🔬 mesoscale physics

Turning Number and the Geometry of Disorder-Induced Topology

This paper proposes a refined topological characterization of disordered quantum systems by introducing the "turning number"—an invariant describing the geometry of the tangent vector of momentum-space Hamiltonian maps—which reveals new topological phase transitions in the Su-Schrieffer-Heeger model that occur even when the system remains spectrally gapped.

Original authors: Lars Emmrich, Michael Potthoff

Published 2026-10-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lars Emmrich, Michael Potthoff

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern physics, scientists have long sought to categorize the different states of matter. Traditionally, this was done by looking for patterns in how atoms arrange themselves or how they break specific symmetries, a framework that successfully explained magnets and crystals. However, a newer frontier emerged with the discovery of topological phases. These are states of matter that cannot be distinguished by local patterns but are instead defined by global, unbreakable properties, much like how a coffee mug and a donut are fundamentally the same shape because both have a single hole, regardless of how you stretch or squish them. For decades, physicists believed they had a complete map of these phases for materials that are perfectly clean and do not interact with one another. But the real world is rarely perfect; it is full of impurities and disorder. The question remained: does this messiness destroy the delicate topological order, or does it hide new, unexpected structures that our old maps simply cannot see?

A team of researchers at the University of Hamburg has now uncovered a hidden layer of geometry within these disordered systems, revealing that the standard tools used to classify matter are insufficient when randomness is introduced. They focused on a specific model of a one-dimensional chain of atoms, known as the Su-Schrieffer-Heeger model, which is famous for its ability to host topological states. In a perfect, clean version of this chain, the material's state is determined by a simple count of how many times a specific mathematical path winds around a central point. This count, called a winding number, acts as a fingerprint for the phase. The researchers found that when they introduced disorder—randomly varying the energy of the atoms along the chain—the standard winding number still worked, but it failed to tell the whole story. It could not distinguish between a transition driven by the usual physics of clean materials and a completely different kind of transition driven by the disorder itself.

To solve this puzzle, the team looked deeper into the geometry of the path traced by the material's quantum state. Instead of just counting how many times the path loops around the center, they examined the path's own shape and how its direction changes as it moves. They introduced a new measure, which they call the turning number. This value tracks how the tangent vector—the arrow pointing in the direction of the path's motion—rotates as one travels along the loop. In a clean system, this turning number remains constant and uninformative. However, in the presence of strong disorder, the researchers discovered that the turning number can change even when the material remains stable and the energy gap does not close. This happens not because the path crosses the center, but because the path momentarily stops moving in a specific direction, creating a sharp cusp or a "stop" in its flow before continuing.

The researchers used powerful computer simulations to map out the behavior of this system across a wide range of disorder strengths and hopping energies. They found that the phase diagram of the material is far more complex than previously thought. While the standard winding number divides the material into two main regions, the turning number reveals a rich internal structure within those regions. Specifically, they identified a transition regime where the turning number flips from one value to another, driven entirely by the disorder. This transition occurs when the tangent vector of the quantum path vanishes at certain points, a phenomenon that has no counterpart in the clean, disorder-free world. The simulations showed that this new structure is robust and appears consistently when the disorder is strong enough, suggesting that the geometry of the quantum map itself carries a topological signature that was previously invisible.

Crucially, the team demonstrated that this new classification is not just a mathematical curiosity but a real physical distinction. They showed that in the clean limit, the transition between different phases is marked by the closing of an energy gap, a dramatic event where the material's properties change abruptly. In the disordered regime, however, the transition happens differently. The energy gap can remain open and the material can stay insulating, yet the turning number still changes. This indicates that the disorder is not merely blurring the lines between phases but is actively creating a new kind of topological order. The researchers also found that the boundary where this new transition occurs can be predicted by analyzing how the disorder strength competes with the energy of the atoms, providing a clear theoretical guide for where to look for these effects.

The findings suggest that our understanding of topological matter needs to be expanded beyond the traditional classification of winding numbers. The geometry of the quantum map, specifically how its direction changes, offers a new source of topological structure that is particularly relevant for systems with disorder. This work implies that there are likely other hidden geometric features in more complex, higher-dimensional materials that could similarly be revealed by looking at the shape of their quantum paths rather than just their final position. By focusing on the geometry of the map itself, the researchers have opened a new door for understanding how randomness can generate order, offering a refined lens through which to view the quantum world.

Technical Summary: Turning Number and the Geometry of Disorder-Induced Topology

Problem and Motivation
The traditional classification of topological phases relies on homotopy invariants of gapped Bloch Hamiltonians, which remain unchanged under continuous deformations preserving the bulk gap and symmetries. While this "tenfold way" provides a complete classification for noninteracting, clean systems, it does not resolve geometric properties of the Hamiltonian map that can change without altering the homotopy class. For interacting or disordered systems, topological characterization is often extended using Green-function invariants (e.g., the Ishikawa-Matsuyama invariant) or the topological Hamiltonian, $H_{\text{top}}(k) = -G^{-1}(k, i\omega=0)$ . However, these approaches primarily focus on homotopy classes and do not fully capture the geometric structure of the map $k \mapsto H_{\text{top}}(k)$ , particularly in regimes where Green-function zeros (rather than poles) drive topological transitions. The authors investigate whether the geometry of the momentum-space Hamiltonian map itself carries robust topological structure beyond conventional winding numbers, specifically in the context of disorder-induced transitions.

