Thermal nature of confining strings

TL;DR

The paper shows that in the static flux tube of the massive Schwinger model, entanglement drives emergent thermality as the interquark separation approaches the string-breaking distance $d_c$, with the interstitial reduced density matrix becoming nearly thermal and the entanglement spectrum becoming highly mixed. Using a lattice Hamiltonian formulation and a Jordan–Wigner mapping to spins, the authors analyze energy scales, energy density, charge distribution, and chiral condensate, then quantify entanglement via entanglement entropy and the spectrum. They find a pronounced peak in entanglement measures and a closing gap near $d_c M_s \simeq 7$, accompanied by a thermal overlap with an effective temperature that peaks at this point, indicating a microscopic thermalization transition within the flux tube. These results link confinement, entanglement, and emergent thermality in a stationary strongly-coupled gauge theory and motivate extensions to higher-dimensional QCD-like systems to understand hadronization temperatures without external baths.

Abstract

We investigate the quantum statistical properties of the confining string connecting a static fermion-antifermion pair in the massive Schwinger model. By analyzing the reduced density matrix of the subsystem located in between the fermion and antifermion, we demonstrate that as the interfermion separation approaches the string-breaking distance, the overlap between the microscopic density matrix and an effective thermal density matrix exhibits a pronounced, narrow peak, approaching unity at the onset of string breaking. This behavior reveals that the confining flux tube evolves toward a genuinely thermal state as the separation between the charges grows, even in the absence of an external heat bath. In other words, one cannot tell whether a reduced state of the subsystem arises from a surrounding heat bath or from entanglement with the rest of the system. The entanglement spectrum near the critical string-breaking distance exhibits a rapid transition from the dominance of a single state describing the confining electric string towards a strongly entangled state containing virtual fermion-antifermion pairs. Our findings establish a quantitative link between confinement, entanglement, and emergent thermality, and suggest that string breaking corresponds to a microscopic thermalization transition within the flux tube.

Figures (9)

• Figure 1: Cartoon of our setup using $N=20$ staggered sites as an example. The yellow shaded region exemplifies a centered subsystem of size $L=8$ in the middle of the chain. The first row shows the vacuum state without external source which is our reference state. The second row contains the external sources at separation $d=1\,a$, the third at $d=3\,a$, etc.
• Figure 2: Ground-state potential $V(d)$ (top) and normalized mass gap of the first excited state $E_1(d)/M_s$ (bottom) as functions of the separation $d\cdot M_s$ between the external static charges ($M_s\equiv E_1(0)$ is the mass of the (pseudo)scalar boson). The potential rises linearly at short distances, reflecting confinement. We define the critical separation $d_c$ (shown by the vertical dashed line) as the distance at which the derivative of the potential drops by 80% of its plateau value; this gives the critical distance of $d_c\cdot M_s \simeq 7$. At this distance, the energy stored in the flux tube reaches the threshold for pair creation. The linear increase is elucidated by the red dashed line which is a linear fit with $V/M_s\sim 0.193\, d\cdot M_s$. The plateau level is $V/M_s\sim 1.09$. The mass gap exhibits a minimum at the same distance, providing an independent signature of string breaking.
• Figure 3: Spatial profiles of the ground-state energy density (top), pressure (middle), and charge density (bottom) as functions of the separation $d\cdot M_s$ between the external charges (vertical axis) and the spatial coordinate along the lattice spin chain (horizontal axis). See Fig. \ref{['setup_fig']} for the setup. The energy density illustrates the formation and subsequent decay of the confining flux tube, while the charge distribution shows the emergence of a dynamical fermion–antifermion pair that screens the external sources. At large separations, the flux tube disappears and two neutral meson-like states remain.
• Figure 4: Electric-field energy (top) and spatially averaged condensate (bottom) of the central 8 (blue circles) and 12 (red squares) sites, respectively, shown relative to their vacuum values. The electric energy increases with the separation between the static charges, reaching a maximum near the critical distance $d_c M_s \simeq 7$, where the system becomes unstable to pair creation. The chiral condensate exhibits the opposite trend, being suppressed inside the flux tube and attaining maximal reduction at the same distance, signaling partial restoration of chiral symmetry within the confined region.
• Figure 5: Entanglement entropy as a function of the charge separation $d \cdot M_s$ for different partitions of the system. Top: Half-chain entropy capturing the total entanglement between the left and right halves of the lattice. Bottom: Entanglement entropy of centered subsystems of length $L$, measured relative to the vacuum -- hence the region of negative $S_{EE}$.