Diamagnetism and density-wave order in the pseudogap regime of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$

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 ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$, and propose experiments to clarify the value of this dimensionless ratio using existing samples and techniques.

Figure 1

Sample system configuration of the model of Ref. [17] when $T/{\rho }_s=0.18$ (below the Kosterlitz-Thouless SC transition $T_c)$, and with a representative set of model parameters (that will be defined in Sec. 2), namely $L_{\mathrm{PBC}}=64,\lambda =1,g{a}^2=0.4,g^{\prime }{a}^2=0$, and $w{a}^2=-0.2$. We show a configuration of the variables $n_{i\alpha }$ on a subregion of the $64\times 64$ lattice and employ two different visualization techniques. In the upper plot we use shading to represent the strength of the CDW order parameter, with darker shading corresponding to stronger CDW magnitude $\sqrt{|{\Phi }_{ix}{|}^2+{\left|{\Phi }_{iy}\right|}^2}$ on site $i$, and color to represent the orientation of the variables in the SC plane. The white arrows here correspond to the magnitude and orientation of the SC order ${\Psi }_i$. The lower plot uses color to represent the magnitude of the CDW order parameter and black arrows to illustrate the streamlines of ${\Psi }_i$. In both plots, a vortex-antivortex pair is visible in the upper left and lower right corners. The enhancement of CDW correlations in the vicinities of these vortices is also visible.

Figure 2

The CDW correlation length ${\xi }_{\mathrm{cdw}}$ extracted from Monte Carlo simulations. We illustrate two different methods for extracting the correlation length for a representative set of parameters. The first method fits the structure factor $S\left(q_x\right)$ to the shifted Lorentzian function of Eq. (18) for each $T$, as illustrated in the inset for $T=0.5$. Error bars come from the covariance matrix of the least-squares fit. The second method is to calculate ${\xi }_{\mathrm{cdw}}$ from Eq. (19), with error bars corresponding to the statistical Monte Carlo error.

Figure 3

Monte Carlo calculations of the linear diamagnetic susceptibility $M/B$. We plot $sM/\left(a^{2}B\right)$ for a representative set of parameters and various $L_{\mathrm{OBC}}$, with $M/B$ calculated from Eq. (20) in a Monte Carlo simulation on a system with open boundary conditions. The dashed line is the location of the Kosterlitz-Thouless transition, which is determined by calculating the helicity modulus associated with superconducting correlations (see Ref. [17]).

Figure 4

The CDW correlation length ${\xi }_{\mathrm{cdw}}^{\mathrm{Xray}}$ extracted from x-ray scattering experiments [12] on ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.67}$ as a function of $T$. ${\xi }_{\text{cdw}}^{\text{Xray}}\left(T\right)$ is extracted from Lorentzian fits to background-subtracted x-ray scattering data. The inset illustrates this fit for data at $T=80\text{K}$.

Figure 5

Comparison of $S_{{\Phi }_x}$ calculated in Monte Carlo simulations to x-ray data from CDW scattering experiments [12] on ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.67}$. We illustrate results that demonstrate the effects of varying $g$ (top), $g^{\prime }$ (middle), and $w$ (bottom) in our model. Note that, following the procedure of Ref. [17], there are two fitting parameters for each set of parameters $\lambda ,g{a}^2,g^{\prime }{a}^2$, and $w{a}^2$: The value of ${\rho }_s$ as well as the vertical scaling factor are both adjusted to make the Monte Carlo and experimental curves match in the vicinity of the peak.

Figure 6

Comparison of $R\left(T\right)$ calculated in Monte Carlo simulations to data from x-ray scattering experiments [12] and torque magnetometry experiments [4] on YBCO. We illustrate results for the same parameter sets as in Fig. 5. For each set of parameters $\lambda ,g{a}^2,g^{\prime }{a}^2$, and $w{a}^2$, the rescaling factor for the Monte Carlo data along the $T$ axis is determined from the $S_{{\Phi }_x}$ vs $T$ fit. The shading for $R\left(T\right)$ accounts for statistical Monte Carlo errors as well as the uncertainty in the method for extracting ${\xi }_{\mathrm{cdw}}$ (see Fig. 2). The dashed red line in each plot corresponds to an extrapolation to $L_{\mathrm{OBC}}=\infty$, which is performed by a least-squares fit to a quadratic polynomial of $1/L_{\mathrm{OBC}}$.