Commit 222ce131 authored by Sebastian Greschner's avatar Sebastian Greschner

response and supmat

parent dbab07d8
File added
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msy/m/n/10 ^^X \OML/cmm/m/it/10 J[]$ \OT1/cmr/m/n/10 an in-ter-
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\OT1/cmr/m/n/9 ler, N. Gold-man, and F. Grusdt, arXiv preprint
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......@@ -14,12 +14,21 @@
\citation{pollmann2012symmetry}
\citation{Cardarelli2017}
\citation{Pollmann2010,Pollmann2012,Pollmann2012A}
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\@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).}}{2}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {IV}Two-leg cylinder}{2}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {VII}Phase diagram}{2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {VIII}Extended data on the string tension}{2}{}}
\@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, $\chi =80$). }}{3}{}}
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\@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)}}{3}{}}
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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 $\mu =0$.}}{3}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {VII}Phase diagram}{3}{}}
\bibdata{supmatNotes,references}
\bibcite{Shastry1981}{{1}{1981}{{Shastry\ and\ Sutherland}}{{}}}
\bibcite{Anderson1987}{{2}{1987}{{Anderson}}{{}}}
......@@ -35,24 +44,17 @@
\bibstyle{apsrev4-1}
\citation{REVTEX41Control}
\citation{apsrev41Control}
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\@writefile{lof}{\contentsline {figure}{\numberline {S7}{\ignorespaces Excitation gap $\Delta \epsilon $ (in units of $J_x$) for the (a) 1D toy model, (b) 2-leg (c) 4-leg ladder model within the vacuum gauge-sector at $\mu =0$ for different system sized $L=12$ (crosses), $L=24$ (circle), $L=36$ (plus-symbols). The dotted lines indicate the estimated phase-transition position from Tab.\nobreakspace {}I{}{}{}\hbox {}. (d) Excitation gap as function of the number of legs for $J_y / J_x=1$ (center of the D-phase) and $J_y/J_x=1.8$ (Sy-phase).}}{4}{}}
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......@@ -115,6 +115,35 @@ As mentioned in the main text the ground states can be understood from a $J_y\ll
%%
%%
\begin{figure*}[tb]
%
\begin{minipage}[t]{.03\linewidth}
\raisebox{3.8cm}[0cm][0cm]{(a)}
\end{minipage}
\begin{minipage}[t]{.26\linewidth}
\includegraphics[height=4cm]{entanglement_spectrum_QLM_TwoLegs_DMRG_OBC_n1.00_k0.00_m0.00_L100_chi100.pdf}
\end{minipage}
%
\begin{minipage}[t]{.03\linewidth}
\raisebox{3.8cm}[0cm][0cm]{(b)}
\end{minipage}
\begin{minipage}[t]{.26\linewidth}
\includegraphics[height=4cm]{entanglement_spectrum_QLM_ThreeLegs_DMRG_OBC_n1.50_k0.00_m0.00_L100_chi200_40values_nolabel.pdf}
\end{minipage}
%
\begin{minipage}[t]{.03\linewidth}
\raisebox{3.8cm}[0cm][0cm]{(c)}
\end{minipage}
\begin{minipage}[t]{.26\linewidth}
\includegraphics[height=4cm]{entanglement_spectrum_QLM_FourLegs_DMRG_OBC_n2.00_k0.00_m0.00_L100_chi100_20values_nolabel.pdf}
\end{minipage}
%
\caption{Mid-system entanglement spectrum of the QLM Eq.~(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).}
\label{fig:S_entanglement}
\end{figure*}
%%
To shed light into the properties of the intermediate Haldane-like phase, we define the variational MPS ground state as $|\Psi\rangle = \sum_\sigma \Lambda \Gamma_\sigma | \sigma \rangle$. The $\Gamma_\sigma$ matrices, considered as an automorphism of $\chi\times\chi$ matrices, fulfill the following simple algebraic relations defined by Gauss' law.
A simple example is given by the following matrices, with the lowest non-trivial bond dimension $\chi=2$:
\begin{align}
......@@ -141,34 +170,6 @@ The string order is given by $O_{SO} = \lim_{|x-x'|\to\infty} \la S^z_x \e^{\ii
As the Haldane-phase in this toy model, the SPT-phase is protected by a $Z_2 \times Z_2$ symmetry, with two $\pi$-rotations in the pseudo-spin space $\e^{\pi S_z}$ and $\e^{\pi S_x}$. Here, $S_z$ is defined as above and a corresponding $\e^{\pi S_x}$ maps the states $\alpha\to\beta$ and $0_\pm\to \tilde{0}_\pm$. For the trial-state of Eq.~\eqref{eq:lgt1dMPS} we may now define, following the discussion of Pollmann and Turner in Ref.~\cite{pollmann2012symmetry}, a generalized order parameter $\mathcal{O}_{Z_2\times Z_2} = \tr\left(U_x U_z U_x^\dagger U_z^\dagger \right) / 2$. The 2-dimensional representations of the symmetry group corresponding to the MPS ansatz $U_x$ and $U_z$ are derived from a generalized transfer matrix and are given by by the actual Pauli-matrices $\sigma_z$ and $\sigma_x$. With this we find $\mathcal{O}_{Z_2\times Z_2}=-1$ indicating the topologically non-trivial character of the phase.
