[Paper Review] Machine learning assisted measurement of local topological invariants

M. D. Caio, Marco Caccin+3 more|arXiv (Cornell University)|Jan 10, 2019

Topological Materials and Phenomena4 references4 citations

TL;DR

This paper proposes a machine learning approach to infer local topological invariants—specifically the winding marker—in one-dimensional chiral systems using experimentally accessible local density of states (LDOS) data from scanning tunneling microscopy. By training a convolutional neural network on LDOS profiles, the method accurately predicts spatially averaged winding markers, enabling non-invasive, high-precision discrimination between topological phases with distinct integer invariants, even in composite systems where transport measurements are infeasible.

ABSTRACT

The continuous effort towards topological quantum devices calls for an efficient and non-invasive method to assess the conformity of components in different topological phases. Here, we show that machine learning paves the way towards non-invasive topological quality control. To do so, we use a local topological marker, able to discriminate between topological phases of one-dimensional wires. The direct observation of this marker in solid state systems is challenging, but we show that an artificial neural network can learn to approximate it from the experimentally accessible local density of states. Our method distinguishes different non-trivial phases, even for systems where direct transport measurements are not available and for composite systems. This new approach could find significant use in experiments, ranging from the study of novel topological materials to high-throughput automated material design.

Motivation & Objective

• To develop a non-invasive method for assessing topological phases in quantum devices, particularly where transport measurements are impractical.
• To address the challenge of measuring local topological invariants—such as the winding marker—directly in solid-state systems.
• To enable discrimination between topological phases with different integer invariants using only experimentally accessible LDOS data.
• To demonstrate that artificial neural networks can learn to infer local topological order from spatial patterns in LDOS, even in disordered or composite systems.

Proposed method

• A supervised machine learning framework is employed, using a convolutional neural network (CNN) to map local density of states (LDOS) profiles to spatially averaged winding markers.
• The training dataset consists of LDOS profiles and corresponding winding marker values computed from a family of one-dimensional Kitaev chain Hamiltonians with chiral symmetry.
• The network is trained to predict the average winding marker over regions of size $ L_c/6 $ centered at $ L_c/4 $ and $ 3L_c/4 $, enabling local phase discrimination.
• The method leverages edge state signatures in LDOS to infer topological order, even when the global invariant is not directly measurable.
• The network is evaluated on both homogeneous and composite systems, with performance quantified via root mean square error (RMSE).
• The approach is robust to disorder and does not require contact deposition, making it suitable for non-destructive testing of topological materials.

Experimental results

Research questions

• RQ1 Can a machine learning model accurately predict local topological invariants from experimentally accessible LDOS data in one-dimensional chiral systems?
• RQ2 Can the model distinguish between topological phases with different integer winding numbers using only LDOS profiles?
• RQ3 How well does the model perform on composite systems composed of regions in distinct topological phases?
• RQ4 To what extent can the neural network generalize to larger or more complex Hamiltonian families?
• RQ5 Is the method non-invasive and suitable for high-throughput material screening in topological quantum device development?

Key findings

• The neural network achieves a root mean square error (RMSE) of 0.289(1) in predicting the average winding marker across a composite system of length $ L_c = 200 $, indicating high predictive accuracy.
• The model successfully distinguishes between trivial and non-trivial topological phases, even when edge states are obscured by disorder.
• The predicted winding marker for one region shows strong correlation with the actual value in that region, while showing no correlation with the opposite region, confirming spatial locality of predictions.
• The method enables discrimination between topological phases with distinct integer invariants (e.g., $ w = 0 $, $ -1 $) without relying on counting edge states.
• The approach is robust to disorder and does not require contact-based measurements, making it ideal for non-destructive testing of fabricated topological devices.
• The framework is extendable to other local topological markers, such as the Chern marker in two-dimensional systems, and can be tailored to specific experimental Hamiltonian families.