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\begin{document}

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\begin{header}

  \begin{minipage} {.08\textwidth}
        \epsfxsize=7.5cm \epsfbox{logo.eps}
  \end{minipage} \hfill
  \begin{minipage}   {.81\textwidth}
     \centerline{{\veryHuge \bfseries\sffamily Band- to Mott-insulator transitions in 1D}}

     \vspace*{1cm}

     \centerline{\Huge \sffamily {\underline{M.~Sekania$^1$}}, A.~P.~Kampf$^1$, G.~I.~Japaridze$^4$, H.~Fehske$^2$, Ph.~Brune$^1$, and G.~Wellein$^3$ }
     \centerline{\LARGE \sffamily $^1$University of Augsburg; $^2$University of Greifswald; $^3$RRZE Erlangen; GERMANY}
     \centerline{\LARGE \sffamily $^4$Institute of Physics, Georgian Academy of Sciences; GEORGIA}
  \end{minipage}\hfill
  \begin{minipage} {.10\textwidth}
    \begin{center}
      \epsfxsize=9.0cm \epsfbox{logo2hg.eps}
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  \end{minipage} \hfill

\end{header}

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  \begin{pbox}
    \section*{Insulator-Insulator Transitions}

    \begin{itemize}
      \item Origin: competition between charge density wave (CDW) formation and local
            Coulomb repulsion.


            \begin{itemize}
              \item[] Sources for CDW formation:
              \item[] - finite range Coulomb interactions;
              \item[] - staggered potential;
              \item[] - electron-phonon coupling.
            \end{itemize}

      \item Defining criteria for insulating phases:

            \hspace{0mm} \begin{tabular}{ll}
                            Band insulator (BI)       & $\Delta_c=\Delta_s$        \\
                            Mott insulator (MI)       & $\Delta_c>0$, $\Delta_s=0$ \\
                            correlated insulator (CI) & $\Delta_c>\Delta_s>0$      \\
                          \end{tabular}

      \item Charge gap: $\Delta_c=E^{N+1}_0+E^{N-1}_0-2E^{N}_0$
      \item[] Spin gap: $\Delta_s=E_0(S^z=1)-E_0(S^z=0)$

      \item[] \begin{description}
                  \item[Note:] $\Delta_c$ is distinct from the optical excitation gap $\Delta_{opt}$: \\
                               $\Delta_{opt}\equiv$ minimal excitation energy $(E_m-E_0)$ in the same particle number sector \\
                               \underline{Selection rule} for optical transitions: \\
                               $\langle0|\hat{j}|m\rangle\not=0$ only if $|m\rangle$ and $|0\rangle$ have different site-parities
               \end{description}

    \end{itemize}
  \end{pbox}
  \begin{pbox}
    \section*{1D Model Hamiltonians}

    \begin{itemize}

      \item{\bf Holstein-Hubbard model (HHM):}
      \vspace{1.0em}

      \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
        \begin{minipage}{210mm}
          \vspace{-5mm}
          \begin{equation}
            H_{HHM} = H_{t-U}-g\omega_0\sum_{i,\sigma}(b^{\dagger}_{i}+b_{i})n_{i,\sigma}+\omega_0\sum_{i}b^{\dagger}_{i}b_{i}
            \nonumber
          \end{equation}
        \end{minipage}}
      }


      \vspace{0.5em}

      \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
        \begin{minipage}{210mm}
          \vspace{2mm}
          \begin{equation}
            H_{t-U} = -t\sum_{i,\sigma}(c^{\dagger}_{i\sigma}c_{i+1\sigma}+H.c.)+U\sum_{i}n_{i\uparrow}n_{i\downarrow}
            \nonumber
          \end{equation}
        \end{minipage}}
      }

      \vspace{0.5em}

      $g=\sqrt{\varepsilon_p/\omega_0}$ ia a dimensionless electron-phonon coupling
      constant and $\omega_0$ denotes the frequency of the optical phonon mode.

      \vspace{0.5em}
      \item{\bf Adiabatic limit of the Holstein-Hubbard model (AHHM):}
      \vspace{1.0em}

      \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
        \begin{minipage}{210mm}
          \vspace{2mm}
          \begin{equation}
            H_{AHHM} = H_{t-U}-\sum_{i,\sigma}\Delta_i n_{i\sigma}+\frac{K}{2}\sum_{i}\Delta_i^2
            \nonumber
          \end{equation}
        \end{minipage}}
      }


      \vspace{0.5em}

      The elastic energy of the lattice is included via a "stiffness constant" $K(\varepsilon_p)$.
      $\Delta_{i}=(-1)^i\Delta$ is a measure of a staggered density modulation.

