We study for random quantum spin systems the energy gap between the ground and first excited states to clarify a relation to the spin-glass-paramagnetic phase transition. We find that for the transverse Sherrington-Kirkpatrick model the vanishing of the averaged gap does not identify the transition. A power-law form of the gap distribution function leads to power-law distributions of the linear, spin-glass, and nonlinear susceptibilities. They diverge at different points, which we attribute to a quantum "Griffiths" mechanism. On the other hand, the classical treatment is justified for the transverse random energy model and the phase transition can be found by a sudden change of the ground state.
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 study effects of random fluctuations in quantum spin-glass systems. We show that the expression of the magnetic susceptibility is divided into the classical and quantum parts. The former can be analyzed by the standard replica method with the static approximation. The latter incorporates dynamical correlations and is determined by the energy gap distribution. By using the many-body interacting Ising spin-glass model with transverse field, we discuss possible mechanisms of the quantum phase transitions.
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, 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, with such phase being characterized by a nontrivial ergodicity breaking; moreover,
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∗>Tf), with both the amplitude of the maximum and the value of
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∗, one for
Tf<T<T∗, and another one for
T>T∗. It is argued that the conventional paramagnetic state corresponds to
T>T∗, whereas the temperature region
Tf<T<T∗ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities.
In this paper we investigate for the infinite-range Ising spin-glass model [i.e., the Sherrington-Kirkpatrick (SK) model] with a transverse field under the static approximation by using the imaginary-time replica formalism. From the investigations we show three important results: First, we show that a replica-symmetric quantum spin-glass phase is stable in most of the area of the spin-glass phase in the temperature-transverse field phase diagram. This confirms the existence of a stable replica-symmetric spin glass phase under the static approximation, which is contrary to some previous results derived without the static approximation where the replica-symmetric solution is always unstable in the whole spin-glass phase. Second, we show our theoretical result for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{nl}}$ which conforms to the experimental result of nonlinear susceptibility measurement by Wu et al. [Phys. Rev. Lett. 71, 1919 (1993)] in a quantum spin glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}.$ Third, in a classical (SK) spin-glass system, we confirm the anomaly in the second temperature derivative of
CH/T near the glass transition temperature
Tg, associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.