Commit 06ee369d authored by Sebastian Greschner's avatar Sebastian Greschner

supmat

parent 7fa580e8
......@@ -3,31 +3,35 @@
\citation{Tschirsich2019}
\citation{Ralko2005}
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\citation{pollmann2012symmetry}
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\@writefile{toc}{\contentsline {title}{Supplementary material to "Deconfining disordered phase in two-dimensional quantum link models"}{1}{}}
\@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 1{}{}{}\hbox {}\unskip \@@italiccorr )}}.}}{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}{}}
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\citation{pollmann2012symmetry}
\citation{Cardarelli2017}
\citation{Pollmann2010,Pollmann2012,Pollmann2012A}
\bibdata{supmatNotes,references}
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\@writefile{toc}{\contentsline {section}{\numberline {III}2-leg ladder}{2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {IV}Two-leg cylinder}{2}{}}
\@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 $\mu =0$.}}{2}{}}
\newlabel{tab:S_transitions}{{I}{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}{}}
\bibdata{supmatNotes,references}
\bibcite{aklt}{{1}{1987}{{Affleck\ \emph {et~al.}}}{{Affleck, Kennedy, Lieb,\ and\ Tasaki}}}
\bibcite{pollmann2012symmetry}{{2}{2012{}}{{Pollmann\ \emph {et~al.}}}{{Pollmann, Berg, Turner,\ and\ Oshikawa}}}
\bibcite{Cardarelli2017}{{3}{2017}{{Cardarelli\ \emph {et~al.}}}{{Cardarelli, Greschner,\ and\ Santos}}}
\bibcite{Pollmann2010}{{4}{2010}{{Pollmann\ \emph {et~al.}}}{{Pollmann, Turner, Berg,\ and\ Oshikawa}}}
\bibcite{Pollmann2012}{{5}{2012}{{Pollmann\ and\ Turner}}{{}}}
\bibcite{Pollmann2012A}{{6}{2012{}}{{Pollmann\ \emph {et~al.}}}{{Pollmann, Berg, Turner,\ and\ Oshikawa}}}
\@writefile{toc}{\contentsline {section}{\numberline {VIII}Extended data on the string tension}{2}{}}
\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}}}
\bibcite{Tschirsich2019}{{5}{2019}{{Tschirsich\ \emph {et~al.}}}{{Tschirsich, Montangero,\ and\ Dalmonte}}}
\bibcite{aklt}{{6}{1987}{{Affleck\ \emph {et~al.}}}{{Affleck, Kennedy, Lieb,\ and\ Tasaki}}}
\bibcite{pollmann2012symmetry}{{7}{2012{}}{{Pollmann\ \emph {et~al.}}}{{Pollmann, Berg, Turner,\ and\ Oshikawa}}}
\bibcite{Cardarelli2017}{{8}{2017}{{Cardarelli\ \emph {et~al.}}}{{Cardarelli, Greschner,\ and\ Santos}}}
\bibcite{Pollmann2010}{{9}{2010}{{Pollmann\ \emph {et~al.}}}{{Pollmann, Turner, Berg,\ and\ Oshikawa}}}
\bibcite{Pollmann2012}{{10}{2012}{{Pollmann\ and\ Turner}}{{}}}
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......@@ -35,13 +39,14 @@
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\@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}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {VIII}Extended data on the string tension}{3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {}References}{3}{}}
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\@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}{}}
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\@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}{}}
......
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......@@ -78,8 +78,15 @@ In this supplementary material we give details on our numerical results for the
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Relation to quantum dimer models}
For the limit of $|\mu| \gg J_x, J_y$, $\mu>0$, charges are pinned to the B-sublattice and the dynamics is reduced to the spins on the links. Indeed, this limit corresponds to a model of tightly packed hard-core dimers, a quantum dimer model on a square lattice.
Traditionally, as in Refs.~\cite{Shastry1981,Anderson1987, Moessner2002, Ralko2005} the dimer-model Hamiltonian contains ring-exchange terms, which - on the square lattice - have been shown to introduce several phase transitions: Phases found and discussed e.g. in Ref.~\cite{Tschirsich2019} are Neel, columnar, or a plaquette-ordered (also dubbed RVB-solid) phase. At the phase transition-point between columnar and plaquette-ordered phase, lies the so called Rokhsar-Kivelson point, discussed in the paper. Here we do not have such strong ring-exchange terms (only in terms from fourth order perturbation) and the physics for the strong-coupling dimer limit is indeed different from the phase diagram of Refs.~\cite{Ralko2005}.
For the limit of $|\mu| \gg J_x, J_y$, $\mu>0$, charges are pinned to the B-sublattice and the dynamics is reduced to the spins on the links. Indeed, this limit corresponds to a model of tightly packed hard-core dimers, a quantum dimer model on a square lattice.
\begin{align}
H_{\mu \gg J_x, J_y} = \sum_\vR \frac{J_\vec{k}^2}{\mu} S^z_{\vec{k}}(\vR) \,+\, \cdots
\end{align}
Traditionally, as in Refs.~\cite{Shastry1981,Anderson1987, Moessner2002, Ralko2005} the dimer-model Hamiltonian contains ring-exchange terms
\begin{align}
H_{QDM} = \sum_\vR (R^+_\vR + R^-_\vR) + \lambda Q_{\vR}^2 \,.
\end{align}
On the square lattice this models has been shown to exhibit several phase transitions as function of $\lambda$: Phases found and discussed e.g. in Ref.~\cite{Tschirsich2019} are Neel, columnar, or a plaquette-ordered (also dubbed RVB-solid) phase. At the phase transition-point between columnar and plaquette-ordered phase, lies the so called Rokhsar-Kivelson point, discussed in the paper. Here we do not have such strong ring-exchange terms (only in terms from fourth order perturbation $\sim J_{x}^2 J_{y}^2/\mu^3$) and the physics for the strong-coupling dimer limit is indeed different from the phase diagram of Refs.~\cite{Ralko2005}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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