      \vspace{0.5em}
      \item{\bf Ionic-Hubbard model (IHM):}
      \vspace{1.0em}

      \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
        \begin{minipage}{210mm}
          \vspace{2mm}
          \begin{equation}
            H_{IHM} = H_{t-U}+\sum_{i,\sigma}(-1)^i\frac{\Delta}{2} n_{i\sigma}
            \nonumber
          \end{equation}
        \end{minipage}}
      }


    \end{itemize}

    \vspace{1.0em}

    $\mathbf{\diamond}$ {\bf Only the half-filled case is considered.}

  \end{pbox}
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  \vspace{-0.1em}
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  \begin{pbox}
    \section*{Simple Cases for the Ionic Hubbard Model}
      \begin{itemize}
        \item $\mathbf{U=0}$:
          \begin{itemize}
            \item[] $\triangleright$ $E_k=\pm\sqrt{4t^2cos^2(k)+(\Delta/2)^2}$;
            \item[] $\triangleright$ $\Delta_c=\Delta_s=\Delta$;
            \item[] {\red $\Rightarrow$} {\blue Band Insulator.}
          \end{itemize}
        \item $\mathbf{t=0}$ {\bf (atomic limit)}:
          \begin{itemize}
            \item[] $\triangleright$ One {\it insulator-insulator transition point} at $U_c=\Delta$
          \end{itemize}
      \end{itemize}
  \end{pbox}


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  \begin{pbox}
    \section*{Ionic Hubbard Model}


    \subsection*{Results for $\mathbf{\Delta_s}$ and $\mathbf{\Delta_c}$}

    \begin{minipage}[t]{0.43\textwidth}
      \vspace{-10.0cm}
      {\bf DMRG} calculations were performed for $\Delta=0.5$, on open chains with
      $L=\{30,40,50,60\}$ (main plot) \\
      $L=\{30,40,50,60,200,300\}$ (inset), and extrapolated to the limit of
      infinite chain length.
    \end{minipage}
    \begin{minipage}[t]{0.57\textwidth}
      \includegraphics[clip=true,width=14.5cm]{figGAP.eps}
    \end{minipage}

    \vspace{-1cm}
    \begin{itemize}
      \item existence of a single insulatot-insulator transition point;
      \item {\red  $\Delta_c>0$ at $U_c(\Delta)$};
      \item origin of transition: level crossing of different site-parity sectors [3,6]\\
            {\red $\mathbf{\Rightarrow}$} $\Delta_{opt}=0$ at transition point.
    \end{itemize}

    \vspace{0.5em}

    \centerline{\includegraphics[clip=true,width=20cm]{figIPD.eps}}

    \vspace{1.0em}


  \end{pbox}
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  \vspace{0.2em}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{pbox}
    \subsection*{Density Distribution}

    \vspace{0.5em}

    \begin{minipage}[t]{0.48\textwidth}
      \includegraphics[clip=true,width=12.5cm]{figCDW.eps}
    \end{minipage}
    \begin{minipage}[t]{0.02\textwidth}
      \vspace{-9.0cm} $\ \ $
    \end{minipage}
    \begin{minipage}[t]{0.48\textwidth}
      \vspace{-9.0cm}
      {\bf DMRG} calculations on a $L=32$ open chain.
      \begin{itemize}
      \item For finite $\Delta$ the CDW persists for {\it arbitrary finite} $U$.
      \end{itemize}
      \vspace{1cm}
      \centerline{\includegraphics[clip=true,width=12cm]{figIPD_1.eps}}
    \end{minipage}


  \end{pbox}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \vspace{0.2em}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{pbox}
    \subsection*{Bond-Charge-Density Distribution}
    \vspace{1.0em}

    \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
      \begin{minipage}{210mm}
        \begin{equation}
          BD(r) = \frac{\sum_{\sigma}\langle c^{\dagger}_{r,\sigma}c^{\phantom{\dagger}}_{r+1,\sigma}
          + H.c.\rangle}{\frac{1}{L}\sum_{r,\sigma}\langle c^{\dagger}_{r,\sigma}c^{\phantom{\dagger}}_{r+1,\sigma}
          + H.c.\rangle}-1\,.
          \nonumber
        \end{equation}
      \end{minipage}}
    }

