Confinement and String Breaking for ${\mathrm{QED}}_2$ in the Hamiltonian Picture

The formalism of matrix product states is used to perform a numerical study of ($1+1$)-dimensional QED—also known as the (massive) Schwinger model—in the presence of an external static “quark” and “antiquark”. We obtain a detailed picture of the transition from the confining state at short interquark distances to the broken-string “hadronized” state at large distances, and this for a wide range of couplings, recovering the predicted behavior both in the weak- and strong-coupling limit of the continuum theory. In addition to the relevant local observables like charge and electric field, we compute the (bipartite) entanglement entropy and show that subtraction of its vacuum value results in a UV-finite quantity. We find that both string formation and string breaking leave a clear imprint on the resulting entropy profile. Finally, we also study the case of fractional probe charges, simulating for the first time the phenomenon of partial string breaking.

Popular Summary

Confinement is one of the key concepts of the standard model that explains why isolated quarks or gluons are not observed in nature. By probing the vacuum with a heavy quark-antiquark pair, one can distinguish two regimes: At short and intermediate distances, a string (i.e., a flux tube) of color electric fields forms, keeping the pair confined. At large distances, on the other hand, the string breaks because of the production of light charged particles out of the vacuum that screen both probe charges. However, a full derivation of this physical picture from the microscopic theory of quantum chromodynamics has been lacking. Here, we perform this task for the simpler case of ${\mathrm{QED}}_2$: quantum electrodynamics in two dimensions (one time and one spatial dimension, i.e., the Schwinger model).

We consider two probe charges and perform numerical simulations of the probed vacuum within the framework of tensor network states. We numerically simulate the modified vacuum quantum state in the presence of two probe charges, which allows us to obtain a detailed picture of the different aspects of the confinement mechanism. We also succeed, for the first time, at simulating the theoretically predicted phenomenon of partial string breaking in the case of fractional probe charges. We perform our simulations using the powerful tensor network state framework. This approach, although mainly developed in the context of condensed-matter physics, is quite universal since it essentially exploits the general structure of quantum entanglement in large systems. One of the main advantages of the tensor network approach is that it provides direct access to the quantum state, which allows us to study confinement and string breaking in terms of local observables, such as the electric field, the interquark potential, and the charge density. We conclude that string breaking leaves a characteristic imprint on the entanglement entropy.

We expect that our findings will pave the way for studies of string breaking in higher-dimensional models.

Figure 1

(a) String tension ${\sigma }_Q$, (b) electric field per site, (c) comparison with the strong-coupling result Eq. (3.3) (dashed line) for $m/g=0.125$ and $m/g=0.25$, and (d) comparison with the weak-coupling result Eq. (3.4) (dashed line) for $m/g=1$, 2, 4. Inset: Zoom-in on the $m/g=4$ curve.

Figure 2

(a) $m/g=0.25$, $Q=0.45$. Fit of the form $(-1/6)\log (1/\sqrt{x})+A+C/\sqrt{x}$ to $S_Q(x)$ and $S_0(x)$. Inset: Linear fit to $\mathrm{\Delta }{S}_Q(x)$. (b) $\mathrm{\Delta }{S}_Q$ for different values of $m/g$.

Figure 3

$m/g=0$. (a) Potential for $Q=1$ and $Q=1.5$ compared with exact result in the continuum Eq. (4.2). Inset: Convergence for $x\to +\infty$ to Eq. (4.2) for $Q=1$. (b) Distribution of fermion charge for $Q=1.5$ for different separation lengths of the quark and antiquark for $x=100$. The results are compared with the exact result Eq. (4.3).

Figure 4

$m/g=0$, $Q=1$. Spatial profile of $\mathrm{\Delta }{S}_Q$ for different values of $L$ and scaling to the continuum limit ($x\to +\infty$). (a) $Lg=0.85$, (b) $Lg=15.65$.

Figure 5

$Q=1$, $x=100$. (a) Quark-antiquark potential for different values of $m/g$. (b) Comparison of potential with nonrelativistic limit result (dashed line) for $m/g=0.25$, 0.75, 1, 2. (c) The total charge of the light fermions on the negative axis $Q_{-}(L)$ for different values of $m/g$. (d) Electric field for $m/g=0.75$. (e) Charge distribution for $m/g=0.75$. For $Lg=17.3$, we compare with the charge distribution of the nonrelativistic meson state (full red line). (f) Comparison of the charge density of the left cloud (full line) with that of the nonrelativistic meson state, Eq. (4.5) (dashed line) for $Lg=17.3$, now for $m/g=0.125$, 0.5, 1, 2.

Figure 6

$m/g=2$, $Q=1$. $\mathrm{\Delta }{S}_Q(z)$ for different values of $L$ and scaling to the continuum limit. (a) $Lg=5.25$, (b) $Lg=10.95$.

Figure 7

Cartoon picture of string breaking in the nonrelativistic limit $m/g\to \infty$. The electric field gets successively screened by light quarks (antiquarks) that bind to the external charges with charge $+Q$ ($-Q$). This leads to the formation of two mesons: one meson existing of the heavy quark and light antiquarks with charge $+1$ and one meson existing of the heavy antiquark and the light quarks with charge $-1$. (a) $Q=4.5$: The remaining electric field in between with net charge $-0.5$ confines the two meson configurations asymptotically. (b) $Q=5$: The electric field is entirely screened and the meson configurations are deconfined.

