Classically emulated digital quantum simulation for screening and confinement in the Schwinger model with a topological term

We perform digital quantum simulation, using a classical simulator, to study screening and confinement in a gauge theory with a topological term, focusing on ($1+1$)-dimensional quantum electrodynamics (Schwinger model) with a theta term. We compute the ground state energy in the presence of probe charges to estimate the potential between them, via adiabatic state preparation. We compare our simulation results and analytical predictions for a finite volume, finding good agreements. In particular our result in the massive case shows a linear behavior for noninteger charges and a nonlinear behavior for integer charges, consistently with the expected confinement (screening) behavior for noninteger (integer) charges.

Figure 1

Results for $V_f(\ell )$ with $n_{\text{shots}}={10}^5$. The statistical uncertainties are smaller than the markers. The curves (in colors similar to and lighter than the data markers) represent $V_f^{(0)}(\ell )$ on the left and $V_f^{(0)}(\ell )+V_f^{(1)}(\ell )$ on the right with the corresponding volumes. The dashed curve (purple) on the right panel depicts $V_f^{(0)}(\ell )+V_f^{(1)}(\ell )$ for $N=101$, $Lg=40$.

Figure 2

Left: results for $V_f(\ell )$ with $n_{\text{shots}}=4\times {10}^5$. The error bars represent the statistical uncertainties. The solid (cyan) and dotted (magenta) curves represent $V_f^{(0)}$ and $V_f^{(0)}+V_f^{(1)}$, respectively. Right: the finite-volume analog ${\sigma }_f$ of the string tension for $m/g=0.2$. The error bars represent the standard uncertainties from the fit. The dashed (green) and dotted (magenta) curves represent ${\sigma }_{\text{Coulomb}}/g^2$ and ${\sigma }^{(1)}/g^2$, respectively. The band region represents the finite-volume counterpart of ${\sigma }^{(1)}/g^2$.

Figure 3

Results for $[E({\theta }_0,q,\ell )-E(0,0,0)]/g^{2}L$ with $N=15,m/g=0.2,ag=0.4$ and $n_{\text{shots}}={10}^5$. The statistical uncertainties are much smaller than the markers. The solid (black) and dashed (magenta) curves represent the results of the mass perturbation theory with $q=0$ and ${\sigma }^{(1)}({\theta }_0/2\pi )/g^2$, respectively.

Figure 4

Density plots of the eigenenergy of $H_A(t)/g$ for the first excited state relative to the ground state (both states in the $Q=-1$ sector) computed for $N=15$, $m_0/g=0.5$, $m/g=0.15$, $ag=0.4$, ${\theta }_0=0$, and $q=1$. Each panel shows the result for the indicated value of $\ell$.

Figure 5

Left: dependence of the quantum simulation result for $V_f(\ell )$ on the adiabatic time $T$ for $N=21$, $m/g=0.25$, $q=1$, ${\theta }_0=0$, and for the fixed value $\delta t=0.3g^{-1}$. The dashed lines represent the results calculated by exact diagonalization. Right: dependence of the quantum simulation result for $E({\theta }_0)/g^{2}L=E({\theta }_0,q=0,\ell =0)/g^{2}L$ on ${\theta }_0$ and $\delta t$ for $N=15$, $m/g=0.05$, $m_0/g=0.55$, $T=99g^{-1}$ and for a small value $a=0.1g^{-1}$. The exact diagonalization (QuSpin) result is represented by the solid curve.

Figure 6

Potential $V_f(\ell )$ for several values of $m/g$, and for $N=15$, $q=0.25$, ${\theta }_0=0$. The blue crosses represent the results of quantum simulation without statistical uncertainties. The black solid curves represent the $\mathcal{O}(m)$ analytic prediction $V_f^{(0)}(\ell )+V_f^{(1)}(\ell )$.

Figure 7

The values of $(d/d\ell )(V_f^{(0)}+V_f^{(1)})/g^2$ at $\ell =L/2$ for $L=(N-1)a$, $m/g=0.2$, and $ga=0.4$, with the indicated values of $N$. We also plot ${\sigma }^{(1)}/g^2$ given in (17).