Methodology
The study focuses on the one-dimensional chiral Su-Schrieffer-Heeger (SSH) model with binary potential disorder. The system consists of spinless fermions on a chain with intracell ( $t_1$ ) and intercell ( $t_2$ ) hoppings, subject to random on-site potentials $\pm W/2$ .

Topological Hamiltonian: The authors utilize the disorder-averaged single-particle Green's function $G(k, \omega)$ to define the topological Hamiltonian $H_{\text{top}}(k) = -G^{-1}(k, 0)$ . Due to chiral symmetry, $H_{\text{top}}(k)$ is constrained to the form $\mathbf{d}(k) \cdot \boldsymbol{\sigma}$ , where the Bloch vector $\mathbf{d}(k) = (d_x(k), d_y(k))$ traces a closed loop in the plane as $k$ traverses the Brillouin zone (BZ).
Invariants:
- Winding Number ( $\nu$ ): The standard homotopy invariant counting the winding of $\mathbf{d}(k)$ around the origin.
- Turning Number ( $\tau$ ): A geometric invariant defined as the winding number of the tangent vector $\dot{\mathbf{d}}(k) = \partial_k \mathbf{d}(k)$ . This is a regular-homotopy invariant, requiring the "immersion condition" $\dot{\mathbf{d}}(k) \neq 0$ to hold throughout the BZ.
Numerical and Analytical Approaches:
- Numerical simulations were performed by averaging over $N=1000$ (or $2000$) disorder configurations for system sizes $L=200$ to $600$.
- Asymptotic perturbative expansions were derived for the limits of weak intercell hopping ( $t_2 \to 0$ ) and strong disorder ( $W \to \infty$ ) to predict transition regimes.
- Weyl's inequality was employed to establish rigorous bounds for the existence of a spectral gap.

Key Contributions and Results
The paper identifies a new topological structure arising from the geometry of the $H_{\text{top}}(k)$ map, characterized by the turning number $\tau$ .

Disorder-Driven Transitions via Zeros: In the clean limit ( $W=0$ ), the topological transition at $t_1=t_2$ is driven by the closing of the spectral gap (a pole in $G$ ), changing $\nu$ from 0 to 1 while $\tau$ remains constant at $+1$ . In the strong-disorder regime, the transition at $t_1=t_2$ is driven by Green-function zeros (poles of the self-energy). While $\nu$ still jumps from 0 to 1, the turning number $\tau$ changes from $-1$ to $+1$ . This demonstrates that $\tau$ distinguishes between pole-driven and zero-driven transitions, a distinction invisible to $\nu$ .
Immersion-Driven Transitions: The authors identify a regime where $\tau$ changes while the spectral gap remains open and $H_{\text{top}}(k)$ remains finite. This occurs when the immersion condition is violated, i.e., $\dot{\mathbf{d}}(k) = 0$ for some $k$ .
- Phase Diagram: The extended phase diagram in the $(t_2/t_1, W)$ plane reveals regions with distinct $\tau$ values within a fixed $\nu$ sector. Specifically, for $t_2 < t_1$ and strong disorder, the system transitions from $\tau = +1$ (clean-like) to $\tau = -1$ (strong-disorder-like).
- Intermediate Phase: Perturbative analysis and numerical results for $W=2.5$ reveal an intermediate phase with $\tau = 0$ (a "figure-eight" loop) between the $\tau = +1$ and $\tau = -1$ regimes. This phase is bounded by critical values $t_{c1}$ and $t_{c2}$ where the loop develops cusp singularities at time-reversal-invariant momenta ( $k=0, \pi$ ).
Asymptotic Prediction: Theoretical analysis predicts the crossover scale for the turning-number transition. For strong disorder ( $W \gg t_1$ ), the transition regime is located around $t_{2,c} \simeq 4t_1^3 / W^2$ . This analytical estimate aligns quantitatively with numerical results for $W \gtrsim 3$ .
Symmetry Constraints: The paper analyzes the constraints imposed by chiral and time-reversal symmetries. While the winding number $\nu$ is constrained by the exchange symmetry $t_1 \leftrightarrow t_2$ (implying $\nu(t_2, t_1) = 1 - \nu(t_1, t_2)$ ), no such symmetry relation exists for $\tau$ . Furthermore, time-reversal symmetry dictates that transitions involving $\Delta \tau = \pm 2$ can occur either directly (via singularities at generic $k$ ) or via an intermediate $\tau=0$ phase (via singularities at $k=0, \pi$ ).

Significance
The paper establishes that the geometry of momentum-space Hamiltonian maps, specifically the turning number of the Bloch loop, serves as an additional source of topological structure beyond conventional homotopy classifications. This geometric refinement:

Distinguishes between topological transitions driven by Green-function poles (clean limit) and those driven by Green-function zeros (strong disorder).
Reveals topological phases and transitions that are invisible to the standard winding number classification, even when the system remains spectrally gapped.
Provides a framework for understanding "immersion-driven" transitions where the tangent vector of the Hamiltonian map vanishes, offering a new perspective on disorder-induced topology in one-dimensional chiral systems.

The authors suggest that these findings motivate the search for analogous $\tau$ -resolved phases in interacting systems and the development of gauge-covariant formulations for higher-dimensional Hamiltonian maps.

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