This Colloquium reviews the 25~year quest to understand the continuous (second-order), mean-field-like phase transition occurring at 17.5~K in URu2Si2. About ten years ago, the term ``hidden order'' (HO) was coined and has since been utilized to describe the unknown ordered state, whose origin cannot be disclosed by conventional solid-state probes, such as x~rays, neutrons, or muons. The HO is able to support superconductivity at lower temperatures (Tc$\approx${}1.5\,\,K), and when magnetism is developed with increasing pressure both the HO and the superconductivity are destroyed. Other ways of probing the HO are via Rh~doping and large magnetic fields. During the last few years a variety of advanced techniques have been tested to probe the HO state and these attempts will be summarized. A digest of recent theoretical developments is also included. It is the objective of this Colloquium to shed additional light on the HO state and its associated phases in other materials.

Throughout the past three decades, the hidden order (HO) problem in URu₂Si₂ has remained a ‘hot topic’ in the physics of strongly correlated electron systems with well over 600 publications related to this subject. Presently in 2014, there has been significant progress in combining various experimental results embedded within electronic structure calculations using density functional theory (DFT) to give a consistent description of the itinerant behaviour of the HO transition and its low temperature state. Here, we review six different experiments: ARPES, quantum oscillations, neutron scattering, RXD, optical spectroscopy and STM/STS. We then establish the consistencies among these experiments when viewed through the Fermi-surface nesting, folding and gapping framework as predicted by DFT. We also discuss a group of other experiments (torque, cyclotron resonance, NMR and XRD) that are more controversial and are presently in a ‘transition’ state regarding their interpretation as rotational symmetry breaking and dotriacontapole formation. There are also a series of recent ‘exotic’ experiments (Raman scattering, polar Kerr effect and ultrasonics) that require verification, yet they offer new insights into the HO symmetry breaking and order parameter. We conclude with some constraining comments on the microscopic models that rely on localized 5f-U states and strong Ising anisotropy for explaining the HO transition, and with an examination of different models in the light of recent experiments.

This topical review describes the multitude of unconventional behaviors in the hidden order, heavy fermion, antiferromagnetic and superconducting phases of the intermetallic compound URu ₂ Si ₂ when tuned with pressure, magnetic field, and substitutions for all three elements. Such ‘perturbations’ result in a variety of new phases beyond the mysterious hidden order that are only now being slowly understood through a series of state-of-the-science experimentation, along with an array of novel theoretical approaches. Despite all these efforts spanning more than 30 years, hidden order (HO) remains puzzling and non-clarified, and the search continues in 2019 into a fourth decade for its final resolution. Here we attempt to update the present situation of URu ₂ Si ₂ importing the latest experimental results and theoretical proposals. First, let us consider the pristine compound as a function of temperature and report the recent measurements and models relating to its heavy Fermi liquid crossover, its HO and superconductivity (SC). Recent experiments and theories are surmized that address four-fold symmetry breaking (or nematicity), Isingness and unconventional excitation modes. Second, we review the pressure dependence of URu ₂ Si ₂ and its transformation to antiferromagnetic long-range order. Next we confront the dramatic high magnetic-field phases requiring fields above 40 T. And finally, we attempt to answer how does random substitutions of other 5 f elements for U, and 3 d , 4 d , and 5 d elements for Ru, and even P for Si affect and transform the HO. Commensurately, recent theoretical models are summarized and then related to the intriguing experimental behavior.

Resonant ultrasound spectroscopy and machine learning are used to rule out many theories of hidden order in URu ₂ Si ₂ .

Magnetic fields are now available to 100 T (pulsed), 45 T (static) at temperatures below 0.3 K. Such technical developments allow the study and tuning of (quantum) phase transitions, unusual magnetic structures and (high-temperature) superconductors in a variety of quantum materials. An especially important class of strongly correlated electron materials is the heavy Fermi liquids (HFLs) displaying numerous reduced-moment antiferromagnets, quantum critical points, unconventional superconductivity, hidden order (HO) and other mysterious ground states. Among the ‘heavy fermions’, the duality of 5f electrons in uranium-based compounds introduces interesting behavior that can be affected by large magnetic fields. I list a few such heavy fermion materials to be considered: URu₂Si₂ and its tunable hidden state, UBe₁₃ and UPt₃ as very HFL paramagnets that become superconducting, the magnetic superconductors UPd₂Al₃ and UNi₂Al₃, and the ferromagnetic s UGe₂, URhGe and UCoGe. There are also the suggested metamagnetic Fermi-surface reconstructed intermetallic compounds such as UPt₂Si₂ and UCo₂Si₂. Present research attention focuses on the high-field behavior (30–40 T) of URu₂Si₂ and its destruction of HO. These and other U-based systems, e.g. UAu₂Si₂, UIrGe, etc., expand the opportunities of high magnetic field studies far into the future.