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In the adiabatic model, we have a system acquiring a Berry phase due to cyclic slow periodic evolution under a Hamiltonian. While in the topological phases, we talk about systems such as anyons which acquire fractional phases under permutation or exchange trajectories in space.

I am not clear about how the topological phases arise (in general)? Has it the same mechanism as the Berry phase? Is it a special kind of Berry phase characteristic of the topological system? Has the system to follow a path adiabatically and return back to its initial state for such a phase to emerge?

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    $\begingroup$ It would be instructive to do the exercise of how internal gauge fields and external electromagnetic gauge fields combine and give an effective Chern-Simons or BF theory in QH/TI. $\endgroup$ – AHusain Jan 7 at 21:06
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When a quantum system, parametrized by a manifold of classical parameters, evolves along a closed path in the parameter space, its state experiences a unitary transformation, which is called a geometric phase.

In most applications of quantum computing, this parameter space is usually a set of control parameters used to drive the system. More precisely, the system's Hamiltonian depends on a set of parameters, some of them can be controlled by the experimenter to generate an evolution according to the Schrödinger equation: $$\frac{d|\psi(t)\rangle}{dt} = \frac{1}{i\hbar} H(x(t)) |\psi(t)\rangle$$
The Hamiltonian $H(x)$ depends on the parameters (coordinates) $x$ belonging to the control manifold $\mathcal{M}$. The control of the system evolution is performed through the "driving the Hamiltonian" along the trajectories $x(t)$ by the experimenter.

An example of a space of control parameters of a quantum computing system is the parameter space of the laser pulses acting on trapped ion qubits. Here, the control parameters $x$ consist of the laser phases and exposure times.

Adiabatic phase

When the system's initial Hamiltonian has a degenerate ground state $$ H(x(0)) |\psi_n\rangle = E_0 |\psi_n\rangle, \quad n=1, ..., N,$$ Then if we evolve the system very slowly along a closed cycle in the parameter space, specifically if $$\langle\psi_n|\frac{dH}{dt}|\psi_n\rangle T << E_1(t)-E_0(t)$$ Where $T$ is the cycle time period and $E_1$ is the energy of the first excited state. Then, when the above adiabatic condition applies, the system evolves within the degenerate ground state subspace and its final state after the evolution cycle can be approximated by: $$ |\Psi(T) \rangle = e^{\frac{-iET}{\hbar}} Pe^{\frac{i}{\hbar}\oint A_i(x) dx^i} |\Psi(0) \rangle, $$ The term in front of $|\Psi(0) \rangle $ is the geometric phase. Both the initial $|\Psi(0) \rangle $ and the final $|\Psi(T) \rangle $, states are combinations of the ground state vectors; $P$ denotes the time ordering operator, and $A_i$ are the matrices whose elements are given by: $$A^{mn}_i = \langle\psi_m(t)|\frac{d}{dx^i} |\psi_n(t)\rangle $$ Where $|\psi_n(t)\rangle$ are instantaneous eigenvectors of the time varying Hamiltonian: $$ H(x(t)) |\psi_n(t)\rangle = E_0(t) |\psi_n(t)\rangle$$,

The Lie algebra valued differential form $A_i dx^i$ is called the Wilczek-Zee non-Abelian potential. It is a non-Abelian generalization of the electromagnetic vector potential. The Wilczek-Zee potential is also a gauge potential; a change of the (computational) basis that we use for the degenerate subspace is equivalent to a gauge transformation which does not change the value of the geometric phase.

The term $e^{\frac{-iET}{\hbar}}$ is called a dynamical phase. In our case it is just a global phase which does not change the physical state. The dynamic phase in contrast to the geometric phase depends explicitly on the cycle time. If the system is rotated fast, then less dynamical phase is accumulated. The Wilczek-Zee phase in contrast depends only on the trajectory in the parameter space and not on the evolution time (as long as the evolution is not too fast to break the adiabatic theorem).

