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The hole-doped cuprate high temperature superconductors enter the pseudogap regime as their superconducting critical temperature, T c , falls with decreasing hole density. Experiments have probed this regime for over two decades, but we argue that decisive new information has emerged from recent X-ray scattering experiments [1][2][3]. The experiments observe incommensurate charge density wave fluctuations whose strength rises gradually over a wide temperature range above T c , but then decreases as the temperature is lowered below T c . We propose a theory in which the superconducting and charge-density wave orders exhibit angular fluctuations in a 6-dimensional space. The theory provides a natural quantitative fit to the X-ray data, and can be a basis for understanding other characteristics of the pseudogap.The X-ray scattering intensity [4] of YBa 2 Cu 3 O 6.67 at the incommensurate wavevectors Q x ≈ (0.31, 0) or Q y ≈ (0, 0.31), shown in Fig. 1, increases gradually below T ≈ 200K in a concave-upward shape until just above T c = 60K. One possibility is that this represents an order parameter of a broken symmetry, and the correlation length is arrested at a finite value by disorder; however, such order parameters invariably have a concave-downward shape.The temperature range is also too wide to represent the precursor critical fluctuations of an ordering transition. Indeed, there is no ordering transition below T c , and, remarkably, the scattering intensity decreases below T c at a rate similar to that of the rate of increase above superconductivity and charge density wave order [11,12]. The Landau theory introduces a complex field Ψ(r) to represent the superconductivity, and two complex fields Φ x,y (r) to represent the charge order. The latter can represent modulations at the incommensurate 3 wavevectors Q x,y in not only the site charge density, but also modulations in bond variables associated with a pair of sites [12,13]; nevertheless, we will refer to it simply as "charge"order. The free energy is restricted by 3 distinct U(1) symmetries: charge conservation, translations in x, and translations in y, which rotate the phases of Ψ, Φ x , and Φ y respectively.There are also the discrete symmetries of time-reversal and the square lattice point group, and these lead to the following form of the Landau free energy density (we ignore possible anisotropies in the spatial derivative terms):The earlier analysis [9] considered "phase" and "vortex" fluctuations of only the superconducting order, Ψ, and then assumed that the charge order amplitude was proportional to −v |Ψ| 2 , where v > 0 is the competing order coupling: this analysis found a small decrease in charge order with decreasing T , but did not find a prominent peak near T c . Here, we shall provide a theory which is non-perturbative in v, and which includes the thermal fluctuations of both Ψ and Φ x,y self-consistently, and applies over a wide range of temperatures.Our starting assumption is that it is always preferable for the electronic Fermi surface t...

A quantum critical (QC) fluid exhibits universal subleading corrections to the area law of its entanglement entropies. In two dimensions when the partition involves a corner of angle θ, the subleading term is logarithmic with coefficient aα(θ) for the α-Rényi entropy. In the smooth limit θ → π, a1(θ) yields the central charge of the stress tensor when the QC point is described by a conformal field theory (CFT). For general Rényi indices and angles, aα(θ) is richer and few general results exist. We study aα(θ) focusing on two benchmark CFTs, the free Dirac fermion and boson. We perform numerical lattice calculations to obtain high precision results in θ, α regimes hitherto unexplored. We derive field theory estimates for aα(θ), including new exact results, and demonstrate an excellent quantitative match with our numerical calculations. We also develop and test strong lower bounds, which apply to both free and interacting QC systems. Finally, we comment on the near collapse of aα(θ) for various theories, including interacting O(N ) models. CONTENTS

The entanglement entropy (EE) has emerged as an important window into the structure of complex quantum states of matter. We analyze the universal part of the EE for gapless systems put on tori in 2d/3d, denoted by χ. Focusing on scale invariant systems, we derive general non-perturbative properties for the shape dependence of χ, and reveal surprising relations to the EE associated with corners in the entangling surface. We obtain closed-form expressions for χ in 2d/3d within a model that arises in the study of conformal field theories (CFTs), and use them to obtain ansatzes without fitting parameters for the 2d/3d free boson CFTs. Our numerical lattice calculations show that the ansatzes are highly accurate. Finally, we discuss how the torus EE can act as a fingerprint of exotic states such as gapless quantum spin liquids, e.g. Kitaev's honeycomb model.