The optical conductivity ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}) of the two-dimensional Hubbard model on finite clusters of 4\ifmmode\times\else\texttimes\fi{}4 and \ensuremath{\surd}10 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}10 sites is reported. Experimental features found in the high- ${\mathit{T}}_{\mathit{c}}$ cuprate superconductors can be qualitatively reproduced by this model including the presence of mid-infrared spectral weight, a Drude peak whose intensity grows with doping and a total spectral weight that varies slowly as the system is doped away from half filling. We find that to reproduce qualitatively the main experimental features of ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}), we need to work with a coupling constant of the order of the bandwidth, i.e., U/t\ensuremath{\sim}8--10. Results for the negative-U Hubbard model are included for comparison.The optical conductivity ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}) of the two-dimensional Hubbard model on finite clusters of 4\ifmmode\times\else\texttimes\fi{}4 and \ensuremath{\surd}10 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}10 sites is reported. Experimental features found in the high-${\mathit{T}}_{\mathit{c}}$ cuprate superconductors can be qualitatively reproduced by this model including the presence of mid-infrared spectral weight, a Drude peak whose intensity grows with doping and a total spectral weight that varies slowly as the system is doped away from half filling. We find that to reproduce qualitatively the main experimental features of ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}), we need to work with a coupling constant of the order of the bandwidth, i.e., U/t\ensuremath{\sim}8--10. Results for the negative-U Hubbard model are included for comparison.

Physical review. B, Condensed matterMay 1, 199248 citations

Method of spectral densities for the Hubbard model

Wolfgang Nolting

Physics Letters AMar 1, 19726 citations

Quantum-classical simulation of two-site dynamical mean-field theory on noisy quantum hardware

Trevor Keen, Thomas Maier, Steven Johnston +1 more

Abstract We report on a quantum–classical simulation of the single-band Hubbard model using two-site dynamical mean-field theory (DMFT). Our approach uses IBM’s superconducting qubit chip to compute the zero-temperature impurity Green’s function in the time domain and a classical computer to fit the measured Green’s functions and extract their frequency domain parameters. We find that the quantum circuit synthesis (Trotter) and hardware errors lead to incorrect frequency estimates, and subsequently to an inaccurate quasiparticle weight when calculated from the frequency derivative of the self-energy. These errors produce incorrect hybridization parameters that prevent the DMFT algorithm from converging to the correct self-consistent solution. To avoid this pitfall, we compute the quasiparticle weight by integrating the quasiparticle peaks in the spectral function. This method is much less sensitive to Trotter errors and allows the algorithm to converge to self-consistency for a half-filled Mott insulating system after applying quantum error mitigation techniques to the quantum simulation data.Abstract We report on a quantum–classical simulation of the single-band Hubbard model using two-site dynamical mean-field theory (DMFT). Our approach uses IBM’s superconducting qubit chip to compute the zero-temperature impurity Green’s function in the time domain and a classical computer to fit the measured Green’s functions and extract their frequency domain parameters. We find that the quantum circuit synthesis (Trotter) and hardware errors lead to incorrect frequency estimates, and subsequently to an inaccurate quasiparticle weight when calculated from the frequency derivative of the self-energy. These errors produce incorrect hybridization parameters that prevent the DMFT algorithm from converging to the correct self-consistent solution. To avoid this pitfall, we compute the quasiparticle weight by integrating the quasiparticle peaks in the spectral function. This method is much less sensitive to Trotter errors and allows the algorithm to converge to self-consistency for a half-filled Mott insulating system after applying quantum error mitigation techniques to the quantum simulation data.

arXiv:1910.09512Quantum Science and TechnologyMar 6, 202048 citations

Spectral properties of the Hubbard model

Tsutomu Momoi

Physica B Condensed MatterFeb 1, 19951 citation

Optical, magnetic, and single-particle excitations in the multiband Hubbard model for cuprate superconductors

