\section{Details on Exact Diagonalization Study of the Phase Diagram}\label{app:ED}


For a thorough investigation of the model Hamiltonian, we desire (i) to identify the possible quantum phases at zero temperature, (ii) to locate the phase boundaries (i.e., critical points) of the model accurately in the thermodynamic limit, and (iii) to extract critical exponents at continuous phase transitions. 


However, in numerics we are limited to simulating only finite size systems. Thus in order to locate phase boundaries precisely, finite-size scaling of order parameters~\footnote{More precisely, one investigates the finite-size scaling behaviors of an \emph{squared} order parameter $O$ detecting symmetry-breaking, since $\expval{O}=0$ in the ground state of a finite size system, which is symmetric.} or derivatives of energy are often employed. In our case, we can alternatively leverage the fact that the critical points we are analyzing are between quantum phases with different symmetry properties,
like zAFM-VBS, which spontaneously break spin-flip and site inversion respectively, and zAFM-XY, the latter of which does not break any symmetries but exhibits  {characteristic power-law correlations}. 
In such situations, the so-called method of level spectroscopy~\cite{Nomura_1994, OshikawaKT2021} can be employed. 



The key idea is to determine the phase transition point in the thermodynamic limit, by extrapolating from the level crossing of first excited states in finite-size systems, see \figref{fig:schematic}.
The crucial point is that unlike the ground state obtained in finite size systems which is generally symmetric across parameter space, the numerically obtained first excited states from each side of symmetry broken (or quasi-long-range ordered) phases belong to different symmetry sectors, i.e., they transform differently under different symmetry generators of the system.
Therefore, the first excited states from two sides of the phase do not exhibit an avoided level-crossing, 
and instead directly cross.
We can identity such a crossing point as the `critical point' of a finite size system, see \figref{fig:schematic}. 
Then, by performing a finite size scaling of the locations of these crossing points, we can extract the  transition point in the thermodynamic limit with very good accuracy, which often tends to converge very fast with system size.
In the following sections, we elaborate on the method  as well as its theoretical foundations~\cite{DiFrancesco:1997nk}. Finally, we  argue why this method is particularly useful for analyzing the BKT phase transition~\cite{BKT_1, BKT_2}, and also for DQC.















\begin{figure}[!t]
    \centering
\includegraphics[width = 0.6 \textwidth]{fig_schematic.pdf}
    \caption{\label{fig:schematic} Schematic diagram of exact energy eigenvalues for the ground state $\ket{\psi_g}$ as well as the first and second excited states,  $\{ \ket{\psi_e}, \ket{\psi_e'}\}$ across the deconfined quantum critical point $t_c(L)$ in  finite size  systems of size $L$, assuming periodic boundary conditions. Note that at finite size, there is a gap scaling as $1/L$ at the critical point. Although the phases on either side of the phase transition (zAFM and VBS) 
    are expected to spontaneously-break (different) symmetries, note that the ground state $\ket{\psi_g}$ is always unique and hence symmetric.
    For example, deep in the zAFM phase, the ground state is well-captured by the symmetric superposition of two antiferromagnetic classical configurations (i.e., a ``cat state''). Deep in the VBS, the ground state is well-captured by the symmetric superposition of two dimerized singlet patterns (also a ``cat state'').
    One sees that in both instances, $\ket{\psi_g}$ transforms trivially under both spin-flip transformation $\mathbb{Z}_2^x$ and site-centered inversion transformation $\cal I$, and in fact, this property holds true for all values of the tuning parameter $\alpha$.
    Instead, the signature of spontaneous symmetry-breaking shows up in the  excited states: 
    $\ket{\psi_e} \mapsto - \ket{\psi_e}$ under the spin-flip transformation while it remains invariant under site-centered inversion; and conversely $\ket{\psi_e'} \mapsto - \ket{\psi_e'}$ under site-centered inversion but remains invariant under the spin-flip transformation. Therefore, all three states belong to different symmetry sectors. Importantly, there  is therefore necessarily a direct (i.e., non-avoided) crossing between the excited states $\ket{\psi_e}$ and $\ket{\psi_e'}$, whose energy difference to $\ket{\psi_g}$ should go as $\sim 1/L$ as expected at a continuous phase transition; an emergent $U(1)$ symmetry of the DQC implies that this crossing point occurs at the true critical point in the thermodynamic limit. }
\end{figure}


\subsection{Level Spectroscopy Method from State-Operator Correspondence}


The level spectroscopy method can be rigorously understood from conformal field theory (CFT), a long-wavelength description 
appropriate for many systems at criticality.
Consider a one-dimensional periodic system with size $L$ and the CFT Hamiltonian $H_\textrm{CFT}$, a lattice Hamiltonian \emph{exactly} realizing a certain conformal field theory. The celebrated \emph{state-operator correspondence}~\cite{DiFrancesco:1997nk} posits that the energy spectrum of $H_\textrm{CFT}$ is related to the scaling dimensions of the operators appearing in the CFT  via the following relation:
\begin{equation} \label{eq:CFT_correspondence}
    H_\textrm{CFT} \ket{\psi_\alpha} = E^\textrm{CFT}_\alpha \ket{\psi_\alpha} \quad \rightarrow \quad 
    E^\textrm{CFT}_\alpha  = \frac{2\pi v}{L} \qty(\Delta_\alpha-\frac{c}{12}),
\end{equation}
where $\ket{\psi_\alpha}$ is an eigenstate with label $\alpha$, $v$ is the speed of gapless degrees of freedom, $c$ is the central charge, and $\Delta_\alpha$ is the scaling dimension of the corresponding operator. From this one sees that the hierarchy of the excited energy spectrum of critical Hamiltonians with periodic boundary conditions directly corresponds to the hierarchy of the scaling dimensions of the operators of the corresponding conformal field theory. 



However, this relation is only for the \emph{exact} CFT Hamiltonian. In practice, a generic lattice Hamiltonian $H_\textrm{lattice}$ at criticality does not realize the exact CFT Hamiltonian, but instead is the sum of $H_\textrm{CFT}$ and extra irrelevant perturbations $\lambda \int \dd{x} O_\textrm{CFT}(x)$, where $O_\textrm{CFT}(x) = \sum_\beta a_\beta \psi_\beta(x)$ and $\psi_\beta(x)$ is an operator with scaling dimension $\Delta_\beta$. Such perturbations give a correction to \eqnref{eq:CFT_correspondence} for finite-size systems, which can be calculated as such:
\begin{equation}
    E^\textrm{lattice}_\alpha  = E_\alpha^\textrm{CFT} + \lambda \int \dd{x} \bra{\psi_\alpha} O_\textrm{CFT}(x) \ket{\psi_\alpha}.
\end{equation}
The expectation value of operators within states can be obtained through the so-called operator product expansion (OPE), which yields\footnote{This follows from a radial quantization in the CFT through a logarithmic mapping which transforms the plane to the cylinder.} 
\begin{equation}
    \bra{\psi_\alpha} \psi_\beta \ket{\psi_\alpha} = C_{\alpha \beta \alpha}  \qty(\frac{2\pi}{L})^{\Delta_\beta},
\end{equation}
where $C_{\alpha\beta\alpha}$ is the so-called OPE coefficient, which is of order one. We  see that the leading correction to $E_\alpha^\textrm{lattice}$ at large $L$ is dominated by the smallest $\Delta_\beta$  among the symmetry-allowed irrelevant perturbations. After performing the integration, we obtain
\begin{equation} \label{eq:cft_lattice}
    E_\alpha^\textrm{lattice} = \frac{2\pi v}{L} \qty(\Delta_\alpha-\frac{c}{12}) + \frac{C_{\alpha \beta}}{L^{\Delta_\beta - 1}}.
\end{equation}
where the proportionality constant $C_{\alpha \beta}$ depends on the OPE coefficient $C_{\alpha \beta \alpha}$ and the magnitude of perturbation $\lambda$.
Now, assume that we perturb the system away from the critical point by tuning operator $O_t$ carrying scaling dimension $\Delta_t$, with a coefficient strength $\bar{t} = (t-t_c)$ where $t_c$ is the true transition point at thermodynamic limit (which we desire to determine). Let the energy correction from  this tuning operator be $K_{\alpha t}(L) \bar{t}/L$. Then we get
\begin{equation}
    E_\alpha^\textrm{lattice}(t,L) = \frac{2\pi v}{L} \qty(\Delta_\alpha-\frac{c}{12}) + \frac{C_{\alpha \beta}}{L^{\Delta_\beta - 1}} + \frac{K_{\alpha t}(L)}{L} \bar{t} \equiv \frac{2\pi v}{L} \qty(\Delta^\textrm{lattice}_\alpha(t,L)-\frac{c}{12}).
\end{equation}




Now that we understand the energy spectrum of the lattice Hamiltonian at criticality (assumed described by CFT), we can use this structure to identify the critical point $t_c$. The method is applicable for  critical points with two operators $O_a$ and $O_b$, such that (i) they have the same scaling dimensions $\Delta_a = \Delta_b$, and (ii) their corresponding states in the state-operator correspondence separate out in energy  as we tune away from the criticality. If these two conditions are met, their energy crossing point $t$ in a finite size system $L$ can be determined as such: 
\begin{equation}
    \Delta^\textrm{lattice}_a(t,L) =  \Delta^\textrm{lattice}_b (t,L) \quad \Rightarrow \quad t =t_c - \frac{C_{a \beta} - C_{b \beta}}{K_{a t} - K_{b t}} \frac{1}{L^{\Delta_\beta - 2}}.
\end{equation}
In general, $K_{ab}(L) \equiv K_{a t} - K_{b t}$ and $C_{ab}(L) \equiv C_{a \beta} - C_{b \beta}$ are functions of $L$, which would provide a further correction. Perturbative CFT arguments imply that the $L$ dependence of $K$ is determined by the scaling dimension of tuning operator $\Delta_t$ ($\Delta_t < 2$) such that $K_{ab}(L) \sim L^{2-\Delta_t}$. Therefore, the energy crossing point $t$ at system size $L$ is given by 
\begin{equation} \label{eq:finite_scaling}
    t(L) = t_c - \frac{T(L)}{L^{\Delta_\beta - \Delta_t}}, \qquad T(L) \equiv C_{ab}(L) \frac{L^{2-\Delta_t}}{K_{ab}(L)}.
\end{equation}
In most cases, $C_{ab}(L) \sim \textrm{const}$ and $K_{ab}(L) \sim L^{2-\Delta_t}$ and therefore $T(L)$ can be taken as a fixed constant. However, if the CFT has a marginally irrelevant perturbation, $C_{ab}(L)$ and $T(L)$ can acquire a logarithmic dependence on $L$. In practice, for small systems we study here the logarithmic dependence is so small that its presence barely affects the precise determination of $t_c$. 
Generally, there is more than one irrelevant operator in the lattice critical Hamiltonian, and the finite-size correction will contain contributions from all of them.
To summarize, assuming a phase transition to be identified has an underlying CFT description and two operators with the same scaling dimensions but which move oppositely in energy upon tuning away from criticality, one can precisely locate the critical points using \eqnref{eq:finite_scaling}.

