{'fraction_non_alphanumeric': 0.0431961680818637, 'fraction_numerical': 0.018419333768778575, 'mean_word_length': 4.452516619183286, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Imaging the Fano Lattice to 'Hidden Order' Transition in URu2Si2

Within a Kondo lattice, the strong hybridization between electrons localized in real space (r-space) and those delocalized in momentum-space (k-space) generates exotic electronic states called 'heavy fermions'. In URu2Si2 these effects begin at temperatures around 55K but they are suddenly altered by an unidentified electronic phase transition at To = 17.5 K. Whether this is conventional ordering of the k-space states, or a change in the hybridization of the r-space states at each U atom, is unknown. Here we use spectroscopic imaging scanning tunnelling microscopy (SI-STM) to image the evolution of URuSi2 electronic structure simultaneously in r-space and k-space. Above To, the 'Fano lattice' electronic structure predicted for Kondo screening of a magnetic lattice is revealed. Below To, a partial energy gap without any associated density-wave signatures emerges from this Fano lattice. Heavy-quasiparticle interference imaging within this gap reveals its cause as the rapid splitting below To of a light k-space band into two new heavy fermion bands. Thus, the URu2Si2 'hidden order' state emerges directly from the Fano lattice electronic structure and exhibits characteristics, not of a conventional density wave, but of sudden alterations in both the hybridization at each U atom and the associated heavy fermion states.

conductance dI/dV(r, E) ≡ g(r, E) is proportional to the local density of electronic states LDOS(r, E). However, there are two possible channels for these tunnelling electrons (either to rspace many-body states or to k-space delocalized states), so the measured conductance g(r, E) is best parameterized as a two-channel Fano spectrum 5 (Fig. 1b): (1) Here E is the electron energy, ς is the ratio of the probabilities of tunnelling into the r-space many-body state or into the delocalized k-space states, ε 0 is the energy of the many-body state and Γ is proportional to its hybridization strength.
A periodic array of such atoms in a crystal lattice, each having r-space states, the spins of which are screened by hybridization with the k-space bands, is often referred to as a Kondo lattice [1][2][3] . Eventually as T approaches zero, this hybridization process generates the famous 'heavy fermion' electronic states 1,[8][9][10][11] . In the simplest picture, an initial light k-space band k E is split by such hybridization into two heavy bands: (2) Here f k ε is the renormalized energy of the f-states after they establish coherence, while the strength of their hybridization processes is represented by k Ṽ such that where N(E F ) is the density of states at the Fermi level (see chapter 10 of ref. 1). The type of band-structure in equation (2), which is a signature of such r-space/k-space hybridization, is shown schematically in Fig. 1c for a hole-like E k . We note how the two heavy bands become widely separated in k-space from the original light band (dashed line) where the avoided crossing occurs within the range of hybridization (arrows in Fig. 1c). No phase transition is expected with falling T because this band structure emerges smoothly from the Kondo screening of the magnetic lattice.
For real materials, the Kondo lattice screening processes must be more intricate because the initial unscreened r-space state can exhibit a complex spin manifold and its screening can be due to multiple bands. But despite these complexities, a heuristic hypothesis for SI-STM observables in a Kondo lattice might include a generalization of the Fano spectrum of equation (1) (Fig. 1b) to a 'Fano lattice'-an array of such Fano spectra with the same periodicity as the atoms the localized states of which are undergoing the hybridization. Indeed, where the r-space electronic characteristics of atoms undergoing such Kondo screening has been predicted using either dynamical mean-field theory 12 or large-N expansion theory 13,14 , the electronic structure can exhibit characteristics of such a Fano lattice.

