Entanglement asymmetry has emerged in recent years as a practical quantity to study phases of matter. We present the first study of entanglement asymmetry in gauge theories by considering the chiral anomaly of the analytically solvable massless Schwinger model at both zero and finite temperatures. At zero temperature, we find the asymmetry exhibits logarithmic growth with system size. At finite temperature, we show that it is parametrically more sensitive to chiral symmetry breaking than the corresponding local order parameter: While the chiral condensate decays exponentially, the asymmetry decreases only logarithmically. This establishes the entanglement asymmetry as a promising tool to probe (finite-temperature) phase transitions in gauge theories.

Germany

Adrien Florio

France

Sara Murciano

Entanglement asymmetry in gauge theories: Chiral anomaly in the finite temperature massless Schwinger model

Understanding real-time dynamics of interacting quantum fields in curved space-time remains a major theoretical challenge. We employ tensor network methods to study such dynamics using interacting scalar and gauge theories in 1+1 spacetime dimensions, subject to a quench modeling a homogeneously expanding gravitational background. The models considered are the scalar $\textit{λϕ}$ theory and the Schwinger model, i.e. a Dirac fermion coupled to a U(1) gauge field which is equivalent via bosonization to a scalar field with a cosine self-interaction. In the free scalar limit, both theories reproduce known analytical results, providing a nontrivial numerical validation of bosonization in curved spacetime for the Schwinger model. Our central finding is that self-interactions lead to a suppression of gravitational particle production compared to the free-field case, as evidenced by two-point functions and the spectra of produced particles. We further examine the behavior of entanglement generation and find that interactions suppress entanglement growth in the $\textit{λϕ}$ theory, while in the Schwinger model, the interplay between suppressed particle production and enhanced inter-particle correlations leads to more complex entanglement behavior. Our results pave the way for further explorations of nonperturbative quantum real-time dynamics of interacting scalar and gauge theories in arbitrary gravitational backgrounds.

United States

Evan Budd

David Frenklakh

Swagato Mukherjee

Quantum dynamics of cosmological particle production: interacting quantum field theories with matrix product states

We present a detailed numerical and analytical study of the out-of-equilibrium dynamics of Model G, the dynamical universality class relevant to the chiral phase transition. We perform numerical 3D stochastic (Langevin) simulations of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></math> critical point for large lattices in the chiral limit. We quench the system from the high-temperature unbroken phase to the broken phase and study the nonequilibrium dynamics of pion fields. Strikingly, the nonequilibrium evolution of the two-point functions exhibits a regime of growth, a parametrically large enhancement, and a subsequent slow relaxation to equilibrium. We analyze our numerical results using dynamic critical scaling and mean-field theory. The growth of the two-point functions is determined by the nonlinear dynamics of an ideal non-Abelian superfluid, which is a limit of Model G that reflects the broken chiral symmetry. We also relate the nonequilibrium two-point functions to a long-lived parametric enhancement of soft pion yields relative to thermal equilibrium following a quench.

Italy

Eduardo Grossi

Aleksas Mazeliauskas

Slovenia

Alexander Soloviev

Derek Teaney

Quenching through the QCD chiral phase transition

We discuss a universal nonequilibrium enhancement of long-wavelength Goldstone bosons induced by quenches to the broken phase in Model G—the dynamical universality class of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> antiferromagnet and the chiral phase transition in QCD. Scaling arguments for the coarsening dynamics describing the formation of the chiral condensate predict a parametric enhancement in the infrared spectra of Goldstone bosons, a prediction confirmed by stochastic simulations of the transition. The details of the enhancement are determined by the nonlinear dynamics of a superfluid effective theory, which is a limit of Model G reflecting the broken <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> symmetry. Our results translate to a parametric enhancement of low-momentum pions in heavy-ion collisions at the LHC, which are underpredicted in current hydrodynamic models without critical dynamics.

Supercooled Goldstone Bosons at the QCD Chiral Phase Transition

We investigate the emergence of thermalization in a quantum-field-theoretic model mimicking the production of jets in QCD: the massive lattice Schwinger model coupled to external sources. Specifically, we compute the expectation values of local operators as functions of time and compare them to their thermal counterparts, quantify the overlap between the evolving density matrix and the thermal one, and compare the dynamics of the energy-momentum tensor to predictions from relativistic hydrodynamics. Through these studies, we find that the system approaches thermalization at late times and elucidate the mechanisms by which quantum entanglement drives thermalization in closed field-theoretic systems. Our results show how thermodynamic behavior emerges in real time from unitary quantum dynamics.

