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arXiv:2510.23919 · Physical review. D/Physical review. D.

Thermal nature of confining strings

Sebastian Grieninger, Dmitri E. Kharzeev, Eliana Paula Romero Marroquin

Published Jan 12, 2026• 2 citations
arXivHTMLPDF

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.

DOI
10.1103/v8mv-185x

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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.
Understanding the dynamics of bound state formation is one of the fundamental questions in confining quantum field theories such as Quantum Chromodynamics (QCD). One hadronization mechanism that has garnered significant attention is the breaking of a string initially connecting a fermion and an antifermion. Deepening our understanding of real-time string-breaking dynamics with simpler, lower dimensional models like the Schwinger model can improve our understanding of the hadronization process in QCD and other confining systems found in condensed matter and statistical systems. In this paper, we consider the string-breaking dynamics within the Schwinger model and investigate its modification inside a thermal medium, treating the Schwinger model as an open quantum system coupled to a thermal environment. Within the regime of weak coupling between the system and environment, the real-time evolution of the system can be described by a Lindblad evolution equation. We analyze the Liouvillian gaps of this Lindblad equation and the time dependence of the system's von Neumann entropy. We observe that the late-time relaxation rate decreases as the environment correlation length increases. Moreover, when the environment correlation length is infinite, the system exhibits two steady states, one in each of the sectors with definite charge-conjugation-parity (CPCPCP) quantum numbers. For parameter regimes where an initial string breaks in vacuum, we observe a delay of the string breaking in the medium, due to kinetic dissipation effects. Conversely, in regimes where an initial string remains intact in vacuum time evolution, we observe string breaking (melting) in the thermal medium. We further discuss how the Liouvillian dynamics of the open Schwinger model can be simulated on quantum computers and provide an estimate of the associated Trotter errors.
String breaking, the process by which flux tubes fragment into hadronic states, is a hallmark of confinement in strongly-interacting quantum field theories. A suite of quantum complexity measures is examined using Matrix Product States to characterize the string breaking process in the 1+1D Schwinger model. We demonstrate the presence of nonlocal quantum correlations along the string that may affect fragmentation dynamics, and show that entanglement and magic offer complementary perspectives on string formation and breaking beyond conventional observables.
Recent lattice studies have revealed that the color flux tube between static quark–antiquark pairs exhibits an excess entanglement entropy (flux-tube entanglement entropy, FTE2) that scales linearly with the quark separation L. In this paper, we demonstrate similar behavior in a string-net model based on SU(2)k fusion categories, where the nontrivial object j=1/2 (analogous to color charge) cannot exist in isolation due to the fusion rules, naturally exhibiting “confinement”. We compute the entanglement entropy of the flux tube connecting two j=1/2 objects using the microcanonical (equal-weight) prescription S(R)=ln(dim(Hom(R))) and find an entropy density σk=lnd1/2=ln(2cosπk+2). For k=3, the category reduces to the Fibonacci case, yielding an entropy density σ3 = ln φ ≈ 0.4812 (φ is the golden ratio), which is qualitatively comparable in magnitude to the scale inferred from lattice studies and the entropy surface mechanism. Under a thermalization assumption for the fusion-channel degrees of freedom, minimizing the free energy F=(J−Tσk)L yields a confinement–deconfinement transition at (Tc=J/σk), which is first-order-like (tension sign reversal) rather than a continuous critical transition. The parameter k offers a tunable knob, making the SU(2)k family a computable laboratory for entropic confinement. The predicted entropy-density jump can be directly tested in quantum simulator platforms (e.g., Rydberg arrays or superconducting circuits) that realize Fibonacci anyonic models.
Recent lattice studies have revealed that the color flux tube between static quark-antiquark pairs exhibits an excess entanglement entropy (flux tube entanglement entropy, FTE2) that scales linearly with the quark separation L. In this paper, we demonstrate similar behavior in a string-net model based on SU(2)k fusion categories, where the nontrivial object j=1/2 (analogous to color charge) cannot exist in isolation due to the fusion rules, naturally exhibiting “confinement”. We compute the entanglement entropy of the flux tube connecting two j=1/2objects using the topological definition S(R)=ln⁡(dim⁡(Hom(R))) and find an entropy density σk=ln⁡ d1/2=ln⁡(2cos⁡(π/(k+2))). For k=3, the category reduces to the Fibonacci case, yielding an entropy density σ3 = ln φ ≈ 0.481 (φ is the golden ratio), which is qualitatively comparable in magnitude to the scale inferred from lattice studies and the entropy-surface mechanism. Minimizing the free energy F=E-TS yields a confinement–deconfinement transition at Tc=J/σk.The parameter k offers a tunable knob, making the SU(2)k family a computable laboratory for entropic confinement. The predicted entropy-density jump and critical scaling can be directly tested in quantum simulator platforms (e.g., Rydberg arrays or superconducting circuits) that realize Fibonacci anyonic models.
Recent lattice QCD studies have revealed an excess entanglement entropy (flux tube entanglement entropy, FTE 2 ) in the color flux tube between static quark–antiquark pairs, which grows linearly with the quark separation. In this paper, we illustrate a possible analogy in a simplified string-net model based on the Fibonacci fusion category, where the non-trivial object [Formula: see text] (analogous to color charge) cannot exist in isolation due to the fusion rule [Formula: see text], naturally exhibiting “confinement”. We compute the entanglement entropy of the flux tube connecting two [Formula: see text] objects and find an entropy density of [Formula: see text] ([Formula: see text] is the golden ratio), which quantitatively agrees with the lattice QCD measured FTE 2 density (approximately 0.3–0.5). Minimizing the free energy [Formula: see text] at fixed temperature and volume, the competition between the energy term [Formula: see text] and the entropy term [Formula: see text] yields a confinement-deconfinement phase transition at critical temperature [Formula: see text]. This model provides a heuristic demonstration platform as a minimal model for understanding the entropic properties of the QCD flux tube.
How coherent confining strings reorganize into hadronic matter is a central open problem in nonequilibrium gauge dynamics. Using real-time tensor-network simulations of the massive Schwinger model, we identify a collective organization underlying string fragmentation. For intermediate masses, the dynamics simplifies and more than 99%99\%99% of the temporal evolution of the electric-field probability distribution is governed by a single collective mode whose decay tracks the survival probability of the parent string. The extracted survival law exhibits the same Gaussian dependence implied by the Lund world-sheet area law, providing microscopic support for its characteristic area law dependence without assuming the Lund fragmentation mechanism. The same mode controls the transition from the coherent string to the produced mesons, explains the universal non-thermal effective fluid equation of state, and essentially controls the growth of bipartite entanglement through the causal propagation of massive pairs. These results identify collective fragmentation as an organizing principle of coherent string breaking and provide a microscopic perspective on the dynamical origin of a central ingredient of Lund hadronization.