Given the relation in Eq.~\ref{eq:finite_scaling}, we can perform the finite size scaling in the following way.
For a given system size, we extract the level-crossing point $t(L)$ of the first excited states. We choose the operators $O_\alpha$ and $O_\beta$ corresponding to the operators with different quantum numbers with the same (lowest) scaling dimension. Then, using the first expression in  Eq.~\ref{eq:finite_scaling}, we fit the extracted $t(L)$ using the fitting function $t(L) = t_c - a L^{-b}$ with three fitting parameters $t_c$, $a$ and $b$.



\begin{figure}[!t]
    \centering
    \includegraphics[width = 1 \textwidth]{kt_scaling.pdf}
    \vspace{-10pt}
    \caption{\label{fig:KT_data} Behavior of the system for $\alpha=0.2$ and $J_3/J_6 \in [1.0,1.4]$. (a) Second derivative of energy. (b) zAFM order parameters under iDMRG simulations with   bond dimension $\chi=70,100,200,300,400$ (bond dimension increasing with darkness). Note that unlike at the DQCP, neither the order parameter nor the second derivative of energy changes sharply at the transition.
    (c) Correlation functions $C_{XY}$, $C_\textrm{zAFM}$ and $C_\textrm{VBS}$ at $(J_{3}/J_6,\alpha) = (1.435,0.2)$ corresponding to the XY-QLRO phase. All correlation functions exhibit power-law behavior, with the XY correlation function decaying the slowest.
    (d) Low-lying spectrum (i.e., eigenvalues) of $H_\textrm{eff}$ for system size $L=20$. We use the crossing between two energies $W_0^\textrm{triplet}$ and $W_{\pm 1}^\textrm{triplet}$ to extract the location of the BKT transition.
    (e) Finite size scaling of the crossing point $J_c(L) \equiv J_3/J_6$ at $\alpha=0.2$ between the energies corresponding to zAFM and XY operators with $L=16,18,20,22,24$. 
    In order to verify our expectation on leading irrelevant operators, we fitted with a range of $\gamma\equiv \Delta_\beta - \Delta_t$ in \eqnref{eq:finite_scaling}.
    The curve with $\gamma=2$ fits the best, which is the value expected for the BKT transition, giving $J_c(L\rightarrow \infty) = 1.19114(2)$. Logarithmic corrections can modify this value only slightly.
    }
\end{figure}





\subsection{BKT transitions}

To identify the possible quantum phases of the model Hamiltonian as well as their phase transitions, we performed iDMRG simulations, tracking order parameters across different parameter cuts. However, the phase transitions out of the gapless XY phase (either into the zAFM or VBS phase) pose some challenges. In \figref{fig:KT_data}a,b, we plot the second derivative of the ground state energy and behavior of the order parameter across the phase transition between the XY-QLRO phase and zAFM phase. We note that both the second derivative of energy and the order parameter behave smoothly across the phase transition.


In fact, such smooth behavior of the energy and order parameter is consistent with the Berezinskii-Kosterlitz-Thouless~(BKT) transitions~\cite{BKT_1,BKT_2}, a topological phase transition from a gapped to a gapless phase. BKT transitions are famous for their  logarithmic corrections as well as being a transition with \emph{infinite smoothness}, which makes it difficult to identify their precise location based on conventional methods developed for second-order continuous phase transitions. 
However, it turns out that the level spectroscopy method combined with scaling of the finite size crossing points provides a great solution to identify the phase boundaries accurately, due to the cancellation of  logarithimic corrections, which we will elaborate on below.







Let us first understand the nature of  logarithmic corrections at a BKT transition. At such a transition, the presence of a marginally irrelevant operator generates an anomalous dimension for the operators whose corresponding states appear in the energy spectrum. 
Consider the following perturbation $H'$ on top of the Gaussian Hamiltonian, appearing in the continuum description of our model~\eqnref{eq:critical_ham}:
\begin{equation}
H' = \int \frac{\dd x}{2 \pi} \qty[-y_K \qty( (\partial_t \phi)^2 + (\partial_x \phi)^2 ) + y_V \cos 4\phi ].
\end{equation}
The coefficients $y_K$ and $y_V$ appear in front of  marginal operators and flow under RG according to 
\begin{equation}
    d y_K/d l = - y_V^2, \quad d y_V/dl = - y_K y_V, \quad l = \log L, 
\end{equation}
where $L$ is the length scale of the RG flow. In order to understand the BKT transition, we introduce new variables $s = y_K - \eta y_V$ and $t = y_K + \eta y_V$ such that the BKT transition is characterized by $s=0$ (Note that there are two such boundaries corresponding to $\eta = \pm 1$. For $y_V >0$ the transition is at $\eta = 1$ and for $y_V < 0$ the transition is at $\eta = -1$). 
At the BKT transition, $t$ can still be non-zero; in fact, it obeys the following RG flow: 
\begin{equation}
    d t/dl = - t^2/2 \quad \Rightarrow \quad t \sim 1/\log L. 
\end{equation}
This implies that even if $s=0$, $\abs{y_{K,V}} \sim 1/\log L$, which vanishes slowly as $L \rightarrow \infty$ but nonetheless gives a significant correction to the physical quantities at finite $L$. (Such a correction can be considered as a shift in the Luttinger parameter from the value $1/2$ in our model; note that for a classical BKT transition, $y_V$ is for the perturbation $\cos 2 \phi$ and the Luttinger parameter at the transition is 2.)


In terms of $y_K$ and $y_V$, the scaling dimensions of the operators corresponding to the VBS, zAFM, and XY order parameters are~\cite{OshikawaKT2021}:
\begin{align} \label{eq:log_correction}
    \Delta_{XY} &\approx \frac{1}{2} - \frac{1}{4} y_K \nonumber \\
    \Delta_\textrm{zAFM} &\approx \frac{1}{2} + \frac{1}{4} y_K + \frac{1}{2} y_V \nonumber \\
    \Delta_\textrm{VBS} &\approx \frac{1}{2} + \frac{1}{4} y_K - \frac{1}{2} y_V,   
   
   
\end{align}
where $\Delta_{XY} \equiv \Delta_{e^{\pm i \theta}}$, $\Delta_{zAFM} \equiv \Delta_{\cos 2\phi}$, and $\Delta_{VBS} \equiv \Delta_{\sin 2\phi}$ as defined in Section \ref{app:bosonization}. 
As $y_{K,V} \sim 1/\log L$, for any finite size system scaling dimensions acquire significant corrections, and so do corresponding excited state energies in \eqnref{eq:CFT_correspondence}. 
However, a careful examination shows that for $\eta=1$ ($y_K = y_V \sim 1/\log L$), anomalous dimensions for $\Delta_{XY}$ and $\Delta_\textrm{VBS}$  match, and for $\eta = -1$ ($y_K = - y_V$), anomalous dimensions for $\Delta_{XY}$ and $\Delta_\textrm{zAFM}$ match:
\begin{align}
       
       
                &y_K = y_V \quad \Rightarrow \quad \Delta_{VBS} = \frac{1}{2}  - \frac{1}{4} y_V = \Delta_\textrm{XY} \quad \textrm{(VBS to XY-QLRO transition)} \nonumber\\
        &y_K = -y_V \quad \Rightarrow \quad \Delta_{zAFM} = \frac{1}{2}  + \frac{1}{4} y_V = \Delta_\textrm{XY} \quad \textrm{(zAFM to XY-QLRO transition)} 
\end{align}
This matching is not a coincidence, but is rather due to the enhancement of the $U(1)$ symmetry to $\mathrm{SU}(2)$ at the BKT transition. For the zAFM to XY-QLRO transition, the zAFM and X, Y components of the XY order parameter transform as a vector under this symmetry. Similarly, for the VBS to XY-QLRO transition, the VBS and XY order parameters form a vector. At the transition point all excited states can be labeled by an $\mathrm{SU}(2)$ spin quantum number $S$; we denote by $V$ the singlet $S=0$, and by $W$ the triplet $S = 1$.  Therefore, even in the presence of logarithmic corrections,
the crossing for the pair of (zAFM, XY) energy levels can be identified as the BKT transition point for the zAFM-XY transition, and the crossing for the pair of (VBS, XY) energy levels can be identified as the BKT transition point for the VBS-XY transition.


In \figref{fig:KT_data}d, we numerically plot the energy levels through the zAFM-XY transition (for a particular sweep), where we observe a crossing between the first and second excited states, which correspond to the zAFM/XY CFT operators through the operator-state correspondence. Although we expect all excited states for VBS, zAFM, and XY order parameters to have the same energy in thermodynamic limit, logarithmic corrections in \eqnref{eq:log_correction} separate the VBS excited state from the others.



To  proceed, we obtain the energy spectrum of a given Hamiltonian by exact diagonalization technique (for larger system sizes, one can perform finite DMRG for excited states). Then, by examining their quantum numbers, we can identify the corresponding CFT operators in the given excited states energy spectrum. 
In \eqnref{eq:symmetry}, we elaborate on how bosonic fields $(\theta, \phi)$ transform under the microscopic symmetries of the system. Since CFT operators corresponding to the order parameters are represented in terms of $(\theta, \phi)$, we can immediately identify their quantum numbers under the spin rotation in $z$-direction $S^z_\textrm{tot}$, translation ($k$), bond-centered inversion ($\cal I_{\rm bond}$), and spin-flip ($g \equiv \prod_i X_i$) symmetries as in \tabref{tab:order_qm}.
\begin{table}[!h]
    \centering
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{|>{\centering}m{8em}|*{4}{>{\centering\arraybackslash}m{4em}|}}
\hline 
Operators  &  $S^z_\textrm{tot}$ & $k$ & $\cal I_{\rm bond}$ & $g$  \\
\hline 
VBS  & $0$ & $\pi$ & 1 & 1 \\
\hline
zAFM  & $0$ &$\pi$ & -1 & -1 \\
\hline
XY  & $\pm 1/2$ &$\pi$ & -1 & N/A \\
\hline
\end{tabular}
\caption{\label{tab:order_qm} Quantum numbers carried by CFT operators corresponding to each order parameters. Since we block-diagonalize the Hamiltonian in total spin-$z$ and momentum quantum numbers first, spin-flip quantum number is well-defined only in the zero total spin-$z$ sector.}
\end{table}

However, it is not enough to identify CFT operators through the excited states energy spectrum. Note that the states corresponding to the CFT primary operators are generated by inserting operators with respect to the groundstate. Therefore, an eigenstate corresponding to a certain order parameter will carry the sum of quantum numbers of the groundstate and the order parameter. 