    \vspace{1.5em}


    \begin{minipage}[t]{0.48\textwidth}
      \includegraphics[clip=true,width=12.5cm]{figBDW.eps}
    \end{minipage}
    \begin{minipage}[t]{0.02\textwidth}
      \vspace{-1cm}$\ \ $
    \end{minipage}
    \begin{minipage}[t]{0.48\textwidth}
      \includegraphics[clip=true,width=12.5cm]{figDCB.eps}
    \end{minipage}

    \begin{itemize}
      \item {\bf\it Open boundaries lead to {\it strong} Friedel-like oscillations
            of the bond-charge-density}
      \item $U<U_c$ {\blue\bf (BI)} reduction of bond-charge-density oscillation
      \item $U>U_c$ {\red\bf (CI)} enhancement of bond-charge-density oscillation
    \end{itemize}

    {\red $\mathbf{\Rightarrow}$} possible bond-order wave (BOW) in the CI phase [4,5]



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  \begin{pbox}
    \section*{Holstein-Hubbard Model}

    \begin{minipage}[t]{0.1\textwidth}
      {\bf U=0}
    \end{minipage}
    \begin{minipage}[t]{0.88\textwidth}
      \begin{itemize}
         \item Ground state: CDW {\blue Band insulator} [9,10] for:
               \begin{itemize}
                  \item[] $\triangleright$ $\omega_0\rightarrow 0$ $\varepsilon_p\not= 0$ or
                  \item[] $\triangleright$ $\omega_0>0$, $g>g_c(\omega_0)$.
               \end{itemize}
         \item $\omega_0>0$, $g<g_c(\omega_0)$: quantum phonon fluctuation {\it destroy}
               long range CDW order in the ground state
      \end{itemize}
    \end{minipage}


    \subsection*{Charge and Spin Structure Factors {$[11,12]$}}
    \vspace{0.5em}

    \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
      \begin{minipage}{210mm}
        \begin{eqnarray}
          S_c(\pi) & = & \frac{1}{L}\sum_{j,\sigma\sigma'}(-1)^j\langle (n_{i\sigma}-\frac{1}{2})(n_{i+j\sigma}-\frac{1}{2}) \rangle \nonumber \\
          S_s(\pi) & = & \frac{1}{L}\sum_{j,\sigma\sigma'}(-1)^j\langle S^{z}_{i}S^{z}_{i+j} \rangle, \ \ \ S^{z}_{i}=\frac{1}{2}(n_{i\uparrow}-n_{i\downarrow})  \nonumber
        \end{eqnarray}
      \end{minipage}}
    }

    \vspace{1.0em}

    \begin{itemize}
      \item {\bf HHM} Lanczos results on an 8-site ring, {\green adiabatic}
      (triangles) and {\red non-adiabatic} (squares) regimes.
      \item {\bf AHHM} {\color[rgb]{0.5,0.0,0.6} Lanczos} (on an te ring) and {\blue DMRG} (on an open 64-site chain) results
      for $K=0.74$. \\
    \end{itemize}

    \vspace{-0.5em}

    \begin{minipage}[t]{0.53\textwidth}
      \includegraphics[clip=true,width=14cm]{figSCSS.eps}
    \end{minipage}
    \begin{minipage}[t]{0.02\textwidth}
      \vspace{-10.0cm} $\ \ $
    \end{minipage}
    \begin{minipage}[t]{0.43\textwidth}
      \vspace{-10.5cm}
      Upper inset: finite-size scaling of $S_c(\pi)$ for various $U$; \\ \\
      lower inset: $U$-dependence of the kinetic energy $E_{kin}$. \\ \\
      Open (closed) simbols correspond to ground state with site-parity $P=-1$ (+1).
    \end{minipage}


    \begin{itemize}
      \item $U/2\varepsilon_p \sim 1$: $S_c$ vanishes ({\blue BI} CDW disappears) and antiferromagnetic
            spin correlations get enhanced ({\green MI})
      \item $S_c$ and $S_s$ for $L=8$ for the AHHM ($K=0.74$) and HHM ($\omega=0.1; \ \varepsilon_p=0.7t$)
            {\it agree perfectly}

    \end{itemize}

  \end{pbox}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \vspace{-0.2em}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{pbox}

