Session U1: Invited Session: Hidden Order in URu2Si2 and Possibly Related Compounds

Symmetry Breaking in the Hidden-Order Phase of URu$_2$Si$_2$

In the heavy fermion compound URu$_2$Si$_2$, the hidden-order transition occurs at 17.5\,K, whose nature has posed a long-standing mystery. A second-order phase transition is characterized by spontaneous symmetry breaking, and thus the nature of the hidden order cannot be determined without understanding which symmetry is being broken. Our magnetic torque measurements in small pure crystals reveal the emergence of an in-plane anisotropy of the magnetic susceptibility below the transition temperature [1], indicating the spontaneous breaking of four-fold rotational symmetry of the tetragonal URu$_2$Si$_2$. In addition, our recent observation of cyclotron resonance allows the full determination of the electron-mass structure of the main Fermi-surface sheets, which implies an anomalous in-plane mass anisotropy [2] consistent with the rotational symmetry breaking. These results impose strong constraints on the symmetry of the hidden order parameter.\\[4pt] [1] R. Okazaki {\it et al.,} Science {\bf 331}, 439 (2011).\\[0pt] [2] S. Tonegawa {\it et al.,} Phys. Rev. Lett. {\bf 109}, 036401 (2012).            

Neutron scattering study of URu$_2$Si$_2$ magnetic properties: from hydrostatic pressure to uniaxial stress

Since the discovery of the unusual magnetic and superconducting properties of URu$_2$Si$_2$ in 1985 by Palstra~[1], this heavy fermion has been extensively studied. A ``Hidden Order'' evidences by bulk properties like specific heat, has been found below $T_0$=17.8K. Neutron scattering in this case is an efficient probe for the study of this compound as large magnetic excitations and an irremovable tiny antiferromagnetic moment are present in this sample. Even though the tiny antiferromagnetic moment aligned along the $c$-axis at $Q_0$ is only $\sim0.01 \mu_B$, the magnetic excitations seem to be associated to a large magnetic moment of $\sim$ 1 $\mu_B$ and show two minimums at $Q_0$=(1,0,0) but also at $Q_1$=(0.6,0,0). These magnetic responses have been intensively studied in normal conditions by Broholm~[2,3] and our group[4], but also versus magnetic field~[5], and more recently under hydrostatic pressure~[6]. The result of these experiments seem to indicate that the Hidden Order is linked to the excitation at $Q_0$ and not to the excitation at $Q_1$. We will present the revisited magnetic properties of URu$_2$Si$_2$ under uniaxial stress along the $a$-axis~[7,8]. Both elastic and inelastic contributions have been measured versus the constraints. In the HO state, as the constraint increases, the AF gap excitation at $Q_0$ decreases and the tiny moment increases: it seems also that there is a relation between both parameters. On the other hand, the excitation gap at $Q_1$ is slightly increasing. From our measurement we infer a critical pressure of $\sim$ 0.33~GPa, with a large increase of the antiferromagnetic moment. This behavior is very similar to results under hydrostatic pressure. Combining hydrostatic pressure, uniaxial stress along the $a$-axis and neutron Larmor diffraction measurements, that gives the lattice distribution of our URu$_2$Si$_2$ crystal, we conclude that the magnetic exchange integrals are dominated by the lattice parameter $a$ and not the ratio $c/a$ as usually believed. \\ \noindent[1]~T.~T.~M. Palstra, \textit{et al.}, Physical Review Letters {\bf 55}, 2727 (1985).\\ \noindent[2]~C. Broholm, \textit{et al.}, Physical Review Letters {\bf 58}, 1467 (1987).\\ \noindent[3]~C. Broholm, \textit{et al.}, Physical Review B {\bf 43}, 12809 (1991).\\ \noindent[4]~F. Bourdarot, \textit{et al.}, Journal of the Physical Society of Japan {\bf 79}, 064719 (2010).\\ \noindent[5]~F. Bourdarot, \textit{et al.}, Physical Review Letters {\bf90}, 067203 (2003).\\ \noindent[6]~A. Villaume, \textit{et al.}, Physical Review B {\bf 78}, 012504 (2008).\\ \noindent[7]~M. Yokoyama, \textit{et al.}, Physical Review B {\bf 72}, 214419 (2005).\\ \noindent[8]~F. Bourdarot, \textit{et al.}, Physical Review B {\bf 84}, 184430 (2011).   

