String-breaking statics and dynamics in a (1+1)D SU(2) lattice gauge theory

Abstract

String breaking is at the core of hadronization models of relevance to particle colliders. Yet, studies of string-breaking dynamics rooted in quantum chromodynamics remain fundamentally challenging. Tensor networks enable sign-problem-free studies of static and dynamical properties of lattice gauge theories. In this work, we develop and apply a tensor-network toolkit based on the loop-string-hadron formulation of an SU(2) lattice gauge theory in 1+1 dimensions with dynamical fermions. We apply this toolkit to study static and dynamical aspects of strings and their breaking in this theory. The simple, gauge-invariant, and local structure of the loop-string-hadron states and constraints removes the need to impose non-Abelian constraints in the algorithm, and allows for a systematic computation of observables at increasingly large bosonic cutoffs, and toward the infinite-volume and continuum limits. Our study of static strings yields a determination of the string tension in the continuum and thermodynamic limits. Our study of dynamical string breaking, performed at a fixed lattice spacing and system size, illuminates underlying processes at play during the quench dynamics of a string. The loop, string, and hadron description offers a systematic and intuitive way to diagnose these processes, including string expansion and contraction, endpoint splitting and particle shower, chain scattering events, and inelastic processes resulting from string dissociation and recombination, and particle production. We relate these processes to several features of the dynamics, such as energy transport, entanglement-entropy production, and correlation spreading. This work opens the way to future tensor-network studies of string breaking and particle production in increasingly complex lattice gauge theories.

Figures (20)

• Figure 1: The site-local Hilbert space of the LSH formulation is characterized by the following quantum numbers: a loop $n_l$, a fermion sourcing one unit of flux to the left, called an incoming string $n_i$, a fermion sourcing one unit of flux to the right, called an outgoing string $n_o$. Shown are a few examples of the basis states $\ket{n_l,n_i,n_o}_r$ at a given site $r$, along with their pictorial representation. An empty circle at an even (odd) site denotes the absence (presence) of a hadron while a doubly filled site at odd (even) denotes the absence (presence) of a hadron. Open incoming and outgoing strings are the endpoint of nonlocal closed string operators. The states in the left have zero loop quantum number while those in the right can take any positive integer values up to a specified cutoff.
• Figure 2: (a) a 6-site MPS representation composed of tensors with a physical leg, $p_r$, and two virtual bonds, $a_{r-1}$ and $a_r$, at site $r$. (b) a 6-site MPO representation consisting of tensors with two physical legs $p_r,p'_r$ and two virtual bonds $b_{r-1},b_r$ (c) Each local MPS tensor has a block-diagonal structure imposed by the super-selection rules between the local charge $(Q_r,q_r)$ associated with the physical legs, and charges $(\tilde{Q}_{r-1},\tilde{q}_{r-1})$ and $(\tilde{Q}_r,\tilde{q}_r)$ associated with the left and right virtual bonds, respectively. The rule is given by $\tilde{Q}_r=Q_r+\tilde{Q}_{r-1}$ and $\tilde{q}_r=q_r+\tilde{q}_{r-1}$. (d) The local MPO tensors inherit an analogous block-diagonal structure imposed by the super-selection rule $\tilde{Q}_r = \tilde{Q}_{r-1}+(Q_r-Q'_r)$ and $\tilde{q}_r = \tilde{q}_{r-1}+(q_r-q'_r)$.
• Figure 3: (a) Static potential as a function of $gl$ for cutoff values $j_{\rm max}\in\{1,\frac{3}{2},2\}$ and lattice-spacing values $ga\in\{0.08,0.1,0.12\}$. Solid markers correspond to system volume $gL=16$ while hollow markers correspond to system volume $gL=14$. An MPS-error truncation extrapolation is already performed but the resulting error bars are too small to be visible. Values are computed at a fixed $\frac{m}{g}=0.5$. (b) Static potential plotted against the distance, $gl$, between static charges in the thermodynamic limit at a fixed $\frac{m}{g}=0.5$. The lines show fits to all contiguous regions in $gl$ with 3 to 5 points in the linear part of the potential between $gl\approx 3$ and $gl\approx 7$. Inset displays the weighted average of the slopes of all these lines used to estimate the string tension at various values of $ga$. The continuum estimate, in turn, is the weighted average of these string-tension estimates at various values of $ga$, and yields the value $\frac{\kappa}{g^2}=0.330(3)$.
• Figure 4: Values of the vacuum-subtracted total excitation, string-in, string-out, baryon, loop, and symmetrized flux expectation values ($\langle\hat{n}(r)\rangle_s$, $\langle\hat{\tilde{n}}_{i}(r)\rangle_s$, $\langle\hat{\tilde{n}}_{o}(r)\rangle_s$, $\langle\hat{n}_b(r)\rangle_s$, $\langle\hat{n}_l(r)\rangle_s$, and $\langle\hat{\tilde{N}}(r)\rangle_s$, respectively) for the ground state in the presence of static charges separated by a distance $gl=3$ (left) and $gl \approx 8$ (right) for $(\frac{m}{g},N,ga,j_{\rm max},D)=(0.5,128,0.12,2,200)$. Static charges are inserted at sites $\{r_1, r_2\}=\{48, 73\}$ and $\{r_1, r_2\}=\{26, 93\}$ for $gl=3$ and $gl \approx 8$, respectively. The bottom row denotes the same plots as in the top row but with a smaller range along the y-axis to reveal finer features of the matter excitations.
• Figure 5: Values of the total excitation, string-in, string-out, baryon, loop, and symmetrized flux expectation values ($\langle\hat{n}(r)\rangle$, $\langle\hat{\tilde{n}}_{i}(r)\rangle$, $\langle\hat{\tilde{n}}_{o}(r)\rangle$, $\hat{n}_b(r)$, $\langle\hat{n}_l(r)\rangle$, and $\langle\hat{\tilde{N}}(r)\rangle$, respectively) for the interacting vacuum for two different bare fermion masses. The other parameters are $N=128$, $x=16$, and $j_{\rm max}=\frac{5}{2}$.