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arXiv:cond-mat/9605111 · Physical review. B, Condensed matter

Quantum Griffiths singularities in the transverse-field Ising spin glass

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

Published Aug 1, 1996• 93 citations
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Abstract

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.

DOI
10.1103/physrevb.54.3336

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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 \ensuremathΓc{\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}\ensuremathΓc​ for the order-disorder phase transition at T=0, the average nonlinear susceptibility \ensuremathχnl{\mathrm{\ensuremath{\chi}}}_{\mathrm{nl}}\ensuremathχnl​ diverges in the paramagnetic phase for \ensuremath{\Gamma} well above \ensuremathΓc{\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}\ensuremathΓc​, as is also demonstrated in the accompanying paper by Rieger and Young. In d=3, the linear susceptibility remains finite at \ensuremathΓc{\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}\ensuremathΓc​, and while Griffiths singularity effects are certainly observable in the paramagnetic phase, the nonlinear susceptibility appears to diverge only rather close to \ensuremathΓc{\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}\ensuremathΓc​. 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 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 the ±J transverse-field Ising spin-glass model at zero temperature on d-dimensional hypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around the strong-field limit. In the SK model and in high dimensions our calculated critical properties are in excellent agreement with the exact mean-field results, surprisingly even down to dimension d=6, which is below the upper critical dimension of d=8. In contrast, at lower dimensions we find a rich singular behavior consisting of critical and Griffiths-McCoy singularities. The divergence of the equal-time structure factor allows us to locate the critical coupling where the correlation length diverges, implying the onset of a thermodynamic phase transition. We find that the spin-glass susceptibility as well as various power moments of the local susceptibility become singular in the paramagnetic phase before the critical point. Griffiths-McCoy singularities are very strong in two dimensions but decrease rapidly as the dimension increases. We present evidence that high enough powers of the local susceptibility may become singular at the pure-system critical point.
We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one dimension. At the critical point the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single continuously varying exponent, ${z}^{\ensuremath{'}}$, which diverges at the critical point, as in one dimension. Consequently, the zero temperature susceptibility diverges for a range of parameters about the transition.
We examine the quantum phase transition at zero temperature between paramagnetic and spin-glass-ordered phases as the strength of a uniform transverse field, \ensuremath{\Gamma}, is varied. At the critical point, ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$, the spin autocorrelation function decays with time, t, as ${\mathit{t}}^{\mathrm{\ensuremath{-}}2}$. As the transition is approached from the paramagnetic phase the nonlinear susceptibility diverges as {\ensuremath{\Vert}ln(\ensuremath{\Gamma}-${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$)\ensuremath{\Vert}/(\ensuremath{\Gamma}-${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$)${\mathrm{}}}^{1/2}$.
The interplay between quantum fluctuations and disorder is investigated in a quantum spin-glass model, in the presence of a uniform transverse field $\mathrm{\ensuremath{\Gamma}}$, as well as of a longitudinal random field hi{h}_{i}hi​, which follows a Gaussian distribution characterized by a width proportional to $\mathrm{\ensuremath{\Delta}}$. The interactions are infinite-ranged, and the model is studied through the replica formalism, within a one-step replica-symmetry-breaking procedure; in addition, the dependence of the Almeida-Thouless eigenvalue ${\ensuremath{\lambda}}_{\mathrm{AT}}$ (replicon) on the applied fields is analyzed. This study is motivated by experimental investigations on the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound, where the application of a transverse magnetic field yields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic susceptibility ${\ensuremath{\chi}}_{3}$, which have led to a considerable experimental and theoretical debate. We have analyzed two physically distinct situations, namely, $\mathrm{\ensuremath{\Delta}}$ and $\mathrm{\ensuremath{\Gamma}}$ considered as independent, as well as these two quantities related, as proposed recently by some authors. In both cases, a spin-glass phase transition is found at a temperature Tf{T}_{f}Tf​, with such phase being characterized by a nontrivial ergodicity breaking; moreover, Tf{T}_{f}Tf​ decreases by increasing $\mathrm{\ensuremath{\Gamma}}$ towards a quantum critical point at zero temperature. The situation where $\mathrm{\ensuremath{\Delta}}$ and $\mathrm{\ensuremath{\Gamma}}$ are related $[\mathrm{\ensuremath{\Delta}}\ensuremath{\equiv}\mathrm{\ensuremath{\Delta}}(\mathrm{\ensuremath{\Gamma}})]$ appears to reproduce better the experimental observations on the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound, with the theoretical results coinciding qualitatively with measurements of the nonlinear susceptibility ${\ensuremath{\chi}}_{3}$. In this later case, by increasing $\mathrm{\ensuremath{\Gamma}}$ gradually, ${\ensuremath{\chi}}_{3}$ becomes progressively rounded, presenting a maximum at a temperature T∗{T}^{*}T∗ (T∗>Tf)({T}^{*}>{T}_{f})(T∗>Tf​), with both the amplitude of the maximum and the value of T∗{T}^{*}T∗ decreasing gradually. Moreover, we also show that the random field is the main responsible for the smearing of the nonlinear susceptibility, acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temperature T∗{T}^{*}T∗, one for Tf<T<T∗{T}_{f}<T<{T}^{*}Tf​<T<T∗, and another one for T>T∗T>{T}^{*}T>T∗. It is argued that the conventional paramagnetic state corresponds to T>T∗T>{T}^{*}T>T∗, whereas the temperature region Tf<T<T∗{T}_{f}<T<{T}^{*}Tf​<T<T∗ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities.