Abstract
Spontaneous symmetry breaking in physical systems leads to salient phenomena at all scales, from the Higgs mechanism and the emergence of the mass of the elementary particles, to superconductivity and magnetism in solids. The hiddenorder state arising below 17.5 K in URu_{2}Si_{2} is a puzzling example of one of such phase transitions: its associated broken symmetry and gap structure have remained longstanding riddles. Here we directly image how, across the hiddenorder transition, the electronic structure of URu_{2}Si_{2} abruptly reconstructs. We observe an energy gap of 7 meV opening over 70% of a large diamondlike heavyfermion Fermi surface, resulting in the formation of four small Fermi petals, and a change in the electronic periodicity from bodycentred tetragonal to simple tetragonal. Our results explain the large entropy loss in the hiddenorder phase, and the similarity between this phase and the highpressure antiferromagnetic phase found in quantumoscillation experiments.
Introduction
The heavyfermion material URu_{2}Si_{2} undergoes a secondorder phase transition at T_{HO}=17.5 K to a socalled hiddenorder (HO) state whose order parameter is still unknown after almost 30 years of continued research^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}. Macroscopically, this transition is characterized by a large entropy loss and an energy gap of ~10 meV in the density of states at the Fermi level^{3,5,6,23}. Microscopically, the paramagnetic (PM) state above T_{HO} in URu_{2}Si_{2} is characterized by the existence of weak quasielastic spin fluctuations at the commensurate wave vector Q_{0}=(1,0,0) (reciprocal lattice units in the simpletetragonal zone) and strong damped magnetic excitations at the incommensurate wave vector Q_{1}=(1±0.4,0,0). The HO transition is accompanied by the development of an energy gap of ~4.5 meV in the spin excitations at Q_{1} and the onset of a sharp inelastic signal (gap ~2 meV) of magnetic excitations at Q_{0}^{4,10,11}.
From an electronic structure standpoint, angleresolved photoemission spectroscopy (ARPES) experiments detected Fermisurface pockets at the Γ, Z and X points of the Brillouin zone in the PM state^{8,14}. In addition, highresolution ARPES and scanning tunnelling microscopy experiments demonstrated that a Fermisurface instability of itinerant heavy quasiparticles takes place at the transition^{14,16,17,18,22}. However, a complete determination of the heavyfermion Fermisurface of URu_{2}Si_{2}, of its changes across the HO transition and a direct observation of the location and momentum dependence of the HO gap around the Fermi surface are still pressing unsolved questions. Here, by using highresolution ultralowtemperature ARPES over the threedimensional (3D) reciprocal space of URu_{2}Si_{2}, we provide a direct experimental answer to these puzzles.
Results
HO gap and related Fermisurface reconstruction
Figure 1a serves as a convenient guide to the 3D reciprocal space in URu_{2}Si_{2}, to help navigate through the data presented in this work. Measurements at fixed photon energy (hν) give the electronic structure along the a−a plane at fixed k_{‹001›}, whereas measurements as a function of photon energy provide the electronic structure in the a−c plane (see Methods). All throughout this paper, the lowtemperature spectra were taken at 1 K. URu_{2}Si_{2} undergoes a superconducting transition at ~1.2 K (ref. 1), showing a superconducting gap of the order of 0.15 meV at 1 K (ref. 25), at least 30 times smaller than our resolution (Methods). Hence, any changes that might be induced by the superconducting transition do not affect our discussions. We indeed checked that apart from thermal broadening, the spectra at 1 and 5 K are identical (see Supplementary Fig. 1 and Supplementary Note 1).
