Quantum oscillations in non-Fermi liquids: Implications for high-temperature superconductors

We address quantum oscillation experiments in high-$T_c$ superconductors and the evidence from these experiments for a pseudogap versus a Fermi liquid phase at high magnetic fields. As a concrete alternative to a Fermi liquid phase, the pseudogap state we consider derives from earlier work within a Gor'kov-based Landau level approach. Here the normal state pairing gap in the presence of high fields is spatially nonuniform, incorporating small gap values. These, in addition to $d$-wave gap nodes, are responsible for the persistence of quantum oscillations. Important here are methodologies for distinguishing different scenarios. To this end we examine the temperature dependence of the oscillations. Detailed quantitative analysis of this temperature dependence demonstrates that a high-field pseudogap state in the cuprates may well “masquerade” as a Fermi liquid.

Figure 1

(a) A density plot of ${\left|\Delta \left(\mathit{r}\right)\right|}^2$ for a pseudogap “blurred vortex” model (Ref. [19]). This indicates that the normal state gap structure (representing preformed pairs) is inhomogeneous as a result of high magnetic fields. (b) The density plot of ${\left|\Delta \left(\mathit{q}\right)\right|}^2$ for the $d$-wave, $n=8$ system used throughout this Rapid Communication. Here $\mathit{q}$ is based on a magnetic translation group. Both plots are normalized to ${\Delta }_{\max }=1$.

Figure 2

(a) A plot of the density of states at the Fermi surface vs $H$ (with $\hslash e/mc=1$) for the 2D, high-field system with self-energy given by Eq. (1), with $\Delta =10$, $\mu =50$, $\gamma /\Delta =0.5$, and ${\gamma }^{\prime }=0.2$. The black line shows the $d$-wave $\Delta \left(\mathit{q}\right)$ used throughout this Rapid Communication, while the dashed purple line shows a constant $\Delta \left(\mathit{q}\right)$. The inset shows the probability distribution function of $\Delta \left(\mathit{q}\right)$ values (normalized so that ${\sum }_{\mathit{q}}\Delta {\left(\mathit{q}\right)}^2=1$) for the high-field, $n=8$ $d$ wave used here (black line) and for comparison a distribution of $\Delta \left(\mathit{k}\right)$ for zero-field $d$ wave, $\Delta \left(\mathit{k}\right)=cos{k}_x-cos{k}_y$ (blue dots). (b) Example spectral functions for the system, with variable $\Delta \left(\mathit{q}\right)$. In both the main plot and the inset, $\Delta =10,4,1$ for the solid black, dashed red, and dash-dotted blue curve, respectively. In the main plot, the self-energy in Eq. (1) is used with $\gamma =5$ and ${\gamma }^{\prime }=0.2$, while in the inset Eq. (20) of Ref. [16] is used with $\Gamma =4$ and $\gamma =0.5$.

Figure 3

(a) A plot of the rescaled temperature dependence of the oscillation amplitude, $a\left(T\right)/a_0$, vs $T{m}^{*}$. The green solid line is for Fermi-liquid theory; the black dashed line represents the present pseudogap model with physical parameters $\Delta =10\hslash {\omega }_c$ (Ref. [35]), $\gamma /\Delta =0.5$ (consistent with Fermi arcs; Refs. [23] and [32]), and ${\gamma }^{\prime }=0.2\hslash {\omega }_c$. The red dotted line represents an unphysical case for comparison, with small $\Delta =\hslash {\omega }_c$ and $\gamma /\Delta =0.1$. The inset shows the residual between the calculation and Fermi-liquid theory. (b) The nonoscillating density of states for the systems in (a). The $\Delta =10\hslash {\omega }_c$ case is quite flat within the range $-1\lesssim \omega \lesssim 1$. In contrast, the more unphysical $\Delta =\hslash {\omega }_c$ case shows a significant curvature of the density of states. The normalization $\hslash {\omega }_c=1$ is used here.