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Anyons Braid Statistics in Quantum Mechanics

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Anyons Braid Statistics in Quantum Mechanics Anyons Braid statistics in quantum mechanics Julian Lang May 4, 2018 Abstract This report contains the information that was presented in my talk about Anyons in the Pros- eminar: Algebra, Topology and Group Theory in Physics by Prof. Matthias Gaberdiel (2018). It contains two parts. In the first part a theoretical explanation of what anyons are and why they can exist is given, in the second part the cyon is discussed as a physical representation of a system that shows fractional statistic. It thereby closely follows the ideas and derivation from the book Anyons from A. Lerda [1] 1 1 Theoretical discussion of the anyon An important idea for the discussion of anyonic statistics is the fact that particles in quantum mechan- ics are indistinguishable. This is due to the fact that there cannot be made any observable distinction between two physical states that differ only by the exchange of indistinguishable particles. This means there exists no experiment by which it is possible to determine if particles were exchanged or not. In classical physics, however, it is possible to follow the trajectories of the particles and by doing so, at least in principle distinguish the particles from one another. In quantum mechanics this is not possible since the particles positions and velocities are fundamentally not determined between measurements. It is known that in 3 dimensions of space this indistinguishability leads to the restriction that only bosons or fermions can exist. Although this result will be reproduced in this report, we will see that in two dimensions there exist additional solutions. 1.1 Dependence on the dimensionality of space By performing a simple thought experiment we can already see that we should expect a different behaviour in 2 space dimensions versus in 3 or more dimensions of space. The idea is as follows: Let us represent the exchange of two particles through a rotation of particle 2 around particle 1 by π radians followed by a translation such that the starting and final configuration is the same. Now let’s perform the same operation in a different way by rotating around −π radians instead. The following figure shows these two possible exchange operations: Figure 1: Example of two possible ways of exchanging particles. We can realize that these two different operations are actually the same operation in 3 or more space dimensions since they only differ by the definition of the coordinate system. The paths of the particles are also equivalent or homotopic since we can continuously deform one into the other by rotating it through the 3rd dimension. In two dimensions this operation cannot be performed. The only way to continuously deform the two paths into each other is by moving the particles through each other. This brings us to the hard-core hypothesis, which states that two particles cannot occupy the same location in space. This hypothesis will be assumed to be true in the following derivation of anyonic statistics. By assuming this hypothesis to hold, the two operations of our thought experiment are no longer equivalent in 2 dimensions. We can therefore expect a different behaviour in 2 dimensions. In order to determine more rigorously how exactly the case in 2 dimensions differs from the case in 3 dimensions it is useful to introduce some additional definitions. 1.2 Configuration space The configuration space of N particles is the set of all configurations the particles can be in. Since we N consider indistinguishable particles it is different from Rd (where d is the dimension of space) since configurations that differ only by the labelling of indices are identical. Therefore we have to divide 2 by the symmetric group of N elements SN . By adding the already discussed hard-core hypothesis the configuration space of N indistinguishable particles is given by N Rd − ∆ N N Rd Md = , with ∆= (r1, ..., rN ) ∈ : ri = rj for some i 6= j . SN We note that exchanging particles like in our thought experiment can be represented by a loop in the configuration space. On those loops we can now apply concepts of Topology in order to try and find all possible different ways of exchanging particles. 1.3 Fundamental group One possible way of classifying all possible loops is by looking at homotopy classes. Each homotopy class contains all loops that can be continuously deformed into each other. Therefore one homotopy class contains equivalent ways of exchanging particles. The set of all homotopy classes is called the fundamental group, where the product of two homotopy classes is defined by chaining loops together. N The fundamental group will be denoted as π1(Md ) in the following text. We see that we can find all possible different ways of exchanging particles by determining the fundamental group of the configuration space. 1.4 The braid group Before going any further let us shortly recall the struc- ture of the braid group. The braid group BN of N strands is the group that represents possible braids of these N strings. Its fundamental elements σi represent the braiding of the two strings i and i + 1 as shown in figure 2. A general element of the braid group is then formed by chaining those fundamental braids to- gether (an example of this can be seen in a later fig- Figure 2: Representation of the funda- ure: 6). By looking at these graphical representa- mental elements of the braid group tions of the braid group we can find its defining rela- tions: BN = hσ1,...,σN−1i, with σiσj = σjσi for |i − j|≥ 2 and σiσi+1σi = σi+1σiσi+1. 2 We note that in contrast to the symmetric group of N elements SN the relation σi = 1 is not fulfilled by the fundamental elements of the braid group. We will see that this difference between BN and SN will lead to the difference in statistics of particles in two versus three dimensions. 1.5 The fundamental group for two particles Let us now consider the case of a two particle system. The configuration space of two particles can be separated into the product of two parts: centre of mass and relative motion. R = r1 + r2 ∈ R , r = r1 − r2 ∈ R \ {0}. Since the centre of mass is symmetric under permutations of particles its configuration space is simply R. The space describing the relative motion, however, takes a more complicated form as we 3 have to identify the points −r and r with the same point in our configuration space, because they 2 only differ by the ordering of indices. This space will be denoted as rd. Our configuration space for two particles therefore takes the form: 2 R 2 Md = × rd . Let us now determine the fundamental group of this space in two and three dimensions of space. 1.5.1 Two dimensional space In two dimensions the space of the relative motion of the particles is a plane. However, we have 2 to identify the points −r and r as discussed earlier. This means that r2 can be represented by the upper half plane with points along the x-axis identified with the counterpart mirrored along the y-axis. Because of the hard-core hypothesis we also remove the origin. By merging the identified points along 2 the x-axis we can continuously deform our space r2 into a cone that is missing the tip, as is illustrated in the figure below: 2 Figure 3: Illustration of the space r2 and possible loops that can be drawn in that space. On this cone we can draw different loops. For example we can draw loops on the side of the cone that are contractible to a point. Furthermore we can draw the loop q (see figure 3), which winds around the cone once. This loop is not contractible as the only way to contract this loop would be to move it above the tip of the cone, which is impossible since we removed it. Therefore the loop q is part of a different homotopy class with respect to the contractible loops. When we square the loop q, i.e. perform it twice, we end up with a loop that is still not contractible but also different from loop q, because it winds around the cone twice. By repeating this operation we can form an infinite amount of different homotopy classes that differ from one another by winding numbers around the cone. In addition to that we can also define negative winding numbers by performing loop q in the opposite direction which is the inverse of q since the chaining together will give back a contractible loop. We therefore see that the fundamental group is the group of integers, where the integer refers to the winding number of the loop around the cone with negative numbers associated to windings in the opposite direction. This is also isomorphic to the braid group B2. The braid group has only one fundamental braid: the braid σ1 that braids 1 and 2 around each other. The number of times 1 and 2 are braided around each other can be associated with the winding number of the loop. Contractible loops are thereby associated with the identity element and negative winding numbers with the inverse −1 2 ∼ braid σ1 . We conclude that π1(M2 ) = B2. 4 1.5.2 Three dimensional space 2 In three dimensions it is useful to further split the space r3 into the relative direction vector of the r particles /|r| and the absolute distance between the particles |r|, the latter of which is indifferent under permutations. By again identifying points that represent the same point in the configuration 2 space we see that the space r3 has the structure of a half sphere with opposite points on the bottom plane identified (ignoring the symmetric part under permutations |r| which just scales the half sphere).
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