Real-time dynamics in $2+\mathrm{1D}$ compact QED using complex periodic Gaussian states

We introduce a class of variational states to study ground-state properties and real-time dynamics in $(2+1)$-dimensional compact QED. These are based on complex Gaussian states which are made periodic to account for the compact nature of the $U\left(1\right)$ gauge field. Since the evaluation of expectation values involves infinite sums, we present an approximation scheme for the whole variational manifold. We calculate the ground-state energy density for lattice sizes up to $20\times 20$ and extrapolate to the thermodynamic limit for the whole coupling region. Additionally, we study the string tension both by fitting the potential between two static charges and by fitting the exponential decay of spatial Wilson loops. As the ansatz does not require a truncation in the local Hilbert spaces, we analyze truncation effects which are present in other approaches. The variational states are benchmarked against exact solutions known for the one plaquette case and exact diagonalization results for a ${\mathbb{Z}}_3$ lattice gauge theory. Using the time-dependent variational principle, we study real-time dynamics after various global quenches, e.g., the time evolution of a strongly confined electric field between two charges after a quench to the weak-coupling regime. Up to the points where finite-size effects start to play a role, we observe equilibrating behavior.

Figure 1

Benchmark of the variational ground state energy for one plaquette against the value of the exact ground state, given by the Mathieu function with the lowest characteristic value. The inset shows the relative error of the variational ground state energy with respect to the exact ground-state energy.

Figure 2

Ground-state energy density extrapolated to the thermodynamic limit. The available lattice sizes are $8\times 8$ for couplings $g^2\ge 1.0,14\times 14$ for $g^2=0.8,0.9$ and $20\times 20$ for $g^2\le 0.7$.

Figure 3

Finite-size scaling for the ground-state energy density at $g^2=0.5$ (a) and $g^2=2.0$ (b). For $g^2=2.0$, the ground-state energy density for $L=8,7,6$ is fitted according to Eq. (25). The remaining data points correspond to $L=5,4,3$. For $g^2=0.5$, lattice sizes of $L=20,18,16$ are used for the fit, the remaining data points correspond to $L=14,12,10$.

Figure 4

The static potential $V\left(d\right)$ of two charges separated by a distance $d$ at $g^2=2.0$. The data points are computed on an $8\times 8$ lattice as the difference between the ground-state energy with the respective static charge configuration and the ground-state energy without static charges. The red line is a fit to the potential according to Eq. (26) with $\sigma =1.001$ and $b=0.146$.

Figure 5

The data points show different spatial Wilson loops $\langle W(R_1,R_2)\rangle$ in the ground state at $g^2=0.5$, computed on a $20\times 20$ lattice, as a function of the area $R_1\times R_2$. The maximally used edge length of a Wilson loops is 10 ($R_1,R_2\le 10$), with a maximum difference between the edges of one ($|R_2-R_1|\le 1$). The red line is a fit to the exponential decay of Wilson loops according to Eq. (27) with $\sigma =0.013,a=0.132$, and $c=0.349$.

Figure 6

String tension fitted via the static potential (blue) and via the decay of spatial Wilson loops (orange). For larger couplings ($g^2\ge 1.5$), the static potential fit performs better than the fit of Wilson loops and agrees with the strong-coupling prediction $g^2/2$. For small couplings ($g^2\le 1.4$), Wilson loop fits are more suitable. The more reliable method is shown with full data points while data points of the other method are made transparent.

Figure 7

String tension in the weak-coupling regime. While the Wilson loop fits show exponential decay of the string tension close to the theoretical value (${\nu }_0=0.318$ compared to ${\nu }_{0,\mathrm{theo}}=0.321$), the static potential fits become unreliable for couplings $g^2\le 0.6$.

Figure 8

Comparison of the ground-state energy density on a $3\times 3$ lattice without static charges, computed for a ${\mathbb{Z}}_3$ lattice gauge theory by exact diagonalization (orange) and for the full $U\left(1\right)$ theory by minimizing the variational energy (blue). The inset shows the variance of the electric field on a link in the variational ground states.

Figure 9

Benchmark of the variational time evolution of the $1\times 1$ Wilson loop after a quench from $g^2=2.5$ to $g^2=4.0$ on a $3\times 3$ lattice. It is compared with the time evolution of ${\mathbb{Z}}_3$ lattice gauge theory computed by exact diagonalization (the truncation from $U\left(1\right)$ to ${\mathbb{Z}}_3$ should only play a minor role in the strong-coupling regime).

