Size Winding: Geometry, Topology & Dynamics

• Size winding is a phenomenon defining the directional accumulation of winding in systems, quantifying how geometric, topological, and dynamical traits scale in polymers, lattice paths, and quantum operators.
• It employs methods from continuum geometry, spectral theory, combinatorics, and statistical mechanics to compute precise scaling laws and winding distributions, such as logarithmic variance and non-Gaussian moment ratios.
• Understanding size winding enhances models of phase transitions, DNA mechanics, and quantum chaos while informing topological invariants and optimizing protocols in quantum teleportation and polymer confinement experiments.

Size winding encompasses the interplay between geometric, topological, and dynamical aspects of "winding"—the directional accumulation or linking of a curve, edge, operator, or other structure—where the “size” of the system, walk, winding region, or operator itself serves as a key parameter. In diverse contexts ranging from polymer physics and quantum spin chains to lattice path enumeration and quantum chaos diagnostics, size winding measures how winding properties are constrained, scale, or manifest as system size, operator complexity, or spatial extent increases. Methods to compute, constrain, or optimize size winding bridge continuum geometric invariants, spectral theory, combinatorics, statistical mechanics, and quantum dynamics.

1. Directional Net Winding and Polar Writhe in Open Polymers

The directional net winding framework is central in modeling the geometry of open-ended elastic polymers such as DNA under force or torque (Prior et al., 2010). Here, the winding number is a directional invariant quantifying how two curves—most commonly the central axis and an offset or edge curve of a ribbon—wind around each other in a selected direction (typically the $z$-axis: the axis of applied forces).

Mathematically, the net winding $W$ is decomposed into:

• Local winding—the integral along the chain of the angular rate of a ribbon vector in the transverse (xy) plane:

$$W_\ell = \frac{1}{2\pi} \int_{z_{\min}}^{z_{\max}} \left[ \psi'(z) + \phi'(z) \right] dz,$$

where $\psi$ and $\phi$ are appropriate Euler angles.

• Non-local winding—accounting for windings arising from distant sections of the chain separated by turning points along $z$.

A major technical advance is a new derivation of the polar writhe: an expression that distinguishes contributions from "north" (upward) and "south" (downward) segments via monopole vector potentials $\mathbf{A}^N$ and $\mathbf{A}^S$, capturing both local and global coiling. The local polar writhe for an upward segment is: $$W_p^\ell = \frac{1}{2\pi} \int (1 - \cos\theta)\, \phi'(z)\, dz,$$ with analogous terms for downward sections.

Crucially, the net winding is invariant under end-restricted ambient isotopies—deformations that vanish at the endpoints—which ensures topological consistency in physical models. This approach rectifies limitations in prior Gauss linking number and Fuller-based frameworks by encoding directionality and self-avoidance at fixed boundaries. In applications (notably single-DNA experiments), the net winding serves as the appropriate conserved constraint in partition function calculations, enabling self-avoiding, analytically tractable models that naturally feature both local and non-local geometric winding contributions.

2. Statistical Scaling of Winding with System Size: Polymers and Lattice Walks

The statistical properties of winding, particularly its scaling with polymer or walk length, provide insight into how the interplay of geometry, topology, and self-avoidance dictates the winding behavior as size increases. This has been explored for both self-avoiding and random walks (SAW and RW):

• Three-dimensional SAWs around a bar show a non-Gaussian winding angle distribution:

$$p(\theta, L) = \frac{1}{(\ln L)^{\alpha}} f\left( \frac{\theta}{(\ln L)^{\alpha}} \right), \qquad \alpha \approx 0.75,$$

with the variance scaling as $\langle\theta^2\rangle \sim (\ln L)^{2\alpha}$, and the fourth-moment ratio $$\gamma = \langle\theta^4\rangle/\langle\theta^2\rangle^2 \approx 3.74$$, indicative of heavy (non-Gaussian) tails (Walter et al., 2011). This scaling is intermediate between the 2D random walk and plane SAW cases.

