Theory of the Optical Conductivity of Spin Liquid States in One-Dimensional Mott Insulators

The low-energy dynamical optical response of dimerized and undimerized spin liquid states in a one-dimensional charge transfer Mott insulator is theoretically studied. An exact analysis is given for the low-energy asymptotic behavior using conformal field theory for the undimerized state. In the dimerized state, the infrared absorption due to the bound state of two solitons, i.e., the breather mode, is predicted with an accurate estimate for its oscillator strength, offering a way to detect experimentally the excited singlet state. The effects of external magnetic fields are also discussed.

Figure 1

Optical conductivity $\sigma (\omega )$ of 1D Mott insulators. We take the unit $J=\mathcal{C}=a=1$. (a) $\sigma (\omega )$ of the $XY$ spin chain for $g{u}_0=0.31$ (dimerized) and that for $g{u}_0=0$ (undimerized; $K=2$) at $T=\frac{1}{4\pi }$. Note that $\eta =\pi {\eta }_1/\sqrt{2}$ when $K=2$. (b) Low-energy behavior of $\sigma (\omega )$ of the undimerized $XXZ$ chain for $K=1$, $\frac{4}{3}$, and 2 at $T=\frac{1}{4\pi }$, which is obtained from Eq. (9) by putting $q=0$. The spinon velocity is given by $v=\pi Ja/2=\pi /2$ when $K=1$. Because of the interaction effect, $\sigma (\omega )$ is enhanced at low energies as $K$ is decreased. Note that CFT description is valid only if $\omega$, $T\ll J=1$. In the limit of $\omega \gg T$, $\sigma (\omega )$ for $K=1$ approaches to the constant value $2\pi {\eta }_1^2$. (c) Low-energy behavior of $\sigma (\omega )$ of the dimerized Heisenberg chain with $M=0.2$ and $v=\pi /2$ at $T=0$. Delta function peak is located at $\omega =\sqrt{3}Mv$. The $s\mathrm{\text{−}}\overline{s}$ continuum starts at $\omega =2Mv$. Comparing (b) and (c), the continuum intensity is transferred to the breather peak.

Figure 2

Optical conductivity $\sigma (\omega )$ of the field-induced TL liquid with magnetization $M_z=0.1$ for $K=1$, $\frac{4}{3}$, and 2 at $T=\frac{1}{8\pi }$. We use the unit $v=a=1$. The location of ${\omega }_c=2\pi M_{z}v/a$ is indicated. The imaginary part of ${\tilde{G}}^R(q,\omega )$ is nonzero in the shaded region of the right figure. Note again that CFT description is valid only if $\omega$, $T\ll v=1$.