Quantum Griffiths Singularities in the Transverse-Field Ising Spin Glass

We report a Monte Carlo study of the effects of {\it fluctuations} in the bond distribution of Ising spin glasses in a transverse magnetic field, in the {\it paramagnetic phase} in the $T\to 0$ limit. Rare, strong fluctuations give rise to Griffiths singularities, which can dominate the zero-temperature behavior of these quantum systems, as originally demonstrated by McCoy for one-dimensional ($d=1$) systems. Our simulations are done on a square lattice in $d=2$ and a cubic lattice in $d=3$, for a gaussian distribution of nearest neighbor (only) bonds. In $d=2$, where the {\it linear} susceptibility was found to diverge at the critical transverse field strength $Γ_c$ for the order-disorder phase transition at T=0, the average {\it nonlinear} susceptibility $χ_{nl}$ diverges in the paramagnetic phase for $Γ$ well above $Γ_c$, as is also demonstrated in the accompanying paper by Rieger and Young. In $d=3$, the linear susceptibility remains finite at $Γ_c$, and while Griffiths singularity effects are certainly observable in the paramagnetic phase, the nonlinear susceptibility appears to diverge only rather close to $Γ_c$. These results show that Griffiths singularities remain persistent in dimensions above one (where they are known to be strong), though their magnitude decreases monotonically with increasing dimensionality (there being no Griffiths singularities in the limit of infinite dimensionality).

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