%%
\begin{figure*}[tb]
%
\begin{minipage}[t]{.03\linewidth}
\raisebox{3.8cm}[0cm][0cm]{(a)}
\end{minipage}
\begin{minipage}[t]{.26\linewidth}
\includegraphics[height=4cm]{entanglement_spectrum_QLL_TwoLegs_OneRung_SpinHalf_DMRG_upup_L200_chi200.pdf}
\end{minipage}
%
\begin{minipage}[t]{.03\linewidth}
\raisebox{3.8cm}[0cm][0cm]{(b)}
\end{minipage}
\begin{minipage}[t]{.26\linewidth}
\includegraphics[height=4cm]{entanglement_spectrum_QLL_ThreeLegs_OneRung_SpinHalf_DMRG_updown_m0.00_L100_chi200.pdf}
\end{minipage}
%
\begin{minipage}[t]{.03\linewidth}
\raisebox{3.8cm}[0cm][0cm]{(c)}
\end{minipage}
\begin{minipage}[t]{.26\linewidth}
\includegraphics[height=4cm]{entanglement_spectrum_QLL_FourLegs_OneRung_SpinHalf_DMRG_upup_L100_chi100.pdf}
\end{minipage}
%
\caption{Mid-system entanglement spectrum of the QLM Eq.~(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).}
\label{fig:S_entanglement}
\end{figure*}
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{2-leg ladder}
......@@ -189,7 +190,6 @@ For $\mu=0$ a symmetry protected topological phase~(SPT) emerges. This phase can
\begin{minipage}[t]{.7\linewidth}
\includegraphics[width=.99\linewidth]{nolabel_parity_order_string_order_QLM_TwoLegs_DMRG_OBC_n1.00_k0.00_m0.00_L100_chi200_leg0_gamma_list_span20.pdf}
\vspace{1pt}
% \includegraphics[width=.99\linewidth]{parity_order_string_order_QLM_TwoLegs_DMRG_OBC_n1.00_k0.00_m0.00_L100_chi200_leg0_gamma_list_span20.pdf}
\end{minipage}
%
......@@ -199,7 +199,6 @@ For $\mu=0$ a symmetry protected topological phase~(SPT) emerges. This phase can
\begin{minipage}[t]{.7\linewidth}
\includegraphics[width=.99\linewidth]{nolabel_parity_order_string_order_QLM_ThreeLegs_DMRG_OBC_n1.50_k0.00_m0.00_L100_chi200_leg0_gamma_list_span20.pdf}
\vspace{1pt}
% \includegraphics[width=.99\linewidth]{parity_order_string_order_QLM_ThreeLegs_DMRG_OBC_n1.50_k0.00_m0.00_L100_chi200_leg0_gamma_list_span20.pdf}
\end{minipage}
%
......@@ -207,7 +206,6 @@ For $\mu=0$ a symmetry protected topological phase~(SPT) emerges. This phase can
\raisebox{3.1cm}[0cm][0cm]{(c)}
\end{minipage}
\begin{minipage}[t]{.7\linewidth}
% \includegraphics[width=.99\linewidth]{nolabel_parity_order_string_order_QLM_FourLegs_DMRG_OBC_n2.00_k0.00_m0.00_L100_chi100_leg0_gamma_list_span20.pdf}
\includegraphics[width=.99\linewidth]{parity_order_string_order_QLM_FourLegs_DMRG_OBC_n2.00_k0.00_m0.00_L100_chi100_leg0_gamma_list_span20.pdf}
\end{minipage}
%
......@@ -282,37 +280,35 @@ 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}
\includegraphics[width=.8\linewidth]{supmat_figx_energy_QLM_FourLegs_PBC_m0_L40_logscale.pdf}
%
\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.
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).
$E_0$ 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}
%
\begin{figure}[tb]
\includegraphics[width=.99\linewidth]{gap_legs.pdf}
\caption{Excitation gap $\Delta \epsilon$ (in units of $J_x$) for the (a) 1D toy model, (b) 2-leg (c) 4-leg ladder model within the vacuum gauge-sector at $\mu=0$ for different system sized $L=12$ (crosses), $L=24$ (circle), $L=36$ (plus-symbols). The dotted lines indicate the estimated phase-transition position from Tab.~\ref{tab:S_transitions}.
(d) Excitation gap as function of the number of legs for $J_y / J_x=1$ (center of the D-phase) and $J_y/J_x=1.8$ (Sy-phase).}
\label{fig:gap_legs}
\end{figure}
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.
This observation can be understood from the calculation of excitation gap $\Delta\epsilon$ as depicted in Fig.~\ref{fig:gap_legs}. Here we show for various geometries and models discussed in the main text the energy gap to the first excited state within the same gauge sector as the ground-state. The data clearly shows that the D-phase exhibits a large excitation gap $\Delta\epsilon \sim 0.4 J_x$.
Interestingly, this value stays remarkably constant changing the number of legs.
For $J_x/J_y \to 0$ this gap decreases. For the 4-leg ladder system, with partially restored x-y symmetry this is also the case for $J_y/J_x\to 0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Extended data on the string tension}
......
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