Mean Chiral Displacement in 1D Systems

• Mean chiral displacement (MCD) is a dynamical observable that extracts half the winding number from the evolution of a quantum state in one-dimensional chiral systems.
• It is implemented via methods such as photonic quantum walks, driven-dissipative lattices, and cold-atom experiments, showing robustness to disorder and boundary effects.
• The MCD’s quantization in the long-time limit provides clear dynamical signatures of topological phase transitions and insights into environmental couplings.

The mean chiral displacement (MCD) is a bulk dynamical observable that enables direct extraction of topological invariants in one-dimensional chiral systems from single-particle quantum dynamics. Introduced as a general probe for the winding number in bipartite lattices, MCD is robust to disorder, insensitive to initial conditions, and applicable to static, driven, or dissipative realizations. It is central to experimental protocols in topological photonics, quantum walks, and cold-atom systems, offering clear dynamical signatures of topological transitions and environmental interactions (Pavan et al., 10 Dec 2025, Cardano et al., 2016, Villa et al., 2023, D'Errico et al., 2020, Sinha et al., 10 Mar 2025).

1. Mathematical Definition, Operator Structure, and Symmetry

Consider a one-dimensional bipartite lattice indexed by cells $n$, each hosting two sublattice states $|A_n\rangle$ and $|B_n\rangle$. The chiral symmetry operator $\Gamma$ acts as

$$\Gamma |A_n⟩=+|A_n⟩, \quad \Gamma |B_n⟩=-|B_n⟩,\quad \Gamma^2=1, \quad \{\Gamma,H\}=0.$$

The canonical position operator in unit-cell basis is

$$\hat{x} = \sum_{n=1}^N n (|A_n\rangle\langle A_n| + |B_n\rangle\langle B_n|).$$

For a quantum state $|\psi(0)\rangle$ localized at $n=0$, the time-dependent MCD is given by

$$\text{MCD}(t) = \langle \psi(0)|\,\Gamma\, U^\dagger(t)\,\hat{x}\, U(t)\,|\psi(0)\rangle,$$

where $U(t)=e^{-iHt}$. Equivalently (Cardano et al.), for spinor-resolved inputs,

$$\langle\Gamma x\rangle(t) = \sum_{\nu=A,B} \langle 0,\nu|e^{iHt}\, \Gamma\, \hat{x}\, e^{-iHt} |0,\nu\rangle.$$

The key symmetry is that $\Gamma$ anticommutes with the Hamiltonian $H$, ensuring particle–hole symmetry and that the long-time dynamics are governed by chiral winding properties rather than trivial localization (Pavan et al., 10 Dec 2025, Cardano et al., 2016).

2. Connection to Topological Invariants and Long-Time Limit

In translationally invariant systems (e.g., SSH model), one can diagonalize $H$ in momentum space: $H(k) = E_k\,\mathbf{n}(k)\cdot\boldsymbol{\sigma},$ where $E_k$ and $\mathbf{n}(k)$ encode sublattice hopping and other chiral parameters. The winding number $\eta$ is

$$\eta = \frac1{2\pi} \int_{-\pi}^{\pi} (n_x \partial_k n_y - n_y \partial_k n_x) \, dk,$$

with explicit values depending on hopping ratios (trivial for $v>w$, topological for $v<w$). The MCD approaches

$\langle\Gamma x\rangle(\infty) = \frac{\eta}{2},$

in the long-time limit, i.e., bulk dynamics encode half the winding number, or equivalently, a multiple of the Zak phase (Pavan et al., 10 Dec 2025, Cardano et al., 2016, D'Errico et al., 2020, Villa et al., 2023, Sinha et al., 10 Mar 2025).

3. Dynamical Probes and Experimental Protocols

Quantum Walks with Photons: In discrete-time photonic quantum walks, OAM encodes the lattice index and polarization the coin state. The walk operator $U$ is implemented via waveplates and q-plates. After $t$ steps, the state is projected onto $\pm$ chiral eigenstates and spatially resolved. The measured MCD,

$C(t) = \sum_m m [P_{+,m} - P_{-,m}],$

approaches the winding number after a small number of steps without band filling or edge interrogation. Robustness to dynamical disorder is demonstrated by preserving MCD quantization under random protocol fluctuations (Cardano et al., 2016, D'Errico et al., 2020, Sinha et al., 10 Mar 2025).

Driven-Dissipative Photonic Lattices: Under weak uniform loss $\gamma$ and coherent drive, the steady-state wavefunction $|\psi_\omega\rangle$ satisfies $$(\omega + i\gamma - H)|\psi_\omega\rangle = |s\rangle$$. The steady-state MCD at frequency $\omega$,

$$C_\mathrm{ss}(\omega) = \langle \psi_\omega| \Gamma\,x |\psi_\omega\rangle,$$

is frequency-integrated to extract the winding number up to $O(\gamma^2)$ corrections. Applicable to synthetic frequency dimensions (modulated ring resonators), where only intensity measurements are required (Villa et al., 2023).

