Pseudogap opening and formation of Fermi arcs as an orbital-selective Mott transition in momentum space

We present an approach to the normal state of cuprate superconductors which is based on a minimal cluster extension of dynamical mean-field theory. Our approach is based on an effective two-impurity model embedded in a self-consistent bath. The two degrees of freedom of this effective model can be associated to the nodal and antinodal regions of momentum space. We find a metal-insulator transition which is selective in momentum space. At low doping, quasiparticles are destroyed in the antinodal region, while they remain protected in the nodal region, leading to the formation of apparent Fermi arcs. We compare our results to tunneling and angular-resolved photoemission experiments on cuprates. At very low energy, a simple description of this transition can be given using rotationally invariant slave bosons.

Figure 1

(Color online) The Brillouin zone is divided into two patches $P_{+}$ (inside the inner blue square) and $P_{-}$ (between the two squares). Dashed line is the free $\left(U=0\right)$ Fermi surface at $\delta =0.1$ for $t^{\prime }/t=-0.3$. $P_{+}$ (respectively, $P_{-}$) encloses the nodal (respectively, antinodal) region.

Figure 2

(Color online) Partial density of states of the two patches $P_{+}$ (solid blue curve with circles) and $P_{-}$ (solid red curve with squares) and total density of states (dashed curve); $t^{\prime }/t=-0.3$.

Figure 3

(Color online) Real part of the even and odd self-energies at $\omega =0$, extrapolated from the results for $\beta =200$, $U=2.5$. Solid lines are the slave-boson (RISB) solution, while the symbols are the CTQMC results. The dotted line is the lower-band edge of the $P_{-}$ patch represented in Fig. 2.

Figure 4

(Color online) Averages occupancies of the even and odd orbitals, obtained with RISB method (dashed lines) and CTQMC (solid lines with circles). $\beta =200$, $U=2.5$.

Figure 5

(Color online) Quasiparticle residues of the even and odd orbitals obtained with RISB method (dashed lines) and CTQMC (solid lines with circles). $\beta =200$, $U=2.5$.

Figure 6

(Color online) Real parts of the self-energies at $\omega =0$, extrapolated from CTQMC results at $\beta =200$, as a function of $U$ at a fixed doping $\delta =8\%$. Dotted line is the lower-band edge of the $P_{-}$ patch represented in Fig. 2.

Figure 7

(Color online) Spectral function $A_{-}\left(\omega \right)$ for the odd orbital, obtained with Padé approximants (see Appendix ), for various dopings at $\beta =200$. A vertical shift of 0.3 has been added between the curves for clarity. Inset: Zoom of the same curves at low frequencies (no shift added).

Figure 8

(Color online) Gap to positive coherent excitations in the odd orbital; obtained from Eq. (28).

Figure 9

(Color online) Spectral function $A_{+}\left(\omega \right)$ for the even orbital, obtained with Padé approximants (see Appendix ), for various dopings at $\beta =200$. A vertical shift of 0.2 has been added between the curves for clarity. Inset: Zoom of the same curves at low frequencies (no shift added).

Figure 10

(Color online) Total spectral function $A_{\text{tot}}\left(\omega \right)$ for various temperatures at $\delta =0.08$. A vertical shift of 0.3 has been added between the curves for clarity.

Figure 11

(Color online) (a) STM curve for various temperatures at $\delta =0.08$, in arbitrary units. A vertical shift has been added between the curves for clarity. (b) DOS measured by STM experiments on Bi2212 with $T_c=83\text{K}$. Figure reprinted with permission from Ref. 51 [Fig. 22b]. Copyright 2007 by the American Physical Society.

Figure 12

(Color online) (a) Imaginary part of the self-energy ${\Sigma }_{+}\left(\omega \right)$ for the even orbital. A vertical shift of 0.1 has been added between the curves for clarity. (b) Zoom over the low-frequency region (no shift added).

Figure 13

(Color online) (a) Imaginary part of the self-energy ${\Sigma }_{-}\left(\omega \right)$ for the odd orbital. A vertical shift of 0.2 has been added between the curves for clarity. (b) Zoom over the low-frequency region (no shift added).

Figure 14

(Color online) Doping dependences of (a) the real part of ${\Sigma }_{\pm }\left(0\right)$, (b) the even- and odd-orbital occupations, and (c) $\mu -{\Sigma }_{-}\left(0\right)$. Arrows in (c) indicate the critical doping where the odd band becomes insulating. The results are obtained for different patching schemes as shown in (d) using the RISB method.