Figure 1b presents the Fermi surface at 1 K in the a−a plane around Γ. The data show two central squarelike Fermisurface sheets surrounded by four smaller halfcircles, henceforth called ‘Fermi petals’, along the (100) and equivalent directions. The central squares originate from an ‘M’shaped heavy band, shown in the raw spectra of Fig. 1c, which is cut by the Fermi level slightly below the tips of the ‘M’, thus crossing E_{F} twice^{22}. The Fermi petals represent one of the crucial new discoveries of this work. As seen from Fig. 1c,d, they correspond to small electron pockets and present a gap of Δ_{P}≈5 meV with respect to the holelike wings of the band forming the inner squares. The distance between the centre of two opposite petals is 0.91±0.01 Å^{−1}. It is worth noting that this corresponds, in reciprocallattice units, to the incommensurate excitation vector Q_{1}=(0.6,0,0) observed in inelastic neutron scattering experiments^{4,10} (see Methods). Thus, our data suggest a possible link, which should be investigated in future theoretical and experimental works, between the electronic structure around the petals and the incommensurate inelastic spin excitations at Q_{1}. More important, as shown in Fig. 1e,f, in the PM state at 20 K the gap forming the Fermi petals is not observed anymore within our resolution. The heavy band there now crosses E_{F}, giving rise, as we will see next, to a larger Fermi surface in the PM state.
Figure 2a,b present a complete view of the changes at the inplane Fermi surface over the Γ, Z and X points—c.f. pink plane in Fig. 1a. At 1 K, in Fig. 2a, one sees the Fermi petals around Γ. The transition to the PM state, in Fig. 2b, comes with the formation of a large diamondlike Fermi surface of diagonals ~Q_{1} along (100) and (010), instead of the petals present in the HO state. In fact, Fig. 2c–f show that the petals along (100) originate from the anticrossing of two bands occurring at k_{‹100›}=±Q_{1}/2. In the HO state, Fig. 2c,e, the anticrossing gap is Δ_{P}≈5 meV, as already described in Fig. 1. In the PM state, within our resolution, the above bands disperse through E_{F} approximately at their crossing point (±Q_{1}/2) and the gap forming the petals is not observed anymore: it has either closed or shifted above E_{F}. Fig. 2g–j show that this dramatic reconstruction of the Fermi surface is accompanied by the closing, in the PM state, of a large gap Δ_{HS}≈7 meV around ‘hotspots’ of Fermi momenta k_{F}≈±0.3 Å^{−1} in the (110) direction. As seen in Fig. 2a, the ‘hotspots’ are actually ‘hot arcs’, gapping all the region between two petals in the ordered state.
Figure 3 shows the associated changes in the electronic structure around Z, both along the (010) and (110) directions— Fig 3a–d and e–h, respectively. It is noteworthy that at 1 K, the spectra at Z, Fig. 3a,c,e,g, are essentially identical to the spectra at Γ, Fig. 2c,e,g,i. In both cases, one observes the central Mshaped band giving rise to the two inner Fermi squares, the small Fermi petals formed by the twoband anticrossing at k_{‹100›}=±Q_{1}/2, with an anticrossing gap of 5 meV and the gap of 7 meV at hotspots of momenta k_{F}≈±0.3 Å^{−1} along (110). At 20 K, Fig. 3b,d,f,h, the states around Z disperse through E_{F}, closing the previous gaps. Owing to the broad line shapes at this temperature, we cannot determine the precise shape of the Fermi sheet around Z in the PM state. Note however, from Fig. 3b,d, that the Fermi momenta around Z along (010) are now ~0.06 Å^{−1} smaller than Q_{1}/2, their value around Γ along the same direction in the PM state. As we will discuss in detail later, these observations indicate that the electronic structures around Z and Γ become equivalent in the HO state.
The spectral gaps at the hotspots and Fermi petals, Δ_{HS} and Δ_{P}, are further analysed and summarized in Fig. 4. The raw energy distribution curves at the hotspots at 20 and 1 K are shown by the red and blue circles in Fig. 4a. The continuous lines under these data are fits using a Lorentzian spectral function (shadowed red and blue curves) plus a linear background, cutoff by a Fermi–Dirac distribution at the corresponding temperature, all convolved to a Gaussian of fullwidth at half maximum=5 meV, representing the experimental resolution broadening. This analysis shows that with respect to the Lorentzian at 20 K, the Lorentzian at 1 K sharpens and develops a gap Δ_{HS}=6±1 meV. The leadingedge gap of ~2 meV results merely from resolution broadening. For comparison, the orange and green circles in Fig. 3a show the raw spectra along (110) at the outer square, at 20 and 1 K, respectively. At this kpoint, the spectra sharpen but do not develop a gap in the HO state, as demonstrated by the leading edge of the raw data and the fits (orange and green continuous lines and shadowed Lorentzians).