    \subsection*{Optical Conductivity for HHM}

    \vspace{0.5em}

    \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
      \begin{minipage}{210mm}
        \begin{equation}
          \sigma^{reg}(\omega)=\frac{\pi}{L}\sum_{m\not= 0}\frac{|\langle\psi_0| \hat{j} |\psi_m\rangle|^2}{E_m-E_0}
          \delta(\omega-E_m+E_0) \nonumber
        \end{equation}
      \end{minipage}}
    }

    \vspace{0.5em}

    current operator $\hat{j}=-iet \sum_{i\sigma} (c^{\dagger}_{i\sigma} c_{i+1 \sigma} - c^{\dagger}_{i+1 \sigma} c_{i\sigma})$

    \vspace{1.0em}

    \begin{minipage}[t]{0.53\textwidth}
      \includegraphics[clip=true,width=14cm]{figOPC.eps}
    \end{minipage}
    \begin{minipage}[t]{0.02\textwidth}
      \vspace{-12.5cm} $\ \ $
    \end{minipage}
    \begin{minipage}[t]{0.43\textwidth}
      \vspace{-12.5cm}
      $L=8$, $\omega_0\!=\!0.1t$ and $g^2\!=\!7$. \\ \\
        \begin{tabular}{llll}
          $\bullet$ & $U=0$           & $\Delta_{opt}>0$;      \\
          $\bullet$ & $U\sim U_{opt}$ & $\Delta_{opt}\sim 0$; \\
          $\bullet$ & $U=3t$          & $\Delta_{opt}>0$.     \\
        \end{tabular}\\ \\
      Dashed lines: normalized integrated spectral weights $S^{reg}(\omega)$.\\
      Lower two panels: $\sigma^{reg}$ for pure Hubbard chain $g=0$ (dotted lines)
    \end{minipage}

    \vspace{-0.5em}

    {\blue  $\mathbf{\Rightarrow}$ $U<U_{opt}$: particle-hole excitations across the BI gap}\\
    {\red   $\mathbf{\Rightarrow}$ $U=U_{opt}$: optical gap closes (level crossing)}\\
    {\green $\mathbf{\Rightarrow}$ $U>U_{opt}$: excitations can be related to those of the Hubbard} \\
    {\white $\mathbf{\Rightarrow}$} {\green model (MI)}\\

  \end{pbox}

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  \begin{pbox}

    \subsection*{$\mathbf{\Delta(U,K)}$ and $\mathbf{\Delta_{cr}(U)}$}

    \vspace{0.5em}

    \begin{minipage}[t]{0.48\textwidth}
      \includegraphics[clip=true,width=12.5cm]{figDCR.eps}
    \end{minipage}
    \begin{minipage}[t]{0.02\textwidth}
      \vspace{-9.0cm} $\ \ $
    \end{minipage}
    \begin{minipage}[t]{0.48\textwidth}
      \vspace{-9.0cm}
      {\bf AHHM,} $\Delta(U,K)$:\\
      $L=8$ ring {\red (triangles)}\\
      $L=64$ open chain {\blue (stars)}\\
      {\bf IHM,} level crossing line $\Delta_{cr}(U)$:
      $L=8$ ring {\green (diamonds)}\\
      extrapolated data to $L=64$
      from $L=8,10,12,14$ ring (circles)
    \end{minipage}



  \end{pbox}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \vspace{0.1em}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{pbox}
    \section*{Conclusions}

    \vspace{-0.5em}

    \subsection*{Ionic Hubbard Model}

      \vspace{0.5em}
      \centerline{\includegraphics[clip=true,width=20cm]{figIPD_F.eps}}
      \vspace{0.5em}

      \begin{itemize}
        \item $U_c(\Delta)$: $\Delta_{opt}=0$ $\Delta_c=\Delta_s>0$
        \item The insulator-insulator transition on finite chain results from a groun-state
              level crossing of the site-parity sectors with $P=\pm1$.
              \begin{itemize}
                \item $\ ${\blue BI}: unique ground state with $P=+1$;
                \item $\ ${\red  CI}: two-fold degenerate ground state (CDW+BOW).
              \end{itemize}
      \end{itemize}

    \vspace{-1.0em}

    \subsection*{Holstein-Hubbard Model}

      \begin{itemize}
        \item Non-adiabatic regime $S_c$, and $S_s$ indicate continuous {\blue BI}-{\green MI} transition

