Random field disorder and charge order driven quantum oscillations in cuprates

In the pseudogap regime of the cuprates, a period-2 charge order breaks a ${\mathbb{Z}}_2$ symmetry, reflecting a broken translational symmetry. Therefore, the interaction of charge order and quenched disorder due to potential scattering, can, in principle, be treated as a random field Ising model. A numerical analysis of the ground state of such a random field Ising model reveals local, glassy dynamics in both two and three dimensions. The dynamics are treated in the glassy limit as a heat bath which couples to the itinerant electrons, leading to an unusual electronic non-Fermi-liquid. If the dynamics are strong enough, the electron spectral function has no quasiparticle peak and the effective mass diverges at the Fermi surface, precluding quantum oscillations. In contrast to charge density, $d$-density wave order (reflecting staggered circulating currents) does not directly couple to potential disorder, allowing it to support quantum oscillations. At fourth order in Landau theory, there is a term consisting of the square of the $d$-density wave order parameter, and the square of the charge order. This coupling could induce parasitic charge order, which may be weak enough for the Fermi liquid behavior to remain uncorrupted. Here, we argue that this distinction must be made clear, as one interprets quantum oscillations in cuprates.

Figure 1

Current pattern for period-8 DDW, reproduced from Ref. [21]. The wave vector $\mathbf{Q}=(3\pi /4a,\pi /a)$, where $a$ is the lattice constant [21, 22]. In Landau theory, it can couple to a CDW of period $2\mathbf{Q}$. The relative magnitudes of the currents are depicted by the thickness of the arrows in the legend. Note the antiphase domain wall structure.

Figure 2

Scaling of $P_{\mathrm{dw}}$ in the one-dimensional case. The collapse works as expected for $x={(J/\sigma )}^2/L={\zeta }^2/L$; the logarithmic correction factor $(1+AlnL)$ is found to have little effect: $A=0\pm 0.3$. The best-fit parameters to Eq. (3) are $x_0=0.0596\pm 0.0007,\lambda =-0.0103\pm 0.0005,$ and $\theta =0.034\pm 0.002$. All points correspond to averages performed over 2048 disorder realizations.

Figure 3

Scaling of $P_{\mathrm{dw}}$ in $2\text{D}$ and $3\text{D}$. All points are averages over 2048 disorder realizations, except in $3\text{D}$ for $L=40$ and $L=80$, with 1024 realizations, and for $L=160$, with 512 realizations.

Figure 4

Leading-order (one-loop) self-energy graph. The fermion couples to the bath of RFIM fluctuations.

Figure 5

Spectral density as a function of frequency relative to the chemical potential, with wave vector $k=k_F$. Several values of the exponent $\psi$ are plotted. The relative scale of each curve is arbitrary: the unspecified prefactor ${\mathrm{\Sigma }}_0$ is not included.

Figure 6

Phase diagram for the stacked problem. The $3\text{D}$ isotropic case corresponds to the dashed diagonal line $J_z=J_{\parallel }$. For each value of $J_z$ considered, the numerically identified phase transition is a red $\times$. The mean-field result Eq. (14) is the red line (including an approximate scale factor). The ordered phase is above the $\times$ symbols.