On the other hand, as shown in Fig. 4b for the data along (010), while the spectra at the outer square only sharpen in the HO state, the spectra at the petals’ centroid both sharpen and develop a doublepeak structure. Empirically, we found that a function representing two Lorentzianbroadened, asymmetric singularities, separated by a gap Δ_{P}, reproduces the 1 K data at the petals much better than any fit using a double Lorentzian (see Supplementary Fig. 2 and Supplementary Note 2). For both types of fits, the peaktopeak gap is Δ_{P}=5±1 meV. At 20 K, the double structure disappears and the local spectrum at the petals’ centroid is essentially described by a featureless Fermi–Dirac distribution.
Symmetry change of the electronic structure
The HO transition is also accompanied by conspicuous symmetry changes of the electronic structure. Note first, from Fig. 2a,b, that although at 1 K the spectral weights of the Fermi surfaces around Γ and Z are similar, at 20 K they are inequivalent, being more pronounced along (100) at Γ, and enhanced along (110) at Z. In fact, as noted earlier, in the HO state the heavyband dispersions at Γ and Z are identical, but become dissimilar in the PM state. This suggests that below T_{HO}, the electronic structure becomes consistent with a periodicity given by the commensurate vector Q_{0}. It is noteworthy that these changes occur on heating the sample above T_{HO}, and thus are not related to possible different photoemission selection rules at 50 and 31 eV, which are temperatureindependent.
Consider next the Brillouinzone edge along (110). At hν=50 eV, this corresponds to a point, henceforth noted , located slightly below X, within 10% of the zone size along k_{z}—see the inset of Fig. 2b. Fig. 5a,b show zooms around the leftmost points of the Fermi surface maps in Fig. 2a,b. Although defining a precise shape of the Fermi surface around is difficult, as at this point the bands near E_{F} are very heavy^{22}, the data in Fig. 5a,b show a reconstruction of the spectral weight taking place across the transition, which we discuss below.
To try to rationalize the changes around , we represent by dashed ovals, in Fig. 5a, Fermi pockets compatible with the observed distribution of spectral weight at 20 K and matching the local twofold symmetry of the bodycentred tetragonal (BCT) zone. In the HO state, we expect the electronic structure to fold into a simpletetragonal Brillouin zone, such that all the points become fourfold symmetric, and mutually perpendicular ovals on nearestneighbour corners fold on each other. This results in an outer circle and an inner fourring buckle, compatible with the Fermisurface spectral weight observed at 1 K in Fig. 5b. It is noteworthy that a portion of the outer circle is also observed in the upperleft Fermi surface at 1 K in Fig. 2a.
To check the above analysis, we plot in Fig. 5c,d the momentum distribution curves for the spectral weight at E_{F} along lines 1 (parallel to Z−X−Z), 2 (parallel to Γ−X−Γ or ‹110›), 3 (upper buckle) and 4 (left buckle) of Fig. 5a,b. It is noteworthy that for the cuts along lines 1 and 2, the spectral intensities are symmetric with respect to both at 20 K (red lines) and 1 K (blue lines), as these cuts correspond to highsymmetry directions in both the BCT and simpletetragonal (ST) zones. A gap in spectral intensity at the is evident at 1 K, but not at 20 K. By contrast, along cuts 3 and 4, the spectral intensity is asymmetric with respect to at 20 K, corresponding to the distribution of spectral weight at E_{F} observed in Fig. 5a, but it becomes symmetric with respect to at 1 K, compatible with an ST symmetry and the buckles observed in Fig. 5b.
Note, from Fig. 5a,b, that the observed spectral intensity at 1 K is not symmetric with respect to the ST Brillouin zone for all cuts parallel to lines 3 and 4. In fact, as one can see from these figures, the spectral weight of the folded buckles at 1 K is largest where the original, unfolded features, were located at 20 K. This is qualitatively compatible with the coherence factors describing the momentum distribution of the spectral function in a standard bandfolding or nesting scenario^{26,27}. However, as it is presently unknown whether the HO state corresponds to a conventional density wave, and how the coherence factors should be calculated, a quantitative explanation of the observed distribution of spectral weight at E_{F} lies beyond the scope of this work.
