Real-Time Dynamics in U(1) Lattice Gauge Theories with Tensor Networks

Tensor network algorithms provide a suitable route for tackling real-time-dependent problems in lattice gauge theories, enabling the investigation of out-of-equilibrium dynamics. We analyze a U(1) lattice gauge theory in ($1+1$) dimensions in the presence of dynamical matter for different mass and electric-field couplings, a theory akin to quantum electrodynamics in one dimension, which displays string breaking: The confining string between charges can spontaneously break during quench experiments, giving rise to charge-anticharge pairs according to the Schwinger mechanism. We study the real-time spreading of excitations in the system by means of electric-field and particle fluctuations. We determine a dynamical state diagram for string breaking and quantitatively evaluate the time scales for mass production. We also show that the time evolution of the quantum correlations can be detected via bipartite von Neumann entropies, thus demonstrating that the Schwinger mechanism is tightly linked to entanglement spreading. To present a variety of possible applications of this simulation platform, we show how one could follow the real-time scattering processes between mesons and the creation of entanglement during scattering processes. Finally, we test the quality of quantum simulations of these dynamics, quantifying the role of possible imperfections in cold atoms, trapped ions, and superconducting circuit systems. Our results demonstrate how entanglement properties can be used to deepen our understanding of basic phenomena in the real-time dynamics of gauge theories such as string breaking and collisions.

Popular Summary

The quantum description of the real-time dynamics of many-body systems is one of the most fundamental problems at the frontier of several fields in physics. One classic example of such challenges is the description of string breaking in quantum electrodynamics in which the electric flux string that connects a particle-antiparticle pair breaks to produce the particle-antiparticle pair in a manner very much akin to confinement in nuclear forces. Despite its apparent simplicity, this dynamical process is very difficult to describe in a fully quantum-mechanical manner. Here, by exploiting recent algorithmic developments in the field of gauge-invariant tensor networks, we investigate the real-time dynamics of string breaking in a quantum link formulation of the one-dimensional U(1) lattice gauge theory.

Our simulations characterize paradigmatic aspects of string breaking, such as the Schwinger mechanism, as well as the propagation of entanglement in the system. We focus on a variety of mass and electric-field couplings, and we are able to show how the wave fronts of entanglement and matter propagation are tied together during this process. In addition, we describe simple-meson scattering problems in real time, where the formation of a quantum ebit of entanglement takes place after collisions.

Our results demonstrate how tensor network methods can faithfully describe the real-time dynamics of processes of interest in high-energy physics. We expect that our findings will open up a new perspective on our understanding of these processes from the point of view of entanglement theory.

Figure 1

Left panel: Hilbert space and gauge-invariant states of the QLM. (i) In the quantum link formulation, the gauge fields defined on the links are described by spins (in our case, $S=1$). (ii) Staggered fermions represent matter and antimatter fields on a lattice bipartition: On the even (odd) bipartition, a full (empty) site represents a particle (antiparticle). (iii) Hilbert space and gauge-invariant states of the QLM. The Gauss law, Eq. (3), constrains the number of possible states at each lattice site. Notice that the Gauss law depends on the lattice site because of the staggered fermions. Middle panel: Cartoon states for the different stages of the string-breaking dynamics (see text). Here, the leftmost site is an odd one. Right panel: Sample simulation for the electric-field dynamics when quenching an initial string state (B region) connecting two charges, and surrounded by the vacuum (A regions) for $m=0=g$. Primary string breaking takes place in four stages (C–F), until an antistring is created in place of the original string. The latter also decays during the secondary string breaking. The shaded areas represent the wave fronts estimated from entanglement entropies (see Sec. 4), which are directly related to the electric-field evolution.

Figure 2

Tensor network representations of a many-body quantum system. (a) Any quantum state can be described via a tensor $\mathrm{\Psi }$ of exponential dimension $d^N$, where $d$ is the dimension of the local Hilbert space and $N$ the number of sites. (b) In a MPS representation, the wave function is characterized by $N$ local tensors $A$, each one of dimensions $m^{2}d$, where $m$ is the maximum Schmidt rank allowed between different bipartitions. (c) Gauge-invariant tensor network: The gauge-invariant state can be represented via a tensor network state where a MPO imposes the gauge invariance while a variational MPS accommodates for the detailed description of the wave function.

Figure 3

Real-time evolution of a string of electric flux of length $L=20$ embedded in a larger lattice (of length $N=100$) in the vacuum state (the initial cartoon state is sketched on the left side of the figure with the notation of Fig. 1). The electric flux real-time evolution (column A) is shown with the evolution of the mass excitations (column B) and the evolution of the bipartite von Neumann entropy (column C) for $m=0$, $g=0$ (line 1), $m=0.25$, $g=1.25$ (line 2), and $m=3$, $g=3.5$ (line 3). The von Neumann entropy is calculated using a bipartition of the system defined via a cut between lattice sites $x$ and $x+1$.

Figure 4

Mean electric field of the central six lattice sites as a function of time $\tau$ for the electric coupling $g=0.00$, 0.25, 0.50, 0.75, 1.00, 1.05, 1.10, 1.15, 1.20, 1.25, 1.50 (orange to light blue) for $m=0$ (top panel) and $m=0.25$ (bottom panel). Primary (secondary) string breaking occurs when the mean electric field crosses the zero line from positive (negative) values.

