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 mechanics 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 structure 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 together (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). On this half sphere again different loops can be drawn. Closed loops which are contractible, like the loop q1, and non contractible loops like q2 (see figure 4). Analogous to the case in two dimensions we can ask what happens, if we perform the loop q2 twice. For that let us consider the loop drawn in red in figure 4.

Figure 4: Examples of contractible (q1) and non contractible loops (q2). The red loop is homotopic to 2 q2 and to q1.

This loop is in fact a closed loop going through the points x and y, since opposite points are identiﬁed. If we now continuously deform this loop by moving the point x counter clockwise along the 2 ground plane, we obtain the loop q2. Therefore we see that these two loops are in the same homotopy class. By moving the point x in the clockwise direction until it merges with point y, we can contract 2 2 the loop. Therefore the loop q2 is contractible. We conclude that r3 has only two homotopy classes. 2 Therefore the fundamental group π1(M3 ) is isomorphic to the cyclic group of two elements Z2. This is also isomorphic two the symmetric group of two elements S2, where we identify contractible loops 2 with the identity and non contractible loops with the permutation of 1 and 2 σ12. Then σ12 = 1 2 reﬂects the fact that q2 is contractible. 2 ∼ We conclude that in three dimensions of space and for two particles π1(M3 ) = S2.

1.6 The fundamental group in general 2 ∼ Since the result that π1(M3 ) = S2 emerges from the fact that we had one free dimension in which we 2 were able to rotate points and deform the curve q2, we can suspect that fundamental group is also isomorphic to the symmetric group in higher dimensions. In fact it has been shown (e.g. by E. Artin in 1926 [4]) that this holds true and the distinction between braid and symmetric group for 2 and 3 or more dimensions holds even for N particles in general, i.e.:

N ∼ SN for d ≥ 3 π1(Md ) = . (BN for d = 2 Therefore we recover our result from the previous thought experiment that there is a fundamental diﬀerence when exchanging particles in 2 versus in higher dimensional space. In the following we will look at an example of 3 particles in order to clarify the importance of the structure of the braid group in this context.

5 1.7 Example: braiding for 3 particles

Let us look at a configuration of three particles in two dimensional space at some time t0, as shown in figure 5. At some later time t1 the particles are back in the same configuration they started in up to a permutation of indices. We can define azimuthal angles between the particles on the plane given −1 yi−yj by ϕ = tan ( ). The first configuration is then described by ϕ12 = 0, ϕ13 = η, ϕ23 = ξ. The ij xi−xj configuration at time t1 has different azimuthal angles ϕ12 = ξ + π, ϕ13 = η + π, ϕ23 = π.

Figure 5: An example conﬁguration of three particles. After some time the particles are in there original conﬁguration, but the particles 1 and 3 were exchanged.

This change in azimuthal angles and permutation of particles can be represented by the braiding σ1σ2σ1 (see figure 6(a)). This can be seen by looking at a side view of the paths the particle take from t0 to t1. By looking only at the final configuration of the particles, however, the braiding σ1σ2σ1 is 3 not the only solution. For example the braiding σ1σ2σ1 (illustrated in figure 6(b)) gives the same final configuration. The winding or azimuthal angles of the particles, however, are different for the second braiding since particles 1 and 2 performed an additional 2π rotation around each other. Therefore the correct winding angle is ϕ12 = η+3π. We therefore see, that only by looking at the configuration of the particles we cannot know all the information we need to determine how the particles were exchanged. We need to look at the actual braiding of the particles in order to determine the winging angles and thereby the change to the wave function. This is very different to the case of three dimensions where, since the fundamental group is the group SN and the braiding of the particles doesn’t matter, the final and starting configurations are enough to know the effect on the wave function.

1.8 The exchange operator Now it is possible to discuss how the exchange of particles does affect the wave function. For that let N us define some exchange operation α ∈ π1(Md ). This operation is applied to the wave function by the exchange operator P (α). Since P describes a symmetry of the system it has to fulfil the product rule P (α2)P (α1)= P (α2α1) as it can be shown that if the operator picks up a phase, this phase has to be independent of the state and is of such a form that we can redefine the operator to include this phase. Hence, fulfilling the stated product rule. Furthermore it follows that 1 = P (αα−1)= P (α)P (α−1) ⇒ P (α−1)= P (α)−1. Therefore the operator P possesses all the properties needed for a representation of a group. By additionally assuming that our wave function is scalar valued in C we arrive at the result N that P is a one dimensional complex representation of the fundamental group π1(Md ). Therefore in order to find out how the exchange of particles affects the wave function we need to compute those representations.

