Continuous magnetic phase transition in artificial square ice

Abstract

Critical behavior is very common in many fields of science and a wide variety of many-body systems exhibit emergent critical phenomena. The beauty of critical phase transitions lies in their scale-free properties, such that the temperature dependence of physical parameters of systems differing at the microscopic scale can be described by the same generic power laws. In this work we establish the critical properties of the antiferromagnetic phase transition in artificial square ice, showing that it belongs to the two-dimensional Ising universality class, which extends the applicability of such concepts from atomistic to mesoscopic magnets. Combining soft x-ray resonant magnetic scattering experiments and Monte Carlo simulations, we characterize the transition to the low temperature long range order expected for the artificial square ice system. By measuring the critical scattering, we provide direct quantitative evidence of a continuous magnetic phase transition, obtaining critical exponents which are compatible with those of the two-dimensional Ising universality class. In addition, by varying the blocking temperature relative to the phase transition temperature, we demonstrate its influence on the out-of-equilibrium dynamics due to critical slowing down at the phase transition.

Advances in artificial spin ice

“…The exploration of the phase diagrams in artificial spin systems has only recently begun, spurred on by the creation of systems with coupled superparamagnets in thermodynamic equilibrium and the possibilities for characterization with lowenergy muon spin spectroscopy 11 , as well as neutron and synchrotron X-ray scattering. In particular, the use of indirect measurement techniques such as resonant soft X-ray scattering 12,97,151,152 , neutron scattering and reflectometry [100][101][102]153 , and X-ray photon correlation spectroscopy 82,154 can contribute to further understanding of the long-range and short-range spatial correlations in these frustrated systems. The low-temperature phases in artificial kagome spin ice have yet to be observed, and there are many different universality classes of phase transitions to explore, not only Ising but also Kosterlitz-Thouless 22,93,155 and Potts transitions 94 , to name a few.…”

“…'Methods' image is adapted with permission from reF. 12 , APS. 'Dynamics' image is adapted from reF.…”

“…Therefore, artificial square ice does not have an extensive degeneracy, and it is of interest to clarify the nature of the phase transition to the low-energy state 90,91 . Indeed, in recent X-ray scattering experiments, first measurements of critical behaviour across the phase transition have been obtained, providing evidence of a continuous transition belonging to the two-dimensional Ising universality class 12 . In addition, it may be possible to alter the nature of the phase transition by modifying the energies of the vertex configurations, for example by using an exchange-bias field 92 .…”

“…In this Review, we focus on what has happened in the field of artificial spin ice since two major reviews in 2013 (reFs 3,4 ), which provide a useful foundation for a reader unfamiliar with the field. In particular, in frustrated arrangements of nanomagnets, there has been a wide variety of interesting phenomena discovered, including emergent magnetic monopoles 5,6 , vertexbased frustration 7,8 , chiral dynamics 9 and phase transitions [10][11][12] . The existence of emergent phenomena, which arise from the collective behaviour of the nanomagnets, underscores why artificial spin ices can be considered as metamaterials.…”

Colloquium : Ice rule and emergent frustration in particle ice and beyond

“…They should perhaps be called "Topology Breaking" systems because for any non-zero temperature, violations of the topological constraints (the ice-rule) in form of monopoles substantially changes the physics. For a dramatic example: while the antiferromagnetic ground state of artificial square ice (Morgan et al, 2011;Nisoli et al, 2010;Porro et al, 2013;Wang et al, 2006;Zhang et al, 2013) would seem to be well described by the Rys F-model, the ordering transition of the latter is in the Kosterlitz Thouless class (Kosterlitz and Thouless, 1973;Lieb, 1967a), whereas in "real" square ice the transition is simply second order in the Ising class (Levis et al, 2013;Sendetskyi et al, 2019;Wu, 1969) precisely because monopoles break the topological constraint.…”