In order to identify the quantum numbers of the groundstate across the phase transition, we can perform a perturbative analysis. Consider the groundstate manifold in the e.g. zAFM symmetry broken phase. In the fixed-point limit, the groundstate manifold simply consists of two symmetry broken states, $\ket{\uparrow \downarrow \uparrow ...}$ and $\ket{\downarrow \uparrow \downarrow ...}$. However, when we perturb the system, two symmetry broken states become hybridized, forming symmetric and anti-symmetric combinations as eigenstates of the system.  
From degenerate perturbation theory, we can show that two ground states are connected at order  $(L/2)$. If $L=4n$, since couplings are all positive, the matrix element generated is always negative. Therefore, the symmetric superposition has lower energy. However, for $L=4n+2$, the $2n+1$-th order perturbative contribution connecting two ground states is positive, and the anti-symmetric superposition has a lower energy. Since two ground states are exchanged under translation, bond-centered inversion, and spin flips, the symmetric superposition carries trivial quantum numbers while the anti-symmetric superposition carries $-1$ quantum numbers for those symmetry operators. The quantum number carried by the groundstate is summarized in \tabref{tab:gs_qm}. 
\begin{table}[!h]
    \centering
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{|>{\centering}m{8em}|*{4}{>{\centering\arraybackslash}m{4em}|}}
\hline 
System size  &  $S^z_\textrm{tot}$ & $k$ & $\cal I_{\rm bond}$ & $g$  \\
\hline 
$L=4n$  & $0$ & $0$ & 1 & 1 \\
\hline
$L=4n+2$  & $0$ & $\pi$ & $-1$ & $-1$ \\
\hline
\end{tabular}
\caption{\label{tab:gs_qm} Quantum numbers carried by the groundstate for different system sizes.}
\end{table}
Therefore, using \tabref{tab:order_qm}  and \tabref{tab:gs_qm}, one can obtain the quantum numbers of the state corresponding to a desired CFT operator in any finite-size system. For example, when $L=4n+2$, the VBS state will carry the quantum number $(S^z,k,{\cal I_{\rm bond}}, g) = (0,0,-1,-1)$. Based on this understanding, we can perform exact diagonalization of the Hamiltonian, which is block-diagonalized by symmetry quantum numbers, to identify energies corresponding to specific CFT operators. 



Finally, we discuss the result of finite-size scaling analysis for BKT transitions using \eqnref{eq:finite_scaling}. 
At the BKT transition, $\Delta_\beta=4$ is the lowest scaling dimension of the  irrelevant symmetry-allowed perturbations, which break the emergent $\mathrm{SU}(2)$ symmetry and can thus lift the the degeneracy of zAFM and XY order parameters. We find two such operators with Lorentz spin $\ell = \pm2$ and $S=2$: $O^{ab}_{3,1} =(L_{-2} J^a_{-1} \bar{J}^b_{-1} + a \leftrightarrow b- {\rm trace})|0\rangle$ and  $O^{ab}_{1,3} = (\bar{L}_{-2} J^a_{-1} \bar{J}^b_{-1}  + a \leftrightarrow b - {\rm trace})|0\rangle$, and two operators with Lorentz spin $\ell= \pm 4$ and $S =2$: $O^{ab}_{4,0} = (J^a_{-2} J^b_{-2} +  a \leftrightarrow b - {\rm trace})|0\rangle$ and $O^{ab}_{0,4} = (\bar{J}^a_{-2} \bar{J}^b_{-2}+  a \leftrightarrow b - {\rm trace})|0\rangle$. Here we are using the state operator correspondence and $\mathrm{SU}(2)_1$ current algebra notation, where $J^a_{n}$ ($\bar{J}^a_n$) is the generators of $\mathrm{SU}(2)$ left (right) current algebra, and $L_n$ ($\bar{L}_n$) generators of the left (right) Virasoro algebra.
Inversion symmetry fixes a linear combination of the operators with $\ell = \pm 2$ and a linear combination  of operators with $\ell = \pm 4$. $U(1)$ symmetry picks out the operator with $m = 0$ out of the five $S = 2$ operators. Thus, we find two operators with $\Delta = 4$ that break $\mathrm{SU}(2)$, but preserve the microscopic symmetries, and are not total derivatives (i.e., global descendants). Note in particular that there is no such operator with $\ell = 0$: the only non-descendant with this quantum number is $T \bar{T}$, which is an $\mathrm{SU}(2)$ singlet and does not lift the degeneracy between zAFM and XY operators.


Both $O_{1,3}$, $O_{3,1}$ and $O_{4,0}$, $O_{0,4}$ are expected to acquire anomalous dimensions due to the presence of marginally irrelevant operator at the KT transition. Likewise, the marginal operator driving the KT transition (with coefficient $s$) acquires an anomalous dimension. Thus, we expect logarithmic corrections to $C(L)$ and $K(L)$ in \eqnref{eq:finite_scaling}. However, the effect of logarithmic correction is negligible at small system sizes we examine.
Using ED up to system size $L=24$, which can be done with a standard laptop in a few minutes, we determine the BKT transition points with high accuracy as shown in \figref{fig:KT_data}e.



\begin{figure}[!t]
    \centering
    \includegraphics[width = 0.9 \textwidth]{dqcp_scaling.pdf}
    \vspace{-10pt}
    \caption{\label{fig:ED_DQCP} (a) Exact diagonalization spectrum of the model for $J_3/J_6=0.5$ and $\alpha \in [0.50,0.56]$ at $L=20$. Here we plot first four lowest energies. Each excited state can be labeled by relevant quantum numbers -- how it transforms under the $\mathrm{U}(1)_{xy}$ spin rotation, $g$ spin flip, and ${\cal I}$ site-centered inversion, and accordingly, gives the scaling dimension of the associated CFT operator. (b) Finite size scaling of the crossing point $\alpha_c(L)$ between the VBS and zAFM operators with $L=20,22,24,26,28$. Here, we demonstrate the linear regression between $\alpha_c(L)$ and $1/L^\gamma$ for $\gamma = 2,2.5,3,3.5$. In the ideal scenario, $\gamma =2$, which can change in practice. However, the results are quite insensitive to the choice of $\gamma$;
    we obtain $\alpha_c(L\rightarrow \infty) = 0.530374(3)$. (c) Finite size scaling of the Luttinger parameter. Here, we demonstrate the linear regression between $K_c(L)$ and $1/L^2$, where we obtain $K_c(L\rightarrow \infty) = 0.349061(1)$.} 
\end{figure}

\subsection{Deconfined Quantum Critical Points}

In the main manuscript, we have shown iDMRG simulation results that indicate the presence of a continuous phase transition between VBS and zAFM phases. Since VBS and zAFM break different symmetry groups, such a phase transition corresponds to DQC. From the bosonization analysis in Section \ref{app:bosonization}, we can show that its conformal field theory description is again captured by \eqnref{eq:critical_ham} as in the case of the BKT transitions. However, unlike the BKT transition, the DQCP has no marginal operator that contributes logarithmic corrections to RG flows of operators. Therefore, the general finite-size scaling relation of  \eqnref{eq:finite_scaling} should suffice for the DQCP.



At the DQCP, CFT operators with the lowest scaling dimensions are given by $\cos 2\phi$ and $\sin 2\phi$, which correspond to the zAFM and VBS order parameters respectively. Therefore, the crossing between eigenstates corresponding to the VBS and zAFM operators is expected, as shown in \figref{fig:ED_DQCP}a. However, note that the line of deconfined quantum criticality is characterized by the Luttinger parameter $K$, which is an unknown quantity to be determined. For DQCP, we have $\Delta_\beta = 2 + 4K$ from $\delta L = [(\partial_x \phi)^2+(\partial_t \phi)^2] \cos 4\phi$,
which can lift the degeneracy of ${O}_\textrm{zAFM}$ and ${O}_\textrm{VBS}$.
The scaling dimension of the tuning parameter $\Delta_t = 4K < 2$. 
Therefore, we expect the scaling behavior (\eqnref{eq:finite_scaling}) with $\gamma \equiv \Delta_\beta - \Delta_t = 2$. However, note that this is the expectation from the ideal scenario. In practice, the presence of other irrelevant operators, potentially with larger magnitudes, can alter the scaling behavior at small system sizes in the way dependent on critical exponents including Luttinger parameter $K$. Therefore, we performed finite size scaling with a range of $\gamma$.
It turns out that using different values of $\gamma$ for the fitting does not affect much the resulting thermodynamic limit of the crossing point, as we demonstrate in \figref{fig:ED_DQCP}(b) at $J_3/J_6=0.5$ for the system size upto $L=28$. Therefore, using the level spectroscopy method, we can determine the phase boundary accurately, as depicted in  Fig.~1b of the main text. Further, one can employ the relation \eqnref{eq:CFT_correspondence} at the DQCP to determine the value of the Luttinger parameter. Since $\dim[e^{i2\phi}] = K$ while $\dim[e^{i\theta}] = 1/4K$, it follows from \eqnref{eq:cft_lattice} that at the DQCP, 
\begin{align}\label{eq:fs_lutt} 
    \qty(\frac{E_\textrm{zAFM}-E_\textrm{gs}}{E_{XY}-E_\textrm{gs}})^{1/2} &= \sqrt{ \qty(K + \frac{C_1}{L^{\Delta_\beta-2}} + \dots) \cdot \qty(\frac{1}{4K} + \frac{C_2}{L^{\Delta_\beta-2}} + \dots)^{-1} } = 2K + {\cal O}(1/L^{\Delta_\beta-2}).
\end{align}
By performing a finite-size scaling of the above quantity, we can obtain the Luttinger parameter $K$ as in \figref{fig:ED_DQCP}(c). Note that one can perform the following finite-size scaling for the quasiparticle velocity as well:
\begin{align}\label{eq:fs_vel} 
    \qty(\frac{L}{2\pi})  \sqrt{ (E_\textrm{zAFM}-E_\textrm{gs}) \cdot (E_{XY}-E_\textrm{gs}) } &= v \sqrt{ \qty(K + \frac{C_1}{L^{\Delta_\beta-2}} + \dots) \cdot \qty(\frac{1}{4K} + \frac{C_2}{L^{\Delta_\beta-2}} + \dots) } = \frac{v}{2} + {\cal O}(1/L^{\Delta_\beta-2}).
\end{align}




















\section{Bosonization of Rydberg Hamiltonian}  
\label{app:bosonization}


\subsection{Mapping into Continuum Description}


In this section, we review and apply the technique of bosonization in the context of the Rydberg   Hamitonian $H_\textrm{eff}$ considered in the main text, to derive our field-theoretic description of the system. We first utilize a Jordan-Wigner transformation to map spin-1/2s to spinless fermions; the map is given by $S^z_i = c^\dagger_i c^{\vphantom{\dagger}}_i - 1/2$ and $S^+_i = c^{\vphantom{\dagger}}_i \prod_{i<j}(1-2 c^\dagger_j c^{\vphantom{\dagger}}_j)$. The  effective Hamiltonian, truncated to terms with range of at most next nearest-neighbor interactions, can thus be expressed as follows:
\begin{align}
    H_\textrm{eff} &= -\frac{J_3 }{2} \sum_i \qty( c_i^\dagger c^{\vphantom{\dagger}}_{i+1} + \,\textrm{H.c.}) - \alpha J_3  \sum_i \qty[ c_i^\dagger \qty(n_{i+1} - \frac{1}{2}) c^{\vphantom{\dagger}}_{i+2} + \,\textrm{H.c.} ] \nonumber \\
    & \quad + J_6 \sum_i \qty[ \qty(n_i - \frac{1}{2}) \qty(n_{i+1} - \frac{1}{2}) +\alpha^2 \qty(n_i - \frac{1}{2}) \qty(n_{i+2} - \frac{1}{2}) ], 
\end{align}
where $n_i = c^\dagger_i c^{\vphantom{\dagger}}_i$. 


