Angular Fluctuations of a Multicomponent Order Describe the Pseudogap of YBa ₂ Cu ₃ O _{6+ x}

Abstract

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...

Long-range order and pinning of charge-density waves in competition with superconductivity

“…To facilitate comparison with the results of Ref. 25, we present in Fig. 1a the structure factor S Φx /a 2 = 2S CDW /a 2 as a function of T in a clean layer with and without a magnetic field.…”

“…We begin with the model considered by Hayward et al , 25 for a real 6-dimensional order parameter, equivalent to a complex SC field Ψ = n 1 + in 2 and two complex CDW fields, Φ x = n 3 +in 4 and Φ y = n 5 +in 6 . Here, for the sake of simplicity, we disregard quartic and anisotropic CDW terms, which appear in Ref.…”

Charge modulation as fingerprints of phase-string triggered interference

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