Real-Time Chiral Dynamics

• Real-time chiral dynamics is the study of time-dependent evolution and relaxation of chiral degrees of freedom, such as axial currents and order parameters, under nonequilibrium conditions.
• Advanced methods including lattice simulations, QITE, and AdS/QCD enable precise tracking of pion propagation, chiral instabilities, and anomaly-induced transport phenomena.
• Applications range from high-energy QCD and ultrafast molecular reactions to spintronics, providing crucial insights into phase transitions, chirality control, and dynamic critical scaling.

Real-time chiral dynamics encompasses the evolution, propagation, and relaxation of chiral degrees of freedom—such as axial charge, chiral currents, and chiral order parameters—on their natural timescales, under nonequilibrium or thermal conditions. This concept integrates phenomena from QCD pion quasiparticle propagation and chiral phase transitions to chiral instabilities and anomalous transport in gauge plasma, as well as coherent chiral tunneling in molecular systems and ultrafast chiral responses in driven reactions. Real-time approaches are essential for understanding the interplay of chiral symmetry, its breaking, and anomaly-induced processes in both high energy and condensed matter systems.

1. Theoretical Foundations of Real-Time Chiral Dynamics

The real-time formulation of chiral dynamics situates chiral symmetry and its spontaneous (and explicit) breaking at the forefront of describing collective excitations and transport. In low-temperature QCD, the canonical object is the pion, the pseudo-Goldstone mode of chiral symmetry breaking. The Son–Stephanov expansion around (T, m_q = 0)—with T below the crossover temperature—demonstrates that the real-time two-point correlation functions of axial and pseudoscalar densities possess a sharp pole corresponding to a pion-like excitation. The dynamical pion mode obeys a modified dispersion relation,

$$\omega_{\mathbf{k}}^2 = u(T)^2 (m_\pi^2 + k^2) + O(k^4),$$

where $u(T)<1$ encodes temperature-induced Lorentz symmetry breaking (Brandt et al., 2014, Brandt et al., 2014). At higher energies and in the presence of gauge fields, the chiral anomaly drives nonconservation of the axial current, leading to anomalous transport phenomena such as the Chiral Magnetic Effect (CME), in which an axial chemical potential or imbalance induces an electromagnetic current along an applied magnetic field (Fukushima et al., 2010, Mace et al., 2017, Mace et al., 2016).

2. Real-Time Lattice and Quantum Simulation Methods

First-principles unbiased computations of real-time chiral dynamics rely increasingly on lattice gauge theory, classical-statistical simulation, and digital/quantum algorithms. On the lattice, real-time evolution of Wilson or overlap fermions in discretized SU(N)×U(1) gauge backgrounds allows direct tracking of anomaly-driven axial and vector currents, as well as the build-up and relaxation of chiral imbalances (Mueller et al., 2016, Mace et al., 2016). Technically, the Dirac field operator is expanded in a complete set of numerically evolved mode functions, and currents are formed via operator expectation values. In the (1+1)–dimensional massive Schwinger model, quantum imaginary time evolution (QITE) is used to prepare thermal states and then real-time Suzuki–Trotter decomposition is applied to simulate quenched chiral dynamics (Ikeda et al., 31 Jul 2024, Kharzeev et al., 2020). These approaches quantitatively resolve, for example, the time-dependent generation and damping of CME currents and the eventual restoration of detailed balance via mass-induced chirality relaxation or thermal decoherence.

3. Real-Time Chiral Instabilities and Topological Charge Transfer

A central phenomenon in non-Abelian plasma and QED is the chiral plasma instability: an initial imbalance of right- over left-handed fermions triggers the exponential growth of gauge-field helical modes. In Abelian plasma, this instability channels fermionic chirality into gauge-field (Chern–Simons) magnetic helicity via a self-similar inverse cascade, as shown numerically by real-time lattice simulations with large $e^2N_f$ (Mace et al., 2020, Buividovich et al., 2015). In contrast, in non-Abelian environments, e.g., SU(2) gauge theory at strong coupling, axial charge is absorbed predominantly by topological (sphaleron) transitions that change the Chern–Simons number of the gauge field (Schlichting et al., 2022). The dispersion relation for unstable modes in the linear regime is

$\omega^2 = k^2 \mp \sigma_\chi k,$

with instability for $k < \sigma_\chi$ and maximal growth at $k_{max} = \sigma_\chi/2$.

Table 1: Chirality Transfer Mechanisms (Abelian vs. Non-Abelian) | Plasma Type | Instability Fate | Dominant Mechanism | |------------------|---------------------------|----------------------------| | U(1) (QED) | Inverse, turbulent cascade| Magnetic helicity transfer | | SU(N) (QCD-like) | Sphaleron transitions | Topological Chern-Simons |

In heavy-ion collisions, these dynamics determine the lifetime and possible experimental signatures of axial charge and CME currents, with nonconservation time scales set by the sphaleron diffusion rate $\Gamma_{CS}$ (Grieninger et al., 2023).

