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Illustration of a hypothetical device for studying the quantum Hall effect in a world with four spatial dimensions
Illustration of a hypothetical device for studying the quantum Hall effect in a world with four spatial dimensions. Credit: LMU/MPQ

A researcher at the University of Birmingham shows how a physical phenomenon, first predicted to occur in systems with four spatial dimensions, can be explored in the laboratory.

In our everyday experience, space has three dimensions along which we can move up/down, right/left or forward/backward. But what if our universe was different? Would the laws of physics have new consequences if space had an extra fourth dimension? Now, a way to explore such questions by looking at real-world experiments has been demonstrated by Dr Hannah Price, a researcher at Birmingham, working with an international team led by Professor Immanuel Bloch in Münich and Professor Oded Zilberberg in Zürich. As recently published in the scientific journal Nature, the scientists were able to use a gas of ultracold atoms to observe a dynamical version of a novel type of “quantum Hall effect” that is predicted to occur in four dimensions, opening the way for the exploration of higher-dimensional physics. 

Our understanding of electricity and magnetism has been developed over centuries but only recently have scientists asked whether these building blocks would lead to different effects in a universe with more than three spatial dimensions. This question was inspired by the discovery that the number of dimensions can play a surprisingly important role in how electrical currents flow through a material. In 1980, Klaus von Klitzing was studying a device in which the current-carrying electrons were able to move freely only along two dimensions, say “right/left” and “forward/backward”, while being confined along the third dimension “up/down”. He found that at very low temperatures and under a very large magnetic field, the electrical resistance across the sample only took certain quantized values, unlike if electrons can move freely in all three spatial dimensions. This remarkable phenomenon is now known as the “quantum Hall effect”, and led to von Klitzing being awarded the 1985 Physics Nobel Prize. In fact, the quantization of resistance that he observed is so precise and yet so reproducible across different experimental samples that it now provides one of our best meteorological standards for defining electrical resistance.

Shortly after its discovery, the quantum Hall effect sparked a revolution in our understanding of physical materials. As realized by David Thouless and collaborators, this effect is, in fact, a signature of a new and exotic type of quantum matter: a so-called “topological phase of matter”. In mathematics, topology describes, for instance, how many “holes” a surface has and into what other shapes it can be squished and stretched without tearing the surface. From this perspective, a doughnut and a coffee cup have the same topology, as one can be moulded into the other without poking or closing a hole. Crucially, Thouless and co-workers showed that similar topological ideas describe how electrons behave in a solid. In particular, electrons moving under the influence of a strong magnetic field in two dimensions can form a special quantum phase of matter, which will have the same topological properties across different experimental samples, leading to the reproducible effect observed by von Klitzing. For this breakthrough as well as his earlier work in Birmingham with Mike Kosterlitz on topological phase transitions, Thouless was awarded half of the 2016 Physics Nobel Prize, celebrating the importance of topological ideas in modern physics.

The discovery and exploration of topological phases of matter is currently a key focus of research around the world, with scientists seeking to learn more about these exotic states and to develop new materials with useful properties. Remarkably, we now know that the laws of physics would lead to different types of topological states if our world had more spatial dimensions. While von Klitzing discovered an effect that was special to two dimensions, theoretical physicists have predicted that there would be a new type of quantum Hall effect emerging in four dimensions. Like its 2D cousin, this “4D quantum Hall effect” would be reproducible and precisely quantized, but with even richer electrical properties signifying a new topological phase of matter. However, for many years, this prediction was regarded as a mathematical curiosity, expected to remain out of reach for actual experiments.

Now, a dynamical analogue of the 4D quantum Hall effect has been realised in the laboratory of Immanuel Bloch at LMU/MPQ Münich, working with theoretical physicists, Hannah Price at the University of Birmingham and previously the INO-CNR BEC Center in Trento, and Oded Zilberberg at ETH Zürich. In this experiment, the scientists implemented an earlier proposal from Zilberberg and collaborators that the 4D quantum Hall effect can be explored by looking at a special two-dimensional system called a “topological pump”. This built on a mathematical insight, going back again to David Thouless, that particles moving in lower dimensions can capture aspects of a higher-dimensional topological phase of matter, provided certain properties of the lower-dimensional system are “pumped”, i.e. are forced to dynamically change in time. In this way, the scientists can look at a two-dimensional system and yet learn about the physics of a four-dimensional material. “One way to understand how this works”, explains Hannah Price, “is to think of our 2D system as taking a snap-shot, at each moment, of a slice of a hypothetical 4D quantum Hall system. By pumping the system, we are scanning over all the different slices. Then by adding all the snapshots together, we can see how particles would behave in the 4D quantum Hall effect.”  

To realise a topological pump experimentally, a cloud of atoms is cooled down close to absolute zero and placed in a 2D optical lattice, which acts like a “crystal of light” through which the atoms can move. The lattice is then “pumped” by dynamically varying certain properties of the laser beams forming the optical lattice. This pumping causes the atoms to move and by observing this motion, the scientists were able to extract key signatures of the 4D quantum Hall effect. In particular, they measured topological properties that would characterise this exotic phase of matter in four dimensions.

These results have now been published in the journal Nature alongside complementary work by an American research team, who used a photonic topological pump to show how particles would move across the surface of a 4D quantum Hall system. Together, these papers provide the first experimental look into the physics of higher-dimensional topological phases of matter. An exciting next step would be to go further and create a system that actually behaves like it has four spatial dimensions in the laboratory. As proposed by Hannah Price and collaborators, this could be achieved by exploiting recent experiments that add extra “synthetic dimensions” for atoms. In this approach, the atoms would move up/down, right/left and forward/backward, but also along a fourth “synthetic dimension” corresponding to an atom being in different internal states. Thanks to these advances, scientists are beginning to find out how physics would be different if we lived in a universe with more spatial dimensions and discovering, in this science-fiction world, a wide variety of new quantum phenomena.