Time-Domain Braiding Phase

Updated 15 October 2025

Time-domain braiding phase is a quantum effect in topological systems where quasiparticles acquire nontrivial phase factors through time-ordered exchanges rather than spatial movements.
It underpins experimental protocols, such as pulsed excitations and AC transport, that reveal exotic statistics in systems like anyons, loop excitations, and Majorana zero modes.
The concept connects scaling dimensions, statistical angles, and memory timescales, providing a robust order parameter for detecting topological order through observable transport signatures.

Time-domain braiding phase is a concept at the intersection of topological quantum matter, low-dimensional strongly-correlated systems, and nonequilibrium quantum transport, describing statistical phase factors acquired by quasiparticles or extended objects as a result of “exchanges” that unfold over time rather than spatially. Recent theoretical and experimental developments have elucidated its role as a dynamical manifestation of exchange statistics—especially in systems with anyons, loop excitations, or Majorana zero modes where braiding cannot be interpreted solely in terms of spatial worldlines. In the time domain, the act of braiding is probed by time-resolved evolution, out-of-equilibrium transport, or Hamiltonian deformation, and the accumulated phase can serve as an order parameter for topological order or signal the presence of exotic statistics.

1. Time-Domain Braiding: Definitions and Conceptual Framework

The time-domain braiding phase generalizes the familiar notion of spatial braiding of quasiparticles or defects to scenarios where the primary “exchange” information is imprinted by temporal ordering, nonstationary dynamics, or adiabatic cycles of the ground state. For Abelian anyons in 2+1D, the exchange of two quasiparticles converts the wavefunction by a phase factor $e^{i\theta}$ ; for non-Abelian anyons, a unitary rotation in a ground state manifold is induced. Traditionally, such phases are extracted from spatial braiding: adiabatic exchange paths, interferometry, or the construction of worldlines in spacetime. In the time-domain setting, the phase is imprinted by operations such as:

Sending pulses of quasiparticles into a quantum point contact (QPC) and monitoring the outcome of tunneling events at well-defined times (Ruelle et al., 13 Sep 2024, Lee et al., 2022).
Executing sequence of membrane operator insertions or generalized modular (S-matrix) transformations in Hamiltonians that are time-evolved adiabatically (Jiang et al., 2014).
Inducing local operations or manipulations (e.g., superconducting phase differences, gating, or pulse drives) whose temporal sequence leads to effectively “braided” excitations, as in engineered Majorana setups (Gorantla et al., 2017, Posske et al., 2019).

A crucial principle is that the accumulated phase during these processes is set by both the system's underlying statistics and the full topological character of the dynamical path. Notably, the time-domain braiding phase can serve as a sharp, non-local order parameter for topologically ordered phases, and (in suitable settings) can be extracted from experimentally accessible quantities such as current, noise, or state overlaps.

2. Mathematical Formulations and Theoretical Frameworks

The time-domain braiding phase emerges in several mathematical guises, reflecting the diversity of systems and physical observables:

Spacetime Topological Invariants: In 3+1D, a triple linking number (TLN) $Tlk(M, N, P)$ of three worldsheet membranes quantifies the intrinsic three-loop braiding invariant, directly captured in generalized modular S-matrix elements for topologically ordered phases (Jiang et al., 2014). In TQFT language, this appears as an extra phase factor from cohomological data or higher-form gauge couplings.
Field-Theoretic Actions with Topological Terms: In nonlinear sigma model (NLSM) descriptions, a topological $\Theta$ -term produces a phase $\exp(i\Theta \times \text{instanton number})$ , with the instanton number set by spacetime linking or winding incurred by time-dependent braiding (Bi et al., 2014).
Out-of-Equilibrium Transport and FDT: In fractional quantum Hall edge physics, braiding phases appear in two-point correlators and determine shot-noise and current via explicit relations; for instance, in the nonequilibrium Fluctuation-Dissipation Theorem (FDT) framework, the DC current and noise are related by

$S_\mathrm{tun}(\omega_{\mathrm{dc}}) = -2e^{*2}\cot\theta\,\text{Im}[X^R(\omega_{\mathrm{dc}})]$

with $\theta$ the statistical angle and $X^R$ the retarded response built from forward/backward correlators differing by $e^{\pm 2i\theta}$ for different time orderings (Safi, 12 Oct 2025).

