Spin and chirality orderings of the three-dimensional Heisenberg spin glass are studied under magnetic fields in light of the recently developed spin-chirality decoupling-recoupling scenario. It is found by Monte Carlo simulations that the chiral-glass transition and the chiral-glass ordered state, which are essentially of the same character as their zero-field counterparts, occur under magnetic fields. The implication to the experimental phase diagram is discussed.

Ordering of the Heisenberg spin glass with the nearest-neighbor Gaussian coupling is investigated by equilibrium Monte Carlo simulations in four and five dimensions. Ordering of the mean-field Heisenberg spin glass is also studied for comparison. Particular attention is paid to the nature of the spin-glass and chiral-glass orderings. Our numerical data suggest that, in five dimensions, the model exhibits a single spin-glass transition at a finite temperature, where the spin-glass order accompanying the simultaneous chiral-glass order sets in. In four dimensions, the model exhibits a marginal behavior. Chiral-glass transition at a finite temperature not accompanying the standard spin-glass order is likely to occur, while the critical region associated with the chiral-glass transition is very narrow suggesting that the dimension four is close to the marginal dimensionality.Concerning the point (i), several earlier numerical studies including the high-temperature expansion 16 and the numerical domain-wall renormalization-group calculation 4,17 suggested that d SG ℓ might be close to four. Meanwhile, Anderson and Pond argued that d SG ℓ = 3 18 . First Monte Carlo (MC) simulation on the high-dimensional Heisenberg EA model was performed by Stauffer and Binder 19 . By studying the temporal decay of the EA order parameter, they suggested that a finite-temperature SG order occurred in D = 5 and 6, but not in D ≤ 4 19 . More recently, the 4D Heisenberg EA model was studied by Coluzzi by equilibrium MC simulation 20 . By examining the behavior of the Binder ratio, she suggested the occurrence of a finite-temperature SG transition in D = 4 in contrast to the suggestion of Ref. 19 . There seems to be no consensus as to the point (i).The point (ii) above is closely related to the controversy regarding whether the spin-glass and the chiral-glass orders occur simultaneously or separately in 3D. To the authors' knowledge, concerning the chiral-glass order in D ≥ 4 dimensions, no calculation has been reported so far. In the present paper, we wish to fill this gap. We study both the spin-glass and the chiral-glass orders of the Heisenberg EA model in both 4D and 5D by means of a largescale equilibrium MC simulation. In particular, we simulate larger lattices and lower temperatures than those covered in Ref. 20 . For comparison, a simulation is also performed on the mean-field Heisenberg SK model corresponding to D = ∞.Our data suggest that, in 5D, the model exhibits a single SG transition at a finite temperature reminiscent to the one of the Heisenberg SK model. The chirality orders simultaneously with the spin, but it behaves as the composite operator of the spin, not as the order parameter. The SG ordered state in 5D accompanies a peculiar type of RSB, most probably a one-step-like RSB, which is different in character from the full RSB realized in the SK model. In 4D, by contrast, the model exhibits a pure chiral-glass transition at a finite temperature, not accompanying the standard SG order. The critical region associated wit...

The ordering of the three-dimensional Heisenberg spin glass with the weak random anisotropy in magnetic fields is studied by extensive equilibrium Monte Carlo simulations. Both the spin and the chirality are monitored. We find strong numerical evidence that a replica symmetry breaking transition occurs in the chiral sector, which accompanies the simultaneous spin-glass order. Despite the similarity in the global symmetry, the ordering behavior of the weakly anisotropic Heisenberg spin glass differs significantly from that of the strongly anisotropic Ising spin glass. The obtained phase diagram in the temperature -magnetic field plane is similar to the experimental phase diagram. Our results highlight the importance of the chirality in the spin-glass ordering of the Heisenberg-like spin glass, and support the spin-chirality decoupling-recoupling scenario of spin-glass transitions.

The spin and the chirality orderings of the three-dimensional Heisenberg spin glass with the weak random anisotropy are studied under applied magnetic fields by equilibrium Monte Carlo simulations. A replica symmetry breaking transition occurs in the chiral sector accompanied by the simultaneous spin-glass order. The ordering behavior differs significantly from that of the Ising SG, despite the similarity in the global symmetry. Our observation is consistent with the spin-chirality decoupling-recoupling scenario of a spin-glass transition.Whether or not spin-glass (SG) magnet exhibits a thermodynamic phase transition in applied magnetic fields has been a long-standing issue [1]. This issue is closely related to the fundamental question of whether the SG ordered state in zero field accompanies an ergodicity breaking not directly related to the global symmetry of the Hamiltonian, i.e., the replica symmetry breaking (RSB). Most of numerical studies on SG have focused on the properties of the Ising SG [1]. Since the Ising SG possesses no global symmetry in magnetic fields, the occurrence of a phase transition in fields would necessarily mean the occurrence of RSB. Unfortunately, numerical simulations on the Ising SG have been unable to give a definitive answer concerning the existence of a SG transition in magnetic fields [2,3].Experimentally, some evidence against an in-field transition has been reported for the strongly anisotropic Ising-like SG Fe 0.5 Mn 0.5 TiO 3 [4].In fact, many of real SG materials are more or less Heisenberg-like rather than Ising-like, in the sense that the magnetic anisotropy is considerably weaker than the isotropic exchange interaction [1]. Recent experiments on such weakly anisotropic Heisenberg-like SG suggested the occurrence of an in-field SG transition [5], in apparent contrast to Ref. [4]. Meanwhile, via recent theoretical studies, it has become increasingly clear that the Heisenberg SG possesses an important physical ingredient which is totally absent in the Ising SG, i.e., the chirality [6,7]. In particular, the chirality scenario of Ref. [6,7] claims that the chirality is a hidden order parameter of the SG transition of real Heisenberg-like SG magnets. According to this spin-chirality decoupling-recoupling scenario, in the fully isotropic Heisenberg SG, the spin and the chirality, which are coupled at short length/time scales, are eventually decoupled at long length/time scales, and the system exhibits a chiral-glass transition without the standard SG order. In the weakly anisotropic Heisenberg SG, the Heisenberg spin is "recoupled" to the chirality at these long length/time scales via the random magnetic anisotropy. The SG order of the weakly anisotropic Heisenberg SG is then dictated by the ordering of the chirality. In zero field, some numerical support for such a scenario was reported [8,9,10,11,12], although some other groups claimed that the chiral-glass transition of the isotropic system already accompanied the SG order [13,14].In connection to experiments, in-field order...

Spin and chirality orderings of the three-dimensional Heisenberg spin glass under magnetic fields are studied by large-scale equilibrium Monte Carlo simulations. It is found that the chiral-glass transition and the chiral-glass ordered state, which are essentially of the same character as their zero-field counterparts, occur under magnetic fields. The chiral-glass ordered state exhibits a onestep-like peculiar replica-symmetry breaking in the chiral sector, while it does not accompany the spin-glass order perpendicular to the applied field. Critical perperties of the chiral-glass transition are different from those of the standard Ising spin glass. Magnetic phase diagram of the model is constructed, which reveals that the chiral-glass state is quite robust against magnetic fields. The chiral-glass transition line has a character of the Gabay-Toulouse line of the mean-field model, yet its physical origin being entirely different. These numerical results are discussed in light of the recently developed spin-chirality decoupling-recoupling scenario. Implication to experimental phase diagram is also discussed. *