U(1) lattice gauge theory and string roughening on a triangular Rydberg array

TL;DR

This work shows that a triangular Rydberg-atom array can simulate a ($2+1$)D $U(1)$ lattice gauge theory in which confinement is mediated by flux strings. Near a deconfined quantum critical point, these strings exhibit transverse roughening, evidenced by a logarithmic increase of the string width with separation and a universal Lüscher correction to the confining potential, consistent with $\gamma_0 = \pi/24$. The authors demonstrate, using tensor-network methods, that the roughening emerges from first-order plaquette-like processes in the underlying Rydberg Hamiltonian and persists under realistic truncations, with quench dynamics revealing large fluctuations and possible string breaking. The results establish a concrete, experimentally accessible platform to study confinement, string fluctuations, and non-equilibrium string dynamics in ($2+1$)D, with direct relevance to high-energy and condensed-mmatter physics.

Abstract

Lattice gauge theories (LGTs) describe fundamental interactions in particle physics. A central phenomenon in these theories is confinement, which binds quarks and antiquarks into hadrons through the formation of string-like flux tubes of gauge fields. Simulating confinement dynamics is a challenging task, but recent advances in quantum simulation are enabling the exploration of LGTs in regimes beyond the reach of classical computation. For analog devices, a major difficulty is the realization of strong plaquette interactions, which generate string fluctuations that can drive a roughening transition. Understanding string roughening -- where strong transversal functions lead to an effective restoration of translational symmetry at long distances -- is of central importance in the study of confinement. In this work, we show that string roughening emerges naturally in an analog Rydberg quantum simulator. We first map a triangular Rydberg array onto a (2+1)D U(1) LGT where plaquette terms appear as first-order processes. We study flux strings connecting static charges and demonstrate that, near a deconfined quantum critical point, the string exhibits logarithmic growth of its transverse width as the separation between charges increases, along with the universal Lüscher correction to the confining potential -- both signatures of string roughening. Finally, we investigate the real-time dynamics of an initially rigid string, observing large fluctuations after quenching into the roughening regime, as well as string breaking via particle-pair creation. Our results indicate that rough strings can be realized in experimentally accessible quantum simulators, opening the door to detailed studies of how strong fluctuations influence string-breaking dynamics.

Figures (13)

• Figure 1: (a) Rydberg atoms trapped in optical tweezers arranged on a triangular geometry with lattice spacing $a$. The figure depicts a $2/3$-ordered configuration of Rydberg excitations, corresponding to the LGT bare vacuum on the dual hexagonal lattice. Atoms in the region marked in light blue have been flipped ($\ket{g}\leftrightarrow \ket{r}$). In the LGT picture, this atomic configuration corresponds to a positive $Q = +2$ (red dot) and a negative $Q = -2$ (blue dot) charge connected by two rigid strings of electric field (dark blue lines) and separated by a distance $R=\sqrt{3}(d+1/3)a\approx5.77a$, corresponding to $d=3$. Inset: charges and strings reside on the sites and links of the dual hexagonal lattice, respectively. Strings form in correspondence of flipped dimers (gray ellipses) relative to the vacuum. The six-site unit cell is highlighted in gray. (b) Equivalence between the atomic configurations and the corresponding gauge-invariant states for two sites of the unit cell, highlighted in blue and orange in (a), respectively. Global spin flips leave the configurations unchanged, and the configurations associated to the remaining four sites are obtained through $C_3$ rotations. (c) First order processes in the Rabi frequency $\Omega$ induce the formation of string loops and charged loops in the bare vacuum. (d) Third order processes in $\Omega$ lead to transversal fluctuations of the string. First order processes in $\Omega$ lead to string breaking. (e) Schematic ground-state phase diagram for the triangular Rydberg array. In the absence of defects (static charges), the system hosts two ordered phases---the $1/3$- and the $2/3$-phase---separated by a DQCP (yellow line). Inset: introducing two static charges (red and blue circle) in the $2/3$ phase generates two strings of electric field (dark blue lines) connecting them. Deep in the phase, the string is 'rigid' with a width that remains constant as the distance between the two charges increases. Approaching the DQCP, the string becomes 'rough' with its width increasing logarithmically with distance. Based on the distance the string can break at different values of $\Delta/U$, leading to the creation of pairs of dynamical charges.
• Figure 2: (a)-(b) Illustration of the definition of the string width. The width $w^2(x)$ of the string connecting two static charges separated by a distance $R$ along the $x$-direction, shown in (b), is computed from the string profile in (a) using Eq. \ref{['eq:w2_x']}. Due to the choice of $y_0$, $w^2(x)$ exhibits oscillations, revealing three sublattices. (Left) For small $\Delta/U$, when the string is rough, $w^2(x)$ displays a parabolic shape with maximum at $R/2$. (Right) For large $\Delta/U$, when the string is rigid, $w^2(x)$ remains constant within each sublattice. (c) String width at $R/2$ as a function of $\Delta/U$ for one sublattice of (b), estimated from a parabolic fit in $x$ (corresponding black dashed line in (b)). For large $\Delta/U$, $w^2(R/2)$ remains constant as $R$ increases, while, for small $\Delta/U$, $w^2(R/2)$ increases with $R$. Results obtained using a finite MPS with bond dimension $\chi=900$.
• Figure 3: (a) String width $w^2(R/2)$ as a function of $R$ for three representative $\Delta/U$ in Fig. \ref{['fig:Fig2_Width_profile']}(c). For small $\Delta/U$, e.g., $\Delta/U=3.188$ and $3.198$, $w^2(R/2)$ increases with R, consistent with the expected logarithmic dependence; for large $\Delta/U$, e.g., $\Delta=3.3$, it remains constant; while for intermediate values there is a crossover between the two behaviors. Red lines indicate fits to $A\log{R}+C$. (b) Estimate of the fit parameter $A$ as a function of $\Delta/U$ for different $\Omega/U$, extracted via the analysis described in App. \ref{['sec:App_procedure_fit']}.
• Figure 4: (a) The confining potential $V(R)$ as a function of the distance $R$ between two static charges, shown for two representative values of $\Delta/U$. Red dashed lines indicate fits of the potential according to Eq. \ref{['eq:V(R)_gamma']}. Insets: examples of corresponding string profiles $\epsilon(x,y)$ [Eq. \ref{['eq:energy_density']}] along with the static and dynamical charges $|Q|$. (b)-(c) Fit parameters of $V(R)$ as function of $\Delta/U$ for three values of $\Omega/U$, extracted via the analysis described in App. \ref{['sec:App_procedure_fit']}. (b) The string tension $\sigma$ decreases with decreasing $\Delta/U$, i.e., when approaching the DQCP. (c) The $\gamma$ correction vanishes for large $\Delta/U$ (gray region) and becomes finite near the DQCP. The dashed vertical lines indicate the values of $\Delta/U$ used in (a). Results obtained using a finite MPS with bond dimension $\chi=900$.
• Figure 5: Universal Lüscher correction to the linear confining potential as a function of $\Delta/U$ for three values of $\Omega/U$, obtained by combining the fit parameters of the string width $A$ [Fig. \ref{['fig:Fig3_width_log']}(b)] and confining potential, $\sigma$ and $\gamma$, [Fig. \ref{['fig:Fig4_potential']}(b) and (c)] according to Eq. \ref{['eq:gamma_0']}, as detailed in App. \ref{['sec:App_gamma']}. The dashed horizontal line indicate the universal value $\gamma_0=\pi/24$ for a rough string in ($2+1$)D.