Sebastian Grieninger

Dmitri E. Kharzeev

Andrea Palermo

China

Shuzhe Shi

Thermalization from quantum entanglement: Jet simulations in the massive Schwinger model

The possible link between entanglement and thermalization, and the dynamics of hadronization are addressed by studying the real-time response of the massive Schwinger model coupled to external sources. This setup mimics the production and fragmentation of quark jets, as the Schwinger model and quantum chromodynamics (QCD) share the properties of confinement and chiral symmetry breaking. By using simulations of quantum dynamics on classical hardware, we study the entanglement between the produced jets, and observe the growth of the corresponding entanglement entropy in time. This growth arises from the increased number of contributing eigenstates of the reduced density matrix with sufficiently large and close eigenvalues. We also investigate the physical nature of these eigenstates, and find that at early times they correspond to fermionic Fock states. We then observe the transition from these fermionic Fock states to mesonlike bound states as a function of time. In other words, we observe how hadronization develops in real time. At late times, the local observables at midrapidity (such as the fermion density and the electric field) approach approximately constant values, suggesting the onset of equilibrium and approach to thermalization.

Kazuki Ikeda

Dmitri Kharzeev

Vladimir Korepin

Kwangmin Yu

Quantum real-time evolution of entanglement and hadronization in jet production: Lessons from the massive Schwinger model

We consider a <math><msub><mi>Z</mi><mn>3</mn></msub></math> gauge theory coupled to three degenerate massive flavors of fermions, which we term Quantum Z(3) Dynamics, QZD. The spectrum can be computed in <math><mn>1</mn><mo>+</mo><mn>1</mn></math> dimensions using tensor networks. In weak coupling the spectrum is that of the expected mesons and baryons, although the corrections in weak coupling are nontrivial, analogous to those of nonrelativistic QED in <math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions. In strong coupling, besides the usual baryon, the singlet meson is a baryon-antibaryon state. For two special values of the coupling constant, the lightest baryon is degenerate with the lightest octet meson, and the lightest singlet meson, respectively.

Andreas Weichselbaum

Semeon Valgushev

Robert D. Pisarski

Mass gaps of a <math><msub><mi>Z</mi><mn>3</mn></msub></math> gauge theory with three fermion flavors in <math><mn>1</mn><mo>+</mo><mn>1</mn></math> dimensions

We consider the fermionic (logarithmic) negativity between two fermionic modes in the Schwinger model. Recent results pointed out that fermionic systems can exhibit stronger entanglement than bosonic systems, exhibiting a negativity that decays only algebraically. The Schwinger model is described by fermionic excitations at short distances, while its asymptotic spectrum is the one of a bosonic theory. We show that the two-mode negativity detects this confining, fermion-to-boson transition, shifting from an algebraic decay to an exponential decay at distances of the order of the de Broglie wavelength of the first excited state. We derive analytical expressions in the massless Schwinger model and confront them with tensor network simulations. We also perform tensor network simulations in the massive model, which is not solvable analytically, and close to the Ising quantum critical point of the Schwinger model, where we show that the negativity behaves as its bosonic counterpart.

Two-fermion negativity and confinement in the Schwinger model

We present a detailed study of the finite momentum dynamics of the <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> critical point of QCD, which lies in the dynamic universality class of “model G.” The critical scaling of the model is analyzed in multiple dynamical channels. For instance, the finite momentum analysis allows us to precisely extract the pion dispersion curve below the critical point. The pion velocity is in striking agreement with the predictions relation and static universality. The pion damping rate and velocity are both consistent with the dynamical critical exponent <math><mi>ζ</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math> of model G. Similarly, although the critical amplitude for the diffusion coefficient of the conserved <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> charges is small, it is clearly visible both in the restored phase and with finite explicit symmetry breaking, and its dynamical scaling is again consistent with <math><mi>ζ</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math>. We determine a new set of universal dynamical critical amplitude ratios relating the diffusion coefficient to a suitably defined order parameter relaxation time. We also show that in a finite volume simulation, the chiral condensate diffuses on the coset manifold in a manner consistent with dynamical scaling, and with a diffusion coefficient that is determined by the transport coefficients of hydrodynamic pions. Finally, the amplitude ratios (together with other nonuniversal amplitudes also reported here) compile all relevant information for further studies of model G both in and out of equilibrium.

Dynamics of the <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> critical point in QCD: Critical pions and diffusion in model G

The Schwinger process in strong electric fields creates particles and antiparticles that are entangled. The entropy of entanglement between particles and antiparticles has been found to be equal to the statistical Gibbs entropy of the produced system. Here we study the effect of quantum interference in sequences of electric pulses, and show that quantum interference suppresses the entanglement entropy of the created quantum state. This is potentially relevant to quantum-enhanced classical communications. Our results can be extended to a wide variety of two-level quantum systems.