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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.

We calculate the universal contribution to the α-Rényi entropy from a cubic trihedral corner in the boundary of the entangling region in 3 + 1 dimensions for a massless free scalar. The universal number, vα, is manifest as the coefficient of a scaling term that is logarithmic in the size of the entangling region. Our numerical calculations find that this universal coefficient has both larger magnitude and the opposite sign to that induced by a smooth spherical entangling boundary in 3 + 1 dimensions, for which there is a well-known subleading logarithmic scaling. Despite these differences, up to the uncertainty of our finite-size lattice calculations, the functional dependence of the trihedral coefficient vα on the Rényi index α is indistinguishable from that for a sphere, which is known analytically for a massless free scalar. We comment on the possible source of this α-dependence arising from the general structure of (3 + 1)-dimensional conformal field theories, and suggest calculations past the free scalar which could further illuminate the general structure of the trihedral divergence in the Rényi entropy.

We study a quasi-2D classical Landau-Ginzburg-Wilson effective field theory in the presence of quenched disorder in which incommensurate charge-density wave and superconducting orders are intertwined. The disorder precludes long-range charge-density wave order, but not superconducting or nematic order. We select three representative sets of input parameters and compute the corresponding charge-density wave structure factors using both large-N techniques and classical Monte Carlo simulations. Where nematicity and superconductivity coexist at low temperature, the peak height of the charge-density wave structure factor decreases monotonically as a function of increasing temperature, unlike what is seen in X-ray experiments on YBa2Cu3O6+x. Conversely, where the thermal evolution of the charge-density wave structure factor qualitatively agrees with experiments, the nematic correlation length, computed to one-loop order, is shorter than the charge-density wave correlation length.

Clear experimental evidence of charge density wave correlations competing with superconducting order in YBCO have thrust their relationship with the pseudogap regime into the spotlight. To aid in characterizing the pseudogap regime, we propose a dimensionless ratio of the diamagnetic susceptibility to the correlation length of the charge density wave correlations. Using Monte Carlo simulations, we compute this ratio on the classical model of Hayward et. al. (Science 343, 1336 (2014)), which describes angular fluctuations of a multicomponent order, capturing both superconducting and density wave correlations. We compare our results with available data on YBa2Cu3O6+x, and propose experiments to clarify the value of this dimensionless ratio using existing samples and techniques.

Universal scaling terms occurring in Rényi entanglement entropies have the potential to bring new understanding to quantum critical points in free and interacting systems. Quantitative comparisons between analytical continuum theories and numerical calculations on lattice models play a crucial role in advancing such studies. In this paper, we exactly calculate the universal two-cylinder shape dependence of entanglement entropies for free bosons on finite-size square lattices, and compare to approximate functions derived in the continuum using several different ansatzes. Although none of these ansatzes are exact in the thermodynamic limit, we find that numerical fits are in good agreement with continuum functions derived using the AdS/CFT correspondence, an extensive mutual information model, and a quantum Lifshitz model. We use fits of our lattice data to these functions to calculate universal scalars defined in the thin-cylinder limit, and compare to values previously obtained for the free boson field theory in the continuum.