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\title{Topological Mott Insulator at Quarter Filing in the Interacting Haldane Model: supplementary information}
%% Notice placement of commas and superscripts and use of &
%% in the author list

%\author{ Peizhi Mai$^{1}$, Ben Feldman$^{2,3,4}$, and Philip W. Phillips$^{1,*}$ }

\author[1]{Peizhi Mai}
\author[2,3,4]{Ben Feldman}
\author[1,*]{Philip W. Phillips}
\affil[1]{Department of Physics and Institute of Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA}
\affil[2]{Geballe Laboratory of Advanced Materials, Stanford, CA 94305, USA}
\affil[3]{Department of Physics, Stanford University, Stanford, CA 94305, USA}
\affil[4]{Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA}


\begin{document}

%\author{ Peizhi Mai$^{1}$, Ben Feldman$^{2,3,4}$, and Philip W. Phillips$^{1,\dagger}$ }

%\affiliation{$^1$Department of Physics and Institute of Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA}
%\affiliation{$^2$Geballe Laboratory of Advanced Materials, Stanford, CA 94305, USA}
%\affiliation{$^3$Department of Physics, Stanford University, Stanford, CA 94305, USA}
%\affiliation{$^4$Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA}


\maketitle
\clearpage
\section{ Hamiltonian with Interaction-Independent Basis}
The analytically solvable Hamiltonian in Eq.~(1) of the main text provides a simple model for the interacting Haldane electrons. However, the interaction term has a simple form (leading to the simple solution) only in the basis that diagonalizes the non-interacting Hamiltonian.  This means that the interaction term depends on the choice of $t'$, $\psi$ and $M$. The natural question is whether or not it is possible to write the interaction term in a similarly simple form independent of basis without surrendering the essential solution outlined in the text. The answer is yes. We can write the  Hamiltonian as 
\beq
\begin{split}
H&=\sum_{{\bf k},\sigma}\big[(\varepsilon_{+,{\bf k}}-\mu)n_{+,{\bf k}\sigma}+(\varepsilon_{-,{\bf k}}-\mu)n_{-,{\bf k}\sigma}\big]
+U^\prime\sum_{\bf k}(n_{+,{\bf k}\uparrow}+n_{-,{\bf k}\uparrow})(n_{+,{\bf k}\downarrow}+n_{-,{\bf k}\downarrow}). \label{HHK2}
\end{split}
\eeq
The quantity $n_{+,{\bf k}\sigma}+n_{-,{\bf k}\sigma}$ is a trace in the Haldane orbital space and is therefore independent of the orbital basis, that is, the interaction term is the same for all combinations of $t'$, $\psi$ and $M$. This Hamiltonian is also analytically solvable, since the interaction does not mix the original non-interacting eigenstates. We can write the Green function as