6 (a) The braiding σ1σ2σ1 can represent (b) Alternative braiding for exchanging the exchange of particles 1 and 3 particles 1 and 3. Gives the same final configuration but the winding angles are different.

Figure 6: Two diﬀerent possibilities for the representation of exchanging particles 1 and 3 through the braid group.

1.9 Representation of the fundamental group N ∼ Starting with three or more dimensions of space, we already know that π1(Md ) = SN . Therefore tak- ing into account that we only look at one dimensional representations and applying the representation 2 property of P to the condition σi = 1 we get that:

2 P (σi) = 1 ⇒ P (σi)|ψi = ±|ψi. We obtain the result that the wave function can only act symmetric or antisymmetric under the exchange of two particles. This represents the fact that in three dimensions of space only bosons (symmetric) and fermions (antisymmetric) exist. Rewriting the eigenvalues as ±1= eiνπ gives us the result: ν = 0, 1 where ν is called the statistics of the particle. I.e. bosons have statistics ν = 0 and fermions have statistics ν = 1.

N ∼ 2 Now to the case of two dimensions: π1(M2 ) = BN . Although the condition σi = 1 is no longer valid for the braid group we can still apply the representation property on the condition σiσi+1σi = σi+1σiσi+1 from which follows that:

iνπ P (σi)= P (σi+1) ∈ C ⇒ P (σi)|ψi = e |ψi. Since we lack any additional condition we have no further restrictions on the statistics of the particles. Therefore any statistic ν is in principle allowed. This is why Wilczek named them anyons (from anyons) in his original paper from 1982 [2]. They are not restricted to simply fermionic or bosonic behaviour but can be anything in between.

7 2 Example: The cyon

One example of a physical realization of an anyon is the cyon. The cyon is a composite system consisting of an infinitely long and thin solenoid and a charged point particle that moves in the magnetic field generated by the solenoid (see figure 7). If we orient the solenoid along the z-axis the system is symmetric along this direction. We are only interested into the motion in the x-y plane and will ignore the (free) motion in the z-direction. The solenoid creates a magnetic field that, since the solenoid is infinitely long, vanishes outside the solenoid. Choosing a symmetric gauge for the vector potential we get:

φ A = e with B = ∇ × A = φδ2(r) , 2πr φ

2 where φ = B d x is the flux of the magnetic field. Figure 7: The cyon com- ZZ posite system, consisting of We can now remember that the Lagrangian of a charged particle 2 a solenoid and a particle. (charge e) in an external magnetic field is given by L = 1/2 mv +e/c v·A. From this we can calculate the Hamiltonian of the cyon system and determine the canonical momentum p:

1 e 2 ∂L e H = v · p −L = p − A where p = = mv + A . 2m c ∂v c Now knowing the Hamiltonian for the particle moving in the ﬁeld of the solenoid we can calculate the angular momentum of the particle and thereby determining the spin of the cyon system.

2.1 Spin of the cyon We start by identifying the canonical angular momentum of the system: e eφ J = r × p = r × mv + r × A = J + , c c k 2πc where Jk = r × mv is the kinetic angular momentum of the particle. We see that the canonical and e kinetic angular momentum diﬀer by a term c r × A. Using the Gauss theorem for the electric ﬁeld produced by the particle ∇E = eδ2(x − r) we get:

1 1 J = J + d2x x · E(t, x)B(t, x)+ d2x ∇[E(t, x) x × A] . c k c c Z Z Inserting the magnetic ﬁeld B = φδ2(r) into the ﬁrst term, we see that it equals zero and we are left with:

1 2 Jc = J + d x ∇[E(t, x) x × A] . k c Z This means that Jc and Jk diﬀer only by an integral over a divergence which is a surface term. eφ This surface term, however, is non zero since, as we saw earlier, it has a value of 2πc . Nonetheless it has been argued (e.g. by A.S. Goldhaber, R. Mackenzie 1988 [3]) that this diﬀuse angular momentum which is concentrated on the boundary of the system (which may be arbitrarily far away from the particle-solenoid system) can be neglected when describing local phenomena, like in our case the spin of the cyon system.

