A high-temperature series expansion for the Edwards and Anderson spin-glass order-parameter susceptibility is computed for Ising spins on hypercubic lattices with nearest-neighbor interactions. The series is analyzed by Pad\'e approximants with Rudnick-Nelson-type corrections to scaling. The results agree with the first-order $\ensuremath{\epsilon}$ expansion of Harris, Lubensky, and Chen. The critical exponent ${\ensuremath{\gamma}}_{Q}$ increases monotonically with decreasing dimension, $d$ , for $d<6$ , and apparently tends to infinity at $d=4$ ; however, the critical temperature does not appear to go to zero at $d=4$ .A high-temperature series expansion for the Edwards and Anderson spin-glass order-parameter susceptibility is computed for Ising spins on hypercubic lattices with nearest-neighbor interactions. The series is analyzed by Pad\'e approximants with Rudnick-Nelson-type corrections to scaling. The results agree with the first-order $\ensuremath{\epsilon}$ expansion of Harris, Lubensky, and Chen. The critical exponent ${\ensuremath{\gamma}}_{Q}$ increases monotonically with decreasing dimension, $d$, for $d<6$, and apparently tends to infinity at $d=4$; however, the critical temperature does not appear to go to zero at $d=4$.

Physical Review LettersApr 4, 1977186 citations

Series expansions for the Ising spin glass in general dimension

Lior Klein, Joan Adler, Amnon Aharony A +2 more

We have developed 15th-order high-temperature series expansions for the study of the critical behavior of the Ising spin glass with nearest-neighbor exchange interactions each of which assumes the values \ifmmode\pm\else\textpm\fi{}J randomly. Series for the Edwards-Anderson spin-glass susceptibility (${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$) and two of its derivatives with respect to the ordering field have been evaluated for hypercubic lattices in general dimension, d. These extend previous general-dimension series by five terms. Certain measurable universal amplitude ratios have been estimated from the new series. Accurate critical data for d=5 and the first reliable estimates of the exponent \ensuremath{\beta} for d=4 and 5, are given. We quote \ensuremath{\gamma}=1.73\ifmmode\pm\else\textpm\fi{}0.03, 2.00\ifmmode\pm\else\textpm\fi{}0.25, and 2.${7}_{\mathrm{\ensuremath{-}}0.6}^{+1.0}$ and \ensuremath{\beta}=0.95\ifmmode\pm\else\textpm\fi{}0.04, 0.9\ifmmode\pm\else\textpm\fi{}0.1, and 0.7\ifmmode\pm\else\textpm\fi{}0.2 in 5, 4, and 3 dimensions, respectively. Our results provide a smooth extrapolation between the mean-field results above six dimensions and experiments and simulations in physical dimensions. We relate our calculated derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ to measurements of derivatives of the magnetization with respect to a uniform magnetic field.We have developed 15th-order high-temperature series expansions for the study of the critical behavior of the Ising spin glass with nearest-neighbor exchange interactions each of which assumes the values \ifmmode\pm\else\textpm\fi{}J randomly. Series for the Edwards-Anderson spin-glass susceptibility (${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$) and two of its derivatives with respect to the ordering field have been evaluated for hypercubic lattices in general dimension, d. These extend previous general-dimension series by five terms. Certain measurable universal amplitude ratios have been estimated from the new series. Accurate critical data for d=5 and the first reliable estimates of the exponent \ensuremath{\beta} for d=4 and 5, are given. We quote \ensuremath{\gamma}=1.73\ifmmode\pm\else\textpm\fi{}0.03, 2.00\ifmmode\pm\else\textpm\fi{}0.25, and 2.${7}_{\mathrm{\ensuremath{-}}0.6}^{+1.0}$ and \ensuremath{\beta}=0.95\ifmmode\pm\else\textpm\fi{}0.04, 0.9\ifmmode\pm\else\textpm\fi{}0.1, and 0.7\ifmmode\pm\else\textpm\fi{}0.2 in 5, 4, and 3 dimensions, respectively. Our results provide a smooth extrapolation between the mean-field results above six dimensions and experiments and simulations in physical dimensions. We relate our calculated derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ to measurements of derivatives of the magnetization with respect to a uniform magnetic field.