Cold Atom/Spin-1/2 Rotors: The distinction between open and periodic boundary conditions significantly impacts MCD measurements; edge localization of wavepackets or momentum wrapping introduces systematic deviations from the ideal quantized plateau, but edge states themselves reflect bulk–edge correspondence (Motsch et al., 12 Jan 2026).

4. Extensions: Disorder, Quenches, Floquet and Active Matter

Disordered and Quasiperiodic Chains: In non-uniform SSH models, averaging MCD over translation and input sites converges to the non-commutative real-space winding number—this holds for Anderson and topological Anderson transitions. Measurement protocols involve tuning the wavelength to effectively scan propagation time and accounting for bulk averaging over multiple samples/input sites (Sinha et al., 10 Mar 2025).

Quenched Hamiltonians and Dynamic Transitions: When the system undergoes abrupt Hamiltonian quenches (e.g., between distinct topological phases), the MCD rapidly re-equilibrates to reflect the post-quench winding number, with transient oscillations decaying as $O(t^{-1/2})$. This real-time sensitivity allows tracking dynamically induced topological transitions from single-particle wavefunction evolution (D'Errico et al., 2020).

Floquet Topological Phases: Periodically-driven models (double-kicked quantum rotors, synthetic dimensions) admit exactly analogous definitions of MCD, with time-averaged plateaus at $W/2$ tracking Floquet winding numbers. The boundary-driven deviations identify and characterize edge-localized Floquet states (Motsch et al., 12 Jan 2026).

Active Matter and Stochastic Models: In chiral active Brownian particles subject to jerk (jcABPs), the mean displacement trajectory generalizes MCD to stochastic environments. The interplay of chirality, persistence, and jerk produces complex damped or exploding Lissajous patterns, analytically characterized by the time scales $\tau_P$, $\tau_C$ (chirality), $\tau_J$ (jerk), and their impact on trajectory spiraling. Recovery of standard MCD signatures occurs for vanishing jerk (Jose et al., 25 Aug 2025).

5. Impact of Environmental Coupling and Cavity Effects

Coupling the SSH chain to single-mode cavities via inter-cell hopping induces a nontrivial dynamical renormalization of the inter-cell hopping amplitude,

$w_{\rm eff} = w \exp(-\alpha/2),$

where $\alpha = 2g^2/\omega_0^2$ and $\omega_0$ is the cavity frequency. In the anti-adiabatic (high frequency) regime ($\omega_0 \gg v,w$), the cavity coupling drives a discontinuous MCD jump as $w_{\rm eff}/v$ crosses unity, signifying a topological phase transition dynamically readable via MCD. At intermediate frequencies, retardation and dissipation broaden the transition, resulting in a smooth crossover in MCD from $0$ to $1/2$ as cavity parameters are tuned (Pavan et al., 10 Dec 2025).

6. Practical Considerations: Boundary Conditions, Robustness, and Error Sources

Boundary Effects: Finite-size systems and imposed boundaries introduce oscillatory and systematic errors in the measured MCD via wavepacket reflections and edge state trapping. Correction protocols include increasing system size, fitting or subtracting boundary-induced momentum jumps, and bulk-averaging over multiple initial site injections. In photonic and cold-atom systems, this ensures robust topological signatures (Motsch et al., 12 Jan 2026, Sinha et al., 10 Mar 2025).

Robustness to Disorder: The MCD remains quantized in the presence of chiral-preserving disorder, both static and dynamical, provided the gap is not closed. Ensemble averaging is essential to recover bulk values in finite samples (Cardano et al., 2016, Sinha et al., 10 Mar 2025).

Extensions and Limitations: MCD protocols generalize to higher dimensions (vectorized chiral displacements), alternative symmetry classes (AIII, BDI), Floquet systems, and nonlinear extensions (self-induced topology via Kerr nonlinearity). Limiting factors are fabrication inhomogeneity, wavelength-dependent coupling variations, and minimal impact from next-nearest neighbor couplings (Sinha et al., 10 Mar 2025, Villa et al., 2023).

---

The mean chiral displacement thus provides a unifying, experimentally accessible dynamical observable for resolving bulk topological invariants, tracking environment-induced transitions, and mapping intricate effects of disorder, boundaries, and driven-dissipative physics across quantum and classical implementations (Pavan et al., 10 Dec 2025, Cardano et al., 2016, Villa et al., 2023, D'Errico et al., 2020, Sinha et al., 10 Mar 2025, Motsch et al., 12 Jan 2026, Jose et al., 25 Aug 2025).