4. Chiral Dynamics in the QCD Phase Diagram and Critical Behavior

The nonequilibrium evolution of chiral order parameters near phase transitions is now accessible in real time, both within effective field theory and via AdS/QCD holography. In the chiral limit of QCD, the order parameter obeys Model G dynamics, coupling an O(N) field to conserved charges via stochastic hydrodynamic equations. The real-time functional renormalization group (FRG) demonstrates that both chiral mode and charge diffusion display universal dynamic scaling with exponent $z=d/2$ ($z=3/2$ in $d=3$) (Roth et al., 7 Mar 2024). In AdS/QCD, solving the time-dependent evolution of the chiral condensate across the Columbia plot reveals distinct relaxation patterns: exponential with prethermal plateaus in the first-order region, power-law critical slowing down near second-order criticality, and smooth crossover elsewhere (Tang et al., 29 Jul 2025). Microscopic FRG and AdS/QCD methods provide both scaling functions and explicit thermalization timescales for QCD matter.

5. Ultrafast and Molecular Real-Time Chiral Dynamics

At the molecular and attosecond scale, direct time-resolved probing of chiral dynamics exploits observables such as photoelectron circular dichroism (PECD) and high-harmonic generation (HHG) circular dichroism. In ultrafast pump-probe PECD experiments, the evolution of excited chiral wavepackets is tracked via delay-dependent asymmetries in photoelectron angular distributions, revealing population decay, rotational dephasing, and vibrational relaxation on femtosecond-picosecond timescales (Comby et al., 2016). Synthetic chiral light, as implemented in nonlinear HHG, exploits locally and globally chiral electromagnetic Lissajous figures to probe and reconstruct nuclear rearrangements (e.g., dihedral angle motion) with sub-10 fs precision and large enantio-sensitive signal (Ayuso, 2021). HHG-based time-resolved circular dichroism reveals the evolution and loss of chirality during chemical reactions, tracking the transition from chiral reactant to achiral products in <500 fs, with clear signatures in the sign and magnitude of the emitted high harmonics (Baykusheva et al., 2019). Coherent chiral tunneling in double-well molecular systems—a core model for chiral quantum dynamics—has now been directly measured in the time domain via phase-coherent microwave six-wave mixing, with precise control and readout of the chiral wavepacket oscillations (Sun et al., 9 Dec 2024).

6. Real-Time Chiral Dynamics in Condensed Matter and Spintronics

Chiral molecular systems and chiral crystals exhibit current-induced symmetry breaking and spin–orbit polarization, fundamental to the chirality-induced spin selectivity (CISS) effect. Real-time TDDFT simulations in helical (chiral) wires show that above a critical threshold current, time-reversal symmetry is dynamically broken even in the absence of explicit time-reversal symmetry breaking in the Hamiltonian. This onset correlates with the emergence of nonzero spin and orbital angular momentum, driven by the redistribution of angular degrees of freedom under current flow in a chiral, spin–orbit-coupled background. This process is absent in achiral wires and requires both geometric chirality and finite SOC for spin polarization (Jeong et al., 27 Aug 2025).

Table 2: Chiral Real-Time Effects in Physical Regimes | System | Real-Time Effect | Diagnostic/Observable | |--------------------|-----------------------------|----------------------------------| | QCD (pion sector) | Pion quasiparticle propagation | MEM spectral analysis, $u(T)$ | | Gauge plasmas | Chiral instability, CME | $j^\mu$, $n_5$, Chern–Simons number | | Ultracold matter | Chiral phase relaxation | $\sigma(\nu)$, FRG scaling | | Molecules | Coherent tunneling, PECD, HHG | Enantiomeric excess, CD($t$) | | Chiral wires | Current-driven spin polarization | $S_z$, $L_z$ in real-time TDDFT |

7. Significance, Applications, and Outlook

Real-time chiral dynamics governs the microscopic origin and macroscopic observability of chiral symmetry-breaking effects in QCD (critical phenomena, transport, and hadron structure), electroweak baryogenesis (chirality transfer, magnetogenesis), condensed matter (chiral magnetic and chiral separation effects), and chiral molecular science (enantioselection, ultrafast chirality control). Recent progress maps out a bridge from quantum field theory and nonequilibrium statistical mechanics to highly controlled quantum and attosecond experiments, establishing predictive tools for dynamic critical scaling, ultrafast chiral probing, and device-scale chirality-induced functionalities.

These advances enable a detailed understanding of how chiral symmetry and its breaking or restoration evolve under realistic, time-dependent external fields and thermal effects, encompassing universal scaling near criticality, turbulent chirality transfer, and real-time control and measurement in molecules and materials.