In order to understand the low energy, long-wavelength physics of the ground state of this model at half-filling, we can first start from the non-interacting limit $J_6=0, \alpha = 0$. 
In such a scenario, 
the ground state is one where fermions form a Fermi sea, up to the Fermi wavevector $k_F = \pm \pi/2$. Labeling fermions near $k_F = \pi/2$ as $\psi_R$ (right-propagating) and $k_F=-\pi/2$ as $\psi_L$ (left-propagating), we can express the above Hamiltonian in the continuum limit as $\hat{H} = \int   \hat{h}(x) \dd x$, where the Hamiltonian density $\hat{h}(x)$  is given as
\begin{align}
    \hat{h}(x)/\epsilon  & =
    \sum_{\eta=L,R}\eta \psi_\eta^\dagger(r) (-i \partial_x)  \psi_\eta(r)
    + \xi \, \rho(r) \rho(r+\epsilon) + \alpha^2 \xi \, \rho(r) \rho(r+2\epsilon)
     \nonumber \\
     & + \alpha  \, \sum_{\eta,\eta'=L,R} \qty[ \psi^\dagger_{\eta}(r) \cdot   \rho(r+\epsilon) \cdot \psi_{\eta'}(r+2\epsilon) + \,\textrm{H.c.} ].
\end{align}
Above, $\rho(r) = \sum_{\eta=L,R} \psi^\dagger_\eta(r) \psi^{\vphantom{\dagger}}_\eta(r)$ is the total density of the fermions, and $\epsilon$ is the length scale associated with lattice spacing, $\xi = J_6/J_3$, and we often identify $\eta = R = 1$ and $\eta = L = -1$ following the convention.


Next, we invoke \emph{bosonization} \cite{Giamarchi2004}, a well-established technique that allows us to    express fermionic variables $\psi_{\eta}$ in 1D in terms of bosonic variables $\Phi_{\eta}$: 
\begin{equation}
    \psi_\eta (x)= \hat{F}_\eta  e^{i \eta k_F x} e^{i\Phi_\eta}, \qquad [\Phi_\eta(x),\Phi_{\eta'}(x')] = i \pi  \delta_{\eta,\eta'} \operatorname{sgn}(x-x'),
\end{equation}
where $\hat{F}_\eta$ are so-called Klein factors. One can further introduce two variables $\theta = (\Phi_L + \Phi_R)/2$ and $\phi=(\Phi_L - \Phi_R)/2$, to equivalently express
\begin{equation}
    \psi_\eta (x)= \hat{F}_\eta  e^{i \eta k_F x} e^{i(\theta - \eta \phi)}, \qquad [\phi(x),\theta(x')] = i \frac{\pi}{2} \operatorname{sgn}(x-x').
\end{equation}
Under   bosonization, we therefore obtain the following Hamiltonian, upon keeping the leading order contributions:
\begin{equation} \label{eq:ham_boson}
    H = \frac{u}{2\pi} \int_0^L \dd x \qty[ \frac{1}{K} (\partial_x \phi)^2 + K (\partial_x \theta )^2 ] + g_4 \cos 4\phi + g_8 \cos 8\phi + ...
\end{equation}
where 
\begin{align} \label{eq:LL_parametes}
    u &= \epsilon  \sqrt{ \qty( 1 - \frac{4}{\pi}  \alpha) \cdot \qty( 1 + \frac{4}{\pi}\qty(\xi + \alpha))},  \nonumber \\
    K &= \qty(\frac{1 - \frac{4 \alpha}{\pi}}{1 + \frac{4}{\pi}(\xi +\alpha)})^{1/2}, \nonumber \\
    g_4 &= \frac{J_3}{4\pi^2 \epsilon} \qty( -\xi+ \alpha^2 \xi + 2 \alpha),
\end{align}
where the above parametrization of $(u,K,g_4)$ is correct at most perturbatively in $\xi$ and $\alpha$. 
This is the field theoretic model as presented in the main text. Note that in deriving this field theoretic analysis, we have ignored the long-range interaction terms present in the microscopic Hamiltonian $H_\textrm{eff}$; this is because even if we were to include any long-range interactions between physical operators (e.g.~$\partial \phi$ or $e^{ia \phi}$) which decay as $1/r^n$ with $n \geq 3$, such interaction terms are irrelevant in 1D under the renormalization group flow. Such an expectation is also supported by many other analytical and numerical studies~\cite{Parreira1997, Laflorencie2005, absenceCSB2017}.


\subsection{Symmetry Actions}


We now discuss symmetries  of the system and how they manifest themselves in different representations.
Now, the original microscopic spin Hamiltonian has   translation symmetry $T_x$, spin rotation symmetry $\mathrm{U}(1)_z \times \mathbb{Z}^x_2$ (spin rotation  about the $z$-axis and  spin-flip about the $x$-axis respectively), site-centered inversion ${\cal I}$, and time-reversal symmetry ${\cal T}$. These symmetries act on  microscopic spin variables and bosonic field theory variables in the following way: 
\begin{eqnarray}\label{eq:symmetry}
    T_x:&\quad S^a_i \mapsto S^a_{i+1} \quad & \quad \phi \mapsto \phi+\pi/2, \quad \theta \mapsto \theta+\pi \nonumber \\
    U(1)_z:& \quad \vec{S}_i \mapsto R(\alpha \hat{z}) \vec{S}_{i+1} \quad & \quad \theta \mapsto \theta+\alpha \nonumber \\
    \mathbb{Z}_2^x:& \quad S^{y,z}_i \mapsto -S^{y,z}_i \quad & \quad \phi \mapsto - \phi + \pi/2, \quad \theta \mapsto -\theta \nonumber \\
    {\cal I}:& \quad S^{a}_i \mapsto S^{a}_{-i} \quad & \quad \phi \mapsto - \phi  \nonumber \\
    {\cal T}:& \quad S^{a}_i \mapsto -S^{a}_i \quad & \quad \phi \mapsto - \phi + \pi/2, \quad \theta \mapsto \theta+\pi, \quad i \mapsto -i
\end{eqnarray}
Due to these symmetries, other than higher-order kinetic terms (e.g. $(\partial \phi)^4$, $(\partial \theta)^4$, ...) the only symmetry allowed terms in \eqnref{eq:ham_boson} have the structure of $\cos 4n \phi$, where $n \in \mathbb{N}$. In terms of bosonic fields $(\theta, \phi)$, original spin operators as well as the dimerization operator in continuum can be expressed as (at leading order)~\cite{Giamarchi2004}:
\begin{align} \label{eq:operator_mapping}
    S^x &\sim  \frac{1}{\sqrt{8\pi \epsilon}} \qty( (-1)^x \cos \theta - i \sin \theta \cos 2\phi ), \nonumber \\
    S^y &\sim  \frac{-1}{\sqrt{8\pi \epsilon}} \qty(  (-1)^x \sin \theta + i \cos \theta \cos 2\phi ), \nonumber \\
    S^z &\sim -\frac{1}{\pi} \partial \phi  + \frac{(-1)^x}{\pi \epsilon}\cos 2\phi, \nonumber \\
    D &\sim (-1)^x \vec{S}(x) \cdot \vec{S}(x+1) = \frac{1}{2\pi \epsilon}  \sin 2\phi, 
\end{align}
which is consistent with the symmetry transformation rules in \eqnref{eq:symmetry}.


\subsection{Renormalization Group Analysis}
\label{sec:rg}

Next, let us analyze the possible phases realized by the continuum Hamiltonian Eq.~\eqref{eq:ham_boson} within the renormalization group (RG) framework. If we first ignore the interaction terms $\cos 4n \phi$, then we obtain the following quadratic Hamiltonian
\begin{equation} \label{eq:critical_ham}
    H_0 = \frac{u}{2\pi} \int_0^L \dd x \qty[ \frac{1}{K} (\partial_x \phi)^2 + K (\partial_x \theta )^2 ]. 
\end{equation}
As the corresponding action is Gaussian, $H_0$ describes an exactly solvable model, and the scaling dimension of a generic operator can be derived:
\begin{equation}
    \dim[e^{i(a \phi + b \theta)}] = \frac{a^2 K}{4} + \frac{b^2}{4K} \quad \implies \quad \dim[\cos 4n \phi] = 4n^2 K.
    \label{eq:scaling_dim}
\end{equation}
Additionally, correlation functions are given as
\begin{align}
\label{eq:correlations}
    \expval{S_+(0) S_-(r)} &\sim (-1)^r \frac{C_1}{r^{\frac{1}{2K}}} +  \frac{C_2}{r^{2K + \frac{1}{2K}}} \nonumber \\
    \expval{S^z(0) S^z(r)} \sim \expval{D(0) D(r)} &\sim (-1)^r \frac{C_1}{r^{{2K}}} +  \frac{C_2}{r^{2}}. 
\end{align}
We see that all correlation functions decay as power laws; this is thus a gapless phase called a Luttinger liquid. 

Next, we can analyze the effect of the interactions $\cos 4n \phi$,  treating them as     perturbations away from the quadratic limit. In a one-dimensional system, an operator flows to zero under RG flow if its scaling dimension is larger than 2. When $K > 1/2$, \eqnref{eq:scaling_dim} implies that $\cos 4n\phi$ is irrelevant for all $n\geq 1$  and so the theory remains Gaussian, with   correlation functions decaying  as power laws. From Eq.~\eqref{eq:correlations}, we see that correlation decays slower in the $xy$ channel than the $z$ channel, and hence we identify this as the XY quasi long-range ordered (QLRO) phase. 


When $1/8<K < 1/2$, the $\cos 4 \phi$ term becomes relevant while higher $n$ terms remain irrelevant. Therefore, the $\phi$-field can condense, and the value it takes ($0$ or $\pi/4$), depends on the sign of $g_4$, which is in turn determined by microscopic parameters $(\xi,\alpha)$. Specifically, if $g_4 < 0$, the $\phi$-field condenses at $\phi=0$ and $(-1)^x \expval{S^z} \sim  \cos 2\phi \neq 0$ and the system develops $z$AFM order, breaking $\mathbb{Z}_2^x$ spin-flip symmetry. If $g_4 > 0$, the $\phi$-field condenses at $\phi=\pi/4$ and $D \sim \sin 2\phi \neq 0$ and the system develops VBS order, breaking the site-centered inversion symmetry ${\cal I}$. Note that translation symmetry is broken in both cases.

