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Topological solitons have been studied for decades in classical field theories, and have started recently to impact condensed matter physics. Among those solitons, magnetic skyrmions are two-dimensional particle-like objects with a continuous winding of the magnetization, and magnetic Hopfions are three-dimensional topological solitons that can be formed from a closed loop of a twisted skyrmion string. Whereas intense research is underway with magnetic skyrmions towards a fundamental understanding and potential applications in advanced storage and logic devices, the experimental creation and confirmation of magnetic Hopfions has been elusive so far. Theoretical models suggest that Hopfions can be stabilized in frustrated or chiral magnetic systems, and that target skymions can be transformed into Hopfions by adapting their perpendicular magnetic anisotropy. Here, we present experimental evidence of magnetic Hopfions that were created in magnetic Ir/Co/Pt multilayers shaped into nanoscale disks, which are known to host target skyrmions. The three-dimensional spin texture, which distinguishes magnetic Hopfions from target skyrmions was confirmed by combining two advanced element-specific magnetic X-ray microscopy techniques with about 20-30nm lateral resolution, using X-ray magnetic circular dichroism effect as magnetic contrast mechanism in surface-sensitive X-ray photoemission electron microscopy and bulk-sensitive soft x-ray transmission microscopy. We anticipate that these results will stimulate further investigations of Hopfions with different topologies and their potential application in three-dimensional spintronics devices., Comment: 18 pages, 3 figures

In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models which are in this class have been intensively studied. Here numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlation length are presented for dimensions 3, 4, 5 and 7. Hyperscaling is clearly violated in dimensions 3 and 4 as well as above the upper critical dimension $D=6$. Estimates are obtained for the "violation of hyperscaling exponent" values in the various models., Comment: 5 pages, 9 figures

Simulation data are analyzed for four 3D spin-$1/2$ Ising models: on the FCC lattice, the BCC lattice, the SC lattice and the Diamond lattice. The observables studied are the susceptibility, the reduced second moment correlation length, and the normalized Binder cumulant. From measurements covering the entire paramagnetic temperature regime the corrections to scaling are estimated. We conclude that a correction term having an exponent which is consistent within the statistics with the bootstrap value of the universal subleading thermal confluent correction exponent, $\theta_{2} \sim 2.454(3)$, is almost always present with a significant amplitude. In all four models, for the normalized Binder cumulant the leading confluent correction term has zero amplitude. This implies that the universal ratio of leading confluent correction amplitudes $a_{\chi_{4}}/a_{\chi} = 2$ in the 3D Ising universality class., Comment: 10 pages, 22 figures

Direct measurements of the spin glass correlation function $G(R)$ for Gaussian and bimodal Ising spin glasses in dimension two have been carried out in the temperature region $T \sim 1$. In the Gaussian case the data are consistent with the known anomalous dimension value $\eta \equiv 0$. For the bimodal spin glass in this temperature region $T > T^{*}(L)$, well above the crossover $T^{*}(L)$ to the ground state dominated regime, the effective exponent $\eta$ is clearly non-zero and the data are consistent with the estimate $\eta \sim 0.28(4)$ given by McMillan in 1983 from similar measurements. Measurements of the temperature dependence of the Binder cumulant $U_{4}(T,L)$ and the normalized correlation length $\xi(T,L)/L$ for the two models confirms the conclusion that the 2D bimodal model has a non-zero effective $\eta$ both below and above $T^{*}(L)$. The 2D bimodal and Gaussian interaction distribution Ising spin glasses are not in the same Universality class., Comment: 6 pages, 8 figures

Among the Renormalization Group Theory scaling rules relating critical exponents, there are hyperscaling rules involving the dimension of the system. It is well known that in Ising models hyperscaling breaks down above the upper critical dimension. It was shown by M. Schwartz [Europhys. Lett. {\bf 15}, 777 (1991)] that the standard Josephson hyperscaling rule can also break down in Ising systems with quenched random interactions. A related Renormalization Group Theory hyperscaling rule links the critical exponents for the normalized Binder cumulant and the correlation length in the thermodynamic limit. An appropriate scaling approach for analyzing measurements from criticality to infinite temperature is first outlined. Numerical data on the scaling of the normalized correlation length and the normalized Binder cumulant are shown for the canonical Ising ferromagnet model in dimension three where hyperscaling holds, for the Ising ferromagnet in dimension five (so above the upper critical dimension) where hyperscaling breaks down, and then for Ising spin glass models in dimension three where the quenched interactions are random. For the Ising spin glasses there is a breakdown of the normalized Binder cumulant hyperscaling relation in the thermodynamic limit regime, with a return to size independent Binder cumulant values in the finite-size scaling regime around the critical region., Comment: 11 pages, 22 figures