The hidden-order state of URu 2 Si 2
After the Kondo screening of the magnetic sublattice has begun to alter the electronic structure of URu 2 Si 2 , the unexpected phase transition is signified by sudden changes at T o = 17.5 K in bulk properties such as resistivity, magnetic susceptibility and specific heat [15][16][17] , in the k-space electronic structure 18 , and by the emergence of a partial gap in the optical conductivity 19,20 and point contact measurements 21,22 . The new phase therefore is not 'hidden'-rather, it is the identity of its order parameter that is unknown. At T << T o , this phase is a heavy fermion system with an effective electron mass m* ≈ 25m e (refs 15, 16) that eventually becomes superconducting below 1.5 K. A transition to an antiferromagnetic state also occurs at T N ≈ 17.5 K, but neutronscattering experiments reveal 23 that the ordered moment is far too small (0.03µ B , where µ B is the Bohr magneton) to account for the large entropy loss at T o . Advanced experimental techniques have recently improved our understanding of the changes in the electronic and magnetic excitations at T o . The spin excitations 24 above T o are gapless and overdamped at incommensurate wavevectors. Below T o , sharp magnetic modes 25 appear near (1 ± 0.4,0)π/a 0 above a ω ≈ 4 meV energy gap. And, although angle-resolved photoemission cannot determine empty-state or rspace electronic structure, it does reveal 26 that, for the filled states above T o , a light band crosses E F near (0, ±0.3)π/a 0 ; (±0.3, 0)π/a 0 while, below T o , this evolves into a heavy band below E F . Thus, the screening by light Ru-based d-electron bands of the f-electrons at each U atom apparently begins to alter the URu 2 Si 2 electronic structure somewhere near T ≈ 55 K (refs 15, 16). Then, however, the transition at T o = 17.5 K dramatically alters the density of states at the Fermi level 15,16,19,23 , the electronic band structure 26 and the spin excitation spectrum 23,24 . Among the theoretical proposals to explain this transition to a hidden-order state are two basic classes.
The first class considers k-space susceptibilities of the Fermi surface to a conventional density wave state 15,25,[27][28][29] , while the second class considers ordering of the r-space states at U atoms, with the corresponding alteration (via hybridization changes) to the band structure 12,30-34 .
However, despite two decades of research, which of these classes correctly describes the hiddenorder transition in URu 2 Si 2 remains unknown.

Conventional density wave versus d-f hybridization change
In conventional metallic systems, the opening of an energy gap ∆ below some critical temperature usually occurs as a result of a diverging susceptibility of the k-space states surrounding E F . The pre-existing band structure is unaltered except within a few ±∆ of E F .
Exemplary of such transitions are those to Peierls charge-density-wave or spin-density-wave states 35 . Here r-space modulations in charge and/or spin appear at a fixed wavevector Q*which is the same for empty and filled gap-edge states, and the associated gap always spans E F . In Kondo lattice systems, in contrast, the opening of a gap near E F is due to a profoundly different phenomenon [1][2][3]11 . Hybridization between the r-space and k-space electrons splits a light band into two heavy bands approximately separated in energy by the hybridization range (k Ṽ in equation (2)) and segregated widely in k-space (Fig. 1c). To understand the hidden-order transition in URu 2 Si 2 we therefore face an elementary issue: does it occur because of a diverging Fermi-surface susceptibility of the k-space heavy fermions towards a conventional ordered state, or because of a sudden alteration of the many-body r-space states at each U atom with the associated changes in hybridization? To distinguish between these two fundamentally different situations requires determination of the electronic structure as temperature is reduced to T 0 .

SI-STM studies of URu 2 Si 2
For this purpose, we introduce SI-STM techniques to examine simultaneously the r-space/kspace electronic structure evolutions in URu 2 Si 2 . Here the tip-sample differential tunnelling conductance dI/dV(r, V) ≡ g(r, E = eV) at locations r and sample-bias voltage V yields an image proportional to the local density of electronic states LDOS(r, E), where e is the electron charge.
Moreover, k-space electronic structure elements can be determined simultaneously by using Fourier transform scanning tunnelling spectroscopy [36][37][38][39][40][41][42] . The pure compound URu 2 Si 2 exhibits one excellent cleave surface that is Si-terminated, whereas the alloy U 0.99 Th 0.01 Ru 2 Si 2 , which we use for quasiparticle interference studies, exhibits a different excellent cleave surface that is Uterminated. Each of these two types of cleave surface has the same lattice periodicity and orientation, which is highly distinct from that of the Ru layer (inset in Fig. 2b). They can be distinguished from each other because the U-termination layer exhibits the correct density of impurity states generated by the 1% of U sites upon which Th atoms are substituted (Fig. 3a). In  (1), revealing its excellent parameterization as a Fano spectrum 5 ( Supplementary Fig. 1). In Fig. 2c-e we show the atomically resolved images of the parameters of the Fano spectrum. Here ε 0 (r), Γ(r), and ζ(r) are determined from fitting g(r, E = eV) for each pixel within the yellow box in Fig. 2a. Significantly, we find that the maximum in both ε 0 (r) and ζ(r) and the minimum in Γ(r) occur at the U sites (X in Fig. 2c-e), as would be expected for a Kondo lattice of U atoms. These observations, in combination with theoretical predictions for such a phenomenology [12][13][14] , indicate that the r-space 'Fano lattice' electronic structure of Kondo screening in magnetic lattices can now be visualized.

