SU(2) approach to the pseudogap phase of high-temperature superconductors: Electronic spectral functions

We use an SU(2) mean-field theory approach with input from variational wave functions of the $t\text{−}J$ model to study the electronic spectra in the pseudogap phase of cuprates. In our model, the intermediate-temperature state of underdoped cuprates is realized by classical fluctuations of the order parameter between the $d$-wave superconductor and the staggered-flux state. The spectral functions of the pure and the averaged states are computed and analyzed. Our model predicts a photoemission spectrum with an asymmetric gap structure interpolating between the superconducting gap centered at the Fermi energy and the asymmetric staggered-flux gap. This asymmetry of the gap changes sign at the points where the Fermi surface crosses the diagonal $\left(\pi ,0\right)\text{−}\left(0,\pi \right)$.

Figure 1

Contour plot of the spectral function at the Fermi energy of the pure states, $A_k^{\theta }\left(0\right)$. Doping is 10% and we use a lifetime broadening $\Gamma =0.2\chi$. The dashed line represents the location of the Fermi surface (where the Green’s function changes sign; Ref. 53). From (a) to (d) we have $\text{cos}\theta =0,1/3,2/3,1$. Plot (a) shows the $d$-wave superconductor, (d) displays the pure staggered-flux state.

Figure 2

Scheme of the different regions of the Fermi surface. The SC and SF gaps open near the node ($N$ point; $\varphi =0$) where they are small and do not overlap. The Fermi surface appears as a gapless arc in region I. In region II, the two gaps start to overlap and form an effective gap which is shifted upward in energy (vertical arrows). The effective gap comes down in energy as we move toward the antinode in region II. Exactly at the SU(2) points on the diagonal $\left(0,\pi \right)\text{−}\left(\pi ,0\right)$, the effective gap is symmetric. Beyond the SU(2) points (region III), the midgap is shifted below the Fermi energy.

Figure 3

Schematic evolution of the spectra along a cut parallel to the nodal direction, inside the pocket (through region I in Fig. 2, e.g., cut $b$ in Fig. 6). The dot size is proportional to the spectral weight. From (a) to (d) we have $\text{cos}\theta =0,1/3,2/3,1$. (a) is the superconducting state, (d) is the staggered-flux state.

Figure 4

Same plot as in Fig. 3, but for a cut outside the pocket (through region II in Fig. 2, e.g., cut $c$ in Fig. 6).

Figure 5

Same plot as in Fig. 3, but for a cut outside the SU(2) point (through region III in Fig. 2, e.g., cut $d$ in Fig. 6).

Figure 6

Averaged spectral function at the Fermi energy, $A_{\mathit{k}}\left(0\right)$. Doping is 10% and we use a lifetime broadening $\Gamma =0.2\chi$. The dashed lines are the Fermi surfaces of the SF and SC states, respectively. The full spectra on the cuts $a$ to $d$ are given in Figs. 7, 8, 9, 10.

Figure 7

(Color online) Averaged spectral function along a cut parallel to the nodal direction in the Brillouin zone, cut $a$ in Fig. 6. The spectra are set off in $y$ direction by $k_x$ (in units of $\pi$). $k_x$ goes from 0.25 (lowest curve) to 0.55 (highest curve) in steps of 0.01. The curve at approximately $k_F$ is indicated on the right (red online). We use a lifetime broadening $\Gamma =0.12\chi$. The parameters used are $\Delta \left(\theta \right)=\sqrt{{\left(0.2\text{cos}\theta \right)}^2+{\left(0.25\text{sin}\theta \right)}^2}$ and doping is 10%. The energy $\omega$ is given in units of $2\chi \simeq 200\text{meV}$, the intensities are in arbitrary units.

Figure 8

(Color online) Same plot as in Fig. 7 but on cut $b$ in Fig. 6. $k_x$ goes from 0.15 to 0.45.

Figure 9

(Color online) Same plot as in Fig. 7 but on cut $c$ in Fig. 6. $k_x$ goes from 0.05 to 0.35. An asymmetric effective gap with midgap above the Fermi energy is formed.

Figure 10

(Color online) Same plot as in Fig. 7 but on cut $d$ in Fig. 6. $k_x$ goes from 0 to 0.3. An asymmetric effective gap with midgap below the Fermi energy is formed.