@@ -73,7 +72,7 @@ with fermionic matter within experimental reach for quantum simulations.

Whereas in the large mass limit we observe Ne\'el-like vortex-antivortex and striped crystalline phases,

for small masses there is a transition from the striped phases into a disordered phase whose properties resemble those at the Rokhsar-Kivelson point of the quantum dimer model.

This phase is characterized on ladders by boundary Haldane-like properties, such as vanishing parity and finite string ordering.

Moreover, from studies of the string tension between gauge charges, we find that whereas the stripe phases are confined, the novel disordered phase present clear indications of being deconfined.

Moreover, from studies of the string tension between gauge charges, we find that whereas the striped phases are confined, the novel disordered phase present clear indications of being deconfined.

Our results open exciting perspectives of studying highly non-trivial physics in quantum simulators, such as spin-liquid behavior and confinement-deconfinement transitions,

without the need of explicitly engineering plaquette terms.

\end{abstract}

...

...

@@ -89,7 +88,7 @@ Driven by tremendous progresses in the manipulation and control of ultracold qua

with the long term goal of studying open problems of the early universe, dense neutron stars, nuclear physics or condensed-matter physics~\cite{Kogut1983, Levin2005, Wiese2013}.

Many theoretical proposals~\cite{Cirac2010,Zohar2011,Kapit2011,Zohar2012,Banerjee2012,Zohar2013,Banerjee2013,Zohar2013b,Tagliacozzo2013,Stannigel2014,Kasper2016,Cardarelli2017, Zache2018, barbiero2018} and recent seminal experiments with trapped ions~\cite{Martinez2016}, quantum dimer models in Rydberg-atoms-arrays~\cite{Bernien2017}, lattice modulation techniques~\cite{Clark2018, Goerg2019, schweizer2019}, or atomic mixtures~\cite{Mil2019} have shown first building-blocks of dynamical gauge fields and quantum link models~(QLMs), a generalization of LGT to spin-like link-variables~\cite{Chandrasekharan1997}.

However, the implementation of some building blocks of LGT, such as the ring-exchange corresponding to magnetic field dynamics in analogue implementations of quantum electrodynamics,

require further theoretical and experimental breakthroughs, although there has been progress on isolated plaquettes~\cite{dai2017four} and recent promising proposals~\cite{Celi2019, Bohrdt2019}.

requires further theoretical and experimental breakthroughs, although there has been progress on isolated plaquettes~\cite{dai2017four} and recent promising proposals~\cite{Celi2019, Bohrdt2019}.

% Here there is also a paper of Marcello (Surace et al) on the re-interpretation of Rydberg experiments. We should cite that.

In this paper we show how already the simplest mid-term experimental realizations, without plaquette terms, may be able to explore a wide area of non-trivial phenomena of LGTs. In particular, we report in this Letter that

...

...

@@ -143,13 +142,13 @@ Points depicts the estimated phase transition points, by extrapolating the peak

\paragraph{2D QLM.--} We consider in the following a QLM on a square lattice described by the Hamiltonian

\begin{align}

H = - \sum_{\vR\vRp} J_{\vR\vRp}\left( \psi_\vR^\dagger S^+_{\vR\vRp}\psi_{\vRp} + {\rm h.c.}\right)

H = - \sum_{\vR,\vRp} J_{\vR,\vRp}\left( \psi_\vR^\dagger S^+_{\vR,\vRp}\psi_{\vRp} + {\rm h.c.}\right)

+ \sum_\vR\mu_\vR n_\vR,

\label{eq:QLM}

\end{align}

% mu_r=mu for A and -mu for B. For mu>>0 piling on B, this means that B goes with -mu

where $\psi_\vR^\dagger$ is the fermionic operator at site $\vR$, and $n_\vR=\psi_\vR^\dagger\psi_\vR$.

In our case the gauge field is given by spin-$1/2$ operators $S^+_{\vR\vRp}$ placed at the link between sites $\vR$ and $\vRp$. The

In our case the gauge field is given by spin-$1/2$ operators $S^+_{\vR,\vRp}$ placed at the link between {\color{red}{two neighbouring}} sites $\vR$ and $\vRp$. The

amplitudes $J_{\vR,\vR\pm\vX}=J_x$ and $J_{\vR,\vR\pm\vY}=J_y$ characterize, respectively, the hops along the $x$ and $y$ directions~(Fig.~\ref{fig:sketch}~(a)).