In any case, from the data of Fig. 5, it is clear that in the HO state the spectral weight is pushed to the other side of the ST zone edge, where it was not present in the PM state, strongly suggesting a symmetry change in the electronic structure from BCT to ST below T_{HO}. It is noteworthy that this cannot be a temperature effect, as cooling down should have the opposite effect, namely, sharpening the spectral features and making the asymmetries more pronounced.
The reconstruction of the 3D electronic structure from BCT to ST should also lead to a change of the Fermisurface symmetry in the a−c plane—green plane in Fig. 1a. Figure 6 presents the measurements corresponding to this independent crosscheck. As shown in Fig. 6a, in the HO state the two central Fermi sheets originating from the tips of the Mshaped heavy band present a dispersion along (001). Thus, although in the a−a plane these sheets have a square section, in the a−c plane they form concentric quasicircles that cross almost tangentially at Λ, the midpoint between Γ and Z along (001). This dispersion along (001) demonstrates formally the bulk character of these two bands and, most important, shows that at 1 K the Fermi surface in the a−c plane is symmetric with respect to Λ. This suggests again that the 3D electronic structure shows a ST symmetry in the HO state, regardless of the underlying BCT crystal symmetry.
Incidentally, as the quasicircles described above meet at Λ, they appear to hybridize, forming an inner small ‘Fermi lentil’ of diameter ~0.07±0.02 Å^{−1}, zoomed in Fig. 6b. On the other hand, the Fermi petals present a very weak intensity in these measurements along (001). Their exact dispersion and size in k_{z} are difficult to infer from our data and remain open issues for future works.
At 20 K, Fig. 6c, the a−c Fermi surface reconstructs and is not anymore symmetric with respect to Λ. The clearest feature at this temperature is a large rhombohedral Fermi surface, mostly located in the upper third of the BCT Brillouin zone, elongated along (100). Similar Fermi sheets dispersing quasilinearly towards Γ and Z are also observed. From their momenta at these two points, those Fermi sheets would correspond to the large inplane diamondlike Fermi surface observed in the PM state, described previously.
Figure 6d shows the distribution of spectral weight at the Fermi level at k_{‹100›}=0 along k_{‹001›}, both at 1 and 20 K. From this figure, one also sees that the spectral weight is symmetric with respect to Λ in the HO state, but not in the PM state, further supporting the change of electronic symmetry from BCT at 20 K to ST at 1 K.
Correspondence with thermotransport data
We now compare our data to transport and thermodynamic experiments to show how our measurements provide a unified microscopic picture of all the macroscopic changes induced by the HO transition.
The extremal crosssections, effective masses and electronic contribution to the specific heat γ_{e}=C/T of several Fermisurface sheets in the HO state have been previously determined by deHaas–van Alphen and Shubnikov–deHaas (SdH) measurements^{7,15}. In Table 1 we compare our ARPES data to those transport results (details of the calculations in the Methods). We find an excellent correspondence between our data and the SdH measurements if we identify the outer central square with SdH’s αsheet, the inner central square with the γsheet, the petals with the βsheet and the lentil at Λ with the ηsheet. It is noteworthy that the SdH data as a function of field orientation of ref. 15 show that the crosssectional area of the α, γ and ηsheets remains singlevalued when sweeping the field from (001) to (100), compatible with central Fermi surfaces, while the βsheet splits in two. By crystal symmetry, this sheet must then correspond to a Fermi surface with four noncentral flattened pockets along the main axes of the Brillouin zone, exactly as the petals revealed by our experiments. Thus, our data in the HO state are fully compatible with SdH measurements, while providing a direct microscopic image of the 3D electronic structure of URu_{2}Si_{2}, including the PM state, not accessible through SdH.
