Spin-polaron band structure and hole pockets in underdoped cuprates

We present a variational approach based on the string picture to analyze the internal structure and dispersion of spin polarons with different symmetries in an antiferromagnet. We then use this to discuss the properties of underdoped cuprate superconductor within the “doped insulator” picture. The theory explains the remnant Fermi surface for the undoped compounds, as well as hole pockets, Fermi arcs, high-energy pseudogap, and the midinfrared band in doped materials. Destructive interference between the phases of a photohole near $\Gamma$ and the internal phases of the Zhang-Rice singlet combined with our theory moreover explains the “waterfall” phenomenon.

Figure 1

(Color online) The mechanism of the string effect. Slanted crosses represent links with the contribution to the Ising part of the exchange energy higher by $J/2$ as compared to the Néel state. (b) A string state obtained by a single move of the hole created at site $i$. [(c) and (d)] States obtained, respectively, by two and three consecutive moves.

Figure 2

Dependence of the sign of a given path on the direction of the first move for the (b) $d_{x^2-y^2}$-wave SP, (c)$p_x$-wave SP, and (d) $p_y$-wave SP. (a) For completeness the schematic representation for the $s$-wave SP has also been shown.

Figure 3

(Color online) $\left(\text{a}\right)\to \left(\text{b}\right)$: a string of length 2 starting at $i$ is truncated to a string of length 0 at $j$ by the transverse part of the Heisenberg exchange. $\left(\text{c}\right)\to \left(\text{d}\right)$: a string of length 3 is reduced to one of length 1. $\left(\text{e}\right)\to \left(\text{f}\right)$: a bare hole at site $i$ (string of length 0) is transported to $j$ by $t^{\prime }$ hopping.

Figure 4

(Color online) $\left(\text{a}\right)\to \left(\text{b}\right)$: creation of a bare hole at site $i$ from the Néel state. $\left(\text{c}\right)\to \left(\text{d}\right)$: creation of a string of length 1 starting at $i$ from the Néel $\text{state}+\text{quantum}$ fluctuation. $\left(\text{e}\right)\to \left(\text{f}\right)$: creation of a string of length 2 starting at $i$ from the Néel $\text{state}+\text{quantum}$ fluctuation.

Figure 5

(Color online) Band structure obtained by solving the eigenvalue problem [Eq. (7)]. Also shown is the dispersion of bands observed by Ronning et al. (Ref. 57) in ${\text{Ca}}_2{\text{CuO}}_2{\text{Cl}}_2$ (circles and triangles). A sample Fermi level of a doped system has been chosen as the zero of energy. Vertical arrows label possible optical transitions of a hole at the bottom of the band.

Figure 6

Photoemission spectrum for the half-filled model corresponding to the band structure [Eq. (5)]. In the left panel the spectral weight is computed using only the quantum fluctuation correction [Eq. (15)]; in the right panel the charge-fluctuation correction [Eq. (17)] was used as well.

Figure 7

Top: Fermi surface obtained for 10% hole concentration by rigid filling of the lowest SP band. Bottom: spectral weight of the corresponding SP band along the Fermi surface as a function of the angle $\alpha$.

Figure 8

Top: Fermi surface obtained for 10% hole concentration by rigid filling of the lowest SP band and a contour obtained by extending the “inner part” of the pocket to a free-electron-like Fermi surface. Bottom: energy relative to Fermi energy of the SP band forming the pocket along the free-electron-like Fermi surface as a function of Fermi-surface angle.

Figure 9

Spectral density for hole creation at half-filling along high-symmetry lines in the Brillouin zone. Calculated with (right) and without (left) the contribution from charge fluctuations.

Figure 10

Interpretation of the waterfall phenomenon. Around $\Gamma$ photoholes on oxygen do not couple to the ZRS whence all $t\text{−}J$ derived states have no spectral weight. Photoholes created with this momentum on oxygen instead propagate as (nearly) free-electron states which do not hybridize with the correlated $\text{Cu}d$ orbitals. Moving away from $\Gamma$ the $t\text{−}J$ bands become visible but lose spectral weight as well (dashed potions). The nearly free-electron states cease to exist because they now have appreciable hybridization with the correlated $\text{Cu}d$ orbitals. The higher lying SP bands thus are visible only in a small window in $\mathbf{k}$ space, i.e., the waterfalls.

Figure 11

(Color online) Application of the current operator $j_y$ to a bare hole in the Néel state (a) creates two strings of length 1 [(b) and (c)] with opposite signs.

Figure 12

Optical conductivity at 10% hole concentration.

Figure 13

(Color online) [(a) and (b)] A process which gives rise to the shift of the $s$-wave SP to a second NN site. [(c) and (d)] A process which generates corrections to eigenenergies $E_1$ and $E_1^{\prime }$ of SPs.

Figure 14

[(a) and (b)] A different representation of string states depicted in Figs. 13c, 13d, respectively. (c) A string state contributing to a process which involves hopping to second NN sites. That process gives rise to SP shifts along plaquette diagonals. (d) A string state contributing to a process which involves hopping to NN sites.

Figure 15

[(a) and (b)] A low-order process which involves the action of the transverse part of the exchange term and gives rise to the hybridization of SPs with different symmetry. [(c) and (d)] String states involved in a process mediated by the $XY$ part of the exchange term and resulting in a correction to the eigenenergy of the $d$-wave SP.

Figure 16

[(a) and (b)] Spuriously different string states which are actually identical, which gives rise to the overlap between SPs created at sites $n$ and $l$.