Figure 15

(Color online) Statistical weights of the various dimer cluster eigenstates (labeled as in Table ). $S$ is the intradimer singlet, $1+$ the (spin-degenerate) state with one electron in the even orbital, $E$ the empty state, and $T$ the intradimer triplet. $\beta =200$.

Figure 16

(Color online) Kondo to RKKY crossover as a function of the Coulomb repulsion $U$ at the impurities at half filling. (a) Quasiparticle weights. (b) Even and odd-orbital occupations. (c) Real part of the self-energy. (d) Boson amplitudes for the intradimer singlet (S), the intradimer triplet (T), and the spin-degenerate doublets with one electron and even symmetry $\left(1+\right)$, and three electrons with odd symmetry $\left(3-\right)$.

Figure 17

(Color online) Kondo to RKKY crossover for the two-impurity Anderson model as described by RISB as a function of the impurity level doping ${\delta }^{\ast }=1-n$. Parameters are $U=7D$ and $\overline{t}=0.25D$. (a) Quasiparticle weights. (b) Even- and odd-orbital occupations. (c) Real part of the self-energy. (d) Boson amplitudes for the intradimer singlet (S), the intradimer triplet (T), and the spin-degenerate doublet with one electron and even symmetry $\left(1+\right)$.

Figure 18

(Color online) ${\Delta }_{+}\left(\omega \right)$ for the even orbital obtained with Padé approximants (see Appendix ) for various dopings at $\beta =200$. A vertical shift has been added between the curves for clarity.

Figure 19

(Color online) ${\Delta }_{-}\left(\omega \right)$ for the odd orbital obtained with Padé approximants (see Appendix ) for various dopings at $\beta =200$. A vertical shift has been added between the curves for clarity

Figure 20

(Color online) Real and imaginary parts of dimer (solid lines) and plaquette (symbols) self-energies at $k=\left(0,0\right)$ (black solid line and red diamond) and $k=\left(\pi ,\pi \right)$ (orange solid line and blue diamond) for $\beta =200$ and $\delta =0.08$ (a) and $\delta =0.16$ (b).

Figure 21

(Color online) Comparison between the reconstructed ${\Sigma }_{\left(0,\pi \right)}$ using $M$ interpolation (red solid line) and $\Sigma$ interpolation (blue dashed line) in the dimer and the cluster self-energy of the plaquette calculation at $k=\left(0,\pi \right)$ (black diamonds) for $\delta =0.08$ (a) and $\delta =0.16$ (b). $\beta =200$.

Figure 22

(Color online) Intensity maps of the spectral function $A\left(\mathbf{k},0\right)$ for different doping levels obtained with $M$ interpolation.

Figure 23

(Color online) (a) Normalized intensity $A\left(\varphi ,0\right)/A\left(0,0\right)$ along the Fermi surface vs the angle to the diagonal of the Brillouin zone in degrees ($\varphi =0$ is the node, $\varphi =\pm 45$ the antinode). The nodal intensity $A\left(0,0\right)$ is 0.045 for $\delta =6\%$, 1.66 for $\delta =10\%$, and 4.61 for $\delta =14\%$. $\beta =200$. (b) Angular dependence of the spectral weight along the Fermi surface in ${\text{Ca}}_{2-x}{\text{Na}}_x{\text{CuO}}_2{\text{Cl}}_2$ at $x=0.05$ (black diamonds), $x=0.10$ (red squares), and $x=0.12$ (blue circles) along with data from ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ for $x=0.05$ and $x=0.10$ (open symbols). Figure reprinted from Ref. 64 [Fig. 3b]. Copyright 2005 by Science.

Figure 24

(Color online) Even self-energy on the real frequency axis as obtained using Padé approximants (solid line) and by maximum entropy (dashed line). $\delta =8\%$ and $\beta =200$.

Figure 25

(Color online) Even and odd Green’s functions $G_{\pm }\left(i{\omega }_n\right)$ on the Matsubara axis for various dopings at $\beta =200$.

Figure 26

(Color online) Even and odd self-energies ${\Sigma }_{\pm }\left(i{\omega }_n\right)$ on the Matsubara axis for various dopings at $\beta =200$.

Figure 27

(Color online) Even and odd hybridization functions ${\Delta }_{\pm }\left(i{\omega }_n\right)$ on the Matsubara axis for various dopings at $\beta =200$.