Empirically, we find that the Luttinger parameter $K$ along the DQCP line of our model decreases as we move towards the classical limit $(J_3 \to 0)$.
An interesting question is whether $K$ of the system can decrease to an indefinitely small value while still exhibiting DQC.
Our field theory analysis suggests that the deconfined transition can become destabilized if additional symmetry-allowed perturbations become relevant, such as the $\cos 8\phi$ term (since we would then generally need two independent tuning knobs to ensure both $\cos 4\phi$ and $\cos 8\phi$ vanish). The presence of such a term, if relevant, would give rise to a non-trivial effect. Let $g_8$ be a coefficient for the term $\cos 8\phi$. For $g_8 < 0$, this term is minimized at $\phi= n\pi /4$. Combined with the effect of $g_4 \cos 4\phi$, one would expect a first-order phase transition between zAFM and VBS phases since it acts like a cubic anisotropy at Ising criticality. On the other hand, for $g_8 > 0$, this term is minimized at $\phi = (2n+1)\pi/8$. At such values of $\phi$, both $S^z$ and $D$ take non-zero values. Therefore, together with $g_4 \cos 4\phi$, this term would drive the system away from the DQCP physics into a new phase that breaks both  spin-flip $\mathbb{Z}_2^x$ and site-centered inversion ${\cal I}$ symmetries~\footnote{Note that the above field-theoretic analysis cannot capture the QzAFM phase described in the main text; while the QzAFM phase breaks the translation symmetry in such a way that it is invariant under multiple of $T_x^4$, the spin variables written in terms of $(\phi,\theta)$ in \eqnref{eq:operator_mapping} are always symmetric under $T_x^2$.}.  
The above analysis shows that for either sign for $g_8$, if $\cos 8\phi$ term is relevant, the DQCP physics would disappear. 
Therefore, as $\cos 8\phi$'s scaling dimension is $\dim[\cos 8\phi] = 16 K$, we expect that $K$ along the DQCP line should have a lower bound: $K \geq 1/8$.






\begin{figure}[!t]
    \centering
    \includegraphics[width = 0.97 \textwidth]{lutt_vel.pdf}
    \vspace{-10pt}
    \caption{\label{fig:NNNvsLR} Luttinger parameter $K$ and quasiparticle velocity $v$ extracted by finite size scaling in \eqnref{eq:fs_lutt} and \eqnref{eq:fs_vel} for (a,b) NNN Hamiltonian $H_\textrm{eff}^\textrm{NNN}$ and (c,d) full Rydberg Hamiltonian $H_\textrm{eff}$ along the DQCP line. While $v \rightarrow 0$ for $H_\textrm{eff}^\textrm{NNN}$ as $J_3 \rightarrow 0$, $v$ approaches a finite value for $H_\textrm{eff}$ as $J_3 \rightarrow 0$. } 
\end{figure}




\section{Termination of DQCP along the critical line} \label{app:degeneracy}




In the main text, we have argued that   DQC exists along a line of critical points with varying Luttinger parameters up to a small value of $J_3/J_6$ ($\sim 0.005$). On the other hand, we know that at exactly $J_3/J_6=0$, there is a first-order transition from zAFM to QzAFM phases at a finite value of $\alpha$. 
This means that the DQCP line necessarily  terminates, 
either at some small but non-zero value of $J_3/J_6$, or   exactly only at $J_3/J_6=0$.
In this section, we would like to discuss the possible physical mechanisms behind this termination of the DQCP line, in particular focusing on the role of the range of interactions.



First of all, let us understand the first-order nature of the zAFM-QzAFM transition. Consider the model Hamiltonian truncated up to next-nearest-neighbor (NNN) interactions at $J_3/J_6=0$, which has only Ising-type interactions:
\begin{align}
    H &= \sum_i Z_i Z_{i+1} + \alpha^2 \sum_i Z_i Z_{i+2} \nonumber \\
    &=  \frac{1}{2} \sum_i \qty( Z_i I_{i+1} Z_{i+2} + Z_i Z_{i+1} I_{i+2} + I_i Z_{i+1} Z_{i+2} + I_i I_{i+1} I_{i+2} )    + \qty( \alpha^2-1/2) \sum_i Z_i Z_{i+2} \nonumber \\
    &= 2 \sum_i \qty( P_i^{\uparrow\uparrow\uparrow} + P_i^{\downarrow\downarrow\downarrow}) + (\alpha^2-1/2) \sum_i Z_i Z_{i+2}
\end{align}
where $P^{abc}_i$ is the projection onto the the spin-configuration $(a,b,c)_{i,i+1,i+2}$. With this rewriting of the Hamiltonian, we observe that at $\alpha^2=1/2$, the Hamiltonian is nothing but the sum of local projectors onto up-up-up and down-down-down configurations. Since there are extensively many configurations projected out by $(P_i^{\uparrow\uparrow\uparrow} + P_i^{\downarrow\downarrow\downarrow})$, at $\alpha^2=1/2$, there is an extensive degeneracy scaling like $\text{dim} \sim (\frac{1+\sqrt{5}}{2})^L$ (the degeneracy for given $L$ follows the Fibonacci sequence).
When $\alpha^2< 1/2$, the second term prefers the configuration where every other spins are aligned;  within the degenerate manifold, this corresponds to the up-down-up or down-up-down configuration, which corresponds to zAFM order. When $\alpha^2>1/2$, the second term prefers the configuration where spin direction changes every two sites; within the degenerate manifold, this corresponds to the up-up-down, up-down-down, down-down-up, or down-up-up configuration, which corresponds to quadrupled zAFM (QzAFM) order. 
Since these two classical configurations exhibit level crossing at $\alpha^2=1/2$, 
the phase transition upon tuning $\alpha^2$ must be first-order in nature. 





Now, let us return to the field theoretic description of DQCP. In \appref{app:bosonization}, we pointed out that the DQCP develops an instability toward a first order transition or coexistence when the Luttinger parameter $K$ goes below $1/8$ so that the $\cos 8\phi$ term becomes relevant. However, in the main Fig.2e, we demonstrated that the DQCP line of the truncated model $H_\textrm{eff}^\textrm{NNN}$ has a Luttinger parameter which saturates around $\sim0.28$ in the limit $J_3 \rightarrow 0$ as in \figref{fig:NNNvsLR}(a). In fact, we observed  similar behavior along the DQCP line of the XXZ model (with interactions up to NNN)
\begin{align} \label{eq:XXZ}
    & H_\textrm{eff} = \sum_i \Big[ (X_i X_{i+1} + Y_i Y_{i+1}) + \alpha (X_i X_{i+2}  + Y_i Y_{i+2})  \Big] + \Delta \sum_i \Big[ Z_i Z_{i+1} + \alpha Z_i Z_{i+2}  \Big], \nonumber
\end{align}
which has a qualitatively similar phase diagram. In this case, we numerically observed that as $1/\Delta \rightarrow 0$, along the DQC line between VBS and zAFM phases, the Luttinger parameter does not go below $0.3$. In other words, it appears the DQC line terminates at the point $J_3 =0, \alpha^2 = 1/2$  where the zAFM, QzAFM and VBS phases meet, without the Luttinger parameter $K$ going below $1/8$. This is inconsistent with the termination mechanism described in \appref{sec:rg}. 
However, note that $H^{\rm NNN}_{\rm eff}$ has an extensive degeneracy at $J_3 = 0$ and is, thus, highly fine-tuned. Indeed, we find that as $J_3 \to 0$, the quasiparticle velocity along the DQC line goes to zero, see \figref{fig:NNNvsLR}(b), so at $J_3 =0$ the entire gapless spectrum collapses into the ground state and one obtains an extensive degeneracy.




When we introduce further neighbor interactions, we find that the extensive degeneracy at $J_3 = 0$ gets lifted, and the transition between zAFM and QzAFM phases has a degeneracy of 6 (2 from zAFM and 4 from QzAFM). For example, if we introduce a perturbation $g_3 \sum_i Z_i Z_{i+3}$, then it only lowers the energy of the zAFM state by $g_3$ while it does not affect the energy of the QzAFM state. To compensate this effect, the transition happens at $\alpha_c^2 > 1/2$, and the extensive degeneracy gets reduced to 6 at such $\alpha_c^2$ (still first-order). The first order zAFM to QzAFM transition must then persist for a finite range of $J_3$ and the DQC line must terminate at finite $J_3$. Our current numerics do not allow us to resolve the precise details of  the DQC line termination in this case. It is possible that the zAFM to VBS transition turns first order or broadens into a co-existence region before terminating at the QzAFM phase boundary according to the field-theoretic scenario in section \appref{app:bosonization}. Alternatively, it may remain second order all the way up to a first order transition to the QzAFM phase. Numerically, we find that in the full model $H_{\rm eff}$ the value of the Luttinger parameter along the DQC line becomes as small as $K \approx 0.14$ for the smallest values of $J_3$ where it can be reliably extracted, see \figref{fig:NNNvsLR}(c). This is much smaller than the smallest value $K \approx 0.3$ obtained in the NNN model, leaving the possibility that the field theoretic scenario for the termination of the DQC line with $K$ approaching $1/8$ might be realized here. 








\section{Model Rydberg Hamiltonian with exchange and blockade interactions}
\label{app:model}

In this section we describe how to realize our model Hamiltonian, $H_\textrm{eff} = H_{XY} + H_{ZZ}$, containing both  resonant dipole-dipole spin exchange $H_{XY}$ and van der Waals blockade Ising-like interaction $H_{ZZ}$. First, we explain how the van der Waals blockade gives rise to the Ising couplings relevant to our model. Then, we explain how to adjust the relative strength of $H_{XY}$ and $H_{ZZ}$, i.e. $J_3/J_6$, by controlling the amount of admixture in microwave-dressed Rydberg states. For simplicity, our model Hamiltonian $H_\textrm{eff}$ is designed to contain only $1/R^3$ terms in $H_{XY}$ and $1/R^6$ terms in $H_{ZZ}$, although as we show below, depending on the Rydberg states and admixing schemes chosen, both $1/R^3$ and $1/R^6$ contributions may appear in both $H_{XY}$ and $H_{ZZ}$. While in that case the exact location of the phase boundaries may change, we expect the qualitative features to remain unchanged. As direct exchange (a first order interaction) can generally be made stronger than blockade (a second order interaction), realizing the model as we describe means the tuning $J_3$ can allow access to a broad range of $J_3/J_6$ values in the phase diagram (See \figref{fig:MW}b). If first order and/or second order contributions are present in both $H_{XY}$ and $H_{ZZ}$, the tuning range may be more limited, and we may not access entire phase diagram presented in Fig.1b.

\subsection{Physical Origin of XY and ZZ terms}

Each internal atomic state $\ket{\alpha}, \ket{\beta},....$ of a given atom is specified by quantum numbers $n, l, j, m_j$: principal, angular momentum, total angular momentum, and $z$-projected angular momentum. In general, Rydberg states with the same $n$ but different $l$ have different energy since the amount of screening can change depending on the angular momentum $l$, so-called quantum defect \cite{Browaeys_2016}. Even among the states with the same angular momentum $l$, the degeneracy can be further split by additional perturbation such as external electric/magnetic fields or spin-orbit coupling.

In order to understand the interaction between a pair of atoms in respective Rydberg states, we should first understand their non-interacting energy. Note that the eigenstates of two non-interacting atoms can be written as pair states $\ket{ab}\equiv\ket{a}\otimes \ket{b}$ with energies $E_{ab} = E_{a} + E_{b}$, which form the basis of their Hilbert space.
Then, the two-body interaction terms arise from the dipole-dipole interaction Hamiltonian, which can couple these pair states, with $\hat{n}$ the unit vector along the axis between the atoms, $\vec{d_1}, \vec{d_2}$ the dipole operators for the two atoms \cite{Browaeys_2016}:
\begin{equation}
    V_\textrm{dd} = \frac{1}{4 \pi \epsilon_0} \frac{\vec{d_1} \cdot \vec{d_2} - 3 (\vec{d_1} \cdot \hat{n})(\vec{d_2} \cdot \hat{n})}{R^3}.
\end{equation}
Consider a spin encoded $\ket{\uparrow} = \ket{\alpha}$, $\ket{\downarrow} = \ket{\beta}$ in the bare-atom basis, where throughout, a strong magnetic field creates a Zeeman splitting so that the degeneracy of the Rydberg states of our interest characterized by $(n,l,j,m_j)$ is completely lifted and these energy levels are all isolated.