\beq
\begin{split}
G_{\pm,{\bf k}\sigma}(i\omega_n\rightarrow \omega)&=\frac{(1-\langle n_{+,{\bf k}\bar{\sigma}}\rangle)(1-\langle n_{-,{\bf k}\bar{\sigma}}\rangle)}{\omega-\varepsilon_{\pm,{\bf k}}} 
+ \frac{(1-\langle n_{+,{\bf k}\bar{\sigma}}\rangle)\langle n_{-,{\bf k}\bar{\sigma}}\rangle+(1-\langle n_{-,{\bf k}\bar{\sigma}}\rangle)\langle n_{+,{\bf k}\bar{\sigma}}\rangle}{\omega-(\varepsilon_{\pm,{\bf k}}+U^\prime)}
\\&+\frac{\langle n_{+,{\bf k}\bar{\sigma}}\rangle \langle n_{-,{\bf k}\bar{\sigma}}\rangle}{\omega-(\varepsilon_{\pm,{\bf k}}+2U^\prime)}
\\&=\frac{(1-\langle n_{+,{\bf k}\bar{\sigma}}\rangle)(1-\langle n_{-,{\bf k}\bar{\sigma}}\rangle)}{\omega-\varepsilon_{\pm,{\bf k}}} + 
\frac{\langle n_{+,{\bf k}\bar{\sigma}}\rangle+\langle n_{-,{\bf k}\bar{\sigma}}\rangle-\langle n_{+,{\bf k}\bar{\sigma}}\rangle\langle n_{-,{\bf k}\bar{\sigma}}\rangle}{\omega-(\varepsilon_{\pm,{\bf k}}+U^\prime)}
\\&+\frac{\langle n_{+,{\bf k}\bar{\sigma}}\rangle \langle n_{-,{\bf k}\bar{\sigma}}\rangle}{\omega-(\varepsilon_{\pm,{\bf k}}+2U^\prime)}.
\label{GF2}
\end{split}
\eeq
At zero temperature with a large enough $U^\prime$ ($U^\prime>W_++W_-+\Delta$), for each ${\bf k}$, the state with the same spin would be filled up first to avoid the cost of interaction energy. After that, the state with opposite spin would start to be filled with the interaction energy cost of $2U^\prime$. In this sense, the band structure would be similar to that for the Hamiltonian in Eq.~(1) of the main text if $U=2U^\prime$. Choosing the same example parameter set $t'=0.1$, $\psi=-\pi/2$ and $M=0$ ($W_++W_-+\Delta=6$), we plot the band structure in \figdisp{fig:sup_replotU16} (setting $U^\prime=8$). The system is a topological Mott insulator at $1/4$- (or $3/4$-) filling and a topologically trivial Mott insulator at $1/2$-filling. The band structure is almost the same as that in Fig.~2(d) of the main text if $U=2U^\prime=16$. The only difference is that in \figdisp{fig:sup_replotU16} for Hamiltonian \disp{HHK2}, once the lowest band is filled up, the states of the second lowest band are fixed with no degeneracy. For each ${\bf k}$, the state contains the same spin as that in the lowest band with the same momentum. In Fig.~2(d) of the main text, the degeneracy of the second lowest band remains with a full lowest band. In short, we construct a Hamiltonian \disp{HHK2} for interacting Haldane electrons independent of basis and containing essentially the same physics as Eq.~(1) of the main text under strong correlation.

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.45\textwidth]{./figures/sup_replotU16.pdf}
    \caption{Band structure for the generalized Haldane-HK model in \disp{HHK2} with $t'/t=0.1, \psi=-\pi/2, M=0$ and $U'=8>W_++W_-+\Delta$. $\Delta=6\sqrt{3}t'\sin{\psi}$ is the non-interacting topological gap. $W_{+(-)}$ correspond to the bandwidth of the Haldane upper (lower) band. The blue (or orange)  color represents Chern number $C=-1 ($or $ 1)$. The meshed band consists of only doubly occupied states, while the unmeshed band is singly occupied.}
    \label{fig:sup_replotU16}
\end{figure}

\clearpage

\section{Sign problem of determinantal quantum Monte-carlo simulation for the Haldane-Hubbard model}

The determinantal quantum Monte-carlo (DQMC) simulation suffers from a fermionic sign problem when turning on the Hubbard interaction in the Haldane-Hubbard model. Generally, the average sign decays exponentially as the interaction strength increases or temperature decreases, leading to the exponential increase on the necessary number of measurements to bring down the error bar. For this reason, we are restricted at a relatively high temperature $\beta=3$ while working with a large $U=12$. The average sign varies largely along with the density, as shown in \figdisp{fig:sign}.

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.7\textwidth]{./figures/sup_sign.pdf}
    \caption{Average DQMC sign as a function of density for $t'=0.2, M=0, U=12$ and $\beta=3$ at zero magnetic flux.}
    \label{fig:sign}
\end{figure}


