Mobile impurity approach to the optical conductivity in the Hubbard chain

We consider the optical conductivity in the one-dimensional Hubbard model in the metallic phase close to half-filling. In this regime, most of the spectral weight is located at frequencies above an energy scale $E_{\mathrm{opt}}$ that tends towards the optical gap in the Mott insulating phase for vanishing doping. Using the Bethe ansatz, we relate $E_{\mathrm{opt}}$ to thresholds of particular kinds of excitations in the Hubbard model. We then employ a mobile impurity model to analyze the optical conductivity for frequencies slightly above these thresholds. This entails generalizing mobile impurity models to excited states that are not the highest weight with regards to the SU(2) symmetries of the Hubbard chain, and that occur at a maximum of the impurity dispersion.

Figure 1

Configuration of the integers for the ground state, explicit numbers given are for $L=16,N_{\mathrm{GS}}=10$, and $M_{\mathrm{GS}}=5$.

Figure 2

Configuration of the integers for the particle-hole excitation above the ground state, explicit numbers given are for $L=16,N_{\mathrm{GS}}=10$, and $M_{\mathrm{GS}}=5$.

Figure 3

Particle-hole excitation continuum above the ground state.

Figure 4

Configuration of the integers for the $k\text{−}\mathrm{\Lambda }$ string excited state.

Figure 5

$k\text{−}\mathrm{\Lambda }$ string dispersion for various $U$ and $n$, and particle-hole excitation continuum above this for $U=8$ and $n=0.8$. For small momenta, the $k\text{−}\mathrm{\Lambda }$ string dispersion marks the lower edge of a continuum described by additional excitations, e.g., particle-hole excitations in the charge sector.

Figure 6

Integer configuration for the particle-hole excitation, explicit numbers for $L=16,N_{\mathrm{GS}}=10$, and $M_{\mathrm{GS}}=5$.

Figure 7

Continuum for particle-hole excitation with momentum shifted for clarity.

Figure 8

Integer configuration for the particle-particle excitation, explicit numbers for $L=16,N_{\mathrm{GS}}=10$, and $M_{\mathrm{GS}}=5$.

Figure 9

Continuum for particle-particle excitation with momentum shifted for clarity.

Figure 10

Integer configuration for two hole excited state, explicit numbers for $L=16,N_{\mathrm{GS}}=10$, and $M_{\mathrm{GS}}=5$.

Figure 11

Continuum for two hole excitation with momentum shifted for clarity.

Figure 12

The continuum of lowest-lying excitations of the Hubbard model involving only the charge sector for $n=0.8,U=10$. As the optical conductivity is defined at zero momentum, only the features encountered at $P=0$ are relevant. The $k\text{−}\mathrm{\Lambda }$ string dispersion defines the lower edge of a continuum of excitations involving the $k\text{−}\mathrm{\Lambda }$ string. The contributions to ${\sigma }^{\left(2\right)}\left(\omega \right)$ are shifted by $-2\mu$, in accordance with the spectral representation (18).

Figure 13

The threshold of the $k\text{−}\mathrm{\Lambda }$ string ${\varepsilon }_{k\mathrm{\Lambda }}\left(0\right)$ is shown for various $U$ and $n$.

Figure 14

Optical conductivity (103) for $U=10,n=5/6$ and several values of the cutoff $\mathrm{\Lambda }$. The different curves have been normalized such that ${\sigma }_1({\varepsilon }_{k\mathrm{\Lambda }}+0.5){|}_{k\mathrm{\Lambda }}=1$. For comparison, we show the power-law behavior (106), valid for $\omega \to {\varepsilon }_{k\mathrm{\Lambda }}$. We see that the power law in fact provides a good approximation over the entire range of comparison.

Figure 15

Value of the exponent $\gamma$ in Eq. (106) characterizing the power-law behavior of ${\sigma }_1\left(\omega \right)$ just above the pseudogap, for several values of $U$ and $n$.

Figure 16

Comparison of deconvolved DMRG data with the onset as predicted in Eq. (103), varying the offset and overall scale factor and choosing $\mathrm{\Lambda }=(1-n)\pi /10.$

Figure 17

Contribution to onset of ${\sigma }_1^{\left(2\right)}\left(\omega \right)$ from particle-hole excitation in Eq. (F16) for $U=10$ and $n=5/6$.