow{Q})/(2N){]}^{1/2}$, which equals the order parameter $M_s$ in the thermodynamic limit.

Figure 5

Results for the susceptibility ${\chi }^{\prime \prime }(\overrightarrow{q},\omega )$ as in Fig. 1, but for a density of $p$ strong intradimer couplings with $J_2=3J_1/2$.