Quantum Simulation of the Universal Features of the Polyakov Loop

Lattice gauge theories are fundamental to our understanding of high-energy physics. Nevertheless, the search for suitable platforms for their quantum simulation has proven difficult. We show that the Abelian Higgs model in $1+1$ dimensions is a prime candidate for an experimental quantum simulation of a lattice gauge theory. To this end, we use a discrete tensor reformulation to smoothly connect the space-time isotropic version used in most numerical lattice simulations to the continuous-time limit corresponding to the Hamiltonian formulation. The eigenstates of the Hamiltonian are neutral for periodic boundary conditions, but we probe the nonzero charge sectors by introducing either a Polyakov loop or an external electric field. In both cases we obtain universal functions relating the mass gap, the gauge coupling, and the spatial size, which are invariant under the deformation of the temporal lattice spacing. We propose to use a physical multileg ladder of atoms trapped in optical lattices and interacting with Rydberg-dressed interactions to quantum simulate the model and check the universal features. Our results provide a path to the analog quantum simulation of lattice gauge theories with atoms in optical lattices.

Figure 1

$N_s\mathrm{\Delta }E$ versus $g^2{N}_s^2$ for the gap $\mathrm{\Delta }E$ created by the insertion of the Polaykov loop (lower set) or an external electric field (10 boundary conditions, upper set). Open (filled) markers represent Lagrangian (Hamiltonian) data. The choices of parameters, units, and methods for both of the 24 data sets are explained in the text.

Figure 2

The Polyakov loop (arrows pointing up) and the matter loop (arrows pointing down) composed of matter charges and plaquette quantum numbers for (a) 00BC and (b) 10BC without the Polyakov loop. Additional columns of zeros or ones are implicit. The dotted line indicates the wrapping in the Euclidean time direction.

Figure 3

Multileg ladder implementation for spin-2. The upper part shows the possible $m_z$ projections. Below, we show the corresponding realization in a ladder within an optical lattice. The atoms (green disks) are allowed to hop within a rung with a strength $J$, while no hopping is allowed along the legs. The lattice constants along rungs and legs are $a_r$ and $a_l$, respectively. Coupling between atoms in different rungs is implemented via an isotropic Rydberg-dressed interaction $V$ with a cutoff distance $R_c$ (marked by blue shading).

Figure 4

Quadratic interactions on an asymmetric ladder for $s=2$. The isotropic Rydberg-dressed potential (dashed blue line) is sampled at different distances occurring in the ladder (blue points). Interactions between atoms in different rungs separated by $\mathrm{\Delta }i=|i-i^{\prime }|$ occur in groups. The inset shows the approximate quadratic dependence for $\mathrm{\Delta }i=1$ versus distance $\mathrm{\Delta }m=|m-m^{\prime }|$ within a rung compared to a true quadratic interaction (red solid line). The parameters used are $R_c=a_l=7a_r$.

Figure 5

Data collapse for the quantum magnetic susceptibility of the quantum Ising chain with the known rescaling ${{\chi }^{\text{quant}}}^{\prime }={\chi }^{\text{quant}}{N}_s^{-(1-\eta )}$ versus ${\lambda }^{\prime }=N_s^{1/\nu }(\lambda -1)$. The reduced magnetic field $h^{\prime }=h{N}_s^{15/8}=1$ for all three system sizes.