Commit dbab07d8 authored by Sebastian Greschner's avatar Sebastian Greschner

new supmat

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\newlabel{fig:entanglement_SO}{{3}{3}{}{}{}}
\@writefile{toc}{\contentsline {paragraph}{\numberline {}Edge Haldane order.--}{3}{}}
\newlabel{eq:QLM1D}{{2}{3}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces (a) Average fermionic density and bond configuration of two charges at a distance of $L_D=12$ sites, after substracting the charge-free configuration. We employ DMRG for a cylinder with $L_y=4$ legs and $L=36$ rungs and $\mu =0.4 J_x$. (a) Sy phase\nobreakspace {}($J_y=1.8 J_x$), (b) Sx phase\nobreakspace {}($J_y=0.4 J_x$), (c) D phase\nobreakspace {}($J_y=J_x$). (d) String tension $S_T$ as a function of the distance between the defects $L_D$.}}{3}{}}
\newlabel{fig:config_4T_ST}{{4}{3}{}{}{}}
\@writefile{toc}{\contentsline {paragraph}{\numberline {}String tension.--}{3}{}}
\citation{turner2018scars,feldmeier2019emergent}
\bibdata{lgt2_paperNotes,references}
\bibcite{Wilson1974}{{1}{1974}{{Wilson}}{{}}}
......@@ -71,11 +70,12 @@
\bibcite{Martinez2016}{{19}{2016}{{Martinez\ \emph {et~al.}}}{{Martinez, Muschik, Schindler, Nigg, Erhard, Heyl, Hauke, Dalmonte, Monz, Zoller,\ and\ Blatt}}}
\bibcite{Bernien2017}{{20}{2017}{{Bernien\ \emph {et~al.}}}{{Bernien, Schwartz, Keesling, Levine, Omran, Pichler, Choi, Zibrov, Endres, Greiner, Vuletic,\ and\ Lukin}}}
\bibcite{Clark2018}{{21}{2018}{{Clark\ \emph {et~al.}}}{{Clark, Anderson, Feng, Gaj, Levin,\ and\ Chin}}}
\bibcite{Goerg2019}{{22}{2019}{{G{\"o}rg\ \emph {et~al.}}}{{G{\"o}rg, Sandholzer, Minguzzi, Desbuquois, Messer,\ and\ Esslinger}}}
\@writefile{toc}{\contentsline {paragraph}{\numberline {}String tension.--}{4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces (a) Average fermionic density and bond configuration of two charges at a distance of $L_D=12$ sites, after substracting the charge-free configuration. We employ DMRG for a cylinder with $L_y=4$ legs and $L=36$ rungs and $\mu =0.4 J_x$. (a) Sy phase\nobreakspace {}($J_y=1.8 J_x$), (b) Sx phase\nobreakspace {}($J_y=0.4 J_x$), (c) D phase\nobreakspace {}($J_y=J_x$). (d) String tension $S_T$ as a function of the distance between the defects $L_D$.}}{4}{}}
\newlabel{fig:config_4T_ST}{{4}{4}{}{}{}}
\@writefile{toc}{\contentsline {paragraph}{\numberline {}Conclusions.--}{4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {}Acknowledgments}{4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {}References}{4}{}}
\bibcite{Goerg2019}{{22}{2019}{{G{\"o}rg\ \emph {et~al.}}}{{G{\"o}rg, Sandholzer, Minguzzi, Desbuquois, Messer,\ and\ Esslinger}}}
\bibcite{schweizer2019}{{23}{2019}{{Schweizer\ \emph {et~al.}}}{{Schweizer, Grusdt, Berngruber, Barbiero, Demler, Goldman, Bloch,\ and\ Aidelsburger}}}
\bibcite{Mil2019}{{24}{2019}{{Mil\ \emph {et~al.}}}{{Mil, Zache, Hegde, Xia, Bhatt, Oberthaler, Hauke, Berges,\ and\ Jendrzejewski}}}
\bibcite{Chandrasekharan1997}{{25}{1997}{{Chandrasekharan\ and\ Wiese}}{{}}}
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\@writefile{toc}{\contentsline {abstract}{Abstract}{1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {I}Relation to quantum dimer models}{1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {II}Simplified 1D model}{1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Ground-state energy per site $E / L$ and average entanglement entropy $S_{vN}$ of the simplified 1D model\nobreakspace {}(infinite DMRG simulation, $\chi =80$). The lighter colored curves depict the $\chi =2$ ansatz described in the main text and Eq.\nobreakspace {}\textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces 3{}{}{}\hbox {}\unskip \@@italiccorr )}}.}}{1}{}}
\newlabel{fig:S_1d_m0}{{1}{1}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {S1}{\ignorespaces Ground-state energy per site $E / L$ and average entanglement entropy $S_{vN}$ of the simplified 1D model\nobreakspace {}(infinite DMRG simulation, $\chi =80$). The lighter colored curves depict the $\chi =2$ ansatz described in the main text and Eq.\nobreakspace {}\textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces 3{}{}{}\hbox {}\unskip \@@italiccorr )}}.}}{1}{}}