The transition from the large diamondlike Fermi surface around Γ to the four small petals occurs through the opening of a large gap of Δ_{HS}≈7 meV that suppresses ~70% of this Fermi sheet in the HO state (Table 1). Together with the change in effective mass between the diamond and the petals, this implies a drastic reduction in the electronic contribution to the specific heat, as , where is the momentumaveraged effective mass of the Fermi sheet and S_{F} its maximal crosssectional area^{15}. Hence, for the diamondlike Fermi surface alone, the change in electronic specific heat is γ_{e}^{HO}/γ_{e}^{PM}≈0.4 (Table 1). This agrees well with thermodynamic measurements, which show a dramatic entropy loss with a reduction in the Sommerfeld coefficient of γ_{e}^{HO}/γ_{e}^{PM}≈(65 mJ × (mol × K^{2})^{−1}/112 mJ × (mol × K^{2})^{−1})=0.58 induced by a gap opening of about 10 meV (ref. 3).
Comparison between ARPES and bandstructure calculations
We finally compare, in Fig. 7a,b, the ARPES inplane Fermi surfaces around Γ in the HO and PM states with the Fermi sheets in the largemoment antiferromagnetic (LMAF) and PM phases from the LDA calculations of ref. 12. It is noteworthy that in those calculations the authors assume that the Brillouin zone folds from BCT to ST in the HO state, such that the Fermi surface in the HO state should be similar to the one computed for the LMAF phase.
As can be concluded from these figures, there is a good agreement between the experimental and localdensity approximation (LDA) Fermi surfaces. In the HO state, the inner and outer central squares and the four noncentral Fermi petals observed by ARPES correspond well with the LDA Fermi sheets in the LMAF state. In addition, when going to the PM state, the calculations show a gap closing around (110), as observed in the ARPES measurements at the hotspot (main text), producing in both cases similar diamondlike Fermi sheets—albeit the calculated Fermi sheet appears slightly larger than the experimental one. Thus, these agreements between the ARPES data and LDA calculations are another strong indication that, below T_{HO}, the electronic structure follows an ST symmetry.
On the other hand, as shown in Fig. 7c,e, the experimental electronic structure near E_{F} in the HO phase shows bands dispersing over only a few meV and energy gaps over this same energy scale. By contrast, the purely itinerant LDA calculations, Fig. 7d,f, give dispersions and energy gaps over energy scales at least one order of magnitude larger^{12}.
In fact, a detailed comparison between the experimental band dispersion at 1 K along (100), Fig. 7c, and the corresponding LDA dispersion, Fig. 7d, shows a reasonable agreement between the shapes of the bands near E_{F}, but with an experimental effective mass ten times larger, given by the factor in energy scales needed to overlap the theoretical and experimental dispersions. This large mass renormalization, resulting from the heavy fstates that govern the lowenergy band masses of URu_{2}Si_{2}, is taken into account by other theories or calculations based on a dual localitinerant approach^{13,24,20}. An additional notable difference between the experimental and LDA dispersions along (100) is that at the petals, the data show the anticrossinglike 5 meV gap, while in the calculations the bandcrossing yielding the petals is not avoided, neither in the PM nor in the HO states.
The comparison between the experimental and LDA band dispersions at 1 K along the (110) direction, Fig. 7d, e, shows more marked differences. For instance, although the LDA calculations predict a band anticrossing at around k_{‹110›}=0.4 Å^{−1}, with a gap of ~60 meV separating an upper unoccupied electron pocket and a lower occupied hole pocket with its maximum right at E_{F}, the data show a gap Δ_{HS}≈7 meV with respect to E_{F} at this position, and no sign of a hole pocket, or a band anticrossing, in the vicinity of this kpoint. Note again a factor of ~10 between the energy scales of the measured and calculated dispersions.
Discussion
Other ARPES experiments with lower energy resolution performed only at T<T_{HO} reported a decrease of spectral weight at E_{F} around Γ along the (110) direction, and an increase along (100), with respect to states at binding energies larger than 100 meV (ref. 28). Here, our results unambiguously show that the regions of depleted spectral weight are in fact strongly gapped at E_{F} in the HO state with respect to the PM state, due to the Fermisurface reconstruction from the large diamond to the small petals, and from a BCT to a ST symmetry. Other recent highresolution ARPES experiments also reported a gap opening of 7 meV along the (110) direction, and suggested a change to an ST symmetry in the HO state^{29}, but the survey of the electronic structure was limited to the spectra at the Γ and Z points.
