Modeling a striped pseudogap state

We study the electronic structure within a system of phase-decoupled one-dimensional superconductors coexisting with stripe spin and charge-density-wave order. This system has a nodal Fermi surface (Fermi arc) in the form of a hole pocket and an antinodal pseudogap. The spectral function in the antinodes is approximately particle-hole symmetric contrary to the gapped regions just outside the pocket. We find that states at the Fermi energy are extended whereas states near the pseudogap energy have localization lengths as short as the interstripe spacing. We consider pairing which has either local $d$-wave or $s$-wave symmetry and find similar results in both cases, consistent with the pseudogap being an effect of local pair correlations. We suggest that this state is a stripe-ordered caricature of the pseudogap phase in underdoped cuprates with coexisting spin-, charge-, and pair-density-wave correlations. Lastly, we also model a superconducting state which (1) evolves smoothly from the pseudogap state, (2) has a signature subgap peak in the density of states, and (3) has the coherent pair density concentrated to the nodal region.

Figure 1

(Color online) Unit cell of the SDW with the arrow length and direction proportional to the local spin together with the local $d$-wave paired bonds on two stripes with individual (and generally distinct) phases ${\varphi }_n$ and ${\varphi }_{n+1}$

Figure 2

(Color online) Intensity plot of the spectral weight at the Fermi energy in the first quadrant of the full BZ (units of $\pi /a$ with $a$ the lattice spacing) for $m=t/3$ and $\mu =-1.0t$ (left column) or $\mu =-1.1t$ (right column). Top row [(a) and (b)] is without pairing, middle row [(c) and (d)] with phase-disordered local $d$-wave pairing ${\Delta }_d=t/4$, and bottom row [(e) and (f)] with phase-disordered local $s$-wave pairing ${\Delta }_s=t/3$. Arrows in (a) and (c) indicate cuts shown in Fig. 4. A Lorentzian broadening with width $\eta =0.01t$ was used. All plots have the same absolute color scheme. Calculations are made for a system size $N_x=200$ with periodic boundary conditions and averaged over three realizations of the quenched random phases on the charge stripes in (c)–(f).

Figure 3

(Color online) Spectral weight at the Fermi energy in the pseudogap state symmetrized with respect to the stripe orientation. The parameters are identical to Figs. 2c, 2d, respectively. The “arc” length grows with doping but is in fact restricted by the Bragg planes at $k=\pi /4$ for the present case of period-eight spin stripes.

Figure 4

(Color online) Spectral weight along the three cuts indicated by arrows in Figs. 2a, 2c. The top row (a1,a2,a3) [bottom row (c1,c2,c3)] corresponds to the parameters used in Figs. 2a [Fig. 2c]. The band in (a1) [along arrow 1 in Fig. 2a] is split by the local pair potential ${\Delta }_d=0.25t$ giving a pseudogap of similar magnitude in the antinodal region as seen in (c1) (see also Fig. 7). The cuts in the nodal region (a2,a3,c2,c3) [along arrows 2 and 3 in Fig. 2a] are less affected by the pair potential and there is clearly (see, e.g., c3) a band dispersing through the Fermi energy giving rise to the nodal hole pocket. Note the broadening as a signature of the pseudogap energy scale in near-nodal dispersion (c2). Energy $\omega$ in units of $t$ and with respect to the Fermi energy.

Figure 5

(Color online) Fixed momentum cuts of spectral-function intensity from Figs. 4c1–4c3.

Figure 6

(Color online) The same nodal dispersion as in Figs. 4a2 and 4c2 plotted over a larger energy range. One sees a small flattening of the low-energy dispersion and a broadened high-energy dispersion in c2 as compared to a2 with an abrupt change as a function of energy.

Figure 7

Density of states without (dashed) and with (solid) local pair potential for the same parameters as in Figs. 2a, 2c, respectively. The magnitude of the local pair potential ${\Delta }_d=0.25t$ is clearly reflected in suppression of the low-energy DOS below a pseudogap scale ${\Delta }_0\approx 0.25t$ and is also seen to correspond to the splitting of the antinodal band [Fig. 4c1] around the Fermi energy.

Figure 8

(Color online) Nodal (solid), antinodal (dashed), and total (dotted) low-energy DOS. We find an approximately particle-hole symmetric antinodal pseudogap and a DOS consistent with a band dispersing through the Fermi energy in the nodal region. The nodal weight has a minimum at the edge of the nodal band where the pocket closes. Inset shows the spectral weight [cf. Fig. 2c] at the Fermi energy and the shaded (white) regions of the BZ are integrated over to get the antinodal (nodal) DOS.

Figure 9

(Color online) One-dimensional (direction transverse to stripes) IPR as defined in Eq. (7) versus energy of quasiparticle excitations $E_{\gamma }$ for the same parameters as in Fig. 2c with system size $N_x=200$ (small points) and $N_x=320$ (large points). A large fraction of the states near the pseudogap energy ${\Delta }_0\sim {\Delta }_d=0.25t$ are strongly localized in contrast to states near the Fermi energy. Inset is the corresponding IPR for the same parameters but with the same pairing phase on all stripes resulting in all states being extended Bloch states.

Figure 10

(Color online) Density of states (curves offset for clarity) for different magnitudes of homogeneous $d$-wave pair potentials ${\Delta }_h=0$ (solid), ${\Delta }_h=0.05t$ (dashed), ${\Delta }_h=0.1t$ (dotted). The pseudogap energy is ${\Delta }_0$ and the energy ${\Delta }_1$ signifies the peak from the gapped nodal pocket in the superconducting state. The other parameters are the same as in Fig. 2c: $m=t/3$, ${\Delta }_d=0.25t$, and $\mu =-t$. The thin line shows as comparison the DOS for a system with only stripe order $\left(m=t/3,{\Delta }_d=0,\mu =-t\right)$ and homogeneous $d$-wave potential ${\Delta }_h=0.15t$.

Figure 11

Local density of states in the superconducting state on (solid line) and off (dashed line) a charge stripe. The parameters used for this curve are similar to the dotted line in Fig. 10. The overall collapse of these two curves at low-energy results in real-space homogeneity at low tunneling bias whereas the difference in LDOS at high energy will show up as stripes in the real-space STM field of view.

Figure 12

(Color online) Zero-momentum component of the pair density for parameters identical to the dashed curve in Fig. 10; $m=t/3$, ${\Delta }_h=0.05t$, ${\Delta }_d=0.25t$, and $\mu =-t$. The shading indicates the deviation from zero. The weight is along the tight-binding Fermi surface with highest intensity on the nodal pocket of the pseudogap state. The real part is plotted, there is also a smaller imaginary part (not shown) from the phase-disordered on-stripe pairing.

Figure 13

(Color online) Antinodal spectral weight cut (1 in Fig. 2), for (a) bare-band structure, $m={\Delta }_d={\Delta }_h=0$, (b) pseudogap state $m=t/3$ and ${\Delta }_d=t/4$, and (c) superconducting state $m=t/3$, ${\Delta }_d=t/4$, and ${\Delta }_h=0.1t$ as discussed in the text. [(b) is identical to Fig. 4c1.]