Berry Phases in Adiabatic Evolution: Braiding phases can be understood as Berry (or geometric) phases arising from adiabatic movement in Hamiltonian parameter space, where the phase acquired after a cycle measures the nontriviality of the underlying statistics (Gorantla et al., 2017).
Quantum Statistical Markers in Noise and Correlation Functions: In time-resolved shot-noise experiments with diluted anyon sources, the Fano factor directly encodes the statistical angle—for $\nu=1/3$ Laughlin quasiparticles, $\mathcal F = 3.27$ as predicted by theory for a statistical phase $2\theta = 2\pi/3$ (Lee et al., 2022).

3. Physical Realizations and Measurement Protocols

Several families of experiments and protocols are designed to reveal or utilize time-domain braiding phases:

3.1. Anyon Tunneling, Pulsed Excitations, and Noise Measurement

Experiments in $\nu=1/3$ FQH systems employ triggered pulses to inject anyons toward a QPC. The resulting backscattered current and noise exhibit a memory effect: if the incoming pulse carries a nontrivial braiding phase ( $N=1$ ), the tunneling signal is broadened over a timescale $\tau_\delta \sim 1/(k_BT_{el})$ determined by both temperature and the edge state scaling dimension $\delta$ , in contrast to the case with electrons or pulses with a trivial overall exchange phase ( $N=3$ ), where the signal rapidly decays on the pulse width $W$ (Ruelle et al., 13 Sep 2024).
Weak partitioning and subsequent autocorrelation measurements of highly diluted anyonic beams at a QPC can directly yield the exchange statistical phase through observable noise enhancements, consistent with the time-domain braiding theory (Lee et al., 2022).

3.2. Adiabatic Ground State Evolution and Berry Phase

In 3+1D topologically ordered systems, the time evolution under generalized modular transformations (S-matrix elements), implemented via adiabatic tuning or by explicit membrane operator insertions, produces a dynamical phase exactly set by the triple linking number of loop excitations in the “movie” of space-time evolution (Jiang et al., 2014). The overlap of minimum entropy states (MESs) along a time-dependent path yields the measurable time-domain braiding phase.

3.3. AC-Transport and Phase Response

The nonequilibrium FDT framework enables extraction of the statistical angle $\theta$ from the phase shift $\phi_\omega$ in the AC response of the tunneling current to a weak phase modulation (at frequency $\omega$ ):

$\tan\phi_\omega = \frac{B_{\rm tun,\omega}}{G_{\rm tun,\omega}}$

where $B_{\rm tun,\omega}$ and $G_{\rm tun,\omega}$ are the quadrature and in-phase responses, respectively, and the phase shift is directly related to the statistical angle by $\phi_\omega \simeq -\theta$ for quantum regime and $\delta > 1/2$ (Safi, 12 Oct 2025).

In graphene Fabry–Pérot interferometers, real-time random telegraph noise (RTN) in conductance reveals discrete phase jumps of $2\pi/3$ as the number of localized anyons fluctuates, allowing full time-domain reconstruction of the braiding phases without spatial manipulation (Werkmeister et al., 27 Mar 2024).

4. Time-Domain Braiding in Higher Dimensions and Non-Abelian Settings

In 3+1D, the three-loop braiding phase and Borromean rings/Brunnian braiding become essential signatures of topological order. The accumulated phase is no longer reducible to pairwise linking, but instead encodes higher-order invariants such as the triple linking number or Milnor's $\overline{\mu}$ -invariant (Jiang et al., 2014, Chan et al., 2017). The time-domain realization leverages either adiabatic cycles or dynamical processes where worldsheet topology encodes the nontrivial linking.
For Majorana zero modes in topological superconductors, protocols such as phase-locked high-frequency tunneling drive “virtual” Majorana braiding in the time domain; the effective low-frequency Hamiltonian depends only on the relative drive phase, and the dynamical evolution accumulates the non-Abelian Berry phase (Gorantla et al., 2017). In vortex-based or Josephson architectures, phase manipulation (for instance, through time-varying junction phase differences) can achieve effective “braiding” in the time domain, with readout enabled by charge transfer signatures or conductance oscillations.
Braiding protocols in engineered one-dimensional or photonic systems exploit time-division (period-doubled Majorana modes, or waveguide arrays with adiabatic modulation) to realize effective “exchange” operations in a discretized time-lattice, with resulting geometric or Berry phase accumulation traceable to the time-domain braiding protocol (Bomantara et al., 2017, Noh et al., 2019, Zhang et al., 2021).

