Condensed matter physics in flatland

Adventures in Flatland

In everyday life we think of most objects as having three dimensions. But what would life be like in a two-dimensional world? For one thing, it would be harder to move around. We could no longer step over things but would have to move around them. In 1884 Edwin Abbott published Flatland: A Romance of Many Dimensions, under the pseudonym, A. Square, a satirical novella about social life in Victorian England. People are represented by geometrical objects. Men are represented by shapes such as triangles and hexagons. Women are represented by lines. The social status of men increases with the number sides that their shape has and how many of the sides are of the same length. Abbott’s book created limited interest and was largely forgotten by the 1920s. Interest revived when theoretical physicists started to think about worlds in different dimensions. This interest was stimulated by Albert Einstein’s theories of relativity, that proposed that we live in a four-dimensional world, not a three-dimensional one. Time is the fourth dimension, and there is an intimate and concrete connection between time and space. Attempts to unify gravity with other fundamental forces has led to physicists proposing and studying theories with more than four dimensions.

Changing the number of spatial dimensions leads to different physics because it changes what is mathematically possible. In three dimensions, there were only five highly symmetrical shapes known as Platonic solids (tetrahedron, cube, octahedron, icosahedron, and dodecahedron). In contrast, in two dimensions it is possible to make an infinite number of symmetrical shapes, known as regular polygons, shapes made of straight lines of equal length such as squares or hexagons. Similarly, the number of Bravais lattices differ in two and three dimensions. Changing the number of spatial dimensions changes both what is mathematically possible and what is physically possible.

What would condensed matter physics be like in Flatland? This question received limited attention before the 1970s. Occasionally, theoretical physicists would investigate mathematical models of crystals or magnets in one or two dimensions just because the mathematics was simpler and more tractable than in three dimensions. The goal was to obtain insight into physics in three dimensions. We will consider a famous example, the Ising model. 

In the 1970s, several surprising developments led to significant interest in condensed matter physics in spatial dimensions different from the usual three. First, it became possible to make a wide range of material systems that were two-dimensional. Secondly, theoretical work showed that states of matter, and phase transitions between them, can be qualitatively different in one, two, and three spatial dimensions. And thirdly, considering different numbers of spatial dimensions turned out to be very fruitful for theory, particularly for understanding phase transitions near critical points. 

An extract from "Adventures in Flatland," chapter 5 in Condensed Matter Physics: A Very Short Introduction