We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.
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.
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.
A model of the quantum (transverse) Ising spin glass with a long-range random interaction is proposed and studied here. In this model the random interaction between two spins at a distance
rij apart falls algebraically as ${1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.$ Here, apart from the strength of the quantum fluctuations, the interaction range $\ensuremath{\sigma}$ is also tunable. We have studied the ground-state properties of the model, extending the ``Droplet model'' of the short-range quantum spin glass. The model is also studied using a modified form of the effective Landau action describing the transition of the short-range quantum spin glass. The important features which are due to the long-range interaction are clearly mentioned. Field theoretical renormalization group calculations fail to locate any stable weak-coupling fixed point in the non-mean-field region. The simplest conjecture is that beyond the mean-field region, the critical behavior is governed by the infinite randomness fixed point. We extend the phenomenological scaling relation for the short-range quantum spin glass (based on the assumption of the existence of a dangerously irrelevant operator) to the present long-range interacting case. Most of the possible interesting aspects associated with the quantum transition in the present model are elaborately discussed.
By using real-space renormalization group (RG) methods, we show that spin glasses in a field display a new kind of transition in high dimensions. The corresponding critical properties and the spin-glass phase are governed by two nonperturbative zero-temperature fixed points of the RG flow. We compute the critical exponents and discuss the RG flow and its relevance for three-dimensional systems. The new spin-glass phase we discovered has unusual properties, which are intermediate between the ones conjectured by droplet and full replica symmetry-breaking theories. These results provide a new perspective on the long-standing debate about the behavior of spin glasses in a field.