Quantum oscillations in a bilayer with broken mirror symmetry: A minimal model for ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+\delta }$

Using an exact numerical solution and semiclassical analysis, we investigate quantum oscillations (QOs) in a model of a bilayer system with an anisotropic (elliptical) electron pocket in each plane. Key features of QO experiments in the high temperature superconducting cuprate YBCO can be reproduced by such a model, in particular the pattern of oscillation frequencies (which reflect “magnetic breakdown” between the two pockets) and the polar and azimuthal angular dependence of the oscillation amplitudes. However, the requisite magnetic breakdown is possible only under the assumption that the horizontal mirror plane symmetry is spontaneously broken and that the bilayer tunneling $t_{\perp }$ is substantially renormalized from its ‘bare’ value. Under the assumption that $t_{\perp }=\tilde{Z}{t}_{\perp }^{\left(0\right)}$, where $\tilde{Z}$ is a measure of the quasiparticle weight, this suggests that $\tilde{Z}\lesssim 1/20$. Detailed comparisons with new ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.58}$ QO data, taken over a very broad range of magnetic field, confirm specific predictions made by the breakdown scenario.

Figure 1

Typical Fourier transform of QOs of the magnetic torque for underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.58}(T_c=60\text{K},p\approx 11\%)$ showing the characteristic symmetrically split “three peak” structure. Raw torque data is shown in the inset, taken at $T\approx 1.5\mathrm{K}$, for a field range of 31 to 62.6 T. Structure above 700 T in the Fourier transform is harmonic content.

Figure 2

The Fermi surface of the bilayer system in (a) the absence $\left(t_{\perp }=0\right)$ and (b) the presence $\left(t_{\perp }=0.005t_a\right)$ of an isotropic interlayer tunneling $t_{\perp }$. The parameters used are $t_b=t_a/3$ and $\mu =-2.5306t_a$. Note that we have zoomed in to an area that is one quarter of the full (unreconstructed) Brillouin zone.

Figure 3

QOs of the DOS for very small broadening $\delta =0.0001t_a$ (long lifetimes) and $t_{\perp }=0.005t_a$, in the absence of a Zeeman coupling $\left(\tilde{g}=0\right)$. Panels (a) and (b) show the calculated DOS $\rho$ vs $B$ and $1/B$; panel (c) is the Fourier transform of panel (b). Each peak indicates a characteristic frequency of QOs and the corresponding semiclassical orbits are also illustrated above. The number of equivalent semiclassical orbits, $n$, is indicated below each orbit, and we have explicitly shown the two distinct classes of $\gamma$ orbits. A relatively large range of magnetic field is used $4T<B<1000T$ to capture all of the QO frequencies. The system size is $L_x=2^{23}$.

Figure 4

The evolution of QOs in the DOS $\rho \left(\mu \right)$ for various values of the inverse quasiparticle lifetime $\delta$. The interlayer tunneling $t_{\perp }=0.005t_a$ and the rest of the parameters are detailed at the end of Sec. 2.

Figure 5

The raw Fourier transform of the density of states oscillations as the interlayer tunneling $t_{\perp }$ is increased (left to right), and the inverse lifetime $\delta$ is increased (top to bottom). Larger values of $t_{\perp }$ suppress the central frequencies and enhance the satellite frequencies which correspond to orbits of the true bilayer split Fermi surface. Shorter quasiparticle lifetimes (larger $\delta )$ lead to decreased harmonic content. The field range used here is $20T<B<100T$, with $2^{10}$ data points.

Figure 6

(a) The Fourier transform of the DOS QOs for various polar angles $\theta$ of the magnetic field $\vec{B}$ (different angles have been arbitrarily offset). (b)–(d) Extracted peak heights in the Fourier transform (we have used both amplitude and phase information) versus the polar angle $\theta$. The parameters are $\delta =0.005t_a,t_{\perp }=0.005t_a$. The dashed lines are the theoretical fits of the angular dependence due to spin splitting [Eq. (8)] which is caused by Zeeman effect.