Physical review. B, Condensed matterMay 1, 199159 citations

Griffiths singularities in the disordered phase of a quantum Ising spin glass

Heiko Rieger, A. P. Young

We study a model for a quantum Ising spin glass in two space dimensions by Monte Carlo simulations. In the disordered phase at T=0, rare strongly correlated regions give rise to strong Griffiths singularities, as originally found by McCoy for a one-dimensional model. We find that there are power law distributions of the local susceptibility and local nonlinear susceptibility, which are characterized by a smoothly varying dynamical exponent z. Over a range of the disordered phase near the quantum transition, the local nonlinear susceptibility diverges. The local susceptibility does not diverge in the disordered phase but does diverge at the critical point. Approaching the critical point from the disordered phase, the limiting value of z seems to equal its value precisely at criticality, even though the physics of these two cases seems rather different. \textcopyright{} 1996 The American Physical Society.We study a model for a quantum Ising spin glass in two space dimensions by Monte Carlo simulations. In the disordered phase at T=0, rare strongly correlated regions give rise to strong Griffiths singularities, as originally found by McCoy for a one-dimensional model. We find that there are power law distributions of the local susceptibility and local nonlinear susceptibility, which are characterized by a smoothly varying dynamical exponent z. Over a range of the disordered phase near the quantum transition, the local nonlinear susceptibility diverges. The local susceptibility does not diverge in the disordered phase but does diverge at the critical point. Approaching the critical point from the disordered phase, the limiting value of z seems to equal its value precisely at criticality, even though the physics of these two cases seems rather different. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9512162Physical review. B, Condensed matterAug 1, 1996137 citations

Quantum Griffiths singularities in the transverse-field Ising spin glass

Muyu Guo, R. N. Bhatt, David A. Huse

We report a Monte Carlo study of the effects of fluctuations in the bond distribution of Ising spin glasses in a transverse magnetic field, in the paramagnetic phase in the T\ensuremath{\rightarrow}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 linear susceptibility was found to diverge at the critical transverse field strength ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$ for the order-disorder phase transition at T=0, the average nonlinear susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{nl}}$ diverges in the paramagnetic phase for \ensuremath{\Gamma} well above ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$, as is also demonstrated in the accompanying paper by Rieger and Young. In d=3, the linear susceptibility remains finite at ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$, and while Griffiths singularity effects are certainly observable in the paramagnetic phase, the nonlinear susceptibility appears to diverge only rather close to ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$. 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). \textcopyright{} 1996 The American Physical Society.We report a Monte Carlo study of the effects of fluctuations in the bond distribution of Ising spin glasses in a transverse magnetic field, in the paramagnetic phase in the T\ensuremath{\rightarrow}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 linear susceptibility was found to diverge at the critical transverse field strength ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$ for the order-disorder phase transition at T=0, the average nonlinear susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{nl}}$ diverges in the paramagnetic phase for \ensuremath{\Gamma} well above ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$, as is also demonstrated in the accompanying paper by Rieger and Young. In d=3, the linear susceptibility remains finite at ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$, and while Griffiths singularity effects are certainly observable in the paramagnetic phase, the nonlinear susceptibility appears to diverge only rather close to ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$. 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). \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9605111Physical review. B, Condensed matterAug 1, 199693 citations

Griffiths-McCoy singularities in random quantum spin chains: Exact results

Ferenc Iglói, István A. Kovács

We consider random quantum (tight-binding, XX, and Ising) spin chains in the off-critical region and study their Griffiths-McCoy singularities. These are obtained from the density of states of the low-energy excitations, which is exactly calculated by the Dyson-Schmidt method. In large finite systems, the low-energy excitations are shown to follow the statistics of extremes and their distribution is given by the Fr\'echet form. The relation between the Dyson-Schmidt technique and the strong disorder renormalization group method is also discussed.We consider random quantum (tight-binding, XX, and Ising) spin chains in the off-critical region and study their Griffiths-McCoy singularities. These are obtained from the density of states of the low-energy excitations, which is exactly calculated by the Dyson-Schmidt method. In large finite systems, the low-energy excitations are shown to follow the statistics of extremes and their distribution is given by the Fr\'echet form. The relation between the Dyson-Schmidt technique and the strong disorder renormalization group method is also discussed.

arXiv:0803.2617Physical Review BApr 8, 20084 citations

Random Quantum Magnets with Long-Range Correlated Disorder: Enhancement of Critical and Griffiths-McCoy Singularities

Heiko Rieger, Ferenc Iglói

We study the effect of spatial correlations on quenched disorder in random quantum magnets at and near a quantum critical point. In random transverse-field Ising systems disorder correlations that decay algebraically with an exponent $\ensuremath{\rho}$ change the universality class of the transition for small enough $\ensuremath{\rho}$, and off-critical Griffiths-McCoy singularities are enhanced. We present exact results for 1D utilizing a mapping to fractional Brownian motion and generalize the predictions for the critical exponents and the generalized dynamical exponent in the Griffiths phase to $d\ensuremath{\ge}2$.We study the effect of spatial correlations on quenched disorder in random quantum magnets at and near a quantum critical point. In random transverse-field Ising systems disorder correlations that decay algebraically with an exponent $\ensuremath{\rho}$ change the universality class of the transition for small enough $\ensuremath{\rho}$, and off-critical Griffiths-McCoy singularities are enhanced. We present exact results for 1D utilizing a mapping to fractional Brownian motion and generalize the predictions for the critical exponents and the generalized dynamical exponent in the Griffiths phase to $d\ensuremath{\ge}2$.

arXiv:cond-mat/9904280Physical Review LettersNov 1, 199928 citations

d

, for

d<6

, and apparently tends to infinity at

d=4

; however, the critical temperature does not appear to go to zero at

d=4