ARTICLES PUBLISHED ONLINE: 22 FEBRUARY 2009; CORRECTED ONLINE: 26 FEBRUARY 2009 | DOI: 10.1038/NMAT2395 Hidden order in URu 2 Si 2 originates from Fermi surface gapping induced by dynamic symmetry breaking S. Elgazzar 1 * , J. Rusz 1 , M. Amft 1 , P. M. Oppeneer 1 † and J. A. Mydosh 2 Spontaneous, collective ordering of electronic degrees of freedom leads to second-order phase transitions that are characterized by an order parameter driving the transition. The notion of a ‘hidden order’ has recently been used for a variety of materials where a clear phase transition occurs without a known order parameter. The prototype example is the heavy-fermion compound URu 2 Si 2 , where a mysterious hidden-order transition occurs at 17.5K. For more than twenty years this system has been studied theoretically and experimentally without a firm grasp of the underlying physics. Here, we provide a microscopic explanation of the hidden order using density-functional theory calculations. We identify the Fermi surface ‘hot spots’ where degeneracy induces a Fermi surface instability and quantify how symmetry breaking lifts the degeneracy, causing a surprisingly large Fermi surface gapping. As the mechanism for the hidden order, we deduce spontaneous symmetry breaking through a dynamic mode of antiferromagnetic moment excitations. T he hidden-order state in the uranium material URu 2 Si 2 has been a mystery since its discovery 20 years ago 1–3 . As the temperature is reduced below 100 K, lattice coherence be- tween local uranium f -moments develops and a heavy-fermion liquid starts to form as the uranium moments are dissolved into the Fermi surface. At T 0 = 17.5 K, the hidden-order transition takes place, demonstrated by marked effects in the thermodynamic and transport properties 1–3 . Although the latter properties unam- biguously exhibit sharp anomalies, typical of a magnetic phase transition, microscopic measurements, such as neutron diffraction, muon spin resonance and nuclear magnetic resonance, do not indicate a transition to a well-ordered magnetic phase. Very small antiferromagnetic moments of ∼0.03 μ B have been detected 4 in the hidden order, which by far are too small to explain the large entropy loss and sharp anomalies in the thermodynamic quantities 5,6 . It has been debated vigorously whether these tiny moments are intrinsic to the hidden order or extrinsic, that is, due to stress inhomogeneity; the latter scenario has become widely adopted 7,8 . With pressure, the U moments are resurrected (∼0.4 μ B ) and a transition to an ordered, large-moment antiferromagnetic (LMAF) state occurs. The bulk properties of the hidden-order and LMAF phases are very much alike with similar, continuous changes in the thermodynamic and transport quantities 8–10 . De Haas–van Alphen experiments 11 detect no significant differences between the Fermi surfaces of the hidden-order and LMAF phases and, consistently, neutron scattering finds the same nesting vectors 12 . Nonetheless, the hidden-order and LMAF phases unmistakably have different ground states 8–10 . Below 1.5 K and only out of the hidden order an exotic superconducting state appears, which is the subject of recent interest 13,14 . A variety of theories have been proposed 15–19 to explain the mysterious order parameter of the hidden order. As yet, there is no general agreement as to a model to fully describe the hidden order. 1 Department of Physics and Materials Science, Uppsala University, Box 530, S-751 21 Uppsala, Sweden, 2 II. Physikalisches Institut, Universität zu Köln, Zülpicher Strasse 77, D-50937 Köln, Germany. *On leave from: Faculty of Science, Menoufia University, Shebin El-kom, 32511, Egypt. † e-mail: peter.oppeneer@fysik.uu.se. Experimental investigations such as conductivity measurements 3,9 , Hall effect 20 , infrared spectroscopy 21 , thermal transport 22 and neu- tron scattering 23,24 are consistent with the opening of a Fermi surface gap in both the hidden-order and LMAF phases. As the high- temperature paramagnetic phase possesses a non-gapped Fermi surface, the anticipated marked changes in Fermi surface topology at the paramagnetic to hidden-order/antiferromagnetic transition, although occurring at a small energy scale of about 10 meV, should be traceable by band-structure calculations. A few such calculations have been carried out 25,26 for URu 2 Si 2 . However, these earlier in- vestigations were not yet accurate enough to provide a clear picture neither of the energy dispersions nor the Fermi surface, and in par- ticular, did not attempt to compare the Fermi surfaces for the dif- ferent phases. A materials-specific, microscopic model of how and where the Fermi surface gapping occurs is consequently lacking. We carried out density-functional theory investigations of the energy band dispersions and Fermi surface of URu 2 Si 2 , using two state-of-the-art electronic structure methods (see Supplementary Information). We verified that these two independent full-potential relativistic codes provide precisely the same energy dispersions and Fermi surface. Our aim is to explain first the paramagnetic and LMAF phases and, subsequently, the hidden-order phase. In doing so, we exploit the known similarity of the hidden-order and LMAF phases that has been detected 11 in their Fermi surface and thermodynamic and transport properties 8–10 . In addition, we have investigated the development of a continuous transition from the paramagnetic to LMAF phase by varying the exchange interaction, responsible for the formation of the magnetic moment. In this manner, we gradually transform from a homogeneous small- moment antiferromagnetic (SMAF) state with a total moment of ∼0.03 μ B to the LMAF state. We emphasize that although the SMAF and hidden-order phases do have similarities, we do not state that they are identical, but rather use the spurious SMAF NATURE MATERIALS | VOL 8 | APRIL 2009 | www.nature.com/naturematerials 337