Figure 7

(a) A schematic diagram showing how the in-plane component of the flux enclosed by a given semiclassical orbit $(\delta$ orbit in Fig. 3) is determined. The red curve is the semiclassical orbit, while gray ellipses are the Fermi surfaces. Note that the in-plane directions are in momentum space, while the vertical separation is in real space. The vertical region enclosed (shaded gray) has a (real space) area of $\delta k{\ell }_B^{2}c$. (b) The $\varphi$ dependence of QO amplitude $A_1\left(\varphi \right)$ as in Eq. (9) for various values of the polar angle $\theta$ for $\delta k=0.6417$ defined in our model as well as the case of a larger $\delta k=0.8417$. The anisotropy is clearly more apparent for larger $\theta$ and/or $\delta k$.

Figure 8

The relative amplitude of the QOs at frequency $620T$ for polar angles $\theta =63.5^{\circ }$ and $\theta =51.5^{\circ }$, respectively. The QO amplitude at $\varphi =0$ is maximal and set as the unit 1 for each of the data sets. The solid curve is the theoretical expectation value according to Eq. (9) and the expected $C_4$ rotation symmetry is clearly present. The parameters used in our numerical calculations of the DOS QOs are $t_{\perp }=0.005,\delta =0.001$ and $11T<B<100T$.

Figure 9

The relative amplitude of the QOs at frequency $530T$ for polar angles $\theta ={60}^{\circ }$. The solid curve is the fit to theoretical form given by Eq. (10) with $M=-13.5$. The parameters used in our numerical calculations of the DOS QOs are $t_{\perp }=0.005,\delta =0.001$, and $11T<B<100T$.

Figure 10

Fourier transform of torque quantum oscillation in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.58}$. Analysis of the full field range, from 18.5 to $62.6T$ (red curve), reveals spectral features not present in Fig. 1 but that correspond well with the frequencies shown in Fig. 3. Analysis of the oscillations between 18.5 and $26T$ only (blue curve) shows that spectral weight is shifted away from the main $\gamma$ peak and toward the side lobes. The blue curve has been multiplied by a factor of 10 and truncated at $700T$ for clarity. Note that spectral features below $\approx 150T$ are removed as part of the background-subtraction procedure, and thus this data does not address the possibility of a $90T$ frequency that has been reported in transport measurements [48].

Figure 11

Fourier transforms of QOs in the density of states for two models (a) with mirror symmetry and (b) without mirror symmetry. In (a) we consider identical Fermi surfaces with an interlayer tunneling of the form $t_{\perp }\left(\mathit{k}\right)=t_{\perp }{cos}^2\left(2k_y\right)$, while in (b) the mirror symmetry is weakly broken by considering orthogonal Fermi surfaces with a weak mass anisotropy $\left(t_b=0.95t_a\right)$ and the interlayer tunneling has the same form as before. The bilayer bonding and antibonding Fermi surfaces are almost identical in both cases, yet the QO frequencies are dramatically different: Mirror symmetry forbids breakdown orbits in (a).

Figure 12

Different slopes of $S_k$ versus $\mu$ suggest the effective masses are different for the different orbits. The areas of $\alpha ,\gamma$, and $\epsilon$ orbits are obtained from exact calculations of the Fermi surface, while for $\beta$ and $\delta$ orbits the areas are based on interpolation between the $\alpha ,\gamma$, and $\epsilon$ orbits (shown as the thinner lines). The vertical line is the value of $\mu =-2.5306t_a$ chosen throughout our calculations.

Figure 13

(a) The inset shows two simulated data sets: One is apodized with a boxcar function (black), and the other uses equation (E1) with $\alpha =1.7$ (red). The Fourier transform of the boxcar-apodized data shows multiple side lobes introduced from the sharp cutoff. The data apodized with the Kaiser window has a main peak suppressed by about a factor of two, but with the first side lobe suppressed 20 times more than that in the boxcar data. (b) The inset shows simulated data from equation (E2) before the window is applied. The Fourier transform uses the Kaiser window with $\alpha =1.7$—the same as the red curve in Fig. 13 and in Fig. 10 in the main text. Note that there are no extraneous side lobes.