• Long lattice walks (RW and SAW) in two and three dimensions further reveal size-dependent winding phenomena: for 2D RW, $\langle\theta^2\rangle \propto (\ln N)^2$; for 2D SAW, $\langle\theta^2\rangle \propto \ln N$ with high-order moment ratios tending to the Gaussian limit; for 3D SAW, $\langle\theta^2\rangle \propto (\ln N)^{1.5}$ and ratios remain non-Gaussian at accessible lengths (Hammer et al., 2016). A segmentation approach demonstrates that the variance in winding contributed by successive self-similar segments (of length $2^i$) grows with segment index, leading to anomalous scaling.

This divergence from naive Gaussian scaling in higher dimensions or with self-avoidance underscores the fundamental role of spatial constraints and topology in dictating winding statistics as system size increases.

3. Combinatorial and Spectral Enumeration of Size-Constrained Winding

Combinatorial methods elucidate how winding constraints can be systematically encoded and counted in the configuration space of lattice walks, loops, or polymers:

• For closed polymer loops around a rod, the partition function is constructed by decomposing conformations into sequences of sub-arcs, categorized as crossovers or same-side (T_c or T_s) elements. Recursive combinatoric rules (mimicking Reidemeister moves) generate all topologically valid windings for fixed winding number $w$, and the partition sum is weighted by statistical (Gaussian) arc contributions (Rohwer et al., 2015). This yields explicit formulas—after Laplace/Fourier transformation—for lower and upper bounds on the partition sum, force calculations in geometric confinements, and mean numbers of arc types, all as functions of spatial (size/confinement) parameters.
• Lattice-path enumeration with winding constraints, as in enumerating simple diagonal walks on $\mathbb{Z}^2$ with fixed total winding angle, is achieved via a spectral framework: generating functions for fixed winding (or for cone-limited walks) are encoded as matrix elements of compact self-adjoint operators on Dirichlet space, diagonalized by explicit eigenfunctions involving elliptic integrals (Budd, 2017). The generating functions’ dependence on step count (size) and winding index permits precise asymptotics for area and perimeter in clusters of fixed winding, showing an explicit exponential decay of path counts with both winding and size.

Tables summarizing scaling exponents for winding angle variance:

Model/Context	Scaling of $\langle\theta^2\rangle$	Fourth-moment Ratio $\gamma$
2D RW	$(\ln N)^2$	Gaussian (3)
2D SAW	$\ln N$	Gaussian (3)
3D SAW (bar)	$(\ln L)^{1.5}$, $(\ln L)^{2\alpha}, \alpha \approx 0.75$	3.7 (non-Gaussian)

4. Size Winding and Criticality in Statistical and Quantum Systems

• Winding as a nonlocal order parameter: In the 1D XY model with variable interaction range, the distribution width of the winding number across the system encodes a nonlocal topological probe of the phase transition (Hong et al., 2015). The order parameter $m_q = s_\infty - s$ (with $s$ the normalized winding width) vanishes above and is finite below the critical temperature, capturing mean-field scaling exponents and reflecting system-spanning phase twists.
• Winding statistics in finite-size quantum and statistical rings: For superconducting rings traversing a normal-superconductor transition, a critical circumference $\tilde{C} \sim \xi$ (the Kibble–Zurek correlation length) sets the minimal system size for winding formation (Xia et al., 2020). For $C \gg \xi$, the average winding scales as $\langle|W|\rangle \propto \tau_Q^{-1/8}$ (for quench rate $\tau_Q$), with Gaussian statistics; for smaller sizes, correlations modify this scaling to $\langle|W|\rangle \propto \tau_Q^{-1/5}$, and a trinomial distribution is required to describe allowed winding outcomes.
• Quantum hydrodynamics and winding transport: In quantum spin chains with easy-plane anisotropy, the winding degree of freedom is defined hydrodynamically as the gradient of the spin azimuthal angle, propagating as a conserved transport quantity in the absence of phase slips (Tserkovnyak et al., 2020). Finite system length and dissipative terms (Gilbert damping, phase slip rate) set temporal and spatial scales for winding lifetime, with winding conductivity quantifying the efficiency of topological transport. Topological hydrodynamics becomes accurate only when vorticity flow transverse to the chain is negligible.

