The 2016 Nobel Prize for Physics was awarded to **David Thouless, Duncan Haldan and Michael Kosterlitz** for theoretical discoveries of topological phase transitions and topological states of matter. Topology is a branch of mathematics which classifies bodies according to whether they may be transformed into each another using continual transformation.

Topologically different objects are such objects that cannot be continually deformed. Topological classes are characterized by natural numbers which play an important role in topology. A soccer ball, for example, retains its topological properties even if it is squeezed or kicked. However, we would never be able to gradually deform it into the shape of a car tyre. We would have to make a hole into it and glue its loose ends together. This would be an example of a topological phase transition.

The crystalline solid matter may exist in phases characterized by different values of physical characteristics. For example, iron at room temperature has magnetic properties but when heated to a specific temperature, its magnetic properties are lost. This transition from a magnetic to a non-magnetic state is known as the state transition. Apart from such common phases, crystals may also occur in different topological states. On a macroscopic level, such states are not manifested as conspicuously as magnetism, yet they are not continually transferable to each another and, as such, they are protected in their own way. A topological state in a cadmium, mercury and tellurium compound, for example, which is, in principle, non-conductive, is manifested by the existence of loss-free surface currents. Current losses are prevented by the topological characteristics of surface currents. There are other topological states in solid matter, such as the quantum spin Hall phenomenon. Topological states occur even in acoustics and optics, where sound waves or electromagnetic waves may propagate losslessly over the surface.

In the ’70s of the last century, M. Kosterlitz and D. Thouless were the first ones to have radically changed our understanding of superconductivity by anticipating and explaining the existence of surface conductivity which seemed impossible according to the then existing theory. Later, D. Thouless explained the experiments with thin metallic layers, where conductivity is changed step-wise – the so-called the quantum Hall effect (awarded by Nobel Prize in 1985) – as a manifestation of topological states. At about the same time, D. Haldane explained the properties of atomic magnet chains using topological considerations. In addition, he also predicted the existence of a topological state, the quantum Hall effect in crystalline semiconductors without the action of an external magnetic field. These phenomena were later confirmed experimentally.

This year's Nobel Prize acknowledged the contribution of laureates to the fundamental broadening of our knowledge of the microscopic structure of solids and the discovery of their new, not easily detectable states. The acquired knowledge opens new possibilities of their use in electronics, where lossless surface currents may transfer charge, spin, or information with much lower energy requirements. At the same time, intensive work has been in progress to enable the use of such phenomena in the construction of future quantum computers.