        \item Adiabatic limit ({\bf AHHM})
      \end{itemize}

      \begin{minipage}[t]{0.04\textwidth}
        \vspace{-9.0cm} $\ \ $
      \end{minipage}
      \begin{minipage}[t]{0.46\textwidth}
        \includegraphics[clip=true,width=11.0cm]{figPHD.eps}
      \end{minipage}
      \begin{minipage}[t]{0.02\textwidth}
        \vspace{-9.0cm} $\ \ $
      \end{minipage}
      \begin{minipage}[t]{0.48\textwidth}
        \vspace{-9.0cm}
         \begin{itemize}
           \item $U_{opt}$ $\Delta_{opt}=0$, $\Delta_c=\Delta_s>0$;
           \item $U_{s}$ $\Delta_s=0$, $\Delta_{opt}=\Delta_c>0$;
           \item Weak coupling $U,K^{-1}\ll t$: \\
                 continuous {\blue BI}-{\red CI} and {\red CI}-{\green MI} transitions;
           \item Strong coupling $U,K^{-1}\gg t$: \\
                 discontinuous {\blue BI}-{\green MI} transition
         \end{itemize}
      \end{minipage}

    \vspace{-1.0em}

    {\red \subsection*{Open Questions}}

    {\red
    \begin{itemize}
      \item $\Delta_s>0$ and true {\it long range} bond-order in the {\red\bf CI} {\black phase?}
      \item Do the {\it insulator-insulator} transitions find a continuation {\it away from half filling}?
    \end{itemize}
    }
  \end{pbox}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \vspace{0.2em}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{pbox}
    \section*{References}

    \small{
    \begin{itemize}
    \vspace{-0.4em}
    \item[[1\hspace{-.2em}] ] N.~Nagaosa and J.~Takimoto, J.~Phys.~Soc.~Jpn. {\bf 55}.
    \vspace{-.4em}
    \item[[2\hspace{-.2em}] ] M.~Fabrizio, A.~O.~Gogolin, and A.~A.~Nersesyan,
                              Phys.~Rev.~Lett.~{\bf 83},~2014~(1999)
    \vspace{-.4em}
    \item[[3\hspace{-.2em}] ] N.~Gidopoulos, S.~Sorella, and E.~Tosatti,
                              Eur.~Phys.~J.B {\bf 14}, 217 (2000).
    \vspace{-.4em}
    \item[[4\hspace{-.2em}] ] R.~Resta and S.~Sorella, Phys.~Rev.~Lett. {\bf 74}, 4738 (1995).
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    \item[[5\hspace{-.2em}] ] T.~Wilkens and R.~M.~Martin, Phys.~Rev. B {\bf 63}, 235108 (2001).
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    \item[[6\hspace{-.2em}] ] Y.~Anusooya-Pati, Z.~G.~Soos, and A.~Painelli, Phys.~Rev. B {bf 63}, 205118 (2001).
    \vspace{-.4em}
    \item[[7\hspace{-.2em}] ] M.~E.~Torio, A.~A.~Aligia, and H.~A.~Ceccatto, Phys.~Rev. B {\bf 64}, 121105 (2001).
    \vspace{-.4em}
    \item[[8\hspace{-.2em}] ] J.~B.~Torrance {\it et al.}, Phys.~Rev.~Lett. {\bf 46}, 253 (1981); ibid. {\bf 47}, 1747 (1981).
    \vspace{-.4em}
    \item[[9\hspace{-.2em}] ] E.~Jeckelmann, C.~Zhang, and S.~R.~White, Phys.~Rev. B {\bf 60}, 7950 (1999).
    \vspace{-.4em}
    \item[[10\hspace{-.2em}] ] R.~J.~Bursill, R.~H.~McKenzie, and C.~J.~Hamer, Phys.~Rev.~Lett. {\bf 80}, 5607 (1998);
                              A.~Wei{\ss}e and H.~Fehske, Phys.~Rev. B {\bf 58}, 13526 (1998).
    \vspace{-.4em}
    \item[[11\hspace{-.2em}] ] H.~Fehske, M.~Holicki, and A.~Wei{\ss}e, Adv.~Sol.~State~Phys. {\bf 40}, 235 (2000).
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    \item[[12\hspace{-.2em}] ] B.~B\"auml, G.~Wellein, and H.~Fehske, Phys.~Rev. B {\bf 58}, 3663 (1998);
                              A.~Wei{\ss}e, H. Fehske, G.~Wellein, and A.~R.~Bishop, Phys.~Rev. B {\bf 62}, R747 (2000).

    \end{itemize}
    }

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