| TKNN | |
|---|---|
| Name | TKNN invariant |
| Field | Condensed matter physics |
| Introduced | 1982 |
| Creators | D. J. Thouless, Mahito Kohmoto, M. P. Nightingale, M. den Nijs |
| Related | Integer quantum Hall effect, Chern number, Topological insulator |
TKNN
The TKNN invariant is a topological integer associated with Bloch bands in crystalline solids, introduced in the 1982 paper by D. J. Thouless, Mahito Kohmoto, M. P. Nightingale and M. den Nijs. It provides a rigorous link between band structure, topology and quantized transport, playing a central role in modern Condensed matter physics and the theory of the Integer quantum Hall effect.
The TKNN invariant emerged in the early 1980s during efforts to understand the robust quantization observed in the Integer quantum Hall effect experiments of Klaus von Klitzing and collaborators. The 1982 TKNN paper demonstrated that the Hall conductance of a noninteracting two-dimensional electron gas subject to a strong magnetic field can be expressed as a topological invariant computed from occupied crystalline Bloch states. The idea built on developments in Berry phase theory and earlier work on localization and edge states by researchers at institutions such as Bell Labs and various universities. Principal authors include D. J. Thouless (later a Nobel laureate), Mahito Kohmoto, M. P. Nightingale and M. den Nijs, and the paper rapidly influenced research on topological aspects of band theory and later the discovery of Topological insulator phases.
The TKNN invariant is mathematically identical to the first Chern class (or more specifically the first Chern number) of the U(1) fiber bundle defined by occupied Bloch states over the Brillouin zone, a two-torus for periodic crystals. In practical terms, one computes the integral of the Berry curvature over the Brillouin zone; this integral yields an integer—the TKNN integer—that classifies the filled band configuration. The invariant is gauge‑invariant, stable under continuous deformations that preserve the energy gap, and quantized because it counts topological winding. Connections were clarified with concepts from differential geometry and topology, and through analogies to the Gauss–Bonnet theorem and fiber bundle theory developed in mathematics.
The TKNN invariant provides the microscopic explanation for the exact quantization of the Hall conductivity observed in the Integer quantum Hall effect. For noninteracting electrons at zero temperature, the Hall conductance σ_xy is given by (e^2/h) times the sum of TKNN integers for occupied bands. This result explains plateaus in σ_xy insensitive to disorder and sample details, because topological integers cannot change unless bands cross or the gap closes. The invariant also underpins understanding of robustness against impurities and of the relationship between bulk topology and conducting edge state modes predicted by the bulk–boundary correspondence.
Given a crystal with Bloch Hamiltonian H(k) defined over the Brillouin zone T^2, one chooses a smooth gauge for occupied Bloch eigenstates |u_n(k)>. The Berry connection A_n(k) = i⟨u_n|∇_k u_n⟩ leads to Berry curvature F_n(k) = ∇_k × A_n(k). The TKNN integer C_n is (1/2π) ∫_{T^2} F_n(k)·d^2k, an integer by virtue of the properties of U(1) bundles. In multi‑band systems, the total invariant is the sum over occupied bands. Alternative formulations employ projectors P(k) onto occupied states and express the invariant through traces of P and its derivatives, linking to the noncommutative geometry approach by Connes and to index theorems such as the Atiyah–Singer index theorem.
Beyond explaining quantized Hall plateaus, the TKNN invariant predicts the existence of chiral edge state channels carrying current along sample boundaries; their number equals the total TKNN integer, producing unidirectional transport immune to backscattering. Observations in two‑dimensional electron gases under high magnetic fields at low temperatures provided experimental confirmation via precisely quantized Hall resistance measurements by groups following von Klitzing. Later experiments in engineered systems—cold atoms in optical lattices, photonic crystals, and graphene under moiré potentials—have implemented band structures with nonzero TKNN numbers, verifying the predicted transport and edge phenomena. The invariant also informs the design of devices exploiting dissipationless edge conduction for metrology and potential applications in robust electronics.
The TKNN concept generalizes to symmetry‑protected and interacting systems. For time‑reversal symmetric systems the classification uses Z2 invariants rather than TKNN integers, leading to Quantum spin Hall effect and Topological insulator phases studied by Kane and Mele and by Bernevig, Hughes and Zhang. In higher dimensions, analogous invariants include higher Chern numbers and Chern–Simons forms relevant to 3D topological insulators and Weyl semimetal physics. Extensions handle interacting systems via many‑body Chern numbers, and disorder through noncommutative Chern numbers developed with Bellissard and others. These generalizations preserve the central theme: topological invariants classify phases robust under perturbations and govern observable response coefficients.
Numerical evaluation of the TKNN invariant typically discretizes the Brillouin zone and computes Berry curvature via overlaps of neighboring Bloch states, using methods such as the Fukui–Hatsugai–Suzuki algorithm and Wilson loop techniques. Ab initio band structure codes (e.g., Quantum ESPRESSO, VASP) combined with Wannier interpolation (via Wannier90) permit evaluation in realistic materials. Careful treatment of gauge choices, degeneracies, and finite sampling is required to ensure integer convergence. For disordered or interacting systems, numerical approaches rely on twisted boundary conditions, many‑body Berry phases, and real‑space formulations of the Chern number to compute topological indices robustly.
Category:Condensed matter physics Category:Topological phases of matter