Evolution of density of states at Si-and U-termination surfaces
For U-terminated surfaces (see Fig. 3a), the spatially averaged density of states DOS(E) ∝ <g(E)> spectrum for T > T o is less structured than that of the Si-terminated surface in For the Si-terminated surfaces upon cooling below T o , the overall Fano lineshape of DOS(E) as discussed in Fig. 2c-e is unchanged (Fig. 3c). In the inset to Fig. 3c, we show the evolution of the <g(E)> spectrum between 19 K and 1.7 K. In each case, the red line is the fit to the Fano spectrum at each temperature (excluding the data points in the bias range -7.75 mV to 6.75 mV) while the measured <g(E)> spectra are shown as open squares. Again, by subtracting the fitted Fano spectrum from the <g(E)> at each T value we determine the temperature dependence of the hidden-order DOS(E) modifications (Fig. 3d). At no E value on either surface do these DOS(E) spectra represent a complete gap. Finally, no changes are observed in the high-energy DOS(E) as the temperature falls below T o ( Supplementary Fig. 2), perhaps indicating that the basic Brillouin zone geometry is not altered by the transition.

Heavy f-electron quasiparticle interference imaging
To determine the evolution of k-space electronic structure through T o , we use heavy-electron quasiparticle interference [43][44][45] (QPI) imaging. The Si-terminated surface has proved unproductive for this purpose because the Fourier transform of its g(r, E) images ( Supplementary Fig. 3) are so complex that the multiple bands cannot yet be disentangled. However, in recent studies of heavyfermion QPI in Sr 3 Ru 2 O 7 it was shown that replacing 1% of the Ru atoms by Ti atoms produced intense scattering interference and allowed successful k-space determination 43 . Emulating this approach, we substituted 1% Th atoms on the U sites, which results in crystals usually cleaving at the U layer. The average spectrum on this U-terminated surface develops the narrow resonant DOS(E) structure below the T o (red data between vertical arrows in Fig. 3b), within which we observe intense QPI; see the g(q, E) movies in the Supplementary Information. The 1% Th substitution suppresses T o by only ~1 K (refs 46, 47)and does not alter the basic hidden order phenomenology (refs 46, 47), so the phenomena we report are not caused by our dilute Th doping. Moreover, because the energy scale of DOS(E) alterations is consistent with Th-doped specific heat measurements 46 and because these alterations are already detectable in tunnelling within 1 K below the bulk transition (blue line in Fig. 3b), the electronic structure of the Uterminated surface appears to be bulk representative of the hidden-order phase.
For QPI studies of the hidden-order transition we therefore measure g(r, E = eV) in a 50 nm × 50 nm field of view (FOV) with 250 µV energy resolution and atomic spatial resolution on these U-terminated surfaces (the simultaneous topograph is shown in Supplementary Fig. 4).
In Fig. 4a-f we show simultaneous images of g(r, E) modulations measured at T = 1.9 K for six energies near E F within the energy scale where the resonant feature appears below T o (Fig. 3b).
In Fig. 4g-l we show the six g(q, E) Fourier transforms of each g(r, E) image from Fig. 4a-f ( Supplementary Fig. 5). Four significant advances are already apparent in these unprocessed data. First, the wavevectors of the hidden-order g(r, E) modulations are dispersing very rapidly (within the narrow energy range of DOS(E) modifications in Fig. 3b); this is directly indicative of heavy fermions in the hidden-order state. Second, the magnitude of their characteristic q vectors, which are diminishing towards a small value as they pass through the Fermi energy from below (see Fig. 4i), suddenly jumps to a large value at a few millielectronvolts above E F (see Fig.   4k). It therefore appears that the band supporting QPI is widely split in k-space at this energy centred a few millielectronvolts above E F . Third, the QPI oscillations are highly anisotropic in qspace (Fig. 4g-l). Finally, the most intense modulations rotate by 45° when they pass the energy (compare Fig. 4h and l), indicating a distinct k-space electronic structure for the filled and empty gap-edge states. As we show further in Fig. 5, all of these effects are characteristics of the hidden-order state.

Evolution of k-space structure from Fano lattice to hidden-order state
To determine the k-space electronic structure evolution into the hidden-order state, we measure the temperature dependence of QPI data equivalent to those in Fig. 4 from just above T o down to 1.7 K. In Supplementary Fig. 6 1c) is shown by horizontal arrows in Fig. 5 and appears anisotropic in k-space by a factor of about two (Fig. 5c, d). This k-space structure can also be modelled using equation (2) ( Supplementary Fig. 7). Finally, the DOS(E) changes detected in r-space (Fig. 3) occur within the same narrow energy range and, moreover, are consistent with the gaps deduced from thermodynamics and other spectroscopies (Supplementary section VIII).

Absence of conventional density-wave states
Indications of a conventional density wave would include an energy gap that spans E F , modulations at fixed Q* in topographic images, and modulations at fixed Q* that are the same for empty and filled gap-edge states in g(r, E). Searches for all these phenomena, which must occur if the hidden-order state is a conventional density wave with static wavevector Q* (ref. 35), were carried out. First, high-precision topographic images of both Si-terminated and Uterminated surfaces are acquired and analysed over the same range of temperatures as in Fig. 3 searching for any additional, bias-independent, modulation wavevectors Q* appearing below T o in the Fourier transform of the topograph. Second, we analyse all the g(r, E), in search of modulations at fixed Q* which are the same for empty and filled gap-edge states. Third, we consider the energy gap structure in k-space revealed by Fig. 5c and d. Because these signatures are not observed at any temperature below T o for any topographs or g(r, E) maps ( Supplementary   Fig. 2), and because the observed k-space alterations do not result in a gap spanning E F (Fig. 5c,   d), we conclude that a conventional density wave does not appear to be present at T < T o in URu 2 Si 2 .

Discussion and conclusions
New perspectives on both the general physics of a Kondo lattice (above T o ) and the emergence of the hidden order from this state (below T o ) are revealed by our SI-STM studies of URu 2 Si 2 . First, Kondo screening of the U-atom magnetism appears to begin below 55 K, a deduction now supported by direct imaging of the expected Fano lattice signature [12][13][14] . Here the QPI observations in Figs 4 and 5 revealing a light hole-like band with m* ≈ 8m e cannot account for the m* ≈ 50m e deduced from specific heat measurements at T > T o ; resolution of this issue, which is not relevant to our research objectives here, will require future study. Nevertheless, below T o we observe this band splitting into two new heavy bands, one of which crosses E F with m* ≈ 53 in the (1, 0) direction and m* ≈ 17 in the (1, 1) direction (Fig. 5). Using the geometric mean to estimate the average over k-space, we find m* ≈ 28m e from our QPI data (Figs 4,5) while both thermodynamics 15,16 and ARPES 26 report m* ≈ 25m e for T << T o . In addition, we detect the spectroscopic signature of the transition temperature to within 1 K of the bulk value 46 (Fig. 3b), while the partial gap in the tunnelling density of states (Fig. 3b, d) agrees quantitatively with that derived from bulk specific heat studies 15,16,46