One of the major applications of the adiabatic geometric phase is in the construction of gates in holonomic quantum computers as originally proposed in the seminal paper by Rasetti and Zanardi.

non-Adiabatic phase

In quantum computation, the adiabatic condition is a burden, because it requires the controlled evolution of the system to be slow, enabling the system more time to decohere. Thus, adiabatic quantum evolution, although not excluded as a viable possibility of realizing a quantum computer, is quite constrained. But the fact is, that when the system's evolution is fast, such that the adiabatic theorem does not apply, the system still follows a unitary evolution which can be expressed by a geometric phase. However, this time, it is the responsibility of the experimenter to keep the system in the computational subspace, as the adiabatic theorem ceases to apply. In this case the final state of the system can be expressed by:

$$ |\Psi(T) \rangle = Pe^{\frac{i}{\hbar}(-\int_0^T -E(t) dt + \oint A_i(x) dx^i)} |\Psi(0) \rangle $$

This time, the Hilbert space does not need to be a degenerate ground space of a given Hamiltonian, and the energy matrix in the first term is given by: $$E_{mn}(t) = \langle\psi_m(t)|H(t) |\psi_n(t)\rangle $$ Here, the vectors $|\psi_n(t)\rangle$ are just the (computational) basis vectors that we choose to work with.

The above geometric phase is usually referred to as the (non-Abelian version of the) Aharonov-Anandan phase.

We observe here, there is a very big complication, the dynamical phase became a matrix and it appears inside the time ordering operator. The control of dynamical phases is very complicated and depends on our precise modeling of the system and on our precise speed in controlling the system's evolution. However, there is a trick which helps very much in the non-adiabatic case: If we can control the system along a parallel transport trajectory, in which: $$\langle \Psi(t) |\frac{d}{dt} |\Psi(t) \rangle = 0, $$ Then the dynamical phase becomes again a global phase factor and the geometric phase will be again be given in an expression similar to the Wilczek-Zee case (please see , for example, Sjöqvist).

Topological phase

As stated earlier, the geometric phase does not depend on the speed that we evolve the system in along the trajectory, but it depends on the shape of the trajectory itself. For example, in the Abelian case when the trajectory resides on a Bloch sphere, the geometric phase (a $U(1)$ phase in this case) is just the area enclosed by the trajectory. Thus, a change in the trajectory's shape introduces a change in the geometric phase.

However, when the Wilczek-Zee, or the Aharonov-Anandan potentials are flat, i.e., when the curvature $$F_{ij}= \frac{d A_i }{dx^j} - \frac{d A_j }{dx^i} + [A_i, A_j] = 0$$ vanishes, then the geometric phase does not depend on the exact details of the trajectory, but only on its homotopy class. Thus, if we choose two trajectories enclosing the same holes in the parameter space, we obtain exactly the same geometric phase, which quite a relief, because in this case we do not need to draw exact trajectories in order to generate quantum gates. For example, in braiding two non-Abelian anyons, the geometric phase is topological, and in this case, the anyon trajectories need not to be precise as long one rotates about the other.

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From Wikipedia:

In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states

(emphasis is mine). Clearly, it is a special kind of Berry Phase, determined by the ground state degeneracy of the defining Hamiltonian.

You asked:

Has the system to follow a path adiabatically..

I'm not sure that adiabaticity of the evolution is absolutely necessary (in a similar way to the way that you can find plenty of discussion about non-adiabatic geometric phases), but in practice it's probably a good idea to minimise the occurrence of further excitations that could cause errors in what you're trying to achieve, even if, due to the nature of topological systems, it's quite unlikely that those excitations would contribute anything.

...and return back to its initial state for such a phase to emerge

Well, it wouldn't return to its initial state. The whole point of the topological phase is that you don't want it to return to its initial state. You might return it to its original position, but even that isn't necessary. People talk about braiding anyons -- essentially, swapping their positions.

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