5. Size Winding in Quantum Operator Dynamics and Scrambling

In quantum chaotic many-body systems, size winding appears in the “operator wavefunction”—expansion coefficients of an evolving operator (e.g., Majorana or Pauli string) in an orthonormal basis:

• Phase alignment and size winding: As a local operator $O$ evolves, its support spreads over operators of increasing “size” (i.e., string length or number of non-identity factors). The complex coefficients take the form

$$c_{j_1, \ldots, j_n}(t) = \exp[i(a + b n)] |c_{j_1, \ldots, j_n}(t)|,$$

signifying a phase that winds linearly in size (Zhou et al., 17 Jan 2024, Perugu et al., 29 Sep 2025). The phase-winding distribution $Q(n, t)$ encodes both amplitude and coherent phase and is central for protocols relying on operator growth, such as traversable wormhole teleportation.

• Scramblon effective theory and Lyapunov dependence: The universal phase factor in the “scramblon” propagator at large $N$ (and large $q$ SYK) arises from an extra imaginary-time shift in the contour generating function, yielding a phase factor $\exp(i \varkappa \beta/4)$, where $\varkappa$ is the Lyapunov exponent. This phase shift generates perfect size winding for maximally chaotic systems—regimes where the Lyapunov exponent saturates $2\pi/\beta$ (Zhou et al., 17 Jan 2024).
• Krylov winding: The operator wavefunction in the Krylov basis also develops a phase linear in the Krylov index $n$, a phenomenon termed “Krylov winding” (Perugu et al., 29 Sep 2025). When the Krylov-to-size basis mapping is low rank and the chaos–operator growth bound ($\lambda_L \leq 2\alpha$) is saturated, size winding is strictly linear (otherwise it becomes super-linear). This structure is generic in quantum chaotic systems and connects directly to the underlying quantum information dynamics.
• Implications: Size winding, as an emergent phase coherence in the operator growth process, is not merely a statistical property but governs optimal signal transmission in many-body teleportation protocols. In protocols relying on phase cancellation between system and channel evolution, only proper size-wound operators yield constructive interference and maximal fidelity.

6. Topological Phases and Size-Dependent Winding Numbers

• Exceptionally large winding numbers: In finite-width 2D $d_{xy}$-wave superconductors with a persistent spin texture, the interplay of finite-size-induced hybridization and unidirectional spin–orbit coupling opens gaps in the Andreev bound state (ABS) spectrum (Ikegaya et al., 9 Apr 2025). The system enters a class AIII topological phase, with a winding number

$$N_{1\mathrm{D}} = \frac{i}{4\pi} \sum_{s = \uparrow, \downarrow} \int dk_x \operatorname{Tr}\left[ \Gamma H_{k_x, s}^{-1} \partial_{k_x} H_{k_x, s} \right],$$

scaling with film width and matching the number of propagating channels. The resulting high degeneracy of zero-energy edge states yields robust, quantized, disorder-immune charge transport.

• Physical ramifications: The topological invariants determined by size-dependent winding encode stability of edge modes, quantization of transport, and opportunities for engineered quantum devices in superconducting heterostructures—especially where large degeneracies and perfect charge transfer are desired.

7. Practical Implications and Applications

Winding and size-winding considerations directly impact:

• Single-molecule experiments: Accurate modeling and prediction of DNA behavior under torque/stretching depend on precise winding invariants that enforce self-avoidance and faithfully track twist/writhe partitioning (Prior et al., 2010).
• Polymer partition functions under confinement: Combinatorial enumeration of winding-constrained configurations yields explicit predictions for entropic forces and response to spatial restrictions (Rohwer et al., 2015).
• Topology-driven phase transitions: Winding statistics as nonlocal order parameters sharpen the detection of criticality in 1D systems (Hong et al., 2015) and characterize the scaling crossovers and finite-size effects in the formation and statistics of winding numbers in quantum ring quench transitions (Xia et al., 2020).
• Quantum chaos and information protocols: Size winding as a phase-coherent signature plays a critical role in diagnosing quantum chaos, information scrambling, and optimal conditions for quantum teleportation across many-body channels (Zhou et al., 17 Jan 2024, Perugu et al., 29 Sep 2025).

---

In summary, size winding unifies diverse phenomena, from topological invariance in polymers and complex statistical scaling in lattice models to coherent phase structures in quantum operator dynamics and topological phases in superconducting systems. Across these domains, tracking and characterizing the interaction of winding properties with size and growth processes yields deep insight into the geometry, physics, and information content of complex many-body, statistical, and quantum systems.