The hidden order phase in URu$_2$Si$_2$: Remarkable nesting and spin-orbital hybridization

Aspects of Fermi surface (FS) nesting properties of URu$_2$Si$_2$ are analyzed with particular focus on their implication for the mysterious hidden order phase which occurs at 17.5~K. We show that there exist two Fermi surfaces that exhibit unusually strong nesting at the antiferromagnetic wavevector, $\mathbf{Q}_0$=(0,\,0,\,1). The corresponding energy dispersions fulfill the relation $\epsilon_{1}(\mathbf{k})$=$- \epsilon_{2} (\mathbf{k}\pm \mathbf{Q}_0)$ at eight FS hotspot lines on the surfaces. Notably, the spin-orbital characters of the involved $5f$ states are {\it different}: $j_z$=$\pm$5/2 {\it vs.} $\pm$3/2, and hence the occurring degenerate Dirac crossings are symmetry protected in the nonmagnetic normal state. Pairing of electrons in these two FSs can commence through interaction with a quasiparticle with wavevector $\mathbf{Q}_0$ and exchange of longitudinal angular momentum $\Delta j_z$. Dynamical symmetry breaking through an Ising-like spin-orbital excitation mode at $\mathbf{Q}_0$ with $\Delta j_z$=$\pm$1 induces a hybridization of the two states, causing substantial FS gapping. Concomitant spin and orbital currents in the uranium planes can give rise to a rotational symmetry breaking. The existence of such specifically nested FSs in URu$_2$Si$_2$ is confirmed in recent experiments.\\[4pt] This work has been performed with S. Elgazzar, J. Rusz, Q. Feng, T. Durakiewicz and J.A. Mydosh.   

A Hund's rule mechanism for Hidden Spin-Orbital Density Wave in URu$_2$Si$_2$

It is proposed that the ``Hidden Order'' state of URu$_2$Si$_2$ corresponds to a combined spin-orbital density wave state, which is stabilized by the inter-orbital Hund's rule coupling. The electronic system is described by the underscreened Anderson Lattice Model, in which there are two-fold degenerate f bands which hybridize with a single conduction band. In the normal state, the bands at the Fermi-energy have almost pure 5f orbital characters in accord with the results of first principles electronic structure calculations. The model Fermi-surface has heavy fermion sheets which exhibit interband nesting and intraband nesting with similar wave vectors. The spin-flip terms of the Hund's rule interaction and the interband nesting produces a second-order phase transition which partially gaps the Fermi-surface, and leads to a state with broken spin-rotational invariance without forming a net ordered magnetic moment. The resulting spin nematic phase is consistent with the magnetic torque experiments of Okazaki {\it et al.}. The similarity of the interband nesting and the intraband nesting conditions leads to an adiabatic continuity between the ``Hidden Order'' and Antiferromagnetic phases for small values of the hybridization. The presence of a nearby hybridization gap results in an asymmetric form of the pseudogap caused by the ``Hidden Order'' transition. Precursor fluctuations of the hidden order parameter, above $T_{HO}$, lead to the formation of ``hot spots'' on the Fermi-surface and a depletion of the density of states in the vicinity of the Fermi-energy as is seen by point contact and optical spectroscopies. The amplitude of the precursor fluctuations increase as $T_{HO}$ is driven towards zero, however, the order of the transition switches from second-order to first-order pre-empting the quantum critical point. These results in accord with the change in the order of the transition inferred by Jaime {\it et al.} from measurements of the specific heat in an applied magnetic field. This model might also be applicable to the enigmatic pseudo-gap phases seen in high-temperature superconductors.   

Hastatic Order in URu$_2$Si$_2$

The development of collective long-order via phase transitions occurs by the spontaneous breaking of fundamental symmetries. Magnetism is a consequence of broken time-reversal symmetry while superfluidity results from broken gauge invariance. The broken symmetry that develops below 17.5 K in the heavy fermion compound URu$_2$Si$_2$ has long eluded such identification. In this talk we show that the recent observation of Ising quasiparticles in URu$_2$Si$_2$ results from a spinor order parameter that breaks {\sl double} time-reversal symmetry, mixing states of integer and half-integer spin. Such ``hastatic order'' hybridizes conduction electrons with Ising $5f^2$ states of the uranium atoms to produce Ising quasiparticles; it accounts for the large entropy of condensation and the magnetic anomaly observed in torque magnetometry. Hastatic order also results in a number of predictions for future experiment: a tiny transverse moment in the conduction sea, a collosal Ising anisotropy in the nonlinear susceptbility and a resonant energy-dependent nematicity in the tunneling density of states.