In 1988, physicist F. Duncan Haldane published a paper that significantly influenced condensed matter physics. At the time, the Quantum Hall effect was understood to occur only in two-dimensional materials subjected to powerful external magnetic fields. Haldane proposed a theoretical model for a material, structured like a sheet of graphene, that could exhibit this quantum effect without any net magnetic field.
This work was initially seen as a theoretical curiosity, a construction unlikely to be found in nature. However, it laid the mathematical groundwork for a new class of materials and introduced a novel phase of matter. This insight contributed to Haldane receiving a share of the 2016 Nobel Prize in Physics.
The Quantum Hall Effect Prerequisite
The standard Hall effect is a classic phenomenon in physics. When an electrical current flows through a conducting material and a magnetic field is applied perpendicular to the current, a voltage develops across the material. This Hall voltage is perpendicular to both the current and the magnetic field, arising because the magnetic field deflects electrons to one side of the conductor.
The Quantum Hall Effect (QHE) is the quantum mechanical version of this phenomenon, first observed in the 1980s. It appears in two-dimensional electron systems at extremely low temperatures and in very strong magnetic fields. Under these conditions, the linear relationship of the classical Hall effect breaks down.
The measured Hall resistance jumps between specific, precisely defined values instead of changing smoothly. This phenomenon of resistance taking on only discrete values is known as quantization. It is similar to water flowing down a set of stairs, where it can only rest at the level of each step. Scientists long believed that the intense external magnetic field was a requirement for observing the QHE, as it forces electrons into the quantized orbits that produce the effect.
A Breakthrough Without a Magnetic Field
Haldane’s 1988 model proposed a way to achieve the Quantum Hall Effect without a large, uniform magnetic field. His theoretical framework was based on a honeycomb lattice of atoms, a structure identical to that of graphene. This lattice consists of two distinct sub-lattices of atoms, a feature Haldane used to create the effect.
His innovation was to replace the external magnetic field with a complex, internal magnetic landscape. He theorized that a periodically varying magnetic field could exist within the material itself, with a flux that averaged to zero over each unit cell. This was achieved by introducing specific interactions between atoms that were not immediate neighbors but “next-nearest neighbors,” creating a microscopic magnetic texture.
This internal structure performs the function of an external field by breaking the system’s time-reversal symmetry. This symmetry means the laws of physics governing the system should look the same whether time is running forward or backward. Breaking this symmetry is a requirement for the Hall effect, and by doing so with internal properties, Haldane showed it was possible to generate a “Quantum Anomalous Hall Effect”—a quantized Hall effect without a net external magnetic field.
The Concept of Topological Phases
The Haldane model was the first concrete example of a state of matter now known as a topological insulator, specifically a prototype for Chern insulators. These materials have a defining characteristic: their interior, or “bulk,” acts as an electrical insulator, while their edges are electrically conductive. This behavior is dictated by a mathematical property of the material’s electronic band structure called a topological invariant.
For this system, the invariant is the Chern number, an integer that characterizes the global structure of the electron wavefunctions. Much like the number of holes in an object, this integer cannot be changed by small deformations like stretching or adding impurities to the material.
An analogy for this concept is the difference between a coffee mug and a sphere. A mug and a donut are topologically equivalent because both have one hole and can be smoothly reshaped into one another. A sphere, having no holes, is in a different topological class, just as a material in the Haldane model is distinct from a conventional insulator, forcing conductive states to exist at its boundaries.
From Theory to Experimental Reality
For over two decades, the Haldane model remained a theoretical idea. Graphene, the original inspiration, was not a suitable candidate for realizing the effect. While the model used its honeycomb lattice, the specific magnetic interactions Haldane proposed are too weak in natural graphene to produce a detectable Hall effect.
The experimental breakthrough occurred in 2013. Scientists observed the Quantum Anomalous Hall Effect in a thin film of a magnetic topological insulator: chromium-doped bismuth antimony telluride. By controlling the material’s composition and applying a gate voltage, they tuned it into a state with a perfectly quantized Hall conductance at zero external magnetic field.
This confirmation validated the principles of the Haldane model and solidified the field of topological materials. The robust nature of these topologically protected edge states, which are immune to local imperfections, has opened pathways for new technologies. Potential applications include ultra-low-power electronics (spintronics) and the construction of more fault-tolerant quantum computers.