Deconfined quantum criticality on a triangular Rydberg array

TL;DR

The work demonstrates a deconfined quantum critical point between two Z3-ordered phases in a triangular Rydberg-atom array by combining a field-theoretical continuum description with DMRG on quasi-1D geometries. A 2D effective theory with cos(3φ) and cos(6φ) perturbations maps to a 1D phase field whose critical point exhibits an emergent U($1$) symmetry and a central charge c=1, with critical exponents that depend on the cylinder width through an effective Luttinger parameter K'. The authors corroborate these predictions numerically on infinite cylinders and open ladders, extract scaling exponents, and show how the angular distribution of the order parameter reflects the U($1$) symmetry, providing concrete experimental probes via finite tweezer arrays. They further extend toward experimentally feasible settings, including open geometries with fifth-nearest-neighbor interactions and parameter ranges achievable in current Rydberg platforms, paving the way to observing DQCPs in programmable quantum simulators.

Abstract

Fluctuations can drive continuous phase transitions between two distinct ordered phases -- so-called deconfined quantum critical points (DQCPs) -- which lie beyond the Landau-Ginzburg-Wilson paradigm. Despite several theoretical predictions over the past decades, experimental evidence of DQCPs remains elusive. We show that a DQCP can be explored in a system of Rydberg atoms arranged on a triangular lattice and coupled through van der Waals interactions. Specifically, we investigate the nature of the phase transition between two ordered phases at 1/3 and 2/3 Rydberg excitation density, which were recently probed experimentally in [P. Scholl et al., Nature 595, 233 (2021)]. Using a field-theoretical analysis, we predict both the critical exponents for infinitely long cylinders of increasing circumference and the emergence of a conformal field theory near criticality showing an enlarged U(1) symmetry -- a signature of DQCPs -- and confirm these predictions numerically. Finally, we extend these results to ladder geometries and show how the emergent U(1) symmetry could be probed experimentally using finite tweezer arrays.

Figures (11)

