LETTERS Gapped itinerant spin excitations account for missing entropy in the hidden-order state of URu 2 Si 2 C. R. WIEBE 1,2 *, J. A. JANIK 1,2 , G. J. MACDOUGALL 3 , G. M. LUKE 3 , J. D. GARRETT 4 , H. D. ZHOU 2 , Y.-J. JO 2 , L. BALICAS 2 , Y. QIU 5,6 , J. R. D. COPLEY 5 , Z. YAMANI 7 AND W. J. L. BUYERS 7 1 Department of Physics, Florida State University, Tallahassee, Florida 32306-3016, USA 2 National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306-4005, USA 3 Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada 4 Brockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario L854M1, Canada 5 NIST Center for Neutron Research, Gaithersburg, Maryland 20899-8562, USA 6 Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA 7 CNBC, National Research Council, Chalk River Labs, Chalk River, Ontario K0J 1J0, Canada *e-mail: cwiebe@magnet.fsu.edu Published online: 28 January 2007; doi:10.1038/nphys522 Many correlated electron materials, such as high-temperature superconductors 1 , geometrically frustrated oxides 2 and low- dimensional magnets 3,4 , are still objects of fruitful study because of the unique properties that arise owing to poorly understood many-body effects. Heavy-fermion metals 5 —materials that have high effective electron masses due to those effects—represent a class of materials with exotic properties, ranging from unusual magnetism, unconventional superconductivity and ‘hidden’ order parameters 6 . The heavy-fermion superconductor URu 2 Si 2 has held the attention of physicists for the past two decades owing to the presence of a ‘hidden-order’ phase below 17.5 K. Neutron scattering measurements indicate that the ordered moment is 0.03μ B , much too small to account for the large heat-capacity anomaly at 17.5K. We present recent neutron scattering experiments that unveil a new piece of this puzzle—the spin-excitation spectrum above 17.5 K exhibits well-correlated, itinerant-like spin excitations up to at least 10meV, emanating from incommensurate wavevectors. The large entropy change associated with the presence of an energy gap in the excitations explains the reduction in the electronic specific heat through the transition. The central issue in URu 2 Si 2 concerns the identification of the order parameter that explains the reduction in the specific heat coefficient, γ = C / T , and thus the change in entropy, through the transition at 17.5 K (ref. 6). Numerous speculations about the ground state have been advanced, from quadrupolar ordering 7 , to spin-density wave formation 8 , to ‘orbital currents’ 9 to account for the missing entropy. Here, we present cold-neutron time-of- flight spectroscopy results that shed some light on the ‘hidden- order’ (HO) in URu 2 Si 2 . We have carried out experiments above and below the ordering temperature to measure how the spin excitations evolve. It is clear from our data that above T 0 the spectrum is dominated by fast, itinerant-like spin excitations emanating from incommensurate wavevectors at positions located 0.4a ∗ from the antiferromagnetic (AF) points. From the group velocity and temperature dependence of these modes, we surmise that these are heavy-quasiparticle excitations that form below the ‘coherence temperature’ and play a crucial role in the formation of the heavy-fermion and HO states. The gapping of these excitations, which corresponds to a loss of accessible states, accounts for the reduction in γ through the transition at 17.5 K. Figure 1 shows the excitation spectrum of URu 2 Si 2 at 1.5 K in the H00 plane. The characteristic gaps at ∼2 meV at the AF zone centre (1, 0, 0) and ∼4 meV at the incommensurate wavevectors (0.6, 0, 0) and (1.4, 0, 0) have been known for some time 10 . The incommensurate wavevector corresponds to a displacement of ∼0.4a ∗ from the AF zone centres (that is, where h + k + l = an odd integer, and is thus forbidden in the body-centred-cubic chemical structure). A scenario for this mode- softening at the incommensurate position was previously described with a model based on oscillatory exchange constants between near neighbours (not uncommon for Ruderman–Kittel–Kasuya– Yoshida-type interactions) 10,11 . Figure 2 shows our new neutron results at 20 K in the same H00 plane. Above the phase transition, the sharp spin waves evolve into weak quasielastic spin fluctuations at the zone centre (1, 0, 0), and strong excitations at the incommensurate positions (1 ± 0.4, 0, 0). We have considered the possibility that these incommensurate excitations may be due to magnetovibrational scattering. This can arise from a shift of the phonon excitations at (2, 0, 0) to (1 ± 0.4, 0, 0) as allowed through the neutron scattering cross- section for magnetoelastic coupling 12 . However, with the small moment size of this system, it is improbable that such a scattering process is being observed. It was also originally reported 10 that the incommensurate excitations were just quasielastic fluctuations; constant Q scans on a triple-axis spectrometer resolved a peak at a finite energy of 0.6 THz = 2.5 meV and a decrease in intensity as a function of energy typical of an overdamped response. What was previously unknown was that the quasielastic fluctuations were only the lower limit of a band of high-velocity 96 nature physics VOL 3 FEBRUARY 2007 www.nature.com/naturephysics