Following numerous earlier studies, extensive simulations and analyses were made on the continuous interaction distribution Gaussian model and the discrete bimodal interaction distribution Ising Spin Glass (ISG) models in dimension two (P.H. Lundow and I.A. Campbell, Phys. Rev. E {\bf 93}, 022119 (2016)). Here we further analyse the bimodal and Gaussian data together with data on two other continuous interaction distribution 2D ISG models, the uniform and the Laplacian models, and three other discrete interaction distribution models, a diluted bimodal model, an "anti-diluted" model, and a more exotic symmetric Poisson model. Comparisons between the three continuous distribution models show that not only do they share the same exponent $\eta \equiv 0$ but that to within the present numerical precision they share the same critical exponent $\nu$ also, and so lie in a single universality class. On the other hand the critical exponents of the four discrete distribution models are not the same as those of the continuous distributions, and differ from one discrete distribution model to another. Discrete distribution ISG models in dimension two have non-zero values of the critical exponent $\eta$; they do not lie in a single universality class., Comment: 17 pages, 33 figures

Ising spin glass models with bimodal, Gaussian, uniform and Laplacian interaction distributions in dimension five are studied through detailed numerical simulations. The data are analyzed in both the finite-size scaling regime and the thermodynamic limit regime. It is shown that the values of critical exponents and of dimensionless observables at criticality are model dependent. Models in a single universality class have identical values for each of these critical parameters, so Ising spin glass models in dimension five with different interaction distributions each lie in different universality classes. This result confirms conclusions drawn from measurements in dimension four and dimension two., Comment: 13 pages, 26 figures

Numerical data on scaling of the normalized Binder cumulant and the normalized correlation length are shown for the Thermodynamic limit regime, first for canonical Ising ferromagnet models and then for a range of Ising spin glass models. A fundamental Renormalization Group Theory rule linking the critical exponents for the two observables is well obeyed in the Ising models, but not for the Ising spin glasses in dimensions three and four. We conclude that there is a violation of a standard Josephson hyperscaling rule in Ising spin glasses., Comment: 7 pages, 11 figures

Ising spin glasses with bimodal and Gaussian near-neighbor interaction distributions are studied through numerical simulations. The non-self-averaging (normalized inter-sample variance) parameter $U_{22}(T,L)$ for the spin glass susceptibility (and for higher moments $U_{nn}(T,L)$) is reported for dimensions 2, 3, 4, 5 and 7. In each dimension $d$ the non-self-averaging parameters in the paramagnetic regime vary with the sample size L and the correlation length $\xi(T,L)$ as $U_{nn}(\beta,L) = [K_{d}\xi(T,L)/L]^d$, and so follow a renormalization group law due to Aharony and Harris (1991). Empirically, it is found that the $K_{d}$ values are independent of d to within the statistics. The maximum values $[U_{nn}(T,L)]_{\max}$ are almost independent of L in each dimension, and remarkably the estimated thermodynamic limit critical $[U_{nn}(T,L)]_{\max}$ peak values are also dimension-independent to within the statistics and so are "hyperuniversal". These results show that the form of the spin-spin correlation function distribution at criticality in the large $L$ limit is independent of dimension within the ISG family. Inspection of published non-self-averaging data for 3D Heisenberg and XY spin glasses the light of the Ising spin glass non-self-averaging results show behavior incompatible with a spin-driven ordering scenario, but compatible with that expected on a chiral-driven ordering interpretation., Comment: 10 pages, 22 figures

The critical behaviors of the bimodal and Gaussian Ising spin glass (ISG) models in dimension four are studied through extensive numerical simulations, and from an analysis of high temperature series expansion (HTSE) data of Klein {\it et al.} (1991). The simulations include standard finite size scaling measurements, thermodynamic limit regime measurements, and analyses which provide estimates of critical exponents without any consideration of the critical temperature. The higher order HTSE series for the bimodal model provide accurate estimates of the critical temperature and critical exponents. These estimates are independent of and fully consistent with the simulation values. Comparisons between ISG models in dimension four show that the critical exponents and the critical constants for dimensionless observables depend on the form of the interaction distribution of the model., Comment: 10 pages, 15 figures

A new analysis is given of numerical simulation data on the archetype square lattice Ising Spin Glasses (ISG) with a bimodal ($\pm J$) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian has a non-degenerate ground state so exponent $\eta \equiv 0$ and it has a continuous distribution of energy levels. For the bimodal model, above a size dependent cross-over temperature $T^{*}(L)$ there is a regime of effectively continuous energy levels; below $T^{*}(L)$ there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states. $T^{*}(L)$ tends to zero at very large $L$ leaving only the effectively continuous regime in the thermodynamic limit. We show that in this regime the critical exponent $\eta$ is not zero, so the effectively continuous regime $2$D bimodal ISG is not in the same universality class as the $2$D Gaussian ISG. The simulation data on both models are analyzed using a scaling variable $\tau = T^2/(1+T^2)$ suitable for zero temperature transition ISGs, together with appropriate scaling expressions. Accurate simulation estimates can be obtained for the temperature dependence of the thermodynamic limit reduced susceptibility $\chi(\tau)$ and second moment correlation length $\xi(\tau)$ over the entire range of temperature from zero to infinity. The Gaussian critical exponent from the simulations $\nu = 3.5(1)$ is in full agreement with the well established value from the literature. The bimodal exponent from the thermodynamic limit regime analysis is $\nu = 4.2(1)$, once again different from the Gaussian value., Comment: 10 pages, 18 figures