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International Symposium on Lattice Field Theory (LATTICE)

Understanding thermalization of the magnetic gluons in a non-Abelian gauge theory

Sayak Guin, Harshit Pandey, Sayantan Sharma

Published Apr 14, 2026• 0 citations
Publisher

Abstract

We explore chaotic dynamics in SU(2) gauge theory by extracting the Lyapunov exponent under both in and out-of-equilibrium conditions to understand how chaos develops and influences the approach to thermalization. In thermal equilibrium at very high temperatures, the hard, electric, and magnetic scales are well separated, and one can write an effective theory for the soft gauge fields (magnetic modes) which are classical due to their large occupation number. In the vicinity of the critical temperature $T_c$, $SU(2)$ gauge theory falls within the same universality class as $Z_2$ scalar field theory. Exploiting this correspondence, we determine the Lyapunov exponent in the scalar theory through out-of-time-ordered correlators (OTOC). To probe non-equilibrium dynamics, we prepare the gauge fields in a highly occupied, non-thermal initial state. Under classical Yang–Mills evolution, the system evolves toward a self-similar scaling regime characterized by a scale separation similar to that of the thermal case. Within this regime, we again compute the Lyapunov exponent and compare it to its equilibrium counterpart at similar energy densities. This comparison allows us to quantify the timescale over which the system approaches thermal equilibrium.

DOI
10.22323/1.518.0096

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•May 1, 2019•0 citations
We explore chaotic dynamics in SU(2) gauge theory by extracting the Lyapunov exponent under both in and out-of-equilibrium conditions to understand how chaos develops and influences the approach to thermalization. In thermal equilibrium at very high temperatures, the hard, electric, and magnetic scales are well separated, and one can write an effective theory for the soft gauge fields (magnetic modes) which are classical due to their large occupation number. In the vicinity of the critical temperature TcT_cTc​, SU(2)SU(2)SU(2) gauge theory falls within the same universality class as Z2Z_2Z2​ scalar field theory. Exploiting this correspondence, we determine the Lyapunov exponent in the scalar theory through out-of-time-ordered correlators (OTOC). To probe non-equilibrium dynamics, we prepare the gauge fields in a highly occupied, non-thermal initial state. Under classical Yang–Mills evolution, the system evolves toward a self-similar scaling regime characterized by a scale separation similar to that of the thermal case. Within this regime, we again compute the Lyapunov exponent and compare it to its equilibrium counterpart at similar energy densities. This comparison allows us to quantify the timescale over which the system approaches thermal equilibrium.
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We revisit thermal out-of-time-order correlators (OTOCs) in single-particle quantum systems, focusing on magnetic billiards. Using the stadium billiard as a testbed, we compute the thermal OTOC CT(t)=−⟨[x(t),p]2⟩βC_T(t) = -\langle [x(t), p]^2 \rangle_βCT​(t)=−⟨[x(t),p]2⟩β​ and extract Lyapunov-like exponents λLλ_LλL​ that quantify early-time growth. We map out λL(T,B)λ_L(T, B)λL​(T,B), revealing a crossover from quantum chaos to magnetic rigidity. In parallel, we compute an alternative OTOC built from guiding-center operators, which exhibits qualitatively distinct dynamics and no exponential growth. Our results offer a controlled framework for probing scrambling, temperature dependence, and the interplay of geometry and magnetic fields in quantum systems.
Abstract In this article, we explore dynamical aspects of Out-of-Time-Order correlators (OTOCs) for critical quenches, in which an initial non-trivial state evolves with a CFT-Hamiltonian. At sufficiently large time, global critical quenches exhibit a universal thermal-behavior in terms of low-point correlators. We demonstrate that, under such a quench, OTOCs demarcate chaotic CFTs from integrable CFTs by exhibiting a characteristic exponential Lyapunov growth for the former. Upon perturbatively introducing inhomogeneity to the global quench, we further argue and demonstrate with examples that, such a perturbation parameter can induce a parametrically large scrambling time, even for a CFT with an order one central charge. This feature may be relevant in designing measurement protocols for non-trivial OTOCs, in general. Both our global and inhomogeneous quench results bode well for an upper bound on the corresponding Lyapunov exponent, that may hold outside thermal equilibrium.
We characterize thermalization slowing down of Josephson junction networks in one, two, and three spatial dimensions for systems with hundreds of sites by computing their entire Lyapunov spectra. The ratio of Josephson coupling E_{J} to energy density h controls two different universality classes of thermalization slowing down, namely, the weak-coupling regime, E_{J}/h→0, and the strong-coupling regime, E_{J}/h→∞. We analyze the Lyapunov spectrum by measuring the largest Lyapunov exponent and by fitting the rescaled spectrum with a general ansatz. We then extract two scales: the Lyapunov time (inverse of the largest exponent) and the exponent for the decay of the rescaled spectrum. The two universality classes, which exist irrespective of network dimension, are characterized by different ways the extracted scales diverge. The universality class corresponding to the weak-coupling regime allows for the coexistence of chaos with a large number of near-conserved quantities and is shown to be characterized by universal critical exponents, in contrast with the strong-coupling regime. We expect our findings, which we explain using perturbation theory arguments, to be a general feature of diverse Hamiltonian systems.
We will investigate the notions of thermalization and chaos in the context of quantum field theories. We will first attempt to quantify quantum information spreading through simple lattice systems and then define thermalization as the spread of entanglement across the system. We will then study the notion of operator growth and introduce our main player: the squared commutator and its cousin the OTOC. The rate of growth of the OTOC, it turns out, is identical to the rate of spread of entanglement. We will give a heuristic argument demonstrating this fact. In order the compute the OTOC for quantum field theories we will introduce a contour in the complex plane with two real time folds. After inserting external and interaction vertices at appropriate places in the contour we draw the ensuing Feynman diagrams and do a full pertubative computation of the squared commutator. We sum a special class of diagrams called ladder diagrams which furnishes a recursion relation in the form of an integral equation. The maximal eigenvalue of this equation is defined as a quantum Lyapunov exponent. We carry out this computation for two representative models: a large-N matrix ∅⁴ and QED₃ with a large number of fermionic flavours. In both the cases we compute the Lyapunov exponent as a function of the free parapmeters of the theories.