XY-type exchange interactions originate from the dipole-dipole coupling of pair states $\ket{\alpha \beta}, \ket{\beta \alpha }$, which can be nonzero for example when $\ket{\alpha}$ is an $nS$ state and $\ket{\beta}$ is an $nP$ state. 
As the dipole operator takes non-zero value between the states with different parity, i.e., $\bra{nS}\vec{d}_i\ket{nP} \neq 0$, there can be a dipole-dipole interaction between these states. Therefore, as $E_{\alpha \beta} = E_{\beta \alpha}$, these states couple resonantly and we obtain a direct flip-flop exchange interaction as the following
\begin{equation}
    \bra{\alpha_1 \beta_2} V_{dd} \ket{\beta_1 \alpha_2} \neq 0 \quad \Rightarrow \quad \sim(XX + YY)
\end{equation}

ZZ-type Ising interactions may arise if $V_{dd}$ has nonzero diagonal matrix elements, i.e., $\bra{\alpha} V_{dd} \ket{\alpha} \neq 0$.
However, in order to achieve a wide tunability of $J_3/J^6$, we  try to suppress this type of mechanism and instead rely on a different mechanism to realize the same type of interactions.
Namely, ZZ-type Ising interactions
originate when a pair state such as $\ket{\alpha \alpha}$ has a nonzero dipole-dipole coupling to another pair state $\ket{\gamma \lambda}$ outside of the spin subspace. As the energy difference between $\ket{\alpha \alpha}$ and $\ket{\gamma \lambda}$ is generally large (typically many GHz and much larger than the dipole-dipole coupling strength), this coupling results in a second order energy shift of $\ket{\alpha \alpha}$ of strength $\frac{|\bra{\alpha \alpha}V_\textrm{dd}\ket{\gamma \lambda}|^2}{E_{\alpha \alpha} - E_{\gamma \lambda}} \propto 1/R^6$. This pair state level shift is equivalent to a ZZ term plus single body Z terms. 
The strengh of interactions arising from this mechanism could be substantial, leading to so-called Rydberg blockade.
Note that the dipole-dipole coupling of $\ket{\alpha \alpha}$ with $\ket{\beta \beta}$ gives rise to a negligible Zeeman field.




\subsection{Controlling $J_3/J_6$ with Microwave-mixed Rydberg states}

As the atomic dipole operators in $V_{\textrm{dd}}$ give zero matrix element between atomic states of the same parity, encoding the spins as  microwave-dressed Rydberg states, which gives the controllable admixture of Rydberg states with different parities, can enable tunability of $J_3/J_6$.  The second order processes described above result in a background $1/R^6$ interaction, on top of which we can introduce exchange couplings by using a microwave drive to control the amount of admixture of states of distinct parity. By tuning these first order exchange couplings from zero to much larger than the  second order Ising-type interactions, we can explore a wide range of relative coupling strengths $J_3/J_6$.

We first explain the presence of the background second order terms $H_{ZZ}$. Each Rydberg pair state experiences an energy shift with a corresponding $C_6$ coefficient:
\begin{align} 
    H_{ZZ} & = \sum_{i<j} \frac{C_6^{\uparrow \uparrow}}{d_{ij}^6} n_i^\uparrow n_j^\uparrow + \frac{C_6^{\downarrow \downarrow}}{d_{ij}^6} n_i^\downarrow n_j^\downarrow + \frac{C_6^{\uparrow \downarrow}}{d_{ij}^6} (n_i^\uparrow n_j^\downarrow + n_i^\downarrow n_j^\uparrow ),
\end{align}
where $n^\uparrow, n^\downarrow$ are the projectors $\ket{\uparrow}\bra{\uparrow}$, $\ket{\downarrow}\bra{\downarrow}$, and $d_{ij}$ are the physical distances between the atoms. Substituting $n^\uparrow = \frac{1}{2}(P^R + Z)$, $n^\downarrow = \frac{1}{2}(P^R - Z)$, where $P^R$ is the projector onto the Rydberg spin subspace, gives:
\begin{align}   
    H_{ZZ} & = \frac{1}{4} \sum_{i<j} \frac{(C_6^{\uparrow \uparrow} + C_6^{\downarrow \downarrow} - 2 C_6^{\uparrow \downarrow})}{d_{ij}^6} Z_i Z_j + \frac{1}{4} \sum_{i < j} \frac{(C_6^{\uparrow \uparrow} - C_6^{\downarrow \downarrow})}{d_{ij}^6} (Z_i  P^R_j + P^R_i Z_j) + \frac{1}{4} \sum_{i < j} \frac{(C_6^{\uparrow \uparrow} + C_6^{\downarrow \downarrow} + 2 C_6^{\uparrow \downarrow})}{d_{ij}^6} P^R_i P^R_j.
\end{align}

Considering the 1D geometry we introduced and separating out the terms up to next-to-nearest neighbor from the long range terms $H_{LR}$:
\begin{align}  \label{eq:Rydberg}
    H_{ZZ} = & \frac{1}{4} \frac{(C_6^{\uparrow \uparrow} + C_6^{\downarrow \downarrow} - 2 C_6^{\uparrow \downarrow})}{d_{1}^6} \sum_{i}  (Z_i Z_{i+1} + \alpha^2 Z_i Z_{i+2}) \nonumber \\
    & + \frac{1}{4} \frac{(C_6^{\uparrow \uparrow} - C_6^{\downarrow \downarrow})}{d_{1}^6} \sum_{i} \Big[ (Z_i  P^R_{i+1} + P^R_i  Z_{i+1})  + \alpha^2 (Z_i  P^R_{i+2} + P^R_i Z_{i+2}) \Big] \nonumber  \\
    & + \frac{1}{4} \frac{(C_6^{\uparrow \uparrow} + C_6^{\downarrow \downarrow} + 2 C_6^{\uparrow \downarrow})}{d_{1}^6} \sum_{i}  \Big[P^R_i P^R_{i+1} + \alpha^2 P^R_i P^R_{i+2}\Big] + H_{LR}.
\end{align}

While the full form for $H_{ZZ}$ shown above is necessary for a quantitative treatment when modelling atoms with other internal levels in addition to $\ket{\uparrow}$, $\ket{\downarrow}$, it takes a simpler form when restricting the atomic Hilbert space to the Rydberg spin subspace, in which case each projector $P^R$ becomes an identity operator. The final sum in the expression above is then an overall energy offset and the sums of $\sum_i Z_i$ become total spin operators which just give another energy offset for a fixed magnetization sector. Defining an effective van der Waals coefficient $C_6 \equiv \frac{1}{4} (C_6^{\uparrow \uparrow} + C_6^{\downarrow \downarrow} - 2 C_6^{\uparrow \downarrow})$ so that $J_6 \equiv C_6/d_1^6$, we obtain the model for $H_{ZZ}$ relevant for exploring the DQC ground state phase diagram for fixed magnetization sector:
\begin{align}
    H_{ZZ} = J_6 \sum_{i} \Big[ Z_i Z_{i+1} + \alpha^2 Z_i Z_{i+2} \Big] +H_{LR}^{(ZZ)}
\end{align}






To tune $J_3$ on top of this background, we first choose the two Rydberg spin states such that the dipole moment between two states vanish, e.g. $\ket{nS}$ and $\ket{n'S}$, to encode ``bare'' spin-1/2 $\{\ket{\downarrow}, \ket{\uparrow}\}$. Direct dipolar exchange interaction is then forbidden by selection rules, and for simplicity we also choose states with negligible second order exchange couplings \footnote{appreciable second order exchange couplings \cite{Whitlock_2017} can arise via dipole-dipole coupling of, for example, both $\ket{nSn'S}$ and $\ket{n'SnS}$ to states like $\ket{nPnP}$, if $\Delta E = E_{nSn'S} + E_{n'SnS} - 2 E_{nPnP} \approx 0 $. As this could conspire to give nonzero exchange couplings even when the microwave drives are zero, making it difficult to access the regime of small $J_3/J_6$, for simplicity we assume this is negligible. We can ensure access to the small $J_3/J_6$ regime in a few ways, including either by choosing states such that $\Delta E$ is sufficiently large (for example in alkalis by choosing $\ket{\uparrow}$ to be a $D$ state with very different quantum defect), or such that the dipole coupling on the microwave driven transition is small}. Now, we want to dress the state $\ket{\downarrow}$ in a controlled manner so that the the dressed $\ket{\downarrow}$ has a non-zero dipole moment with $\ket{\uparrow}$. We apply a microwave drive $\Omega_{1}$ with detuning $\delta_{\textrm{MW}}$ to admix another Rydberg state with distinct parity, such as $\ket{m'P}$, into $\ket{nS}$. 
Whereas $\ket{\uparrow} = \ket{n'S}$, $\ket{\downarrow}$ becomes a dressed state
\begin{equation}
    \ket{\downarrow} = c_0 \ket{nS} + c_1 \ket{m'P}, \qquad \abs{c_1}^2 + \abs{c_2}^2 = 1
\end{equation}

As there is a dipolar exchange coupling $\bra{n'Sm'P} V_{dd} \ket{m'Pn'S} = {C_3'}/{d_{ij}^3}$, with the admixture of $\ket{m'P}$ into $\ket{nS}$, there is now also an inherited exchange interaction between $\ket{\downarrow \uparrow}$ and $\ket{\uparrow \downarrow}$ such that $\bra{\uparrow \downarrow} V_{dd} \ket{\downarrow \uparrow} = |c_1|^2 {C_3'}/{d_{ij}^3}$. This leads to a controllable exchange interaction Hamiltonian acting on the Rydberg spins:
\begin{align}
    H_{XY} &= |c_1|^2 \sum_{i<j} \frac{C_3'}{d_{ij}^3} \left( \sigma^+_i \sigma^-_j + \sigma^-_i \sigma^+_j \right),
\end{align}
where $\sigma^+, \sigma^-$ operate on $\ket{\uparrow}, \ket{\downarrow}$. For the ring geometry, we separate the terms up to the next-to-nearest neighbor interactions from the long range terms $H_{LR}$, substitute $\sigma^{\pm} = (X \pm i Y)/2$, and define $J_3(c_1) \equiv \frac{C_3'}{4 d_{1}^3} |c_1|^2$, obtaining:
\begin{align}
    H_{XY} &= J_3 \sum_{i} \Big[  \left( X_i X_{i+1} + Y_i Y_{i+1} \right) + \alpha \left( X_i X_{i+2} + Y_i Y_{i+2} \right) \Big]  + H_{LR}^{(XY)}
\end{align}

Finally, since the spin down $\ket{\downarrow}$ state is an admixture of S and P orbitals, 
the microwave admixing will cause an additional level shift of the pair state $\ket{\downarrow \downarrow}$. Let  $\bra{nSm'P} V_{dd} \ket{m'PnS} = {C_3}/{d_{ij}^3}$. Then, we observe that $\bra{\downarrow \downarrow} V_{dd} \ket{\downarrow \downarrow} = 4 |c_1|^2 |c_0|^2 {C_3}/{d_{ij}^3}$, which corresponds to a ZZ operator of $|c_1|^2 |c_0|^2 {C_3}/{d_{ij}^3}$.
This term gives a $1/R^3$ ZZ interaction denoted $H'_{ZZ}$ as the following:
\begin{equation}
    H'_{ZZ} = 4J_3 \frac{C_3}{C_3'} \abs{c_0}^2 \sum_i \qty[ Z_i Z_{i+1} + \alpha Z_i Z_{i+2} ] + H'^{(ZZ)}_\textrm{LR}, \qquad \abs{c_0}^2 = 1 - \abs{c_1}^2
\end{equation}
whose dependence on the atomic separation $d_{ij}$ is similar to that of $H_{XY}$. Combining the effects from $H_{ZZ}$, $H_{XY}$, and $H_{ZZ}'$ gives us the following tunable total Hamiltonian:
\begin{equation} \label{eq:H_one}
    H_\textrm{tot}(\alpha, \abs{c_1}) = H_{XY} + H_{ZZ} + H_{ZZ}',
\end{equation}
which is a function of the geometric factor $\alpha$ and the amount of admixture $\abs{c_1}$.  
We note that $H_{XY}$ and $H'_{ZZ}$ have tunable, $1/d^3$ decaying strengths, whereas $H_{ZZ}$ has $1/d^6$ decaying strengths.
Compared to the Hamiltonian proposed in the main text, this Hamiltonian is somewhat inconvenient to parametrize; in particular, when working in the regime of small small $J_3/J_6$, the Ising part of $H_{\textrm{tot}}$ is dominated by the $1/R^6$ terms from $H_{ZZ}$, whereas at large $J_3/J_6$, the Ising part is dominated by the $1/R^3$ terms from $H_{ZZ}'$, so that two parameters, the ratio between exchange and Ising couplings and the distance dependence of the interactions, are no longer independent.







