Spectral signatures of modulated $d$-wave superconducting phases

We calculate, within a mean-field theory, the spectral signatures of various striped $d$-wave superconducting phases. We consider both in-phase and antiphase modulations of the superconducting order across a stripe boundary and the effects of coexisting inhomogeneous orders, including spin stripes, charge stripes, and modulated $d$-density wave. We find that the antiphase modulated $d$-wave superconductor exhibits zero-energy spectral weight, primarily along extended arcs in momentum space. Concomitantly, a Fermi surface appears and typically includes both open segments and closed pockets. When weak homogeneous superconductivity is also present, the Fermi surface collapses onto nodal points. Among them are the nodal points of the homogeneous $d$-wave superconductor, but others typically exist at positions that depend on the details of the modulation and the band structure. Upon increasing the amplitude of the constant component, these additional points move toward the edges of the reduced Brillouin zone where they eventually disappear. The above signatures are also manifested in the density of states of the clean, and the disordered system. While the presence of coexisting orders changes some details of the spectral function, we find that the evolution of the Fermi surface and the distribution of the low-energy spectral weight are largely unaffected by them.

Figure 1

(Color online) The DSC configurations considered in this work: (a) site-centered in-phase modulated and (b) site-centered antiphase modulated. (c) Bond-centered antiphase modulated. The magnitude of the order parameter is depicted by the width of the lines, while its sign is given by the colors of the bonds.

Figure 2

(Color online) A sketch of the magnetic and charge distribution in the stripe model. The arrows represent the amplitude and sign of the spin density, and the shading of the circles represents the charge density (the darker the circle is, the higher the charge density is).

Figure 3

(Color online) A schematic of the modulated DDW order parameter. The arrows represent the currents in the system. The gray arrows carry one-half the current of the black ones.

Figure 4

(Color online) [(a)–(c)] The Fermi surface and [(d)–(f)] the low-energy spectral weight of a model corresponding to 1/8 hole-doped ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ in the presence of an antiphase modulated DSC. The Fermi surface is plotted in the first quadrant of the Brillouin zone. The spectral weight is integrated within a 20 meV Lorentzian window centered at zero energy. The amplitude of the modulated DSC is $\Delta =0$ in (a) and (d), $\Delta =0.075t_1$ in (b) and (e), and $\Delta =0.15t_1$ in (c) and (f).

Figure 5

(Color online) The Fermi surface and the low-energy spectral weight in a model of 17% hole-doped ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ with antiphase modulated DSC. The spectral weight was integrated in a similar manner to that in Fig. 4. The results are given for $\Delta =0$ in (a) and (d), $\Delta =0.075t$ in (b) and (e), and $\Delta =0.15t$ in (c) and (f) ($\left|t\right|\simeq 0.6\text{eV}$; see Ref. 38.)

Figure 6

(Color online) The momentum occupation function $n\left(\mathbf{k}\right)$ of a model of 1/8 hole-doped ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ with an antiphase modulated DSC of strength $\Delta =0.075t_1$.

Figure 7

(Color online) The evolution of the nodal points in a model of 1/8 hole-doped ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ with antiphase modulated DSC of strength $\Delta =0.075t_1$ as a function of the amplitude, ${\Delta }_0$, of an additional homogeneous DSC component. The curve depicts the Fermi surface for the case ${\Delta }_0=0$.

Figure 8

(Color online) [(a)–(d)] Fermi surfaces and [(e)–(h)] low-energy spectral weight distributions of a model corresponding to 1/8 hole-doped ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ with antiphase modulated DSC and additional orders. The amplitude of the DSC order parameter is $\Delta =0.1t_1$, and the additional orders are in [(a) and (e)] CDW with ${\varphi }_{\text{CDW}}=0.075t_1$, in [(b) and (f)] spin-stripe order with ${\varphi }_{\text{AF}}=0.075t_1$, in [(c) and (g)] charge and spin-stripe order with ${\varphi }_{\text{AF}}={\varphi }_{\text{CDW}}=0.075t_1$, and in [(d) and (h)] modulated DDW with ${\varphi }_{\text{DDW}}=0.0125t_1$. Note that the reduced BZ is halved in the presence of spin stripes or a modulated DDW, thus complicating a straightforward comparison between the different Fermi surfaces.

Figure 9

(Color online) Averaged density of states in a model of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ with (a) antiphase modulated DSC and (b) in-phase modulated DSC for various magnitudes of the superconducting order.

Figure 10

(Color online) LDOS in a model of ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_4$ with an antiphase modulated DSC of strength $\Delta =0.1t_1$. Atom 0 is located on the symmetry axis of the stripe—the leftmost site in Fig. 1.

Figure 11

(Color online) ${\rho }_{\left(\pi /2,0\right)}$ in a model of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ with (a) antiphase modulated DSC and (b) in-phase modulated DSC for various magnitudes of the superconducting order.

Figure 12

(Color online) FT-LDOS at $\omega =0$ of a model of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_4$ with (a) antiphase modulated DSC order and (b) coexisting antiphase and homogeneous DSC orders. The amplitude of the modulated order is $\Delta =0.1t_1$ and that of the constant component is ${\Delta }_0=0.01t_1$. The results are for an impurity strength of $V_0=150\text{meV}$. The darker regions correspond to higher FT-LDOS and we have indicated high-intensity peaks by circles: Peaks associated with the modulation wave vector are encircled in blue. Peaks that correspond to scattering between the nodal points of the homogeneous DSC are marked in green. Peaks due to scattering between the additional nodal points which appear in the mixed phase are marked in pink. Other high-intensity peaks that are not simply related to scattering between nodal points are designated in red.