It is noteworthy that the extensive gapping of the large diamond Fermi surface, shown by our data, implies the suppression, in the HO state, of a large volume of Fermi momenta available for scattering. This naturally explains the abrupt increase in the electron lifetime below T_{HO} observed by previous thermal transport and ARPES experiments^{14,30}, and studied in detail in another recent ARPES work^{31}.
Our results show that the most prominent macroscopic and microscopic signatures of the HO transition are all the result of a massive reconstruction of a heavyfermion Fermi surface, that we directly image here for the first time. Furthermore, the fact that the periodicity of the electronic structure in the HO state is well described by Q_{0} indicates that this vector, and not Q_{1}, is directly related to the HO parameter—thus, strongly limiting the number of viable theories for the HO transition. The energy gaps at Q_{1} and along the hot arcs would then be an outcome of the above electronic reconstruction. An analogous situation is encountered in electrondoped cuprates, in which bandfolding by a commensurate antiferromagnetic wave vector creates gaps at incommensurate wave vectors^{27}. In addition, our determination of the multiband Fermisurface topology in the HO state should provide crucial input for the experimental and theoretical understanding of the superconducting order parameter in URu_{2}Si_{2} (ref. 32), to which the HO is a precursor. Finally, the similarities and differences between the data, measured in the HO state, and the calculations, formally in the LMAF state, might have a significance in the understanding of the HO transition, and should be further studied experimentally and theoretically. More generally, our results are the first direct determination of a momentumdependent Fermisurface gapping in a heavyfermion system, opening bright perspectives for future studies of exotic classic and quantum phase transitions in these materials.
Methods
Sample preparation and measurements
The highquality URu_{2}Si_{2} single crystals were grown in a triarc furnace equipped with a Czochralski puller, and subsequently annealed at 900 °C under ultrahigh vacuum for 10 days. The highresolution ARPES experiments were performed with a Scienta R4000 detector at the UE112PGM1b (‘1^{3}’) beamline of the Helmholtz Zentrum Berlin—BESSYII using horizontally polarized light from hν=16 eV (total instrument resolution ~4 meV) to hν=50 eV (resolution ~7 meV). The samples were cleaved in situ along the (001) plane at 1 K. The pressure was below 5 × 10^{−11} Torr throughout the measurements. All Fermisurface maps in this work were obtained by integrating the ARPES data over E_{F}±1 meV. The experimental Fermi momenta were determined from fits to momentum distribution curves at E_{F}. The results have been reproduced in five different cleaves.
Lattice parameters and reciprocallattice units
URu_{2}Si_{2} has a BCT crystal structure, of base side a=4.12 Å and height c=9.68 Å. Thus, a*≡2π/a=1.52 Å^{−1}, and the length of the vector Q_{1} is Q_{1}=(1±0.4) × a*, giving 2.135 Å^{−1} or 0.915 Å^{−1}.
3D kspace mapping
Within the freeelectron final state model, ARPES measurements at constant photon energy give the electronic structure at the surface of a spherical cap of radius . Here, m_{e} is the free electron mass, Φ is the work function and V_{0}=13 eV is the ‘inner potential’ of URu_{2}Si_{2} (refs 8, 14, 22). Measurements around normal emission provide the electronic structure in a plane nearly parallel to the a−a plane, as those presented in Figs 1 and 2. Likewise, measurements as a function of photon energy provide the electronic structure in the a−c plane, presented in Fig. 6.
Second derivative rendering
For secondderivative rendering, the raw photoemission intensity maps were convolved with a twodimensional Gaussian of widths σ_{E}=3 meV and σ_{k}=0.02 Å^{−1} for T=1 K, and σ_{E}=4 meV and σ_{k}=0.03 Å^{−1} for T=20 K. The corresponding figures in this work show the negative values of the second derivative along the energy axis, which represent peak maxima in the original energy distribution curves.
Characteristics of Fermi sheets from ARPES data
For a Fermisurface sheet of area S_{F}, we define the average Fermi momentum such that:
as done in SdH works^{15}. For electronlike Fermi sheets (diamond, petals and inner square), we note E_{0} the bottom of the band, directly determined from the ARPES data, and define the average effective mass such that:
For instance, for the inner square (γsheet), we have and E_{0}≈4 meV, which gives . We obtain independent consistent values of the effective masses of the electronlike Fermi sheets through parabolic fits to their dispersions along the (100) and (110) directions.
