Excitonic optical spectra and energy structures in a one-dimensional Mott insulator demonstrated by applying a many-body Wannier functions method to a charge model

We applied a many-body Wannier functions method to theoretically calculate the excitonic optical conductivity spectrum and energy structure in a one-dimensional (1D) Mott insulator at absolute zero temperature with a large system size. Focusing on full charge fluctuations associated with holon and doublon pairs, we employ a charge model, which is interpreted as an effective model for investigating the photoexcitations of a 1D extended Hubbard model under a half-filling of the spin-charge separation. As a result, theoretical spectra with the appropriate broadenings qualitatively reproduce the recent experimental data of ET-${\mathrm{F}}_2\mathrm{TCNQ}$ at 294 K with and without a modulated electric field. Regarding the excitonic energy structure, we found that the excitons, particularly for even-parity, are weakly bound by many-body effects. This is also consistent with the fitting parameters reported in a recent experiment. Thus, the theoretical method presented in this paper is useful for understanding the physical roles of the charge fluctuations in many-body excited states of a 1D Mott insulator.

Figure 1

Schematics of bare bases $|r_M\rangle$ in Eq. (13) of the charge model with $N=6,M_{\max }=N/2=3$. A basis $|r_M\rangle$ contains $M$ pairs of a holon (H) and doublon (D) with the ground state of the Heisenberg model with $N-2M$ sites. Up (down) arrows denote up (down) spins. In addition, $r_{\mathrm{HD}}$ represents the relative distance between H and D, which is defined in the $M=1$ subspace. Therefore, we can choose an integer of $r_1=r_{\mathrm{HD}}\equiv p_1-q_1$ as one of the $r_1$ representations.

Figure 2

Optical conductivity spectra of $N=18,U=10T,V=0,\varepsilon =0$, and $\gamma =0.1T$ for a charge model with $M_{\max }=N/2$ (solid line) and half-filled 1D extended Hubbard model (dotted line). The spectra are computed using a continued fraction method [52].

Figure 3

Schematic representations of the MBWFs method with matrices for a charge model. (a) Unitary transformation, $V^{\mathrm{tr}p}$, from (Bloch) excited states (labeled by $\mu$) to MBWFs (labeled by $r$), where $p=+$ ($-$) denotes the even-parity (odd-parity). (b) Selection rule of $V^{\mathrm{tr}p}$, where $V^{\mathrm{tr}p}$ is required to localize MBWFs on the subspace of $M=1$ by satisfying $r=r_{\mathrm{HD}}$. (c) Selection of $V^{\mathrm{tr}p}$ in this study.

Figure 4

The probability weight in the $M=1$ subspace at $N=18$. Odd-parity Bloch states (a) and corresponding MBWFs (b). Even-parity Bloch states (c) and corresponding MBWFs (d). The filled-in circles are the results with full charge fluctuations ($M_{\max }=N/2$) and empty squares are the results of the HD model [54], as briefly illustrated in Appendix pp1. The solid lines are eye guidelines.

Figure 5

Expectation values of the charge-current operator at $N=18$ (filled-in circles). This size extension is applied through an extrapolation in the direction of increasing $r=r_{\mathrm{HD}}$. The filled-in circles and empty circle, which is one of the extrapolations, construct the expectation values for $N_{\mathrm{ex}}=20$. The inset shows schematics of MBWFs for $N=18$ and odd-parity. Each filled-in diagram localizes at approximately $r=r_{\mathrm{HD}}$ ($1\le r\le N/2-1=8$). Extending the system size to $N_{\mathrm{ex}}=20$, one can naturally obtain the MBWFs of $r=1,\cdots ,I_{\mathrm{ex}}^{-}\equiv N_{\mathrm{ex}}/2-1=9$ by adding an extended MBWF as schematically drawn as a dotted diagram.

Figure 6

Effective models of MBWFs with $V_1=V=2.8T,V_{\alpha \ge 2}=0$. (a) $S_{r^{\prime },r}\equiv {log}_{10}\left(\right|{\left({\tilde{h}}^{-}\right)}_{r^{\prime },r}|/{\tilde{h}}_{\max }^{-})$, where ${\tilde{h}}_{\max }^{-}\equiv max\left(\right|{\left({\tilde{h}}^{-}\right)}_{r^{\prime },r}\left|\right)$ for $N=18$. The entire structure of $S_{r^{\prime },r}$ is almost the same as all of the other effective models of MBWFs calculated in this study. (b) Size-extended odd-parity effective model, ${\tilde{h}}_{\mathrm{ex}}^{-}$, at $N_{\mathrm{ex}}=20$. All $S_{r^{\prime },r}<{10}^{-3}$ terms are ignored in this model. Arrows represent the direction of the size extension. (c1–c4) Results for odd-parity. (d1–d4) Results for even-parity. The values at $N=18$ (filled-in circles) and their size-extended values from $N=18$ to $N_{\mathrm{ex}}=20$ (empty squares).

