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...

We compute the time evolution of the mutual information in out of equilibrium quantum systems whose gravity duals are Vaidya spacetimes in three and four dimensions, which describe the formation of a black hole through the collapse of null dust. We find the holographic mutual information to be non monotonic in time and always monogamous in the ranges explored. We also find that there is a region in the configuration space where it vanishes at all times. We show that the null energy condition is a necessary condition for both the strong subadditivity of the holographic entanglement entropy and the monogamy of the holographic mutual information.Comment: 32 pages, 16 figure

We propose a quantum dimer model for the metallic state of the hole-doped cuprates at low hole density, p. The Hilbert space is spanned by spinless, neutral, bosonic dimers and spin S = 1=2, charge +e fermionic dimers. The model realizes a "fractionalized Fermi liquid" with no symmetry breaking and small hole pocket Fermi surfaces enclosing a total area determined by p. Exact diagonalization, on lattices of sizes up to 8 × 8, shows anisotropic quasiparticle residue around the pocket Fermi surfaces. We discuss the relationship to experiments.T he recent experimental progress in determining the phase diagram of the hole-doped Cu-based high-temperature superconductors has highlighted the unusual and remarkable properties of the pseudogap (PG) metal (Fig. 1). A characterizing feature of this phase is a depletion of the electronic density of states at the Fermi energy (1, 2), anisotropically distributed in momentum space, that persists up to room temperature.Attempts have been made to explain the pseudogap metal using thermally fluctuating order parameters; we argue below that such approaches are difficult to reconcile with recent transport experiments. Instead, we introduce a new microscopic model that realizes an alternative perspective (3), in which the pseudogap metal is a finite temperature (T) realization of an interesting quantum state: the fractionalized Fermi liquid (FL*). We show that our model is consistent with key features of the pseudogap metal observed by both transport and spectroscopic probes.The crucial observation that motivates our work is the tension between photoemission and transport experiments. In the cuprates, the hole density p is conventionally measured relative to that of the insulating antiferromagnet (AF), which has one electron per site in the Cu d band. Therefore, the hole density relative to a filled Cu band, with two electrons per site, is actually 1 + p. In fact, photoemission experiments at large hole doping observe a Fermi surface enclosing an area determined by the hole density 1 + p (4), in agreement with the Luttinger relation. In contrast, in the pseudogap metal, a mysterious "Fermi arc" spectrum is observed (5-7), with no clear evidence of closed Fermi surfaces. However, despite this unusual spectroscopic feature, transport measurements report vanilla Fermi liquid properties, but associated with carrier density p, rather than 1 + p. The carrier density of p was indicated directly in Hall measurements (8), whereas other early experiments indicated suppression of the Drude weight (9-11). Although the latter could be compatible with a carrier density of 1 + p but with a suppressed kinetic term, the Hall measurements indicate the suppression of the Drude weight is more likely due to a small carrier density. Two recent experiments displaying Fermi liquid behavior at low p are especially notable: (i) the quasiparticle lifetime τðω, TÞ determined from optical conductivity experiments (12) has the Fermi liquidlike dependence 1=τ ∝ ðZωÞ 2 + ðcπk B TÞ 2 , with c an order unity constan...

The nature of the pseudogap regime of cuprate superconductors at low hole density remains unresolved. It has a number of seemingly distinct experimental signatures: a suppression of the paramagnetic spin susceptibility at high temperatures, low-energy electronic excitations that extend over arcs in the Brillouin zone, X-ray detection of charge-density wave order at intermediate temperatures and quantum oscillations at high magnetic fields and low temperatures. Here we show that a model of competing charge-density wave and superconducting orders provides a unified description of the intermediate and low-temperature regimes. We treat quantum oscillations at high field beyond semiclassical approximations, and find clear and robust signatures of an electron pocket compatible with existing observations; we also predict oscillations due to additional hole pockets. In the zero-field and intermediate temperature regime, we compute the electronic spectrum in the presence of thermally fluctuating charge-density and superconducting orders. Our results are compatible with experimental trends.

We study how the universal contribution to entanglement entropy in a conformal field theory depends on the entangling region. We show that for a deformed sphere the variation of the universal contribution is quadratic in the deformation amplitude. We generalize these results for Rényi entropies. We obtain an explicit expression for the second order variation of entanglement entropy in the case of a deformed circle in a three dimensional CFT with a gravity dual. For the same system, we also consider an elliptic entangling region and determine numerically the entanglement entropy as a function of the aspect ratio of the ellipse. Based on these three-dimensional results and Solodukhin's formula in four dimensions, we conjecture that the sphere minimizes the universal contribution to entanglement entropy in all dimensions.