\begin{figure}[!t]
    \centering
    \includegraphics[width = 0.8 \textwidth]{Fig_MW.pdf}
    \caption{\label{fig:MW} Mechanism for tuning $J_3$. (a) Rydberg levels of a single atom; many encodings are possible, but we choose $\ket{nS}$ and $\ket{mP}$ states for concreteness to illustrate which levels are dipole coupled. For $\Omega_{\textrm{MW}} = 0$, $\ket{\uparrow} = \ket{n'S}$ and $\ket{\downarrow} = \ket{nS}$, which have no exchange interaction. Tuning $\Omega_{\textrm{MW}} > 0$ admixes components of $\ket{m'P}$ and $\ket{mP}$ into $\ket{nS}$, dressing $\ket{\downarrow}$ and inducing an exchange coupling between $\ket{\uparrow}$ and $\ket{\downarrow}$. (b) Exchange coupling between $\ket{\uparrow \downarrow}, \ket{\downarrow \uparrow}$ dependence on microwave control parameters, assuming a bare exchange strength of 4 times the Rydberg blockade strength, equal dipole matrix elements $\bra{m'P} V_{dd} \ket{nS} = \bra{mP} V_{dd} \ket{nS}$, and $\Omega_{\textrm{MW}}^{(1)} = \Omega_{\textrm{MW}}^{(2)}$. Dashed grey line corresponds to DQCP for $\alpha = 0.5$
   
    }
\end{figure}

\subsection{Cancellation of first order Ising shift due to microwave dressing}



In the previous subsection, we demonstrated the spin-1/2 effective Hamiltonian for encoding scheme using one microwave dressing, and discussed its limitations. In order to overcome these issues, we propose a new scheme where $H_{ZZ}'$ can be cancelled by using a second microwave tone to admix a second state, such as  $mP$, into $nS$, while still maintaining the desired exchange coupling. Since we choose $m<m'<n'$, the transition dipole moment between $\ket{mP}$ and $\ket{n'S}$ is much smaller than the transition dipole moment between $\ket{m'P}$ and $\ket{n'P}$ \cite{Weber_2017}. Therefore, for simplicity we ignore the transition dipole moment between $\ket{mP}$ and $\ket{n'S}$. Like before, we apply microwave drive $\Omega_{2}$  to admix $\ket{mP}$, into $\ket{nS}$ as in \figref{fig:MW}a. 
Whereas $\ket{\uparrow} = \ket{n'S}$, $\ket{\downarrow}$ becomes a dressed state
\begin{equation}
    \ket{\downarrow} = c_0 \ket{nS} + c_1 \ket{n'P} + c_2 \ket{mP},
\end{equation} 
where the coefficients $c_{1,2}$ depends on the strength of microwave drive $\Omega^{(1,2)}_\textrm{MW}$ and detuning $\delta$. One way this can be used to make the ZZ term vanish is following Ref.\,\cite{adjust_dipolar2021}, let us define 
\begin{align}
    \mu_z &\equiv \bra{n'S} d_z \ket{m'P} \nonumber \\
    \mu_+ &\equiv \bra{n'S} d_+ \ket{mP} 
\end{align}
where $d_z$ and $d_+$ are the electric dipole operator in $\hat{z}$ and $+$ polarizations and atomic separations are in $z$-directions (we assume the orbitals are chosen in such a way that dipole moments in other directions vanish). Then, the direct dipolar interaction strength between $\ket{\downarrow}$ at sites $i$ and $j$ is \cite{adjust_dipolar2021}:
\begin{equation}
    \bra{\downarrow_i \downarrow_j} \hat{V}_{dd} \ket{\downarrow_i \downarrow_j} \propto \abs{c_1}^2 \mu_z^2 - \abs{c_2}^2 \mu_+^2
\end{equation}
Therefore, by choosing $\Omega_\textrm{MW}^{(1,2)}$ in such a way that $\abs{c_1} = (\mu_z/\mu_+)\abs{c_2}$, we can always remove the dipolar direct interaction. 

Even more simply, we can choose $\ket{mP}$ to have the same $l, j, m_j$ quantum numbers as $\ket{m'P}$, and admixing both into $\ket{nS}$ with opposite sign detunings as shown in \figref{fig:MW}. $\ket{mP}$ and $\ket{m'P}$ then have identical angular wavefunctions, so the relative strength of their coupling to $\ket{nS}$ is determined by their radial wavefunction matrix elements. As they are admixed with opposite sign, tuning the microwave drives then allows the cancellation of these couplings. This is the scheme we use in the main text, where the effective Hamiltonian is given by 
\begin{equation}
    H_\textrm{tot} = H_{XY} + H_{ZZ},
\end{equation}
which has Ising interactions independent of the degree of admixture (c.f. \eqnref{eq:H_one}).




































\section{State preparation protocols} \label{app:preparation}


In a Rydberg quantum simulator, there are a number of different ways to adiabatically prepare the ground state of $H_\textrm{eff}$ at a desired point in the phase diagram of the model Hamiltonian. One approach is to construct a trajectory inspired by  an adiabatic rapid passage (ARP) \cite{beterov2020} protocol, where each ground state $\ket{g}$ is laser coupled to Rydberg states in ${\cal H}_\textrm{Ryd}$ and then boosted in energy above the Rydberg manifold by adjusting the laser's detuning in the rotating frame, as shown in  Fig.3b of the main text. However, simply transferring all the atoms into the individual subspace ${\cal H}_\textrm{Ryd}$ is not sufficient, as we may want to land in a specific total magnetization sector; in the case of DQC, the sector  with the ground state of  of interest is such that $N^\textrm{gs}_\uparrow=L/2$.

To this end, we introduce a general method that adiabatically ramps $\ket{g}^{\otimes L}$ into a final state in ${\cal H}_\textrm{Ryd}^{\otimes L}$ with a desired magnetization $N_\uparrow$.
It turns out, this can be achieved in a surprisingly simple way: an ARP trajectory where we address $N_\uparrow$ sites with $\Omega_\uparrow$, and $L - N_\uparrow$ sites with $\Omega_\downarrow$. Conceptually, during the adiabatic sweep, the laser $\Omega_\uparrow$ ($\Omega_\downarrow$) can only transfer atoms from $\ket{g}$ into $\ket{\uparrow}$ ($\ket{\downarrow}$), and once transferred, the number of atoms in $\ket{\uparrow}$ is conserved by $H_{\textrm{Ryd}}$. At the end of the trajectory then, when no atoms are left in $\ket{g}$, we are guaranteed to achieve the desired $N_\uparrow$.




To prove this, consider the following Hamiltonian, consisting of three levels in each atom: $\ket{\uparrow}$, $\ket{\downarrow}$, and a ground state $\ket{g}$ as in the main Fig.3. Laser $\Omega_{\textrm{L} \uparrow}$ couples $\ket{g}$ to $\ket{\uparrow}$ on a set of sites $A$ of size $N_A$, and laser $\Omega_{\textrm{L} \downarrow}$ couples $\ket{g}$ to $\ket{\downarrow}$ on a set of sites $B$ of size $N_B$, all with detuning $\Delta_\textrm{L}$:
\begin{align} \label{eq:app_adiabatic}
    H = H_\textrm{eff} + \Omega_{\textrm{L} \uparrow} \sum_{i \in A}(\ket{\uparrow}_i \bra{g}_i + h.c.)   + \Omega_{\textrm{L} \downarrow} \sum_{i \in B}(\ket{\downarrow}_i\bra{g}_i + h.c.) + \Delta_\textrm{L} \sum_{i}\ket{g}_i\bra{g}_i.
\end{align}

At the end of the trajectory where $\Delta_L \gg 1$, we can show that the number of spin up and down states $N_\uparrow$ and $N_\downarrow$ are $N_A$ and $N_B$. To prove this,  note that the following quantities are conserved throughout \textit{any} adiabatic trajectory, where $n^{\uparrow}_i$ is the projector onto $\ket{\uparrow}$ on site $i$:
\begin{align}
    \sum_{i \in A \cup B} n^{\uparrow}_i + \sum_{j \in A} n^g_j = N_A \qquad   \sum_{i \in A \cup B} n^{\downarrow}_i  + \sum_{j \in B} n^g_j = N_B  \qquad \Rightarrow \qquad \textrm{Conserved}
\end{align}
\noindent \emph{Proof:} 
\begin{align}
    \qty[\sum_{i \in A \cup B} n^{\uparrow}_i + \sum_{j \in A} n^g_j, H] &= \qty[\sum_{i \in A \cup B} n^{\uparrow}_i + \sum_{j \in A} n^g_j, H_\textrm{eff}] + \qty[\sum_{i \in A \cup B} n^{\uparrow}_i + \sum_{j \in A} n^g_j, H- H_\textrm{eff}] \nonumber \\
    &= \Omega_{\textrm{L} \uparrow} \sum_{i \in A} \bigg[   n^\uparrow_i +  n^g_i,\,\,  \ket{\uparrow}_i\bra{g}_i + \textrm{h.c.}  \bigg] = 0
\end{align}
where the first term in the RHS of the first line disappears as the DQCP Hamiltonian is block diagonal in Rydberg states and conserves $S^z$ within the manifold. The same holds when $A$ and $B$ are exchanged.



Therefore, if we assume that $n^g_i = 0$ at the end of the trajectory, which we physically expect due to large $\Delta_\textrm{L} \gg 1$, we find that $\sum n_i^\uparrow = N_A$ and $\sum n_i^\downarrow = N_B$ no matter what the trajectory is. This provides a general mechanism to prepare a state with desired filling.