• Figure 1: (a) Neutral atoms trapped on a triangular tweezer array with lattice vectors ${\bf a}_x$ and ${\bf a}_y$, where we depict the laser-coupled electronic levels. (b) Ground-state phase diagram for a cylinder with $N_x=15$ atoms along the $x$ axis and $N_y=9$ atoms along the periodic $y$ axis. (Top) Average occupation number $\langle n_{\bf j} \rangle$ in the $1/3$ phase ($\Delta/U=1.5$), at the DQCP [$(\Delta/U)_{\rm c}=3.158$], and in the $2/3$ phase ($\Delta/U=5$). (Bottom) The $1/3$ ($2/3$) phase spontaneously breaks the $\mathbb{Z}_3$ symmetry: $m=|m|e^{i\phi}$ can point along $\phi=0,2\pi/3,4\pi/3$ ($\phi=\pi/3, \pi, 5\pi/3$), corresponding to a real order parameter $\langle m^3+m^{3\dagger}\rangle=+1(-1)$. At the transition, the $\mathbb{Z}_6$ symmetry is enlarged to U($1$). The histograms show the distribution of $m$ [Eq. \ref{['eq:magnetization']}] over $10^4$ snapshots in the occupation basis when restricting the sites to the "bulk", i.e., excluding the first and the last three rows of atoms along ${\bf a}_x$. Results obtained using a finite MPS with bond dimension $\chi=100$ for $\Delta/U=1.5$ and $\Delta/U=5$, and $\chi=500$ for $\Delta/U=3.158$.
• Figure 2: (a) Mapping from a 2D cylindrical lattice to a 1D field theory model. The phase of the magnetization $m$ [Eq. \ref{['eq:magnetization']}] in each unit cell (of size $3\times 3$) is mapped to a 2D field $\phi(x,y)$ and by dimensional reduction to a 1D field $\phi(x)$. (b)--(e) Analysis of an infinitely long cylinder ($N_x=\infty$) with circumference $N_y=6$ ($l_y=2$ unit cells) for increasing bond dimension $\chi$ [see legend in panel (b)]. (b) The order parameter changes continuously from positive to negative as a function of $\Delta/U$, passing through zero at the critical point at $(\Delta/U)_{\rm c}\approx 3.158$ (dashed vertical line). (c) The correlation length $\xi/a$ as a function of $\Delta/U$ shows a peak at $(\Delta/U)_{\rm c}$, which diverges with $\chi$. Inset: extraction of the central charge $c$ from the scaling $S_{\rm tr}=c\log{(\xi_{\rm tr}/a)}/6$ at $(\Delta/U)_{\rm c}$. (d) Power-law decay of the two-point correlation function $\langle m(r) m(0)\rangle \approx C_1(r)=\rho^2_0 \langle e^{i\phi(r)} e^{-i\phi(0)} \rangle$ at $(\Delta/U)_{\rm c}$, where $m$ is evaluated on unit cells of size $3\times 3\,l_y$. Inset: Luttinger parameter $K'$ as a function of the cylinder circumferences $l_y$, extracted from the power-law decay of $C_1(r)$ [Table \ref{['tab:scaling_exponents']}]. (e) Linear fit of the order parameter and the correlation length (inset) in log-log scale as a function of $\bar{\Delta}=\Delta/U-(\Delta/U)_{\rm c}$, used to extract the critical exponents $\beta$ and $\nu$, respectively. (f) Dependence of the critical exponents on $l_y$. We compare the scaling dimension $x_3$ estimated from (red) $2-1/\nu$ and (blue) $2\beta/(1+\beta)$ to the theoretical prediction (black dots) $x_3= 9 K'/4$ and (black line) $x_3=9 K/(4 \rho^2_0l_y)$, with coefficient $K/\rho^2_0$ obtained from the fit in the inset of (d).
• Figure 3: (a) Unit cell of the infinite ladder ($N_x=\infty$) with $N_y=3l_y=12$ atoms along ${\bf a}_y$. (b) Correlation length $\xi/a$ as a function of $\Delta/U$ for the infinite ladder shown in (a) and increasing bond dimension $\chi$. Inset: extraction of the central charge $c$ from the scaling of $S_{\rm tr}=c\log{(\xi_{\rm tr}/a)}/6$ at $(\Delta/U)_{\rm c} \approx 3.1586$, corresponding to the maximum of the correlation length. (c)-(f) Experimentally feasible system. (c) Lattice and occupation number $\langle n_{\bf j} \rangle$ for $\chi=200$ and $\Delta/U=2.94$. (d) Distribution of the staggered magnetization $m$ [Eq. \ref{['eq:magnetization']}] of the state considered in (c) over $10^4$ snapshots in the occupation basis when restricting the sites to the "bulk" (i.e., excluding the last two "rings" of atoms). (e) Angular distribution of $m$, consistent with a weak potential $V(\phi)$ due to finite size effects SM. (f) Radial distribution of $m$, consistent with a potential $V(\rho)$.
• Figure S1: Phase diagram and phase transitions for the model in Eq. \ref{['eq:Sphi_g3g6_gH']}.
• Figure S2: Analysis of infinitely long cylinders with (first column) $l_y=1$, (second column) $l_y=2$ and (third column) $l_y=3$ unit cells along ${\bf a}_y$. For each of them, we show for different bond dimensions $\chi$: (First row) Power-law decay of the two-point correlation $\langle m({\bf r}) m(0)\rangle \approx C_1(r) =\rho^2_0\langle e^{i\phi(r)} e^{-i\phi(0)}\rangle$ at the transition point $(\Delta/U)_{\rm c}$, where the correlation length is maximal. The Luttinger parameter $K'$, shown in the inset of Fig. 2(d), is extracted from a linear log-log fit (red dashed line). (Second row) Evaluation of the critical exponent $\beta$ (main panels) and $\nu$ (insets), displayed in Fig. 2(f), from linear log-log fits of the order parameter and the correlation length as function of $\bar{\Delta}=\Delta/U-(\Delta/U)_{\rm c}$.