We enforce at each site the Gauss law $[H, G_\vR]=0$ with $G_\vR=\varepsilon_\vR- n_\vR+\sum_{\vec{k}\in\vX,\vY}( S^z_{\vR,\vR+\vec{k}}- S^z_{\vR,\vR-\vec{k}})$,

with $\varepsilon_{\vR\in A}=1$ and $\varepsilon_{\vR\in B}=0$.

...

...

@@ -160,7 +159,7 @@ by means of density matrix renormalization group~(DMRG) techniques~\cite{Schollw

We introduce at this point the ring-exchange operators $R^+_\vR= S_{\vR,\vR+\vX}^+ S_{\vR+\vX,\vR+\vX+\vY}^- S_{\vR+\vX+\vY,\vR+\vY}^- S_{\vR+\vY,\vR}^+$ and $R_\vR^-=(R_\vR^+)^\dagger$.

These operators characterize plaquette states: $R_\vR^-$~($R_\vR^+$) flips a vortex~(antivortex) into an antivortex~(vortex), being zero otherwise,

and $Q_{\vR}=(R_\vR^+R_\vR^-- R_\vR^-R_\vR^+)=1$~(-1) for vortex~(antivortex) and $Q_{\vR}=0$ otherwise~\cite{Arrows}.

%, and $R_\vR^2 = Q_\vR^2$ is $1$~($0$) for flippable~(unflippable) plaquettes.

%, and $R_\vR^2 = Q_\vR^2$ is $1$~($0$) for flippable~(unflippable) plaquettes.

%Note, however, that in contrast to QED realizations or to the quantum dimer model~(QDM), they do not appear explicitly in the Hamiltonian~\eqref{eq:QLM}.

...

...

@@ -171,7 +170,9 @@ and $Q_{\vR}=(R_\vR^+R_\vR^- - R_\vR^-R_\vR^+)=1$~(-1) for vortex~(antivortex) a

{\em Large mass limit.--} First insights are obtained from the limit $|\mu| \gg J_{x,y}$, which, in contrast to the two-leg QLM~\cite{supmat}~\nocite{aklt, Pollmann2010, Pollmann2012, Pollmann2012A}, is different

for $\mu>0$ and $\mu<0$. For $\mu>0$, particles

are pinned in B sites~(Figs.~\ref{fig:sketch}~(b) and (c)).

Local states are characterized by the expectation value of two spin-$1$ operators, $S^z_{\vec{k}}(\vR)= S^z(\vR-\vec{k}, \vR)+ S^z(v,\vR+\vec{k})$, with $\vec{k}=\vX,\vY$.

Local states are characterized by the expectation value of two spin-$1$ operators, $S^z_{\vec{k}}(\vR)= S^z(\vR-\vec{k}, \vR)+$

{\color{red}{$S^z(\vR,\vR+\vec{k})$}},

with $\vec{k}=\vX,\vY$.

For $J_x<J_y$, second-order terms select a ground-state manifold of two states with $S^z_y(\vR)=0$. Fourth-order ring-exchange $\propto J_x^2 J_y^2/ |\mu|^3$~\cite{Cardarelli2017}

favors a configuration of columns of flippable vortex-antivortex and non-flippable plaquettes~(Fig.~\ref{fig:sketch}~(b)). We denote this striped phase Sy.

In a 2D model, a corresponding striped phase Sx of alternating flippable and non-flippable rows of plaquettes is expected for $J_y<J_x$.

...

...

@@ -197,7 +198,7 @@ $\la Q_{\vR} \ra$ and $\la Q_{\vR}^2 \ra$, but a large expectation value of the

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 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).

...

...

@@ -213,14 +214,14 @@ be $0.97$~(see~\cite{supmat} for a more detailed comparison between the DMRG sim

\caption{(a) Parity and string order along the boundary legs of a four-leg ladder, obtained using DMRG for $L_x=100$ rungs and $\mu=0$; (b) largest eigenvalues of the entanglement spectrum, obtained

after dividing the system into two parts along the central rung.}

...

...

@@ -247,7 +248,7 @@ $\beta \to \alpha, \tilde{0}$~(we remove the $\pm$ index). By construction, Gaus

an arbitrary number of intermediate $0$ or $\tilde{0}$ states.

exhibits three ground-state phases (here we neglect a staggered potential term). For $J_y\ll J_x$ the ground state is $\cdots |\alpha\ra |\beta\ra |\alpha\ra |\beta\ra\cdots$, whereas for $J_y\gg J_x$