Figure 5

State diagram of string breaking. The red area shows the parameters where the string breaks and evolves into a negative string ($E_{\mathrm{mean}}<-0.15E_{\max }$). The white area represents the parameters where the mean electric field approaches zero and stays around that value ($|E_{\mathrm{mean}}|<0.15E_{\max }$). And finally, the green area represents the parameters without string breaking ($E_{\mathrm{mean}}>0.15E_{\max }$).

Figure 6

Spreading of the wave front of the electric field for $m=0$ and $g$ increasing from $g=0$ (blue) to $g=1.5$ (orange). A linear fit for $\tau >2$ results in a velocity of $v_{\mathrm{E}}=1.96\pm 0.02$ (dashed green line). Inset (top left): Time needed for the wave front to spread one lattice site as a function of $m$ for $g=0$. Inset (bottom right): Wave front as a function of time evolving from the last site of the original string ($x=x_i$, blue) to the lattice site $x=x_i+16$ (orange) for $g=0$. The wave front is calculated using the electric-field difference ${\mathrm{\Delta }}_E=E(x+2)-E(x)$.

Figure 7

Time needed for the maximal mass production caused by the Schwinger mechanism as a function of the mass for $g=0$, 0.5, 1, 1.5, 2, 2.5, 3, 3.5 (orange to light blue).

Figure 8

Large-mass behavior of the mass-production time scale for $g=0$. The black line fits the data points with ${\tau }_{\max }=(1.54\pm 0.02)/m$. Inset: Log-log plot of the same data.

Figure 9

Time evolution of the electric field, including imperfections of the type $H_I=\xi \sum _x{n}_x(S_{x-1,x}^z+S_{x,x+1}^z)$, with $\xi =0.1$ at the system parameters $m=0$, $g=0$ (left), $m=0.25$, $g=1.25$ (center), and $m=3$, $g=3.5$ (right). The bottom row shows the difference from the result obtained without imperfection.

Figure 10

Bipartite von Neumann entropy $S(x)$ at $m=0$, $g=0$ (solid line), $m=0.25$, $g=1.25$ (dot-dashed line) and $m=3$, $g=3.5$ (dashed line). Partition at the center of the initial string ($x=51$, blue; $x=50$, violet) and in the vacuum ($x=20$, orange).

Figure 11

Estimation of the spreading velocity $v$ from the entanglement entropy $S$. The wave front is defined as when the entropy drops below a certain threshold value with respect to the vacuum. The resulting fit using $v=v_{\mathrm{S}}-a/[{\log }_{10}(\mathrm{\Delta }S)-S_0{]}^2$ gives us an estimate for the spreading velocity of $v_{\mathrm{S}}=(2.0\pm 0.2)$. Inset: Log-log plot of the same data adjusted by $v_{\mathrm{S}}$ and $S_0$.

Figure 12

Scattering of two dynamical mesons using the system parameters $m=0$, $g=8$. The plot illustrates the time evolution of the electric field $E(x)$ as a function of the position $x$. Lower panel: Number of charges $N_{\mathcal{B}}=\sum _{x\in \mathcal{B}}{n}_x$ in the system during the evolution (blue: $\mathcal{B}=\{1,\dots ,32\}$), number of particles present in the center (purple: $\mathcal{B}=16$), and number of charges on either side of the center (coinciding lines, red: $\mathcal{B}=\{1,\dots ,15\}$; orange: $\mathcal{B}=\{17,\dots ,32\}$).

Figure 13

Scattering of two dynamical mesons. Main panel: Entanglement entropy $S(x)$ using a bipartition between sites $x$ and $x+1$ as a function of time. After the scattering, the entropy significantly increases in the system; this is a direct signature of enhanced quantum correlations. Right panel: $S(x)$ at different times (see color bar), showing a clear plateau after the collision, which enlarges as a function of time. The empty circles show the current position of the maxima of the electric field which follow approximately the meson’s center of mass. The dashed line represents $S(x)$ generated by a single meson, while the green bar highlights the difference $\mathrm{\Delta }S$ from the entropy of the colliding mesons (difference between solid and dashed lines at $\tau =120$, $x_i=17$).

Figure 14

Convergence tests of the electric field $E$ for the cases $m=0$, $g=0$ (blue) and $m=0.25$, $g=1.00$ (orange). Left panel: Difference of the computed electric field from the offset $E-E_{\lim }$ at $\tau =8$ as a function of the inverse bond dimension $\chi$. The offset is obtained via a fit according to $E\propto (1/\chi {)}^b+E_{\lim }$. Inset: Truncation error during the time evolution for $\chi =200$. Right panel: Difference of the computed electric field from the offset $E-E_{\lim }$ at $\tau =2$ as a function of the time step $\delta t$. The offset electric field is obtained via a fit according to $E\propto \delta t^b+E_{\lim }$. The exponent $b$ is in perfect agreement with the expected value of 2.

Figure 15

Screening of external static charges after a time $\tau =400$ using $m=0.05$, $g=1$, and $\chi =80$ (blue squares). Two numerical lattice sites are combined to form one physical lattice site because of our staggered-fermion formulation. The dynamical charge density decays exponentially with the distance from the charges; the black line represents the theoretical model with $q(x)=a\exp (-bx)-a\exp (bx)$, where the decay parameter is $b=g/\sqrt{\pi }$ and the scale parameter is $a=\exp (b)-1$.