\newlabel{fig:S_1d_m0}{{S1}{1}{}{}{}}
\newlabel{eq:lgt1dMPS}{{3}{1}{}{}{}}
\citation{pollmann2012symmetry}
\citation{Cardarelli2017}
\citation{Pollmann2010,Pollmann2012,Pollmann2012A}
\bibdata{supmatNotes,references}
\bibcite{Shastry1981}{{1}{1981}{{Shastry\ and\ Sutherland}}{{}}}
\@writefile{toc}{\contentsline {section}{\numberline {III}2-leg ladder}{2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {IV}Two-leg cylinder}{2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {V}Three- and four-leg ladder}{2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {VI}RK states}{2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {VII}Phase diagram}{2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {VIII}Extended data on the string tension}{2}{}}
\bibdata{supmatNotes,references}
\bibcite{Shastry1981}{{1}{1981}{{Shastry\ and\ Sutherland}}{{}}}
\bibcite{Anderson1987}{{2}{1987}{{Anderson}}{{}}}
\bibcite{Moessner2002}{{3}{2002}{{Moessner\ and\ Sondhi}}{{}}}
\bibcite{Ralko2005}{{4}{2005}{{Ralko\ \emph {et~al.}}}{{Ralko, Ferrero, Becca, Ivanov,\ and\ Mila}}}
......@@ -35,22 +35,24 @@
\bibstyle{apsrev4-1}
\citation{REVTEX41Control}
\citation{apsrev41Control}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Mid-system entanglement spectrum of the QLM Eq.\nobreakspace {}(1) of the main text at $\mu =0$ for (a) two-leg ladder, $L=200$ (b) three-leg ladder, $L=100$ (c) four-leg ladder, $L=100$ (DMRG data).}}{3}{}}
\newlabel{fig:S_entanglement}{{2}{3}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Boundary-parity- and string-order of the QLM Eq.\nobreakspace {}(1) of the main text at $\mu =0$ for (a) two-leg ladder, $L=200$ (b) three-leg ladder, $L=100$ (c) four-leg ladder, $L=100$ (DMRG data).}}{3}{}}
\newlabel{fig:S_stringorder}{{3}{3}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {S2}{\ignorespaces Mid-system entanglement spectrum of the QLM Eq.\nobreakspace {}(1) of the main text at $\mu =0$ for (a) two-leg ladder, $L=200$ (b) three-leg ladder, $L=100$ (c) four-leg ladder, $L=100$ (DMRG data).}}{3}{}}
\newlabel{fig:S_entanglement}{{S2}{3}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {S3}{\ignorespaces Boundary-parity- and string-order of the QLM Eq.\nobreakspace {}(1) of the main text at $\mu =0$ for (a) two-leg ladder, $L=200$ (b) three-leg ladder, $L=100$ (c) four-leg ladder, $L=100$ (DMRG data).}}{3}{}}
\newlabel{fig:S_stringorder}{{S3}{3}{}{}{}}
\@writefile{toc}{\contentsline {section}{\numberline {}References}{3}{}}
\newlabel{LastBibItem}{{11}{3}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Schematic phase diagram of the two-leg cylinder model. Color codes depict the von-Neumann bipartite entanglement entropy $S_{vN}$ of the central rung (infinite DMRG simulation, $\chi =80$). }}{4}{}}
\newlabel{fig:S_pd_2torus}{{4}{4}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Comparison of charge and spin configurations obtained from DMRG results for the intermediate phase ($J_x=J_y$, $\mu =0$) (plots (a),(c), and (e)) to the RK state obtained by an equal amplitude overlap of all connected configurations (plots (b), (d), and (f)) for the four-leg cylinder (a) and (b), the four-leg ladder (c) and (d), and the three-leg ladder (e) and (f)}}{4}{}}
\newlabel{fig:S_class_plt}{{5}{4}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {S4}{\ignorespaces Schematic phase diagram of the two-leg cylinder model. Color codes depict the von-Neumann bipartite entanglement entropy $S_{vN}$ of the central rung (infinite DMRG simulation, $\chi =80$). }}{4}{}}
\newlabel{fig:S_pd_2torus}{{S4}{4}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {S5}{\ignorespaces Comparison of charge and spin configurations obtained from DMRG results for the intermediate phase ($J_x=J_y$, $\mu =0$) (plots (a),(c), and (e)) to the RK state obtained by an equal amplitude overlap of all connected configurations (plots (b), (d), and (f)) for the four-leg cylinder (a) and (b), the four-leg ladder (c) and (d), and the three-leg ladder (e) and (f)}}{4}{}}
