[Paper Review] High-order topological insulators from high-dimensional Chern insulators

Ioannis Petrides, Oded Zilberberg|arXiv (Cornell University)|Nov 19, 2019

Topological Materials and Phenomena46 references4 citations

TL;DR

This paper demonstrates that two-dimensional second-order topological insulators emerge as dimensional descendants of a four-dimensional chiral semimetal, using dimensional reduction to link their corner-localized charge quantization to the second Chern number of the ancestor model. The key result is that the quantized corner charge is directly related to the 2nd Chern flux of the 4D parent system, establishing a topological connection between high-dimensional Chern insulators and high-order topological phases via topological pumps.

ABSTRACT

Topological insulators are a novel state of matter that share a common feature: their spectral bands are associated with a nonlocal integer-valued index, commonly manifesting through quantized bulk phenomena and robust boundary effects. In this work, we demonstrate using dimensional reduction that high-order topological insulators are descendants from a chiral semimetal in higher dimensions. Specifically, we analyze the descendants of an ancestor four-dimensional Chern insulator in the limit where it becomes chiral and show their relation to two-dimensional second-order topological insulators. Correspondingly, the quantization of the charge accumulation at the corners of the 2D descendants is obtained and related to the topological indices -- the 1st and 2nd Chern numbers -- of the ancestor model. Our approach provides a connection between the boundary states of high-order topological insulators and topological pumps -- the latter being dynamical realizations of high-dimensional Chern insulators.

Motivation & Objective

• To establish a theoretical connection between high-order topological insulators and high-dimensional topological phases.
• To demonstrate that 2D second-order topological insulators arise as descendants of a 4D chiral semimetal through dimensional reduction.
• To relate the quantized corner charge in 2D systems to the 2nd Chern number of the 4D ancestor model.
• To unify the description of high-order topological insulators and topological pumps via a higher-dimensional topological invariant.

Proposed method

• Applying dimensional reduction from a 4D Chern insulator to a 2D descendant model, mapping the topological invariants across dimensions.
• Using the (4D → 2D) reduction to derive a family of 2D topological pumps that inherit topological properties from the ancestor.
• Defining the 2nd Chern flux via a 5D energy-momentum integral of the Green's function, given by Φ₂ = (3/8π²)∫ d²k d²k̃ [d̂·(∂kₓd̂×∂kᵧd̂×∂k_zd̂×∂k_wd̂)]
• Analyzing the limit where the 4D model becomes chiral (μ₀ → 0), leading to quantized Φ₂ = 1/2 or 0 depending on whether the integration domain encloses a gap-closing singularity.
• Deriving the corner charge q_C in the 2D descendant from the 2nd Chern flux, showing q_C = Φ₂ in the chiral limit.
• Verifying the correspondence between the 2nd Chern flux and corner charge using continuum field theory and explicit model Hamiltonians (Models I, II, III).

Experimental results

Research questions

• RQ1 How do second-order topological insulators in two dimensions arise from higher-dimensional topological phases?
• RQ2 What is the topological invariant in the 4D ancestor model that governs the quantized corner charge in the 2D descendant?
• RQ3 How is the 2nd Chern number of a 4D system related to the corner charge in a 2D high-order topological insulator?
• RQ4 Can dimensional reduction connect high-order topological insulators to topological pumps in lower dimensions?

Key findings

• The 2D second-order topological insulator is the dimensional descendant of a 4D chiral semimetal, with topological properties preserved via dimensional reduction.
• The quantized corner charge in the 2D system is directly proportional to the 2nd Chern flux Φ₂ of the 4D ancestor model.
• In the chiral limit (μ₀ → 0), the 2nd Chern flux Φ₂ takes quantized values of 0 or 1/2, depending on whether the integration domain encloses the gap-closing point.
• The corner charge q_C is exactly equal to the 2nd Chern flux Φ₂ in the chiral limit, establishing a direct topological correspondence.
• The derived expression for Φ₂ matches the corner charge derived from continuum field theory, confirming consistency across methods.
• The results are verified across three distinct 4D Hamiltonian models (I, II, III), showing universal agreement between Φ₂ and q_C.