Now, there is a separate question of whether the trajectory is gapped along the path (except at the end where the gap necessarily has to close since we are preparing a critical state). Here, we make a perturbative argument for the trajectory being gapped. For simplicity, we ignore terms with projectors in \eqnref{eq:Rydberg}, and treat $H_\textrm{DQCP}$ as a small perturbation. First, ignore $H_\textrm{DQCP}$ in \eqnref{eq:app_adiabatic}. Parametrize $(\Delta, \Omega)$ as $\frac{\Delta}{2} = -\eta \cos \theta$ and $\Omega = \eta \sin \theta$. Then, the energy levels of each odd site is given as
\begin{eqnarray}
    & E_1 = -\eta(1+\cos \theta) \quad &\ket{\varphi_1} = \frac{\sin(\theta/2)}{\sin \theta} \cdot \Big[ -(\cos \theta + 1) \ket{g} +  \sin \theta \ket{\downarrow} \Big] = \ket{\downarrow'_\textrm{odd}} \nonumber \\
    & E_2 = 0\quad & \ket{\varphi_2} = \ket{\uparrow} = \ket{\uparrow'_\textrm{odd}}  \nonumber \\
    & E_3 = \eta(1-\cos \theta) \quad & \ket{\varphi_3} =\frac{\cos(\theta/2)}{\sin \theta} \cdot \Big[ -(\cos \theta - 1) \ket{g} +  \sin \theta \ket{\downarrow} \Big]
\end{eqnarray}
and for even site is given as 
\begin{eqnarray}
    & E_1 = -\eta(1+\cos \theta) \quad &\ket{\varphi_1} = \frac{\sin(\theta/2)}{\sin \theta} \cdot \Big[ -(\cos \theta + 1) \ket{g} +  \sin \theta \ket{\uparrow} \Big] = \ket{\uparrow'_\textrm{even}} \nonumber \\
    & E_2 = 0\quad & \ket{\varphi_2} = \ket{\downarrow} = \ket{\downarrow'_\textrm{even}} \nonumber \\
    & E_3 = \eta(1-\cos \theta) \quad & \ket{\varphi_3} =\frac{\cos(\theta/2)}{\sin \theta} \cdot \Big[ -(\cos \theta - 1) \ket{g} +  \sin \theta \ket{\uparrow} \Big]
\end{eqnarray}
If we project into the lower two levels for each site, we have effective two-level system with the following staggering field:
\begin{equation}
    H'_\textrm{stag} = \sum_n (-1)^n \frac{\eta(1+\cos \theta)}{2} Z'_n
\end{equation}
Furthermore, on this effective spin up and down subspace $\ket{\uparrow'}$ and $\ket{\downarrow'}$, the first-order contribution of $H_\textrm{DQCP}$ is given by the following:
\begin{align}
    H'_\textrm{DQCP} &= \qty[\sin(\frac{\theta}{2})]^2  J_3 \sum_i \Big[ (X'_i X'_{i+1} + Y'_i Y'_{i+1}) + \alpha (X'_i X'_{i+2} + Y'_i Y'_{i+2}) + \dots \Big] \nonumber \\
    & + \qty[\frac{1 + (\sin(\theta/2))^2}{2}]^2  J_6 \sum_i \Big[ Z'_i Z'_{i+1} + \alpha^2 Z'_i Z'_{i+2} + \dots \Big] + \textrm{(chemical potential)}
\end{align}
which, in fact, acts like a zAFM Hamiltonian since the effective strength of the $ZZ$ coupling becomes stronger than the $XY$ couplings for $\theta < \pi$. Since the staggering field induces a strong gap on the DQCP-zAFM phase, we can argue that through the trajectory of $\theta \in [0,\pi)$, the system is gapped. 



The ARP-inspired state preparation protocol we describe above is not the only way to prepare the ground state in desired magnetization sector. For example,  if we assume we can prepare any product state in ${\cal H}_\textrm{Ryd}^{\otimes L}$ with high precision, e.g.,~through individually or sub-lattice laser addressing atoms from the ground states, we can  imagine simply beginning  from a state $\ket{\psi_0^\textrm{stag}} = \ket{\uparrow, \downarrow, \uparrow, \downarrow, \dots}$. This can be done by applying a short pulse to the ground state of Rydberg atom array. Consider now implementing a  Hamiltonian with strong alternating $Z$-fields, in addition to the Rydberg Hamiltonian $H_\textrm{eff}$:
\begin{equation}
    H = H_\textrm{eff} + g \sum_n (-1)^n Z_n,
\end{equation}
where $H_\textrm{eff}$ depends on $(J_3/J_6,\alpha)$. We first start from $J_3/J_6 = 0$ and $g > 0$ (assuming $J_6$ is at unity). There, our initial state is a perfect groundstate of the Hamiltonian. Then, we slowly increase the tuning parameter $J_3$ to the critical value for a given $\alpha$ and $J_6$, which can be achieved by modifying the strength of microwave dressing. After which, we  slowly turn off the field $g_2$. Throughout the entire adiabatic path, there is a constant gap except at the DQCP point, which closes as $1/L$.


In both methods,  an explicit bias field is applied that breaks the translation symmetry during the adiabatic evolution, which can lead to  excitations in non-symmetric sectors during  adiabatic evolution. 
In order to overcome this issue, one may envision to implement a symmetric adiabatic evolution. For example, we may start from a symmetric ground state (cat state) of the zAFM phase at $J_3/J_6=0$ for some $\alpha$, and then increase $J_3/J_6$ adiabatically to reach the DQCP ground state. However, this requires the cat state preparation with high fidelity, which is generally nontrivial; note that previously proposed methods of preparing a  cat state  involves an adiabatic trajectory through an Ising quantum phase transition~\cite{cat_state};  as the gap closes there too (in addition to at the DQCP) this entails a significant state preparation time for a high fidelity.
In contrast, our proposed methods involving  an adiabatic evolution in which symmetries are explicitly broken, only encounters a single gap-closing of $\sim 1/L$ near the end of the trajectory, and should thus be more favorable than the symmetric adiabatic evolution protocol. 










 
\begin{figure}[!t]
    \centering
    \includegraphics[width = 0.6 \textwidth]{Fig_sup_orderparam.png}
    \vspace{-10pt}
    \caption{\label{fig:sup_order}  Density plot and radial/angular histograms of measurement outcomes of order parameters, obtained from exact diagonalization at $L=24$ for three different regimes at $\alpha=0.5$: (a) zAFM (b) DQCP (c) VBS. Note that for the radial distribution, the histogram is normalized by the radius so that $N_\textrm{tot} = \int \dd{r} 2\pi r \cdot N(r)$. Therefore, it faithfully represents the radial density of the order parameter. }
\end{figure}
 

\section{Snapshots of Observables in Finite Systems}
\label{app:observable}

In numerical simulations of finite-size systems, we measured the following observables:
\begin{align}
\bar{O}_\textrm{zAFM}& \equiv  \frac{1}{L} \sum_r (-1)^r Z_r \nonumber \\
\bar{O}'_\textrm{VBS} &\equiv \frac{1}{L} \sum_r  (-1)^r \qty[\frac{Z_{r-1}Z_r - Z_{r} Z_{r+1}}{2} ].
\end{align}
Note that the ground state of the system  satisfies the constraint $\sum_i Z_i = 0$. Therefore, if we take a snapshot of the wavefunction in the computational $z$-basis,  the possible values of $\bar{O}_\textrm{zAFM}$ and $\bar{O}'_\textrm{VBS}$ are spaced by multiples of $\frac{4}{L}$.For a given system size $L=2m$,  we thus have
\begin{align}
    \bar{O}_\textrm{zAFM}, \bar{O}'_\textrm{VBS} &\in \frac{1}{L} \cdot [-2m, -2m+4, ..., 2m]
\end{align}
However, note that while $\expval{\bar{O}_\textrm{zAFM}}$ is $\pm 1$ for the ideal zAFM state ($\ket{\uparrow \downarrow \uparrow ...}$), $\expval{\bar{O}'_\textrm{VBS} }$ is $\pm 1/2$ in the ideal VBS state (product state of singlet dimers).  When we compare these observables in a 2d histogram, we have normalized the VBS order parameter by its value in the ideal VBS state. (Strictly speaking, we can fix the relative normalization of ${O}_\textrm{VBS}$ and ${O}_\textrm{zAFM}$ by, for instance, requiring that their distributions have the same second moment. However, the above normalization procedure  already produced distributions with near circular symmetry). In the histogram plot, each data point is  hence  specified by the data set $(\bar{O}_\textrm{zAFM}, 2\bar{O}'_\textrm{VBS})$. After this rescaling, we  obtain a circular distribution of the order parameters $(\bar{O}_\textrm{zAFM}, 2\bar{O}'_\textrm{VBS})$ at the DQCP, as demonstrated in the  Fig.~3(d) of the main text. In fact, already at system size $L=24$, we can observe that the distribution becomes circular (\figref{fig:sup_order}(b)) at the DQCP, whereas the distribution features two isolated peaks in both VBS and zAFM phases (\figref{fig:sup_order}(a,c)). 



In order to identify the rotational invariance of the joint probability distribution, we have shown angular and radial distributions in  Figs.~3(e,f) of the main text. Note that almost uniform angular distribution of $(\bar{O}_\textrm{zAFM}, 2\bar{O}'_\textrm{VBS})$ implies the presence of the emergent $\mathrm{U}(1)$ symmetry, a characteristic feature of DQCP. 
For the radial distribution $\hat{P}(r)$, note that the quantity is calculated by normalizing the number of counts in the bin $[r,r+\delta)$ by $2 \pi (r+\delta/2)$ so that the bias originated from the integration measure is properly removed. Then, the peak location of $\hat{P}(r)$ measures the magnitude of order parameter fluctuation in the system with finite-size $L$, which should decay as $1/L^K$ where $K$ is Luttinger parameter. Although it means that the distribution should shrink to the origin in the thermodynamic limit, the fact that the Luttinger parameter $K$ is small ($\sim$\,$1/3$ for the DQCP plotted in the manuscript Fig.3d) allows us to access a circle with the resolution increasing with the system size. Since the resolution improves as $1/L$ while the radius decays with $1/L^K$ with $K<1$, we can identify the circle better and better as we increase the system size.
Finally, we note that for $L>24$, the data was obtained by simulated measurements in finite DMRG simulations with the periodic boundary condition implemented by the long-range interaction between the first and last sites. In order to improve the statistics, which is limited by the number of measurements ($2 \times 10^4$ for all $L=64,128,192$), we explicitly symmetrize the outcomes with respect to inversion and spin-flip symmetries. 









\section{DMRG simulation of long-range Hamiltonian}\label{app:long_range}


In the Rydberg atom array system, the Hamiltonian has long-range dipolar (XY) and  Van der Waals (ZZ) interactions, which exhibit power-law decaying behavior as $1/r^3$ (XY) and $1/r^6$ (ZZ) respectively. However, an exact representation of a power-law decaying long-range interaction in the DMRG simulation is computationally inefficient, while exponentially decaying interactions can be expressed and computed efficiently in the simulation. Therefore, we approximated these power-law decaying interactions with a set of exponential functions~\cite{LR_Zaletel}; since dipolar and van der Waals interactions  decay quickly with large powers, such an approximation can be made very accurate with only a small number of exponential functions.
In \figref{fig:longrange}, we compare the strengths of the actual interactions and exponentially approximated versions. In the DMRG simulations, we use the sum of four exponential functions, where we find that the integrated difference is less than $1\%$ for both dipolar and $0.01\%$ for Van der Waals interactions, and the difference  decreases for smaller $\alpha$.
Such small differences justify the use of DMRG method with a set of exponentially decaying interactions to represent dipolar and van der Waals interactions. 


\begin{figure}[!t]
    \centering
    \includegraphics[width = 0.9 \textwidth]{LR_fit.pdf}
    \caption{\label{fig:longrange} Comparison between actual dipolar/van der Waals interactions and exponentially fitted approximation using the sum of four exponentials at (a) $\alpha = 0.2$, (b) $\alpha=0.5$, and (c) $\alpha=1.0$. The difference between power-law decaying interactions and their exponential approximations integrated over the entire range is less than 1\% for the range of $\alpha$ we simulated. 
    }
\end{figure}



\end{widetext}






\end{document}





