Figure 1

One-particle spectral functions at zero energy for the period 8 pair density wave system. The left panels show the spectral weight for adding or removing an electron (or hole) at $E=0$ in the extended BZ while the right panels show the same spectral weight folded into the reduced BZ and then repeated, which illustrates the complex shape of the Fermi surface. All panels are at 1/8 doping for (a) $\Delta =0.02$ and (b) $\Delta =0.25$.Reuse & Permissions

Figure 2

Low-lying energy levels for, from top to bottom, modulated $d$ wave ignoring the effects of vortices, modulated $d$ wave including the effects of vortices, and uniform $d$ wave including the effects of vortices. Here $\Delta =0.25$, $\mu =-0.3$, and $L=256$.Reuse & Permissions

Figure 3

The occupation function in the reduced BZ for 3 cases from left to right: (a) $\Delta =0$, $V=0$, and $\mu =-0.23$; (b) $\Delta =0.25$, $V=0$, and $\mu =-0.3$; (c) $\Delta =0$, $V=0.7$, and $\mu =-0.05$. $V$ is the magnitude of the interactions in CDW with periodicity of 4 sites. The reduced BZ for the CDW spans from 0 to $\pi /2$. Here, however, we have folded its FS once for easier comparison to the $\pi$-striped superconductor.Reuse & Permissions

Figure 4

FS for 3 cases from left to right: (a) $\Delta =0$, $V=0$, and $\mu =-0.23$; (b) $\Delta =0.25$, $V=0$, and $\mu =-0.3$; (c) $\Delta =0$, $V=0.7$, and $\mu =-0.05$. $V$ is the magnitude of the interactions in CDW with periodicity of 4 sites. The reduced BZ for the CDW spans from 0 to $\pi /2$. Here, however, we have folded its FS once for easier comparison to the $\pi$-striped superconductor.Reuse & Permissions

Figure 5

Low-energy DOS in the presence of a magnetic field of $L=1024$ for 3 cases: (a) $\Delta =0$, $V=0$, and $\mu =-0.23$; (b) $\Delta =0.25$, $V=0$, and $\mu =-0.3$; (c) $\Delta =0$, $V=0.7$, and $\mu =-0.05$.Reuse & Permissions

Figure 6

Semiclassical motion of a nearly free particle system in the presence of a weak periodic potential (a) and a weak periodic superconducting pairing potential [(b) and (c)]. The direction of the semiclassical motion for particles is shown by black arrows. Holes (shown by red arrows) precess in the opposite direction. The gray area in the center figure is ${(\hslash c/eH)}^2(A_T-A_b)$ where $A_b$ is the area of the small electron pocket in panel (a) and $A_T$ is the area of the original circular FS. Starting from the blue cross in panel (b), the particle can either go over the whole unperturbed circular orbit by tunneling at points G, I, B, and C, or tunnel only at points G and I and Andreev scatter twice at points B and C covering the gray area. Another possible path is to Andreev scatter at points G and I and tunnel at points B and C. However, this path covers the same gray area. The change in the phase of the wave function is $\hslash c{A}_T/eH$ when the particle goes over the whole circular circuit and $\hslash c(A_T-A_b)/eH+\beta$ when it travels around the shaded area, where $\beta$ is the phase shift due to two consecutive Andreev scatterings and is assumed to be relatively field independent. This behavior should be contrasted to that of the linked orbit of Pippard, shown on the left, where the particles orbit around the areas $A_T$ and $A_b$. Thus, as discussed in the text, the areas associated with quantum oscillations in the width of the first LL are different for the periodic potential and the periodic pairing models. Panel (c) shows the closed orbit corresponding to four successive Andreev-Bragg scatterings.Reuse & Permissions

Figure 7

Comparison of the low-energy DOS of a $\pi$-striped superconductor with $\Delta =0.02$ and $\mu =-0.23$ in the presence of a magnetic field of $L=256$ with and without vortices, as described in the text.Reuse & Permissions

Figure 8

Comparison of the low-energy bands for the BdG Hamiltonian of a $\pi$-striped superconductor with $\Delta =0.02$ and $\mu =-0.23$ in the presence of a magnetic field of $L=256$. The solid curves are the bands for the full BdG Hamiltonian, including vortices. The dashed curves are the semiclassical results for no vortices, and the flat lines (dash-dotted lines) are the Landau levels in the limit $\Delta =0$.Reuse & Permissions

Figure 9

The width of the LL closest to $E=0$ as a function of $1/B$ or $L$ for $\Delta =0.02$ and $\mu =-0.23$, corresponding to $1/8$ doping. $1/B$ is written in terms of the lattice constant $a$ and flux quantum ${\phi }_0$. The solid line is a spline fit to the data that shows the oscillatory behavior more clearly. The inset shows the first LL for $L=256$.Reuse & Permissions

Figure 10

Power spectrum associated with the oscillations of the width and position of the first LL for $\Delta =0.02$ at $1/8$ doping. The $x$ axis is rescaled so that it corresponds to area in units of the area of BZ.Reuse & Permissions

Figure 11

(a) Boomerang-shaped FS orbit involving two Andreev-Bragg scatterings and two tunnelings, as shown schematically in Fig. 6b, but for a period 8 modulation. The area of this orbit is denoted $A_T-A_b$ in the text. (b) The corresponding area $A_b$. (c) The area $2A_b-A_T$, corresponding to the difference of figures (a) and (b).Reuse & Permissions

Figure 12

Comparison of the geometrical area $A_b$ and the area associated with quantum oscillations in the width of the first LL for $\Delta =0.02$ vs $\mu$ in the region around $1/8$ doping.Reuse & Permissions

