Heavy fermion compounds are known to exhibit a wide range of ground states encompassing ferromagnetism, anti-ferromagnetism, superconductivity, insulating and a host of others. A number of these compounds also exhibit more than one of these phases simultaneously.
There is one of these heavy fermion materials that stands out among the rest, however, and that is URu2Si2. The reason for this is that there is an unidentified phase transition that occurs in this compound at ~17.5K. What I mean by “unidentified” is that the order parameter is unknown, the elementary excitations are not understood and there is a consensus emerging that we currently may not have the experimental capability to identify this phase unambiguously. This has led researchers to refer to this phase in URu2Si2 as “hidden order”. Our inability to understand this phase has now persisted for three decades and well over 600 papers have been written on this single material. For experimentalists and theorists that love a challenge, URu2Si2 presents a rather unique and interesting one.
Let me give a quick rundown of the experimental signatures of this phase. Firstly, to convince you that there actually is a thermodynamic phase transition that happens in URu2Si2, take a look at this plot of the specific heat as a function of temperature:
In the lower image, one can see two transitions, one into the hidden order phase at 17.5K and one into the superconducting phase at ~1.5K. One can see that there is a large entropy change at the phase transition into the hidden order phase, which makes it all the more remarkable that we don’t know what it going on! I should mention that the resistivity also shows an anomaly going into the hidden order phase both along the a- and c-axis (the unit cell is tetragonal).
Furthermore, the thermal expansion coefficient, , has a peak for the in-plane coefficient and a smaller dip for the c-axis coefficient at the transition temperature. This implies that the volume of the unit cell gets larger through the transition, indicating that the hidden order phase exhibits a strong coupling to the lattice degrees of freedom.
For those familiar with the other uranium-based heavy fermion compounds, one of the most natural questions to ask is whether the hidden order phase is associated with the onset of some sort of magnetism. Indeed, x-ray resonance magnetic scattering and neutron scattering experiments were carried out in the late 80s and early 90s to investigate this possibility. The structure found corresponded to one where there was a ferromagnetic arrangement in the a-b plane with antiferromagnetic coupling along the out-of-plane c-axis. However, this was not the whole story. The magnetic moments were extremely weak (0.02 per Uranium atom) and the magnetic Bragg peaks found were not resolution-limited (correlation length ~400 Angstroms). This means that order was not of the true long-range variety!
Also, rather strangely, the integrated intensity of the magnetic Bragg peak was shown to be linear as a function of temperature, saturating at ~3K (shown below). All these results seemed to imply that the magnetism in the compound was of a rather unconventional kind.
The next logical question to ask was what the inelastic magnetic spectrum looked like. Below is an image exhibiting the dispersion of the magnetic modes. Two different modes can identified, one at the magnetic Bragg peak wavevectors (e.g. (1, 0, 0)) and one at “incommensurate” positions (e.g. 1 0.4, 0, 0). The “incommensurate” excitations exhibit approximately a ~4meV gap while the gap at (1, 0, 0) is about 2meV. These excitations show up with the hidden order and are thought to be closely associated with it. They have been shown to have longitudinal character.
The penultimate thing I will mention is that if one examines the optical conductivity of URu2Si2, a gap of ~5meV in the charge spectrum is also manifest. This is shown below:
And lastly, if one pressurizes a sample up to 0.5 GPa, the URu2Si2 becomes a full-blown large-moment antiferromagnet with a magnetic moment of approximately 0.4 per Uranium atom. The transition temperature into the Neel state is about 18K.
So let me summarize the main observations concerning the hidden order phase:
- Weak short-range antiferromagnetism
- Strong coupling to the lattice
- Dispersive and gapped incommensurate and commensurate magnetic excitations
- Gapped charge excitations
- Lives nearby anti-ferromagnetism
- Can coexist with superconductivity
I should stress that I am no expert of heavy fermion compounds, which is why this is my first real post on them, so please feel free to point out any oversights I may have made!
More information can be found in these two excellent review articles: