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 URu(2)Si(2) these effects begin at temperatures around 55 K but they are suddenly altered by an unidentified electronic phase transition at T(o) = 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 URu(2)Si(2) electronic structure simultaneously in r-space and k-space. Above T(o), the 'Fano lattice' electronic structure predicted for Kondo screening of a magnetic lattice is revealed. Below T(o), 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 T(o) of a light k-space band into two new heavy fermion bands. Thus, the URu(2)Si(2) '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.

. † These authors contributed equally to this project. The existence of electronic symmetry breaking in the underdopedcuprates, and its disappearance with increased hole-density p, are now widely reported. However, the relationship between this transition and the momentumspace ( ⃗ -space) electronic structure underpinning the superconductivity has not been established. Here we visualize the ⃗ =0 (intra-unit-cell) and ⃗ ≠0 (density wave) broken-symmetry states simultaneously with the coherent ⃗ -space topology, for Bi2Sr2CaCu2O8+d samples spanning the phase diagram 0.06≤p≤0.23.We show that the electronic symmetry breaking tendencies weaken with increasing p and disappear close to pc=0.19. Concomitantly, the coherent ⃗ -space topology undergoes an abrupt transition, from arcs to closed contours, at the same pc. These data reveal that the ⃗ -space topology transformation in cuprates is linked intimately with the disappearance of the electronic symmetry breaking at a concealed critical point. 2The highest known superconducting critical temperature Tc (1-3) occurs atop the Tc(p) 'dome' of hole-doped cuprates (Fig. 1A). In addition to the superconductivity, electronic broken-symmetry states (4) have also been reported at low p in many such compounds. Wavevector ⃗ =0 (intra-unit-cell) symmetry breaking, typically of 90 orotational (C4) symmetry, is reported in YBa2Cu3O6+, . Finite wavevector ⃗ ≠0 (density wave) modulations breaking translational symmetry, long detected in underdoped 16), are now also reported in underdoped YBa2Cu3O6+, . Summarizing all such reports in Fig. 1A reveals some stimulating observations.First, although the ⃗ =0 and ⃗ ≠0 states are detected by widely disparate techniques and are distinct in terms of symmetry, they seem to follow approximately the same phasediagram trajectory (shaded band Fig. 1A) as if facets of a single phenomenon (26). The second implication is that a critical point (perhaps a quantum critical point) associated with these broken-symmetry states may be concealed beneath the Tc(p) dome.Numerous earlier studies reported sudden alterations in many electronic/magnetic characteristics near p=0.19 (2,3,27), but whether these phenomena are caused by electronic symmetry changes (28) at a critical point was unknown. 3In ⃗ -space, the hole-doped cuprates also exhibit an unexplained transition in electronic structure with increasing hole density. Open contours or "Fermi arcs" (29)(30)(31)(32) are reported at low p in all compounds studied, while at high p closed hole-like pockets surrounding ⃗ = (±1, ±1) / 0 are observed (33,34). One possibility is that such a transition could occur due to the disappearance of an electronic ordered state, with the resulting modifications to the Brillouin zone geometry altering the topology of the electronic bands (28). 4Our strategy is therefore a simultaneous examination of both the ⃗ -space 11,36) or at the Bragg wavevectors (11,26,35). But the complete doping dependence of these broken-symmetry signatures was unknown. 6To determine the ⃗ -space t...

The identity of the fundamental broken symmetry (if any) in the underdoped cuprates is unresolved. However, evidence has been accumulating that this state may be an unconventional density wave. Here we carry out site-specific measurements within each CuO 2 unit cell, segregating the results into three separate electronic structure images containing only the Cu sites [Cu(r)] and only the x/y axis O sites [O x (r) and O y (r)]. Phase-resolved Fourier analysis reveals directly that the modulations in the O x (r) and O y (r) sublattice images consistently exhibit a relative phase of π. We confirm this discovery on two highly distinct cuprate compounds, ruling out tunnel matrix-element and materials-specific systematics. These observations demonstrate by direct sublattice phaseresolved visualization that the density wave found in underdoped cuprates consists of modulations of the intraunit-cell states that exhibit a predominantly d-symmetry form factor.CuO 2 pseudogap | broken symmetry | density-wave form factor U nderstanding the microscopic electronic structure of the CuO 2 plane represents the essential challenge of cuprate studies. As the density of doped holes, p, increases from zero in this plane, the pseudogap state (1, 2) first emerges, followed by the high-temperature superconductivity. Within the elementary CuO 2 unit cell, the Cu atom resides at the symmetry point with an O atom adjacent along the x axis and the y axis (Fig. 1A, Inset). Intraunit-cell (IUC) degrees of freedom associated with these two O sites (3, 4), although often disregarded, may actually represent the key to understanding CuO 2 electronic structure. Among the proposals in this regard are valence-bond ordered phases having localized spin singlets whose wavefunctions are centered on O x or O y sites (5, 6), electronic nematic phases having a distinct spectrum of eigenstates at O x and O y sites (7,8), and orbital-current phases in which orbitals at O x and O y are distinguishable due to time-reversal symmetry breaking (9). A common element to these proposals is that, in the pseudogap state of lightly hole-doped cuprates, some form of electronic symmetry breaking renders the O x and O y sites of each CuO 2 unit cell electronically inequivalent.Electronic Inequivalence at the Oxygen Sites of the CuO 2 Plane in Pseudogap State Experimental electronic structure studies that discriminate the O x from O y sites do find a rich phenomenology in underdoped cuprates. Direct oxygen site-specific visualization of electronic structure reveals that even very light hole doping of the insulator produces local IUC symmetry breaking, rendering O x and O y inequivalent (10), that both Q ≠ 0 density wave (11) and Q = 0 C 4 -symmetry breaking (11, 12, 13) involve electronic inequivalence of the O x and O y sites, and that the Q ≠ 0 and Q = 0 broken symmetries weaken simultaneously with increasing p and disappear jointly near p c = 0.19 (13). For multiple cuprate compounds, neutron scattering reveals clear intraunit-cell breaking of rotational symmetry (14,15...