Commit 26da5e15 authored by Sebastian Greschner's avatar Sebastian Greschner

modified: fig2.pdf

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	modified:   lgt2_paper.tex
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......@@ -34,7 +34,7 @@
\citation{Cardarelli2017}
\citation{supmat}
\citation{supmat}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Four-leg cylinder at $\mu =0$: (a) Nearest and next-nearest neighbor flippability correlations and (b) fidelity susceptibility with $L=12$, $24$ and $36$ rungs. (c) Local Hilbert space distribution $\nu _k$ of the central rung\nobreakspace {}(see text). (d) Sketched phase diagram of the QLM. Color codes depict the von-Neumann bipartite entanglement entropy $S_{vN}$ of the central rung. Points depicts the estimated phase transition points, by extrapolating the peak positions of the fidelity susceptibility evaluated along the cuts indicated by the dotted lines (DMRG data).}}{2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Four-leg cylinder at $\mu =0$: (a) Nearest and next-nearest neighbor flippability correlations, staggered flippability $O(Sy) = \DOTSB \sum@ \slimits@ _\mathbf {r}(-)^x \delimiter "426830A Q_{\mathbf {r}}^2 \delimiter "526930B $, as well as the expectation value of the ring-exchange $\delimiter "426830A R^+_{\mathbf {r}} \delimiter "526930B $. (b) Fidelity susceptibility with $L=12$, $24$ and $36$ rungs. (c) Local Hilbert space distribution $\nu _k$ of the central rung\nobreakspace {}(see text). (d) Sketched phase diagram of the QLM. Color codes depict the von-Neumann bipartite entanglement entropy $S_{vN}$ of the central rung. Points depicts the estimated phase transition points, by extrapolating the peak positions of the fidelity susceptibility evaluated along the cuts indicated by the dotted lines (DMRG data).}}{2}{}}
\newlabel{fig:4t_cuts_mu0}{{2}{2}{}{}{}}
\citation{supmat}
\citation{Cardarelli2017}
......@@ -48,7 +48,6 @@
\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}}{{}}}
......@@ -73,10 +72,11 @@
\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}}}
\bibcite{schweizer2019}{{23}{2019}{{Schweizer\ \emph {et~al.}}}{{Schweizer, Grusdt, Berngruber, Barbiero, Demler, Goldman, Bloch,\ and\ Aidelsburger}}}
\@writefile{toc}{\contentsline {paragraph}{\numberline {}String tension.--}{4}{}}
\@writefile{toc}{\contentsline {paragraph}{\numberline {}Conclusions.--}{4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {}Acknowledgments}{4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {}References}{4}{}}
\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}}{{}}}
\bibcite{dai2017four}{{26}{2017}{{Dai\ \emph {et~al.}}}{{Dai, Yang, Reingruber, Sun, Xu, Chen, Yuan,\ and\ Pan}}}
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......@@ -125,8 +125,8 @@ $\la n_{\vR}\ra$, the arrow size $\la S_{\vR,\vRp}^z\ra$, and the plaquette colo
\centering
\includegraphics[width=0.99\linewidth]{fig2.pdf}
\caption{Four-leg cylinder at $\mu=0$:
(a) Nearest and next-nearest neighbor flippability correlations and
(b) fidelity susceptibility with $L=12$, $24$ and $36$ rungs.
(a) Nearest and next-nearest neighbor flippability correlations, staggered flippability $O(Sy) = \sum_\vR (-)^x \la Q_{\vR}^2 \ra$, as well as the expectation value of the ring-exchange $\la R^+_{\vR} \ra$.
(b) Fidelity susceptibility with $L=12$, $24$ and $36$ rungs.
(c) Local Hilbert space distribution $\nu_k$ of the central rung~(see text).
(d) Sketched phase diagram of the QLM. Color codes depict the von-Neumann bipartite entanglement entropy $S_{vN}$ of the central rung.
Points depicts the estimated phase transition points, by extrapolating the peak positions of the fidelity susceptibility evaluated along the cuts indicated by the dotted lines (DMRG data).}
......@@ -196,11 +196,6 @@ Whereas for $\mu=0$ for $J_y \ll J_x$~($J_y \gg J_x$) the system is in the Sx~(S
$\la Q_{\vR} \ra$ and $\la Q_{\vR}^2 \ra$, but a large expectation value of the ring-exchange $\la R^+_{\vR} \ra$.
%For a 2D QLM we expect that the critical $J_y/J_x$ for the transition between the intermediate and the Sy phase coincides with $J_x/J_y$ for transition between the intermediate and the Sx phase.
%Although $L_y\ll L_x$, we observe that such a symmetry is approximately fulfilled already for the four-leg ladder case.
%an approximate symmetry between the phase transition points to the Sx and Sy phases, $J_y^c(Sx) \approx 1/J_y^c(Sy)$ .
%As sketched in Fig.~\ref{fig:sketch}~(f), the average spin-configuration of this phase exhibit an alternating pattern of 4 in- or 4 outwards pointing spins (which seemingly violates Gauss' law). While, the expectation value of the flippability $\la R^2_{\vR} \ra$, and $\la Q_{\vR} \ra$ are suppressed, the phase is characterized by large expectation value of the ring-exchange operator $\la R^+_{\vR} \ra$.
A crucial insight on the physics of the intermediate phase is provided by the analysis of the reduced density matrix $\rho_c = \tr |\Psi_0\ra\la \Psi_0|$ for the central rung~(where the trace runs over all other rungs)
in the~(Fock-like) eigenbasis $\phi_k$ of $S^z_{\vR\vRp}$ and $n_{\vR}$.
In Fig.~\ref{fig:4t_cuts_mu0}~(c) we show its diagonal elements $\nu_k = \la \phi_k | \rho_c | \phi_k \ra$, an effective local Hilbert space distribution, sorted by amplitude, for the case of a four-leg cylinder. The Sx and Sy phases are strongly localized in Fock space, i.e. $\nu_k$ has most weight for few basis states. The intermediate phase, however, exhibits a drastically different, much flatter distribution, where many local Fock states contribute with similar weight. The disordered character of the intermediate phase is also witnessed by the entanglement entropy $S_{vN}=-\tr(\rho_c \ln \rho_c)$, which we depict in Fig.~\ref{fig:4t_cuts_mu0}~(d).
......
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