Figure 13

Comparison of the geometrical area, $A_T-A_b$, the boomerang-shaped area in Fig. 11a, and the area associated with oscillations in the position of the first LL, corresponding to the highest frequency peak in Fig. 10, shown as a function of $\mu$.Reuse & Permissions

Figure 14

Power spectrum for oscillations of the position of the lowest LL for small values of the pairing potential amplitude, $\Delta$. As discussed in the text, the peak at 0.0625 corresponds to the area $A_T$, the original FS. The peak at 0.103 corresponds to $A_T-A_b$, the boomerang-shaped area shown in Fig. 11a, while the feature at 0.1065 corresponds to the orbit with area $2A_b-A_T$, shown in Fig. 11c.Reuse & Permissions

Figure 15

Comparison of the low-energy DOS of a $\pi$-striped superconductor in the presence of a magnetic field of $L=1024$ with $\Delta =0.25$ and $\mu =-0.3$ corresponding to $1/8$ doping with and without vortices.Reuse & Permissions

Figure 16

The low-energy DOS for $\Delta =0.4$ and $L=800$ at half filling. Each (double) peak has twice the degeneracy of a LL.Reuse & Permissions

Figure 17

Half width of the peak closest to $E=0$ for different values of $\Delta$ at half filling. The Fermi surfaces for two of the $\Delta$ values in this figure are shown in Fig. 18.Reuse & Permissions

Figure 18

Areas consistent with the quantum oscillations seen in the width of the first peak in the low-energy DOS are shown in red (dark-shaded) for two values of $\Delta$ at half filling. Note that for $\mu =0$ the gray (light-shaded) areas have the same area as the red areas.Reuse & Permissions

Figure 19

Power spectrum associated with the oscillations in the width for $\Delta =0.25$ and $\Delta =0.4$ at half filling. The $x$ axis is rescaled so that it corresponds to area in units of the area of BZ.Reuse & Permissions

Figure 20

Comparison of the geometrical area (red or gray area in Fig. 18) and the area associated with quantum oscillations in the width of the lowest energy peak for different values of $\Delta$ at half filling.Reuse & Permissions

Figure 21

Semilog plot of the width of the first LL for $\Delta =0.4$ at half filling as a function of $1/B$ showing a fairly linear average behavior for not very large fields. This is expected if the broadening is caused by magnetic breakdown. The dashed line is a linear fit to the data.Reuse & Permissions

Figure 22

Power spectrum associated with the position of the first LL for $\Delta =0.6$ and $\mu =-0.5$. The inset shows the position of the first LL for the same parameters.Reuse & Permissions

Figure 23

FS for $\Delta =0.6$ and $\mu =-0.5$. The difference in the area of the gray (light-shaded) and red (dark-shaded) areas gives rise to the strongest peak in the power spectrum of the position of the first LL.Reuse & Permissions

Figure 24

Position of the first LL for $\Delta =0.5$ and $\mu =-0.4$.Reuse & Permissions

Figure 25

FS for $\Delta =0.5$ and $\mu =-0.4$. The red area is associated with short-period oscillations in Fig. 24 for larger magnetic fields and the gray (light-shaded) area is associated with the oscillations in the width of the first LL when magnetic breakdown occurs. The difference in the areas of the two lobes of the figure-eight shape corresponds to long-period oscillations in Fig. 24 at smaller fields. Black (thin) and red (thin) arrows show the two possible semiclassical paths.Reuse & Permissions

Figure 26

Position and width of the first peak for $\Delta =0.25$ and $\mu =-0.3$, corresponding to $\frac{1}{8}$ doping, plotted versus $1/B$.Reuse & Permissions

Figure 27

Power spectrum for $\Delta =0.25$ and $\mu =-0.3$.Reuse & Permissions

Figure 28

FS for $\Delta =0.25$ and $\mu =-0.3$ in the quadrant of the first BZ.Reuse & Permissions

Figure 29

Comparison of the geometrical area and the area associated with quantum oscillations in the width and position of the first LL as a function of $\mu$ for $\Delta =0.2$. The geometrical area is the area corresponding to the red (dark-shaded) region in Fig. 28 in the case of $\Delta =0.2$.Reuse & Permissions

Figure 30

The spectra associated with oscillations in the width and position for $\Delta =0.25$ and $t_2=-0.15$ at $\frac{1}{8}$ doping. The peaks correspond to the gray (light-shaded) and red areas shown in Fig. 31. The results are consistent with those for $t_2=0$.Reuse & Permissions

Figure 31

The areas associated with the first peaks of the position and width spectra in Fig. 30 for $\Delta =0.25$ and $t_2=-0.15$ at $\frac{1}{8}$ doping.Reuse & Permissions

Figure 32

Specific heat versus $1/B$ for $\Delta =0.25$ and $\mu =-0.3$ and $t_2=0$ for different temperatures. Temperatures in units of the hopping term $t$ are shown on the right. Note the $\pi$ phase shift in the oscillatory behavior of specific heat as $T$ increases through $T^{*}\approx 0.003t$.Reuse & Permissions

Figure 33

The oscillatory part of the calculated specific heat for $\Delta =0.25$ and $\mu =-0.34$ with a zero second-nearest-neighbor hopping shown as a function of the magnetic field and temperature. To plot the data, $t=0.16$ eV is chosen.Reuse & Permissions

Figure 34

The oscillatory parts of the specific-heat data by Riggs et al. and calculations for a $\pi$-striped superconductor at $T=1$ K. The left $y$-axis scale is for the experimental data and the right one is for the model.Reuse & Permissions