Comparison of the exact ground state energy89 with the GA and method for the half-filled Hubbard model on a system.
Magnetic excitations at and as a function of for a half-filled cluster: (solid line), (dashed line), and exact diagonalization (filled circles and filled squares). The exact diagonalization results for the excitation at are also reported (unfilled squares).
Sketch of bond- and site-centered stripe structures. Only Cu sites are shown. The dashed lines indicate the antiphase domain boundary.
Cross section of charge- and spin densities for BC and SC stripes in the three-band model. Charge (hole) densities are shown as differences between the stripe solution and the homogeneous antiferromagnet. The lowest panels report the double occupancies on Cu.
Electron mean-field bands for the BC stripe of Fig. 4 measured from the chemical potential. Left (right) panel is in the direction perpendicular (parallel) to the stripe. We also plot the insulating charge transfer gap at momentum measured from the same reference energy (dots).
as a function of (a) and (b) for vertical BC stripes. Sizes are for and for . In (a) we also show the result for (unfilled circles). In (b) we also show the result for site-centered stripes (labeled SC, unfilled symbols) for and one self-trapped hole or “electronic polaron” at . The inset in (b) reports the incommensurability as obtained from the present calculation (line) compared with experimental data from Ref. 12.
Optical conductivity for stripes with the electric field applied perpendicular and parallel to the stripes. computed without the addition of RPA corrections. The corresponding excitations are also indicated in the band structure in Fig. 8 (a); computed within the approach (b).
Electron mean-field bands measured from the chemical potential for momentum in the direction of the stripe (, ). The arrows are the lowest-energy dipole-allowed mean-field transitions in a cluster, labeled by energy and polarization. Notice that the polarization is perpendicular to the stripe. We also plot the energies of the AF insulating bands at momentum (filled dots) measured from the same reference energy.
Optical conductivity labeled by doping, system size and, in the case of stripes, interstripe distance. The units of conductivity are given by with a background dielectric constant (see Ref. 96 and caption of Fig. 12). The curve labeled corresponds to the single-hole solution. For larger dopings the figure is an average over electric field directions parallel and perpendicular to the BC stripes. The inset shows the low-energy spectra excluding the Drude component. We used a Lorentzian broadening of .
Sketch of transition charges (black arrows) and transition currents (gray arrows) for a BC stripe (indicated by the dashed lines) when the collective MIR mode is excited. The transition charges provide a snapshot of the charge oscillation in the collective mode. The black arrows indicate charge increase (up) or decrease (down). The horizontal arrows indicate the associated current fluctuation. Cu and O sites are symbolized by unfilled and filled circles, respectively.
Mean-field energy per stripe cell (8 Cu’s) as a function of the collective phase of the CDW for various doping levels and for stripes in a system.
Experimental67 and theoretical integrated optical conductivity spectral weight integrated up to vs. . The spectral weight is converted to an effective number of electrons as in Ref. 67. The background dielectric constant is our only free parameter and was adjusted to achieve overall agreement of the theoretical and experimental intensities. We also show the computed Drude and regular contributions in each direction. The dashed lines indicate the region of metastability of the solution. The points not joined by lines correspond to the solution, which becomes more stable in that region.
[ Eq. (28)] vs. filling fraction for BC stripes for different values of the next-nearest neighbor hopping . Symbols: calculated energy on a lattice; solid lines: fits as described in the text.
Energy and wave vector dependence of magnetic excitations in the half-filled system as obtained from the approach for and . Squares and triangles correspond to data points from INS experiments on by Coldea et al. 118
Dispersion of magnetic excitations along (left) and perpendicular (right) to an array of BC stripes. We also show the experimental data from Ref. 70.
integrated on the AF magnetic Brillouin zone140 for BC and SC stripes together with the experimental data.70 We have fixed the polarization and intensity renormalization factor as . For details see Ref. 139. For the meaning of the “error” bars see Ref. 70.
Constant energy scans of for BC stripes oriented along the direction. The size of the panels represent the full Brillouin zone. The title indicates the energy of the scan. The intensities are averaged in an energy window for energies and for higher frequencies.
Constant-frequency scans of for BC stripes convolved with a Gaussian . The title indicates the energy or the energy window over which intensities are averaged. In order to compare with the experimental data of Ref. 70 the panels show the average of both horizontal and vertical stripes with the coordinate system rotated by 45° with wave-vector units of .
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