5. Theoretical Principles and Scaling Relations

A central theoretical insight is the strict relationship between the braiding phase, the scaling dimension $\delta$ of quasiparticle tunneling operators, and the temperature-dependent memory timescale $\tau_\delta$ . Notably:

The statistical angle $\theta$ and scaling exponent $\delta$ are intertwined: for Laughlin anyons, $\theta = \pi \delta$ (modulo $\pi$ ) in a Tomonaga–Luttinger liquid (TLL) model. Experimentally, the scaling of shot noise, AC phase response, and the broadening of tunneling features with temperature all reflect these underlying parameters (Safi, 12 Oct 2025, Ruelle et al., 13 Sep 2024).
The nonequilibrium FDT and the properties of correlators retain memory of the underlying braiding via explicit $e^{-2i\theta\,{\rm sign}(t)}$ factors in time-ordered correlators, which distinguish anyonic from fermionic or bosonic processes (Safi, 12 Oct 2025).
In TQFTs describing 3+1D topological orders, only sets of root braiding processes with mutually compatible phases (as assessed by gauge invariance and group structure) are allowed, restricting which time-domain braiding phenomena can coexist (Zhang et al., 2020).

6. Experimental Implications and Outlook

Time-domain braiding phase measurements offer alternatives to conventional interferometry and cross-correlation methods, enabling the direct extraction of both fractional statistics and edge scaling exponents from:

DC backscattering noise and conductance, related by Kramers-Kronig integrals and analytic continuation (Safi, 12 Oct 2025).
AC phase-shift protocols at a single QPC, with robust self-calibration and minimal sensitivity to nonuniversal renormalization.
Time-resolved tunneling or interference measurements, in which the presence (or absence) of a nontrivial phase is seen as a temporal broadening or phase slip, directly mapping fractional statistics into accessible observables (Ruelle et al., 13 Sep 2024, Werkmeister et al., 27 Mar 2024).

Future directions include extending time-domain braiding probes to non-Abelian anyon regimes, multi-quasiparticle coupling (testing permutation group structure via time-ordered cycles), or in complex engineered platforms such as quantum spin liquids, higher-dimensional topological systems, or programmable photonic arrays. The methods described are generalizable, enabling comprehensive studies of topological states’ time-dependent dynamics and robust characterization of statistical phases in both Abelian and non-Abelian settings.

Observable	System/Protocol	Encoded Information
Broadening of tunneling signal ( $\tau_\delta$ )	Fractional QH anyons, pulsed QPC (Ruelle et al., 13 Sep 2024)	Statistical phase $\theta$ , scaling dimension $\delta$
Fano factor $\mathcal F$ in shot noise	Diluted anyon beams, shot noise (Lee et al., 2022)	$2\theta$
Phase slips in AB interference ( $2\pi/3$ )	Graphene interferometer, RTN (Werkmeister et al., 27 Mar 2024)	Exchange (braiding) statistics
AC phase shift $\phi_\omega$ in current	Single-QPC AC probe (Safi, 12 Oct 2025)	$\theta$ and $\delta$
Overlap/phase in adiabatic MES evolution	3+1D topological order (Jiang et al., 2014)	Triple linking invariant (TLN) and higher-order braiding

References

(Jiang et al., 2014): Generalized modular transformations in 3+1D and triple linking braiding.
(Bi et al., 2014): NLSM with topological $\Theta$ -term; geometric/temporal origin of braiding phase.
(Safi, 12 Oct 2025): Nonequilibrium fluctuation–dissipation theorem for anyonic time-domain braiding.
(Ruelle et al., 13 Sep 2024): Direct experimental evidence for delayed tunneling due to time-domain anyonic braiding.
(Lee et al., 2022): Time-domain braiding measured via shot noise in diluted anyon partitioning.
(Werkmeister et al., 27 Mar 2024): Real-time telegraph noise as probe of time-domain braiding in graphene FQH interferometer.
(Gorantla et al., 2017, Bomantara et al., 2017, Noh et al., 2019, Zhang et al., 2021): Majorana and photonic time-lattice braiding protocols.
(Zhang et al., 2020): Classification and compatibility of higher-loop/time-domain braiding in 3+1D.