\newlabel{fig:S_class_plt}{{S5}{4}{}{}{}}
\@writefile{lot}{\contentsline {table}{\numberline {I}{\ignorespaces Estimated critical values of the exchange $J_y$ (in units of $J_x=1$) for the phase transition to the intermediate D phase for $\mu =0$.}}{4}{}}
\newlabel{tab:S_transitions}{{I}{4}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces (a) Charge and bond average configurations of two defects with distance $L_D=2$,$4$,$6$,$8$,$12$,$16$ sites. DMRG-data, $L=36$ rungs, $\mu =0.4 J_x$, $J_y / J_x = 0.4$ (b) Same as (a) but after substracting background without charges.}}{5}{}}
\newlabel{fig:S_pltST_jy0.4}{{6}{5}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces (a) Charge and bond average configurations of two defects with distance $L_D=2$,$4$,$6$,$8$,$12$,$16$ sites. DMRG-data, $L=36$ rungs, $\mu =0.4 J_x$, $J_y / J_x = 1.0$ (b) Same as (a) but after substracting background without charges.}}{5}{}}
\newlabel{fig:S_pltST_jy1.0}{{7}{5}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces (a) Charge and bond average configurations of two defects with distance $L_D=2$,$4$,$6$,$8$,$12$,$16$ sites. DMRG-data, $L=36$ rungs, $\mu =0.4 J_x$, $J_y / J_x = 1.8$ (b) Same as (a) but after substracting background without charges.}}{6}{}}
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\newlabel{LastPage}{{}{6}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {S6}{\ignorespaces Energy error $E_{\rm {err}} = E(\chi ) - E_0$ as a function of the bond dimension $\chi $, for different values of $J_y$ in the disordered region and close to the phase transitions (four-leg cylinder, $\mu =0$, L=40, DMRG results); plot (b) shows the same curves in logarithmic scale. $E_0$ is an estimate of the ground-state energy for a fixed chain length obtained by fitting the data with an exponential curve. }}{5}{}}
\newlabel{fig:energy_vs_chi}{{S6}{5}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {S7}{\ignorespaces (a) Charge and bond average configurations of two defects with distance $L_D=2$,$4$,$6$,$8$,$12$,$16$ sites. DMRG-data, $L=36$ rungs, $\mu =0.4 J_x$, $J_y / J_x = 0.4$ (b) Same as (a) but after substracting background without charges.}}{6}{}}
\newlabel{fig:S_pltST_jy0.4}{{S7}{6}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {S8}{\ignorespaces (a) Charge and bond average configurations of two defects with distance $L_D=2$,$4$,$6$,$8$,$12$,$16$ sites. DMRG-data, $L=36$ rungs, $\mu =0.4 J_x$, $J_y / J_x = 1.0$ (b) Same as (a) but after substracting background without charges.}}{6}{}}
\newlabel{fig:S_pltST_jy1.0}{{S8}{6}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {S9}{\ignorespaces (a) Charge and bond average configurations of two defects with distance $L_D=2$,$4$,$6$,$8$,$12$,$16$ sites. DMRG-data, $L=36$ rungs, $\mu =0.4 J_x$, $J_y / J_x = 1.8$ (b) Same as (a) but after substracting background without charges.}}{7}{}}
\newlabel{fig:S_pltST_jy1.8}{{S9}{7}{}{}{}}
\newlabel{LastPage}{{}{7}{}{}{}}
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......@@ -12,6 +12,9 @@
\usepackage{adjustbox}
\renewcommand*{\thefigure}{S\arabic{figure}}
\DeclareGraphicsExtensions{.eps}
\newcommand{\sebastian}[1]{\textcolor{magenta}{[SG: #1]}}
......@@ -217,9 +220,10 @@ For $\mu=0$ a symmetry protected topological phase~(SPT) emerges. This phase can
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two-leg cylinder}
For the two-leg cylinder we observe a drastically different ground-state phase diagram, as sketched in Fig.~\ref{fig:S_pd_2torus}. Here, the second order ring-change around the cylinder competes with local charge fluctuations in y-direction and, hence, the mechanism described in the main text to stabilize the distinct phases in the large $|\mu|\gg J_x, J_y$ limit is strongly affected.
