L etter Typeset with jpsj3.cls < ver.1.0 > arXiv:1001.2369v1[cond-mat.str-el]14 Jan 2010 Why the hidden order in URu$_{2}$Si$_{2}$ is still hidden — one simple answer Hisatomo H arima ∗ 1 , $^{2}$, Kazumasa M iyake 2 , $^{3}$, Jacques F louquet 2 $^{1}$Department of Physics, Graduate School of Science, Kobe University, Nada, Kobe 657-8501, Japan $^{2}$INAC / SPSMS, CEA-Grenoble and Universit´e Joseph Fourier, 17 rue des Martyrs, 38054 Grenoble, France $^{3}$Division of Materials Physics, Department of Materials Engineering Science, Graduated School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan (Received September 7, 2018) For more than two decades, the nonmagnetic anomaly observed around 17.5 K in URu$_{2}$Si$_{2}$, has been investigated intensively. However, any kind of fingerprintfor the lattice anomaly has not been observed. Therefore, the order has been called ”the hidden order”. One simple answer to why the hidden order is still hidden is presented from the space group analysis. The second order phase transition from I 4 / mmm (No. 139) to P 4$_{2}$ / mnm (No. 136) does not need any kind of lattice distortion in this system, and allows the NQR frequency at Ru-site unchanged. It is compatible with O$_{xy}$ -type anti-ferro quadrupole ordering with Q = (0 , 0 , 1). The characteristics of the hidden order are discussed based on the local 5 f 2 electron picture. KEYWORDS: URu$_{2}$Si$_{2}$, hidden order, quadrupole ordering, Kondo effect, Nambu-Goldstone mode U 4 + valence state (5 f 2 configuration), multipole ordering can be expected as recently discussed, in the physics of Pr skut- terudite systems. 17) A second order phase transition in the solid states causes change of symmetry; gauge symmetry, time reversal symme- try, etc... . On crystal symmetry, it usually accompanies lattice distortion to reach lower symmetry. In exchange for releasing entropy, it becomes a lower energy state in lower tempera- tures. The lattice distortion itself needs loss of energy, so an- other mechanism is necessary to overcome the rising energy in lattice system. The Jahn-Teller e ff ect is known as such a example. The degeneracy of the electronic states is lifted, due to distortion, to obtain the lower energy. Even when the elec- tronic states are not local, such e ff ect is widely discussed in terms of the Jahn-Teller e ff ect. On the experimental side, it seems now well established that the low pressure HO phase is not antiferromagnetic, but that above P$_{x}$ ∼ 0 . 5 GPa through the first order transition long range anti-ferromagnetism (AF) is established with wave vec- tor Q$_{AF}$ = (0 , 0 , 1) corresponding to a lattice doubling along the c axis.$^{18–20)}$Furthermore, the HO phase is characterized by a sharp resonance at energy E$_{0}$ = 1 . 5 meV observed at the wave vector Q$_{O}$ = (1 , 0 , 0) (equivalent to (0 , 0 , 1) in body cen- ter tetragonal lattice). Switching to the AF phase at P$_{x}$ leads to the collapse of the resonance and to the observation of elastic neutron magnetic signal at Q$_{O}$ . 20) Taking into account that the same characteristic vector Q$_{O}$ emerges in the HO and the AF phase as well as the invariance of three Shubnikov de Haas frequencies through P$_{x}$ , 21) the experimental data push to look to a model, where the transition from PM phase to HO or AF ground states corresponds to the same structural transi- tion among the tetragonal classes, while the pressure switch- ing from HO to AF adds a supplementary time reversal break- ing. The magnitude of such a distortion strongly depends on the mechanism to yield lower energy states. Lowering the symmetry changes the crystalline electric field (CEF) and the bonding strengths, through the movements of atoms. The more the lattice is distorted, the much lower energy state the electronic state gets. Therefore, the lattice is sometimes largely distorted, up to 10% of a lattice constant, as 1 T - TaSe$_{2}$. 1) In the ordered CDW phase, the electronic states produce a large gap between the bonding and anti-bonding states, 2) where the strong Coulomb repulsion does not play an important role. On the contrary, when the Coulomb inter- action plays a crucial role, the distortion is kept very small; e.g. only about 0.05% in PrRu$_{4}$P$_{12}$, 3) where the lattice distor- tion itself does not need to gain the lower electronic states. 4) The anti-ferro total-symmetric multipole ordering breaks the translational symmetry to open the gap in the conduction band, then the lattice is slightly, and symmetrically distorted in order to relax multipoles developed on Pr-sites. A previous band structure calculation demonstrates that AF opens a large gap at the Fermi level, then the proposal was that the occurrence of large magnetic fluctuations may preserve a Fermi surface construction at T$_{O}$ in the HO phase similar to that proposed going from PM to AF above P$_{x}$ 16) Recently, by using LDA + DMFT method, an idea quite similar to our proposal has been developed by Kotliar and coworkers, 22) but the selected order parameter of HO is a hexadecapole. In this letter, we propose that the space group in the HO phase is No. 136, then the transition at 17.5 K does not need any kind of lattice distortion, and keeps Ru-site with 4-fold symmetry. Therefore, most of the experimental techniques are unable to detect the characteristic charge distribution in the HO phase. This is one simple answer to why the HO is still hidden. URu$_{2}$Si$_{2}$ shows a large peak in specific heat at 17.5 K ( T$_{O}$ ), indicating a second order phase transition, then the ordered state coexists with superconductivity found below 1.5 K. 5–7) Although the intensive theoretical and experimental studies have been performed, the order parameter of the phase below T$_{O}$ is still unknown, so now it is called the hidden order (HO). Quite di ff erent theoretical proposals have appeared:$^{8–16)}$some neglecting the itinerant character of the 5 f electrons, and some by omitting the local specificity of the U site; in the After a second order phase transition, the lower symmetry must be one of the subgroups of the mother symmetry. The space group in high temperature phase of URu$_{2}$Si$_{2}$ (ThCr$_{2}$Si$_{2}$- E-mail address: hh@kobe-u.ac.jp