(Supplementary Information section VIII).
Thus, there is excellent consistency between the electronic structure of the hidden-order state deduced from thermodynamics, photoemission and our new SI-STM approach. Second, if the hidden order below T o were a conventional density wave, non-dispersive modulations at Q* should appear in the gap-edge states both above and below E F in g(q, E) and in the topographs, but no such phenomena are detected. Moreover, because the high spectral weight gap-edge states of the hidden-order phase (Figs 3 and 4) are at completely different k-space locations below and above E F (Figs 4 and 5) and the indirect gap 3 does not cross E F , these effects are also exclusive of a conventional density wave. Third, the actual evolution of electronic structure corresponds to a light dispersing band crossing E F near k = 0.3(π/a 0 ) above T o that undergoes rapid temperature changes below T o . The result is it splits into two far heavier bands that are widely separated in kspace (Figs 4g, 4l, 5c and 5d) within an energy range coincident with changes in the Fano spectrum near E F (Fig. 3). The dramatic alterations of electronic characteristics 15-26 therefore occur upon the anisotropic heavy-band splitting emerging from the Fano lattice electronic structure. Thus it seems that some sudden modification of the many-body state 12,30-34 periodic with the U atoms ( Fig. 2c-e, 3), along with associated alterations to the r-space/k-space hybridization processes (Figs 4 and 5), is the origin of the hidden order of URu 2 Si 2 . Further development of theoretical models rich enough to capture the complex dual r-space/k-space nature of the hidden order is now required.
More generally, the Fano lattice electronic structure images ε 0 (r), Γ(r), and ζ(r) (Fig. 2c-e) allow the first direct visualization of a Kondo-screening many-body state in magnetic lattice (here with 0 ε ≈ 3-4 meV, centred at the U atoms 12 ), while quasiparticle interference is used to determine a heavy fermion band structure (both above and below E F ). This combined capability to simultaneously visualize the r-space Fano lattice and the k-space heavy fermion structure opens a completely new experimental window onto the physics of multichannel Kondo lattices and heavy fermion physics.

METHODS SUMMARY
Both pure URu 2 Si 2 and U 0.99 Th 0.01 Ru 2 Si 2 samples were grown by the Czochralski method, using a continuous gettered tri-arc furnace under Ar gas starting from stoichiometric amounts of the constituent materials. Laue X-ray diffraction measurements were performed for orientation and to check the single-crystal nature. Our 1 mm × 1 mm × 0.5 mm c-axis normal samples are inserted into a variable-temperature SI-STM system, mechanically cleaved in cryogenic ultrahigh vacuum, and then inserted into the scanning tunnelling microscope head. Atomically flat and clean a-b surfaces consisting of layers of either Si or U atoms (see Figs 2 and 3) are achieved throughout. The cryogenic ultrahigh vacuum conditions enable many months of g(r, E) measurements with atomic resolution and register in the same FOV without any degradation of surface quality. The SI-STM system was stabilized at each operational temperature between 1.7 K and 29 K for a period of about 24 hours before each set of SI-STM measurements. The dI/dV(r, V) measurements were acquired with a standard alternating current lock-in amplifier technique, using bias modulations down to 250 µV RMS at the lowest temperatures. The r-space measurements on the Si surface required sub-atomic resolution on 20 × 20 nm 2 FOV while measurements for k-space determination used 60 × 60 nm 2 FOV to ensure high Fourier resolution of the dispersive scattering interference q-vectors.

Figure 1 A Kondo lattice model and its resulting band structure. a, A schematic
representation of the screening of a localized spin-half state (red) by delocalized k-space electrons (green) caused by the Kondo effect 1-5 . b, A typical Fano tunnelling-conductance spectrum 5 expected near the electronic many-body state depicted in a. c, A schematic representation of the T ≈ 0 band structure expected of a Kondo lattice as in equation (2), with the light hole-like band at high temperature depicted by a dashed line. The approximate hybridization energy range is shown by horizontal arrows.   Starting at energies below E F (g), the predominant QPI wavevectors diminish very rapidly until i; upon crossing a few millielectronvolts above E F , they jump to a significantly larger value and rotate through 45°. Then they again diminish in radius with increasing energy in j, k and l. This evolution is not consistent with a fixed Q* conventional density wave state but is consistent with an avoided crossing between a light band and a very heavy band.  Fig. 4g). A single light hole-like band crosses E F . c, Dispersion of the primary QPI wavevector for T ≈ 5.9 K along the (0, 1) direction (see Fig. 4g). Two heavy bands have evolved from the light band and become well segregated in k-space within the hybridization gap. d, Dispersion of the primary QPI wavevector for T ≈ 5.9 K along the (1, 1) direction (see Fig. 4g). Two heavy bands have evolved from the light band and are again segregated in k-space within the hybridization gap.