Figure 10

Variational time evolution after a quench from $g^2=0.8$ to $g^2=0.5$ for lattice sizes of $8\times 8,10\times 10$, and $12\times 12$. The inset shows the relative error in energy $E$ with respect to the initial energy $E_0$ after the quench.

Figure 11

Variational time evolution after a quench from $g^2=0.6$ to $g^2=0.3$ for lattice sizes of $16\times 16,18\times 18$, and $20\times 20$. The inset shows the relative error in energy $E$ with respect to the initial energy $E_0$ after the quench.

Figure 12

Variational time evolution of the $1\times 1,2\times 2,3\times 3$ and $4\times 4$ Wilson loop after a quench from $g^2=0.5$ to $g^2=4.0$ on an $8\times 8$ lattice. The inset shows the relative error in energy $E$ with respect to the initial energy $E_0$ after the quench.

Figure 13

Variational time evolution on an $8\times 8$ lattice after a quench from $g^2=4.0$ to $g^2=0.5$ with a positive charge placed at ($x_1=2,x_2=4$) and a negative charge at ($x_1=6,x_2=4$). We measure (a) the $1\times 1$ Wilson loop at the origin $W(1,1)$ and (b) the electric field on a link between the two charges $E_1(2,4)$. The red dashed line represents the Coulomb value of the electric field. The inset shows the relative error in energy $E$ with respect to the initial energy $E_0$ after the quench.

Figure 14

Variational time evolution of the electric field on an $8\times 8$ lattice after a quench from $g^2=4.0$ to $g^2=0.5$ with a positive charge placed at ($x_1=2,x_2=4$) (blue dot) and a negative charge at ($x_1=6,x_2=4$) (red dot). The color of the charges is only for graphical illustration (not related to the color bar). The expectation value of the electric field is shown at $t=0.0,t=0.2$ and $t=2.0$. At $t=0.0$, the state is in the variational ground state for $g^2=4.0$ where the electric flux is confined between the two charges. After the quench, the electric field starts to spread over the lattice ($t=0.2$) and equilibrates at the Coulomb value for this charge configuration ($t=2.0$).

Figure 15

Contributions of different orders of $N_{\mathbf{p}}$ configurations to the infinite sums $I_{\text{el}}$ (a) and $I_{\text{mag}}$ (b) appearing in the high ${\gamma }_{\mathbf{k}}$ approximation of the variational energy. Due to the constraint $N_{\mathbf{k}=0}=0$, the sum over all elements of $N_{\mathbf{p}}$ needs to be zero. Every bar represents the summed contributions of all $N_{\mathbf{p}}$ configurations containing a certain number of (1,$-1$) pairs and the remaining entries zero. Orders which are not of this type have a negligible contribution, e.g., ${\left\{N\right\}}_{-2,1,1}$ has a summed contribution to $I_{\text{el}}$ of 0.076 and a summed contribution to $I_{\text{mag}}$ of 0.23.

Figure 16

Contributions of different orders of $N_{\mathbf{p}}$ configurations to the infinite sums $J_{\text{el}}$ (a) and $J_{\text{mag}}$ (b) appearing in the low ${\gamma }_{\mathbf{k}}$ approximation of the variational energy. Due to absence of a constraint, all $N_{\mathbf{p}}$ configurations need to be considered. The orders are organized in groups. $P$ denotes orders which contain a growing number of 1's. Since $N_{\mathbf{p}}$ and $-N_{\mathbf{p}}$ are evaluated together, $P$ also represents orders with a growing number of $-1$'s. $M\left(1\right)P$ contains orders whose nonzero elements are a single $-1$ and a growing number of 1's. The first order in $M\left(1\right)P$ contains a pair of (1,$-1$) as nonzero elements. $M\left(2\right)P$ is structured in the same way as $M\left(1\right)P$ but with two $-1$'s. Analogously for the other groups. The $N_{\mathbf{p}}=\mathbf{0}$ configuration (denoted as $\mathbf{0}$) has vanishing contribution to $J_{\mathrm{el}}$ but a nonzero contribution to $J_{\text{mag}}$.