J. Wagner, Werner R. Hanke, Douglas J. Scalapino

On the basis of exact diagonalizations, a comparative study of two-particle optical and magnetic, as well as single-particle, excitations is presented for a two-dimensional (2D) multiorbital Hubbard model. For reasonable parameter sets appropriate for the cuprate superconductors, the single-particle excitations display strongly correlated states related to the Zhang-Rice Cu-O singlet construction. These states define the gap (to the upper Hubbard band) at half-filling and become partially occupied by doping holes in our 2\ifmmode\times\else\texttimes\fi{}2 unit-cell system. The optical results, which are the first quantitative calculations performed for realistic parameters of the three-band Hubbard model, clearly show three allowed optical transitions: (i) itinerant motion of the Cu-O singlets, having (for doping concentrations x\ensuremath{\ne}0) a spectral Drude distribution around \ensuremath{\omega}=0 with spectral weight proportional to x; (ii) unbinding of the O hole from the Cu spin in the singlet. This gives, in particular, a strong absorption peak due to singlet\ensuremath{\rightarrow}nonbonding oxygen transitions, again with relative weight \ensuremath{\sim}x.It is roughly centered at \ensuremath{\omega}\ensuremath{\sim} ${\mathit{J}}_{\mathrm{Kondo}}$ , where ${\mathit{J}}_{\mathrm{Kondo}}$ is the Kondo coupling and \ensuremath{\Delta} the bare Cu-O charge-transfer energy. It is this singlet unbinding that results in the by far dominant absorption structure between Drude and higher-energy Cu-O transitions and not the often discussed ``mid-infrared'' absorption due to transitions between different singlet states. (iii) Cu-O charge-transfer processes at energy \ensuremath{\sim}\ensuremath{\Delta}+ ${\mathit{U}}_{\mathit{p}\mathit{d}}$ . They show a pronounced excitonic effect due to the p-d interaction ${\mathit{U}}_{\mathit{p}\mathit{d}}$ and have a reduced spectral weight shifted to higher energies for increased dopings. Findings (i)--(iii) are in general accordance with recent experimental data. Our study of the low-energy absorption is complemented with a numerical scaling analysis of the Drude weight in 1D, where, in particular, we find an interesting violation of Lenz's law for 4n-site Hubbard rings. Finally, the magnetic structure factor is calculated for the 2D case. For finite doping it contains a peak at 2 ${\mathit{J}}_{\mathrm{Kondo}}$ , which should be detectable in experiment.On the basis of exact diagonalizations, a comparative study of two-particle optical and magnetic, as well as single-particle, excitations is presented for a two-dimensional (2D) multiorbital Hubbard model. For reasonable parameter sets appropriate for the cuprate superconductors, the single-particle excitations display strongly correlated states related to the Zhang-Rice Cu-O singlet construction. These states define the gap (to the upper Hubbard band) at half-filling and become partially occupied by doping holes in our 2\ifmmode\times\else\texttimes\fi{}2 unit-cell system. The optical results, which are the first quantitative calculations performed for realistic parameters of the three-band Hubbard model, clearly show three allowed optical transitions: (i) itinerant motion of the Cu-O singlets, having (for doping concentrations x\ensuremath{\ne}0) a spectral Drude distribution around \ensuremath{\omega}=0 with spectral weight proportional to x; (ii) unbinding of the O hole from the Cu spin in the singlet. This gives, in particular, a strong absorption peak due to singlet\ensuremath{\rightarrow}nonbonding oxygen transitions, again with relative weight \ensuremath{\sim}x.It is roughly centered at \ensuremath{\omega}\ensuremath{\sim}${\mathit{J}}_{\mathrm{Kondo}}$, where ${\mathit{J}}_{\mathrm{Kondo}}$ is the Kondo coupling and \ensuremath{\Delta} the bare Cu-O charge-transfer energy. It is this singlet unbinding that results in the by far dominant absorption structure between Drude and higher-energy Cu-O transitions and not the often discussed ``mid-infrared'' absorption due to transitions between different singlet states. (iii) Cu-O charge-transfer processes at energy \ensuremath{\sim}\ensuremath{\Delta}+${\mathit{U}}_{\mathit{p}\mathit{d}}$. They show a pronounced excitonic effect due to the p-d interaction ${\mathit{U}}_{\mathit{p}\mathit{d}}$ and have a reduced spectral weight shifted to higher energies for increased dopings. Findings (i)--(iii) are in general accordance with recent experimental data. Our study of the low-energy absorption is complemented with a numerical scaling analysis of the Drude weight in 1D, where, in particular, we find an interesting violation of Lenz's law for 4n-site Hubbard rings. Finally, the magnetic structure factor is calculated for the 2D case. For finite doping it contains a peak at 2${\mathit{J}}_{\mathrm{Kondo}}$, which should be detectable in experiment.

Physical review. B, Condensed matterMay 1, 199197 citations

{\mathit{T}}_{\mathit{c}}

cuprate superconductors can be qualitatively reproduced by this model including the presence of mid-infrared spectral weight, a Drude peak whose intensity grows with doping and a total spectral weight that varies slowly as the system is doped away from half filling. We find that to reproduce qualitatively the main experimental features of ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}), we need to work with a coupling constant of the order of the bandwidth, i.e., U/t\ensuremath{\sim}8--10. Results for the negative-U Hubbard model are included for comparison.

{\mathit{J}}_{\mathrm{Kondo}}

, where

{\mathit{J}}_{\mathrm{Kondo}}

is the Kondo coupling and \ensuremath{\Delta} the bare Cu-O charge-transfer energy. It is this singlet unbinding that results in the by far dominant absorption structure between Drude and higher-energy Cu-O transitions and not the often discussed ``mid-infrared'' absorption due to transitions between different singlet states. (iii) Cu-O charge-transfer processes at energy \ensuremath{\sim}\ensuremath{\Delta}+

{\mathit{U}}_{\mathit{p}\mathit{d}}

. They show a pronounced excitonic effect due to the p-d interaction

{\mathit{U}}_{\mathit{p}\mathit{d}}

and have a reduced spectral weight shifted to higher energies for increased dopings. Findings (i)--(iii) are in general accordance with recent experimental data. Our study of the low-energy absorption is complemented with a numerical scaling analysis of the Drude weight in 1D, where, in particular, we find an interesting violation of Lenz's law for 4n-site Hubbard rings. Finally, the magnetic structure factor is calculated for the 2D case. For finite doping it contains a peak at 2

{\mathit{J}}_{\mathrm{Kondo}}

, which should be detectable in experiment.