Bulk and boundary quantum phase transitions in a square Rydberg atom array

Motivated by recent experimental realizations of exotic phases of matter on programmable quantum simulators, we carry out a comprehensive theoretical study of quantum phase transitions in a Rydberg atom array on a square lattice, with both open and periodic boundary conditions. In the bulk, we identify several first-order and continuous phase transitions by performing large-scale quantum Monte Carlo simulations and develop an analytical understanding of the nature of these transitions using the framework of Landau-Ginzburg-Wilson theory. Remarkably, we find that under open boundary conditions, the boundary itself undergoes a second-order quantum phase transition, independent of the bulk. These results explain recent experimental observations and provide important insights into both the adiabatic state preparation of novel quantum phases and quantum optimization using Rydberg atom array platforms.

Figure 1

(a) Quantum phase diagram: First- and second-order QPTs are denoted by the circle and star markers, respectively. Purple stars mark boundary transitions in systems with OBC. The shaded area marks the region where the system lies in the disordered phase when using PBC but exhibits a boundary-ordered phase with OBC. The gray dashed lines indicate parameter ranges investigated in Figs. 2 and 4. (b) Schematic pictures of three distinct density-wave orders corresponding to the star (orange), striated (green), and checkerboard (blue) phases. Filled sites denote Rydberg excitations while gray sites represent ground-state atoms.

Figure 2

Phase transitions between the disordered, checkerboard, and striated phases. Markers correspond to sizes $L=8$ (circle), 12 (triangle), and 16 (star) with increasing color intensity. The order parameter of the checkerboard [striated] phase across the disordered–checkerboard [checkerboard–striated] boundary shows a second-order QPT in panel (a) [(b)], with its universal collapse presented in the inset. [(c), (d)] The transition between the disordered and striated phases shows distinct signatures of a first-order transition: sharp jumps in the order parameter (c), and the Rydberg density (d) with a double-peaked distribution of QMC measurements at the transition.

Figure 3

Transition between the disordered and star phases. [(a), (b)] Both the order parameter and the Rydberg density converge to sharp step functions for increasing system sizes. (b) Double-peaked distribution of density measurements at the phase boundary indicates a first-order transition.

Figure 4

Mean-field phase diagram of the low-energy Landau theory in Eq. (4), illustrating the disordered, checkerboard, and striated phases as well as the fields condensed in each. The black and red lines represent second-order and first-order QPTs, respectively. The black dots mark the two tricritical points ${\text{T}}_1$ and ${\text{T}}_2$. The numerical minimization was performed taking $g=-u_1=v=-1, u_2=0.75$, and $w=0.5$ in Eq. (4).

Figure 5

Boundary phase transition and its implications for experiments, (a) Comparison of the striated order parameter (color map) obtained from our QMC simulation with OBC against that of the experiment in Ref. [4], showing good agreement. The boundary orders first (lower $\mathrm{\Delta }/\mathrm{\Omega }$, pink stars) and strongly influences the extent of the striated phase, which is extended to a wider range of parameters compared to the bulk behavior on a torus. Orange and green dots denote phase transitions in a system with PBC (Fig. 1). (b) Binder ratio of the boundary order parameter across the transition marked in gray in panel (a). The insets show the boundary ordering (upper left) and universal collapse with 1D exponents (lower right).

Figure 6

Extraction of critical exponents. System sizes are (in increasing color intensity) $8\times 8, 12\times 12, 16\times 16$, and $20\times 20$. [(a), (b)] The order parameter and the Binder ratio for the transition to the checkerboard phase. [(c), (d)] The order parameter and the Binder ratio for the transition to the striated phase from the checkerboard phase. [(e), (h)] Corresponding curves in the top row exhibiting data collapse with the extracted critical exponents.

Figure 7

Effect of truncating interactions at different distances $R_0$ for a system of size $16\times 16$. [(a), (d)] Transition to the star phase at $R_b=1.62$. The slight discrepancy between $R_0=4$ and $R_0=5$ suggests that the transition can be even sharper at $R_0=5$, at this value of the blockade radius. [(b), (e)] Transitions at $R_b=1.45$. For $R_0=2$, the transition is to the star phase, while for $R_0\in \{3,4,5\}$, one obtains a striated phase. [(c), (f)] Transition from the checkerboard to the striated phase for a fixed $\mathrm{\Omega }/\mathrm{\Delta }=2.3$. Taking $R_0=5$ slightly stabilizes the star phase but does not resolve the difficulty of simulating the interface between these two phases.

Figure 8

Seeding procedure for the interface between striated and star phases. Our QMC algorithm struggles in this regime. To estimate the phase boundary, we resort to phase seeding from both sides. [(a), (b)] Effect of the seeding procedure and temperature change on the phase boundary at $L=12$. [(c), (d)] Scaling with the system size at $T=0.002$. The gray region denotes our estimate of the location of the transition point.

Figure 9

RG flow of the theory (E8) in the $(u_0,v_0)$ coupling plane for $D=3, N=2<N_c$. The blue dot represents the stable $O\left(N\right)$-symmetric FPs whereas the other three unstable FPs are marked in red. Owing to the positivity (E9) and condensation conditions, the parameters of our theory are always constrained to lie in the (green/yellow) region to the right of $v_0=-2u_0$ (dashed line) in the upper-half plane. The yellow wedge marks the region from which the stable FP is inaccessible. Figure adapted from Ref. [52].

Figure 10

Schematic RG flow of the tetragonal theory, described by the LGW Hamiltonian (E16) with $M=2, N>1$, in the (a) $u_0=0$, (b) $v_0=0$, and (c) $w_0=0$ planes. In the $v_0=0$ subspace, the stable FP is the cubic one whereas on the other two planes, the $XY$ FP is stable. As before, all the stable (unstable) FPs are marked in blue (red). Figure adapted from Ref. [52].

Figure 11

Allowed region in parameter space of the tetragonal theory (E16), as defined by Eq. (6) for $w_0>0$ (green) and $w_0<0$ (blue). The two planes $v_0=0$ and $v_0+(3/2)w_0=0$ (on which the cubic FPs are stable) are shaded in pink. The blue dot represents the globally stable $XY$ FP; its $v_0$ coordinate is greatly exaggerated here for the sake of visual clarity.