\clearpage

\section{Anti-ferromagnetism in the half-filled system}
To explore the structure of spin correlation at  half filling, the middle peak of Fig.~4 (c) and (d) in the main text, we calculate the zero-frequency spin susceptibility in real and momentum space:
\beq
\begin{split}
S_{m,n}({\bf r},\omega=0)=\frac{1}{N}\sum_{{\bf i}_m,{\bf i}_n}\int_0^{\beta}\langle S^z_{{\bf i}_m+{\bf r}}(\tau) S^z_{{\bf i}_n}(0)\rangle d\tau,\label{rspace}
\end{split}
\eeq
\beq
\begin{split}
S_{m,n}({\bf q},\omega=0)=\sum_{m,n}\sum_{\bf r}S_{m,n}({\bf r},\omega=0)\exp[-i {\bf q}\cdot({\bf r}+\Delta{\bf r}_{m,n})].\label{kspace}.
\end{split}
\eeq
The results  are shown in \figdisp{fig:tp02M0} for the topologically nontrivial case ($t'=0.2,M=0$) and \figdisp{fig:tp0M1} for the topologically trivial case ($t'=0,M=1$) at $\beta=3/t$ and $\beta=5/t$. Note that the smallest magnetic flux $\phi/\phi_0=1/36$ is applied to reduce the finite size effects at lower temperature, and hence the conclusion can be extrapolated to zero field. In \figdisp{fig:tp02M0}, the first row features $S_{m,B}({\bf r},\omega=0)$ ($m=A,B$) indicating that anti-ferromagnetism develops as the temperature decreases. For  strong anti-ferromagnetism, the spin correlation should be ferromagnetic in the intra-sublattice components and anti-ferromagnetic in the inter-sublattice components with the same amplitude. Therefore, we show $S_{B,B}({\bf q},\omega=0)$ and $S_{A,B}({\bf q},\omega=0)$ in the second and third rows, respectively. At $\beta=3/t$, indeed $S_{B,B}({\bf q},\omega=0)$ is ferromagnetic and $S_{A,B}({\bf q},\omega=0)$ is anti-ferromagnetic. But the amplitude yields $|S_{A,B}({\bf q}=0,\omega=0)|<|S_{B,B}({\bf q}=0,\omega=0)|$ leading to a residual ${\bf q}=0$ contribution in the total spin correlation corresponding to the middle peak in Fig.~3(c) in the main text. However, at a lower temperature $\beta=5/t$, the amplitude becomes comparable $|S_{A,B}({\bf q}=0,\omega=0)|\approx|S_{B,B}({\bf q}=0,\omega=0)|$, resulting in a vanishing ${\bf q}=0$ component for the total spin correction, as expected for anti-ferromagnetism. The situation is similar for the topologically trivial case in \figdisp{fig:tp0M1} except that now we are comparing the amplitude $|S_{A,B}({\bf q}=0,\omega=0)+S_{B,A}({\bf q}=0,\omega=0)|$ and $|S_{A,A}({\bf q}=0,\omega=0)+S_{B,B}({\bf q}=0,\omega=0)|$ due to the lifting of the sublattice degeneracy. Similarly, $S_{A,B}({\bf q}=0,\omega=0)$=$S_{B,A}({\bf q}=0,\omega=0)$ for symmetry reasons.

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.7\textwidth]{./figures/sup_tp02M0.pdf}
    \caption{Zero-frequency spin susceptibility at half-filling for the topologically nontrivial case $t'=0.2, M=0$. The left and right columns show the susceptibility at the temperature $\beta=3/t$ and $\beta=5/t$, respectively. The first row features $S_{m,B}({\bf r},\omega=0)$ ($m=$A,B) in real space. The second and third rows display $S_{B,B}({\bf q},\omega=0)$ and $S_{A,B}({\bf q},\omega=0)$ in momentum space, respectively. The interaction strength is $U/t=12$ and the smallest magnetic flux $\phi/\phi_0=1/36$ is applied for all panels.}
    \label{fig:tp02M0}
\end{figure}

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.7\textwidth]{./figures/sup_tp0M1.pdf}
    \caption{Zero-frequency spin susceptibility at half-filling for the topologically trivial case $t'=0, M=1$. The left and right columns show the susceptibility at the temperature $\beta=3/t$ and $\beta=5/t$ respectively. The first row displays $S_{m,B}({\bf r},\omega=0)$ ($m=$A,B) in real space. The second, third rows show $S_{B,B}({\bf q},\omega=0)$, $S_{A,A}({\bf q},\omega=0)$ and $S_{A,B}({\bf q},\omega=0)$ in momentum space, respectively. The interaction strength is $U/t=12$ and the smallest magnetic flux of $\phi/\phi_0=1/36$ is applied for all panels.}
    \label{fig:tp0M1}
\end{figure}

\end{document}
