In Chapter 2.2 of the book, Berry phases are described as a means of calculating geometric phases of anyons. Below, we will give a derivation of the Berry phase for both the case of Abelian anyons and state the more general result for the non-Abelian case.
Derivation of the Abelian Berry Phase
Michael Berry was the first to give a general derivation of the geometric phase in his paper Quantal phase factors accompanying adiabatic change, which is why it is also known as the Berry phase. If we consider a quantum system that is moved slowly through parameter space, then we can expect it to have remained in the same eigenstate because of the adiabatic theorem. However, it will have picked up a total phase factor given by
where is the dynamic phase factor given by
However, our focus is the other part of the total phase factor, the which is the Berry phase. As our system is moving through parameter space, we take a Hamiltonian of the form . Here, the index , where N represents the dimension of our parameter space. Our system will evolve by the time-dependent Schrodinger equation given by
where we have dropped the x parameter from our state vectors because it isn't relevant in our calculations.
By substituting our state vector with phase factors into the Schrodinger equation we can describe our system at the end of its path through parameter space
If we then use the product rule on the left hand side of the above equation and divide by , we get
We then expand the left hand side and end up with an on both sides which cancel to give the expression
By rearranging and taking the inner product of , we get
If we assume that our system is normalized as expected then and we are left with a time derivative of , which in theory could be integrated to give an expression for . However, our wave function only evolves as a function of time because our Hamiltonian is a function of . Therefore, it would be an over simplification of our system to integrate now and it would give us an expression for that equates to zero.
Before we solve for gamma we need to note that
where is indexed from 1 up to N. We can then substitute this into our expression for the time derivative of (whilst again assuming the normalization of our system), to give
We can now integrate the above derivative to get an expression for that does not give us a value of zero. If we integrate in a loop from t=0 to t=T, we get
which is an integral around a closed loop in parameter space. So, we now have an expression for the Berry phase given by
The Berry Connection
There are ways to understand this phase from a topological point of view using the idea of a connection. Let us define the Berry connection as
This implies that the Berry phase can be written as
The Berry connection behaves like a gauge potential in electromagnetism, such that we can make transformations of the form without affecting the overall physics of the system. This is because , so when we integrate to find the Berry phase the additional derivative term will vanish. This gauge invariance makes it difficult to extract any physical information from the system. So just as in electromagnetism, we employ Stoke's theorem to change our expression for the Berry phase so that it's in terms of a field strength or curvature
where S(C) is a is the surface in parameter space bounded by the curve C. The extra index indicates that we are now integrating over a surface instead of around a path. It can also be understood by noting from Stoke's theorem that .
The curvature offers a more intuitive understanding of how the Berry phase forms. We can see that behaves like a flux moving through the surface, S(C) in parameter space. The consequence of this is a topological effect meaning the Berry phase doesn't depend on the specific path taken through parameter space, but only on the flux. This gauge invariance of the Berry phase is part of the reason why anyonic quantum computation is so resilient to environmental interference.
Non-Abelian Berry Phases
The above discussion of the Berry phase is effective for modelling the phases of Abelian anyons but not for the non-Abelian ones. This is because our expression for the Berry phase isn't capable of outputting the unitary matrices needed to describe the statistical evolution of non-Abelian anyons. To do this, we need to generalise the Berry phase to accommodate these more exotic statistics. Our assumption in the derivation of the Berry phase that has limited our description to Abelian anyons only, is that there is no degeneracy of our state. The incorporation of degeneracy complicates the derivation given above because we can no longer use the adiabatic theorem to assume that our system remains unchanged. In fact, the adiabatic theorem is still valid, but now says that when slowly moved through parameter space, our system is bound to remain in the same degenerate subspace of our eigenspace that corresponds to our initial state. Within, this subspace however, the state is free to change.
If we consider this eigenspace to be N-dimensional, then we can show that the non-Abelian Berry connection is given by the matrix
The curvature or field strength can then be calculated as being
Now, if we consider that our non-Abelian system loops around the path C through parameter space such that
where is the non-Abelian Berry phase, also known as the holonomy and is the dynamic phase factor. We should note that does not necessarily equal , but they must both exist in the same degenerate subspace of our Hamiltonian. Using the non-Abelian Berry connection we can express the holonomy as
where denotes the path ordering which is now a necessary component because we have to ensure the operators that move the system around a path in parameter space are in the right order. This is necessary because our non-Abelian system will not evolve by operators that commute with each other and hence must be applied in the correct order.