In a recent paper published in Physical Review B, condensed matter physics one of the top journals, published by American Physical Society, researchers from IIT Delhi considered the topological quantization of Hall conductivity in a two dimensional electron system such as monolayer Graphene when the transverse magnetic field profile is periodically modulated and not uniform.

Using some simple mathematical tools such as Fourier series expansion they showed that such periodically modulated system is a sum of normal Integer Quantum Hall Effect (IQHE) in a uniform magnetic field, a landmark discovery in condensed matter systems and the Anomalous Quantum Hall Effect (AQHE) suggested first by F. D. M. Haldane, namely Quantum Hall Effect without a net magnetic field. Haldane’s discovery particularly led to the discovery a class of materials called Topological Insulators whose current carrying properties can be determined from an abstract mathematical concept called Topological Invariant.

The researchers from IIT Delhi showed that quantized Hall conductivity of their periodically modulated magnetic system is given by the same Topological Invariant as the IQHE. Periodic magnetic modulations on two-dimensional electron systems are imprinted routinely using nano-lithographic techniques. The research therefore opens up the possibility of new type of Topological materials.

“IIT Delhi research opens the possibility of creating new kind of Topological Materials,” says Prof Sankalpa Ghosh of the Physics Department and one of the authors of the paper.

The first author of the paper is IIT Delhi PhD student Manisha Arora.

**Unearthing Topological Invariant**

So what is a Topological Invariant and why should a material scientist be interested in it? Topology is a branch of mathematics that study the geometrical properties that are insensitive to the smooth deformations. For example, two very different objects such as a coffee cup and a doughnut belong to same topological classes and it can be shown that one may be smoothly deformed into other. But a sphere and a doughnut belong to different topological classes since a sphere cannot be smoothly deformed into a doughnut without creating a hole inside it. The integer number, for example, the number of such holes, that characterizes such different topological classes is called Topological Invariant. That such a branch of mathematics can be very useful to understand the conductivity properties of material first became clear with the discovery of Integer Quantum Hall Effect (IQHE) in 1980, a landmark discovery in condensed matter systems awarded with Nobel prize. IQHE occurs in systems with a uniform and large magnetic field and the robust quantization of the Hall conductivity happens because this is given by a Topological Invariant called Chern Number. Subsequently In 1988 Haldane showed that topological quantization of Hall conductivity can also occur effectively in the absence of net magnetic field, leading to anomalous Quantum Hall Effect (AQHE) for which he was also awarded Nobel Prize. Eventually, this idea led to modern day Topological Insulators, one of the most active fields of current condensed matter research.

The Figures below give a schematic of the event that happens and a calculated energy structure of such system.

The work has been published in Physical Review B ( by American Physical Society), Physical Review B, 98, 155425 (2018)