Figure 7

(a) Schematic of the excitonic energy structure of a Mott insulator in the region of low-lying excitation energies. (b, c) Finite-size effects of the relative energies corresponding to the notations in (a) for $V_1=V=2.8T, V_{\alpha \ge 2}=0$. The values of the filled-in and empty circles are calculated using the eigenenergies of the exact diagonalization method for $M_{\max }=N/2$ and ${\tilde{h}}^{\pm }$ in Eq. (36) with $N=12$, 14, 16, and 18, respectively. Both the filled-in and empty circles are fitted with a power series of $1/N$, as shown with the broken lines. The eigenenergies of the size-extended effective model in Eqs. (37, 38, 39, 40, 41, 42) with $N_{\mathrm{ex}}=18$, 20, 24, 28, 34, 42, 54, 76, 128, and 400 give the values of the empty squares. Those of the HD model [54] for $N=12$, 14, 16, 18, 20, 24, 28, 34, 42, 54, 76, 128, and 400 correspond to the filled-in squares. For $N,N_{\mathrm{ex}}>20$, we choose $N$ and $N_{\mathrm{ex}}$ values to keep the intervals in the horizontal axis as equal as possible. In panel (c), the HD model results never quantitatively converge to a plausible value of extrapolating the exact results (filled-in circles) within the limit of $N\to +\infty$.

Figure 8

Optical conductivity spectra with $\varepsilon =0$. The solid line is the experimental spectrum of ET-${\mathrm{F}}_2\mathrm{TCNQ}$ at 4 K, which has ${\sigma }_{\max }=2586$ S/cm at $\omega =0.687$ eV. This spectrum is obtained using the Kramers-Kronig transformation of the reflectivity spectrum with the electric fields of light parallel to the 1D molecular stack. The experimental method of the low-temperature reflectivity measurement was reported in Ref. [60]. The dashed (dotted) line represents the MBWFs spectrum with a size extension from $N=18$ to $N_{\mathrm{ex}}=400$ and $V_1=V=2.8T,V_{\alpha \ge 2}=0$ ($V_1=V=2.4T,V_2=V/2,V_3=V/3,V_{\alpha \ge 4}=0$).

Figure 9

Optical conductivity spectra with and without a modulated electric field. The solid thick lines are experimental data of ET-${\mathrm{F}}_2\mathrm{TCNQ}$ at 294 K, where ${\sigma }_{\max }=1119$ S/cm is observed at $\omega =0.694$ eV (the vertical dashed line). The solid thin lines are MBWFs spectra with size extensions from $N=18$ to $N_{\mathrm{ex}}=400$. (a1–a3) $V_1=V=2.8T,V_{\alpha \ge 2}=0$, and $\varepsilon =1.85\times {10}^{-2}$ ($E_{\mathrm{amp}}=215$ kV/cm, owing to $T=0.107$ eV). $\gamma /T=0.45$ in (a1) and $\gamma /T=0.55$ in (a2). (b1–b3) $V_1=V=2.4T,V_2=V/2,V_3=V/3,V_{\alpha \ge 4}=0$, and $\varepsilon =1.35\times {10}^{-2}$ ($E_{\mathrm{amp}}=163$ kV/cm, owing to $T=0.111$ eV). $\gamma /T=0.45$ in (b1) and $\gamma /T=0.55$ in (b2). Panels (a3) and (b3) are the eigenenergies measured from the ground state energy for $\varepsilon =0$. Filled-in circles (squares) represent eigenenergies with odd-parity (even-parity). Insets are magnified low-lying energy diagrams.

Figure 10

Optical conductivity spectra with and without a modulated electric field. In panels (a) and (b), the solid thick lines are experimental data of ET-${\mathrm{F}}_2\mathrm{TCNQ}$ at 294 K, where ${\sigma }_{\max }=1119$ S/cm is observed at $\omega =0.694$ eV (the vertical dashed line). The solid thin lines are spectra of the HD model with $N=400, V_1=V=2.8T,V_{\alpha \ge 2}=0, \varepsilon =1.77\times {10}^{-2}$ ($E_{\mathrm{amp}}=231$ kV/cm, owing to $T=0.120$ eV), $\gamma /T=0.40$ in panel (a), and $\gamma /T=0.50$ in panel (b). The dotted lines are spectra of the HD model with $N=400, V_1=V=2.4T,V_2=V/2,V_3=V/3,V_{\alpha \ge 4}=0, \varepsilon =1.12\times {10}^{-2}$ ($E_{\mathrm{amp}}=156$ kV/cm, owing to $T=0.128$ eV), $\gamma /T=0.39$ in panel (a), and $\gamma /T=0.44$ in panel (b). (c) Eigenenergies measured from the ground-state energy for $\varepsilon =0$. Empty circles (squares) represent eigenenergies with odd-parity (even-parity) for $V_1=V=2.8T,V_{\alpha \ge 2}=0$. Filled-in circles (squares) represent eigenenergies with odd-parity (even-parity) for $V_1=V=2.4T,V_2=V/2,V_3=V/3,V_{\alpha \ge 4}=0$.

Figure 11

Effective models of MBWFs at $N=18$ (filled-in circles) and their size extended values from $N=18$ to $N_{\mathrm{ex}}=20$ (empty squares) with $V_1=V=2.4T,V_2=V/2,V_3=V/3,V_{\alpha \ge 4}=0$. (a1–a4) Results with odd-parity. (b1–b4) Results with even-parity.

Figure 12

Matrix elements of the effective model of $H_{\varphi }^{\left(\mathrm{C}\right)}$ at $N=18$. The filled-in circles are calculated using Eq. (C15) and are fitted well by Eq. (C17) with a continuous $r$ (dashed line).