In Fig.~\ref{fig:S_pd_2torus} we show numerical results for the phase diagram of the two-leg cylinder as function of $\mu$ and $J_y$. We only observe two clearly distinct phases. In the $J_y\ll J_x$ and $\mu<0$ regime, we observe a VA phase, which is characterized by an ordering of the link-spins in y-direction.
The VA phase which exhibits a quantum phase transition to a VA' like phase, which is adiabatically connected to the Sy phase. A distinct Sx phase is absent.
% xxx correction
For the two-leg cylinder we observe a drastically different ground-state phase diagram, as sketched in Fig.~\ref{fig:S_pd_2torus}. Here, the second order ring-exchange around the cylinder competes with local charge fluctuations in the y-direction and, hence, the mechanism described in the main text to stabilize the distinct phases in the large $|\mu|\gg J_x, J_y$ limit is strongly affected.
In Fig.~\ref{fig:S_pd_2torus} we show numerical results for the phase diagram of the two-leg cylinder as a function of $\mu$ and $J_y$. We only observe two clearly distinct phases. In the $J_y\ll J_x$ and $\mu<0$ regime, we observe a VA phase, which is characterized by an ordering of the link-spins in the y-direction.
The VA phase exhibits a quantum phase transition to a VA' like phase, which is adiabatically connected to the Sy phase. A distinct Sx phase is absent.
\begin{figure}[tb]
\centering
......@@ -238,7 +242,6 @@ Schematic phase diagram of the two-leg cylinder model. Color codes depict the vo
In Figs.~\ref{fig:S_entanglement}~(b) and (c) and ~\ref{fig:S_stringorder}~(b) and (c) we present additional data obtained from our DMRG calculations of the three and four-leg ladder systems at $\mu=0$. The exact twofold degeneracy in the entanglement spectrum is lost for more than two legs in the intermediate D phase. However, we observe the emergence of a distinct gap in the entanglement spectrum between a manifold of low lying and higher entanglement states. At the edge the string and parity order exhibit Haldane-like properties in the D phase~(Fig.~\ref{fig:S_stringorder}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{RK states}
......@@ -279,6 +282,37 @@ In Tab.~\ref{tab:S_transitions} we summarize our results on the phase transition
\end{table}
%%
The robustness of the disordered phase is confirmed in the limit of large bond-dimension.
In Figure \ref{fig:energy_vs_chi} we show the exponential convergence of the ground-state energy for increasing $\chi$, in the zero mass limit.
\begin{figure}[tb]
%
\begin{minipage}[t]{.05\linewidth}
\raisebox{3.1cm}[0cm][0cm]{(a)}
\end{minipage}
\begin{minipage}[t]{.8\linewidth}
\includegraphics[width=.99\linewidth]{supmat_figx_energy_QLM_FourLegs_PBC_m0_L40_deltaE.pdf}
\vspace{1pt}
\end{minipage}
\begin{minipage}[t]{.05\linewidth}
\raisebox{3.1cm}[0cm][0cm]{(b)}
\end{minipage}
\begin{minipage}[t]{.8\linewidth}
\includegraphics[width=.99\linewidth]{supmat_figx_energy_QLM_FourLegs_PBC_m0_L40_logscale.pdf}
\vspace{1pt}
\end{minipage}
%
\caption{
Energy error $E_{\rm{err}} = E(\chi) - E_0$ as a function of the bond dimension $\chi$, for different values of $J_y$ in the disordered region and close to the phase transitions (four-leg cylinder, $\mu=0$, L=40, DMRG results); plot (b) shows the same curves in logarithmic scale.
$E_0$ is an estimate of the ground-state energy for a fixed chain length obtained by fitting the data with an exponential curve.
}
%In (a), $E_{\rm{err}}$ is normalized to $E(40)$.
\label{fig:energy_vs_chi}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Extended data on the string tension}
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