Figure 8

$x=100$. (a) Quark-antiquark potential for $m/g=1$ for different values of $Q$. (b) $Q_{-}(L)$ for $m/g=1$ for different values of $Q$. (c),(d) The same quantities for $m/g=0.5$. (e),(f) The same quantities for $m/g=0.25$

Figure 9

$m/g=0.5$, $x=100$. Left: Charge distribution for $Q=1.75$ (a), $Q=4.5$ (c), and $Q=5$ (e) for different values of the separation length $L$. Right: Electric field for $Q=1.75$ (b), $Q=4.5$ (d), and $Q=5$ (f) for different values of the separation length $L$.

Figure 10

$m/g=0.5$, $Q=4.5$. $\mathrm{\Delta }{S}_Q(z)$ for different values of $L$. We also show the scaling to $x\to +\infty$. (a) $Lg=0.55$, (b) $Lg=2.55$, (c) $Lg=7.35$, and (d) $Lg=13.25$.

Figure 11

$m/g=0.5$, $Lg=15.25$. $\mathrm{\Delta }{S}_Q(z)$ for different values of $Q$ and scaling to $x\to +\infty$. (a) $Q=4.5$, (b) $Q=5$, (c) $Q=1.75$, and (d) $Q=2.5$.

Figure 12

$m/g=0.75$. (a) $Q=0.2$, $x=400$. Distribution of the 10-base logarithm of the Schmidt values ${\lambda }_{q,{\alpha }_q}$ among the eigenvalue sectors $q$ of $L(2n)$. (b) Same as (a) but now for $Q=0.45$. (c) $Q=0.2$. Distribution of the bond dimension among the eigenvalue sectors of $L(2n)$ for different values of $x$. (d) Same as (c) but now for $Q=0.45$.

Figure 13

$Q=0.3$. Continuum extrapolation of the string tension ${\sigma }_Q$ for different values of $m/g$. (a) m/g = 0.125. (b) m/g = 0.3. (c) m/g = 0.5. (d) m/g = 1.

Figure 14

(a) ${\log }_{10}({\mathrm{err}}_{{\sigma }_Q})$ as a function of $Q$. (b) ${\log }_{10}({\mathrm{err}}_{⟨E⟩})$ as a function of $Q$.

Figure 15

$Q=0.3$. Continuum extrapolation of the electric field $⟨E⟩$ for different values of $m/g$. (a) m/g = 0.125. (b) m/g = 0.3. (c) m/g = 0.5. (d) m/g = 1.

Figure 16

$Q=1/2$. Continuum extrapolation of the electric field $⟨E⟩$. (a) $m/g=0.3$, (b) $m/g=0.35$.

Figure 17

$Q=0$. (a) $m/g=0.125$. Fit of the form $(-1/6)\log (1/\sqrt{x})+A+C/\sqrt{x}$ through $S_0(x)$. Inset: Linear extrapolation of $S_0(x)+(1/6)\log (1/\sqrt{x})$ based on the largest five $x$ values, $x=200$, 300, 400, 600, 800, to obtain the coefficients $A$ and $C$. (b) Same as (a) but now for $m/g=0.75$.

Figure 18

$Q=0.3$. Continuum extrapolation of the renormalized half-chain von Neumann entropy for different values of $m/g$. (a) m/g = 0.125. (b) m/g = 0.3. (c) m/g = 0.5. (d) m/g = 1.

Figure 19

Diagrams for the effective action from Eq. (b1). On the first line we have the tree-level and the next-to-leading-order $g^2/m^2$ contribution to $C_0$ Eq. (b3). Evaluation of the first diagram on the second line would give a $g^4/m^4$ correction to $C_0$, while the other diagram would give the leading $g^4/m^4$ contribution to $C_1$.

Figure 20

$m/g=0.25$, $x=100$, $Q=5$, $Lg=10.1$. The stars represent the external charges. Between them the electric background field $-Q=-5$ is applied. (a) 10-base logarithm of the expectation values of some local quantities with the Schwinger vacuum value subtracted. At the boundaries one observes that they are sufficiently small, indicating that we take the nonuniform range wide enough. (b) Maximum and minimum eigenvalues $p_{\max }(n)$ and $p_{\min }(n)$ of $L(n-1)$ we take into account in our numerical scheme on every site. (c) Distribution of the 10-base logarithm of the Schmidt values ${\lambda }_{q,{\alpha }_q}(n)$ among the eigenvalue sectors $q$ of $L(n)$ for $n=150$. (d) Distribution of the of the 10-base logarithm of the Schmidt values ${\lambda }_{q,{\alpha }_q}(n)$ among the eigenvalue sectors $q$ of $L(n)$ for $n=200$.

Figure 21

$m/g=0.25$, $x=100$, $Q=5$, $Lg=10.1$. The stars represent the external charges. Between them the electric background field $-Q=-5$ is applied. (a) Dominant eigenvalue sector of $L(n)$, i.e., eigenvalue $q$ of $L(n)$ with largest $\sum _{{\alpha }_q=1}^{D_q(n)}{\lambda }_{q,{\alpha }_q}(n)$. $D_q(n)$ is taken such that the smallest Schmidt value is around ${10}^{-18}$. (b) ${\max }_q{D}_q(n)$: Largest bond dimension among the eigenvalue sectors of $L(n)$ at every site $n$.