To estimate the mass of the holelike outer square, we fitted the whole Mshaped band around Γ or Z using a threeband model with interband hybridization, introduced in our previous work^{22}, and extracted the band mass around E_{F} from the curvature of the fitted dispersion at E_{F}. As a byproduct, we obtained again a value of the effective mass of the inner electronlike square that agrees with the parabolic estimate for this band. In addition, a simple holelike parabolic fit to the dispersion of the outer square in the vicinity of E_{F} gives an effective mass that is compatible with the result of the full fit to the Mshaped structure.
Finally, from the average Fermi momenta and effective masses, we calculated, following ref. 15, the Sommerfeld coefficient associated to each Fermi sheet as:
where k_{B} is Boltzmann constant and V=49 cm^{3} mol^{−1} is the molar volume of URu_{2}Si_{2}.
It is noteworthy that, for the effective mass and Sommerfeld coefficient of the diamondlike Fermi surface in the PM state, we considered only the electronlike dispersion around Γ and neglected the crossing with the other band at k_{‹100›}=±Q_{1}/2. There is indeed a difficulty in how to estimate, from the present ARPES data, the Sommerfeld coefficient of the diamondlike Fermi sheet in the PM state. The exact calculation depends on the details of the two bands crossing near E_{F}, such as the energy at which they cross with respect to E_{F} or whether they rather maintain the anticrossing gap at 20 K, and on the momentumdependence of this crossing around the Fermi surface. For instance, from our data, the crossing of the bands along (110) is not evident, suggesting that the effect of such crossing on the effective masses along the whole diamond Fermi sheet would be negligible. However, these details remain challenging for future experiments. In any case, the Sommerfeld coefficient of the diamondlike Fermi surface alone cannot exceed the total Sommerfeld coefficient in the PM state observed by thermodynamic measurements, which is ~120 mJ mol^{−1} K^{−2} (ref. 3), or twice our estimate of the Sommerfeld constant for this Fermi sheet. Thus, our calculation of γ_{e}^{HO}/γ_{e}^{PM}≈40% is, at worst, underestimating by a factor of 2 the loss in specific heat for the diamond Fermi surface. This will be attenuated by the fact that a large part of this ‘missing’ factor 2 in the PM state should come from the Fermi sheets around Z and X, whose Sommerfeld coefficients at 20 K are difficult to evaluate from our present data.
Additional information
How to cite this article: Bareille, C. et al. Momentumresolved hiddenorder gap reveals symmetry breaking and origin of entropy loss in URu_{2}Si_{2}. Nat. Commun. 5:4326 doi: 10.1038/ncomms5326 (2014).
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Acknowledgements
We thank C. Pépin, M.A. Méasson, H. Bentmann, F. Bourdarot, S. Burdin, P. Chandra, P. Coleman, T. Durakiewicz, A. Kapitulnik, M. Norman, S. Petit, Y. Sidis, C. Varma and M. Vojta for fruitful discussions, the Helmholtz Zentrum Berlin for the allocation of synchrotron radiation beamtime, and E. Rienks and S. Thirupathaiah for their invaluable help during beamtime. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/20072013) under grant agreement no. 312284. The work at Würzburg University is supported by the Deutsche Forschungsgemeinschaft through FOR1162. A.F.S.S. acknowledges support from the Institut Universitaire de France.
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Project conception: A.F.S.S. and F.R.; ARPES measurements: C.B., F.L.B., H.S. and A.F.S.S.; sample growth and characterisation: P.L.; data analysis: C.B. and F.L.B., with input from A.F.S.S.; interpretation: C.B., F.L.B. and A.F.S.S.; writing of the manuscript: C. B., F.L.B. and A.F.S.S. All authors discussed extensively the results and the manuscript. C.B. and F.L.B. contributed equally to this work.
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Supplementary Figures 13, Supplementary Notes 12 and Supplementary References (PDF 325 kb)
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Bareille, C., Boariu, F., Schwab, H. et al. Momentumresolved hiddenorder gap reveals symmetry breaking and origin of entropy loss in URu_{2}Si_{2}. Nat Commun 5, 4326 (2014). https://doi.org/10.1038/ncomms5326
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