Because the magnetic vector potential has no angular dependence the angular dependence of the wave function describing the particle in the ﬁeld of the cyon is of the form exp(imφ) where m is an

8 integer; assuming that our wave function is single valued (since the vector potential is single valued there is no reason to assume otherwise). Using this we can calculate the eigenvalues of the canonical angular momentum and from that the kinetic angular momentum of the particle:

∂ Jc|ψi = −i~ |ψi = ~m|ψi , ∂φ eφ eφ ⇒ J = J − = ~ m − . k c 2πc hc Therefore the spin of the cyon system is given by:

J (m = 0) eφ s = k = − . ~ hc We note that since the ﬂux of the magnetic ﬁeld is arbitrary the spin of the cyon system can take any value. It is not restricted to half integer or integer numbers as is the case for fermions and bosons.

2.2 Spin statistics relation For fermions and bosons a spin statistics relation exists, that states that bosons or particles with integer spin are described by symmetric wave functions and fermions or particles with half integer spin are described by antisymmetric wave functions. Or more precisely ν = 2s. If this relation still holds true in the cyon case we expect the cyon to be an anyon.

In order to determine the statistics of the cyon we need to look at a two cyon system (see figure 8). When we rotate cyon 1 around cyon 2 by a 2π rotation the wave function picks up a phase due to the Aharonv-Bohm Figure 8: 2 cyons mov- effect. This effect states, that if a charged particle moves through a magnetic ing in the vector poten- vector potential the wave function picks up a phase which depends on the tial of the other. integral over the path of the particle of the vector potential:

e |ψi → exp −i A · dr |ψi , ~c ZΓ where Γ is the path of the particle. Inserting the expressions for A and setting Γ to a closed path over a 2π rotation of cyon 1 around cyon 2 we get the phase:

e e eφ exp −i A · dr = exp −i Bd2r = exp −2πi . ~c ~c hc I ZZ Because cyon 1 moves in the field of cyon 2 but also cyon 2 moves in the field generated by cyon 1 the wave function picks up this phase twice leaving us with a final phase of:

2eφ |ψi → exp −2πi |ψi . hc Comparing this phase with the phase for exchanging particles exp[iνπ] and accounting for the fact that we did a full 2π rotation instead of a single π rotation we get the result:

eφ ν = ⇒ ν = 2s . hc We therefore conclude that the spin statistics relation indeed is still valid for the cyon and the cyon is in general an anyon.

9 3 Conclusion

To summarize we saw that in two dimensions of space particles do not necessarily behave according to bosonic or fermionic statistics, but can have any fractional statistics instead. This is due to the fact, that a fundamental symmetry of rotations is broken in two dimensions and the fundamental group describing the diﬀerent possibilities of exchanging particles is not given by the symmetric group SN but instead has the structure of the braid group BN , which accounts for this symmetry breaking by keeping track of the braiding of the paths of the particles. We also saw, that these fractional statistics can in fact arise in physical systems such as in the given example of the cyon. Although those particles follow fractional statistics, we saw that the at least in the case of the cyon, the spin statistics relation still holds. This means that the cyon in addition to having a fractional statistics also possesses a fractional spin.

On a ﬁnal note I want to thank my supervisor Andrea Campoleoni for his great support and eﬀorts in guiding me through learning and understanding the presented material as well as providing me with the needed resources.

10 List of Figures

1 Example of two possible ways of exchanging particles...... 2 2 Representation of the fundamental elements of the braid group ...... 3 2 3 Illustration of the space r2 and possible loops that can be drawn in that space. . . . . 4 4 Examples of contractible (q1) and non contractible loops (q2). The red loop is homotopic 2 to q2 and to q1...... 5 5 An example configuration of three particles. After some time the particles are in there original configuration, but the particles 1 and 3 were exchanged...... 6 6 Two different possibilities for the representation of exchanging particles 1 and 3 through thebraidgroup...... 7 7 The cyon composite system, consisting of a solenoid and a particle...... 8 8 2 cyons moving in the vector potential of the other...... 9 All figures were originally taken from the book from A. Lerda [1] and subsequently modified.

References

[1] A. Lerda (1992). Anyons. Quantum Mechanics of Particles with Fractional Statistics Berlin Hei- delberg: Springer-Verlag

[2] F. Wilczek (1982). Quantum Mechanics of Fractional-Spin Particles. Phys. Rev. Lett. 48, 114

[3] A.S. Goldhaber and R. Mackenzie (1988). Phys. Lett. 214B, 471

[4] E. Artin (1926). Abh. Math. Sem. Hamburg 4, 47