Relation of Berry phase and winding number I am reading the following article dealing with the properties of Dirac fermion in condensed matter physics : https://arxiv.org/abs/1410.4098
In the page 5 of this article, the formula for the eigenvector of the Dirac fermion in different valley has been given as
\begin{equation}
\left| {\psi_{\xi,\lambda},\vec q} \right\rangle = \frac{1}{\sqrt{2}} \left[ {\begin{array}{*{20}{c}}
{1}\\
{\xi \lambda e^{-i\phi_{\vec q}}}
\end{array}} \right]
\end{equation}
where $tan \phi_{\vec q} = \frac{v_{y}q_{y}}{v_{x}q_{x}}$
The presence of $v_{y}$ and $v_{x}$ is just because we assume that the Dirac fermion has anisotropic velocity.
This article then state that the winding number around a Dirac point is given by
\begin{equation}
{w_{\xi ,\lambda }} = \frac{{\xi \lambda {\mathop{\rm sgn}} \left( {{v_x}{v_y}} \right)}}{{2\pi }}\oint\limits_{{C_i}} {{\nabla _{\vec q}}{\phi _{\vec q}} \cdot d\vec q} 
\end{equation}
In the footnote on page 5 he states that since Berry phase has $2\pi$ ambiguity while winding number does not have such ambiguity. I am confused. Since the formula for evaluating the winding number is just the same as the formula for Berry phase. It seems that winding number carries more information than Berry phase does.
Can anyone elucidate the difference between winding number and Berry phase? I would be very grateful for any suggestions! :)
 A: There are two types of Berry phases. The first type where the Berry phase:
$$\phi_B = \int_{\Gamma} \vec{A}_B \cdot d\vec{r}$$
depends on the integration path $\Gamma$ ($\vec{A}_B $  is the Berry connection). In this case, the topology is probed not by the Berry phase itself, but by the integral of the Berry curvature $B$ over a closed surface $\Sigma$ which is the quantized Chern number. 
$$ C =  \frac{1}{2\pi}\int_{\Sigma}  \vec{B} \cdot \hat{n} dS$$
where $ \vec{B}  = \vec{\nabla} \times \vec{A} $ and $\hat{n}$ is the outer unit vector.
The Chern number is a winding number, because the integral on the Berry connection can be converted into a line integral over the transition functions of the Berry connection. For example, when the surface $\Sigma$  is a sphere, then it can be covered by two coordinate patches on the upper and lower hemisphere, and the relation between the Berry connections on the upper and lower hemisphere is a gauge transformation:
$$ \vec{A}_+  = \vec{A}_-  + \vec{ \nabla}\phi$$
Then using the Stokes theorem we get:
$$ C =  \frac{1}{2\pi}\int_{\Gamma}   \nabla{\phi}  \cdot d\vec{r}$$
(This time $\Gamma$ is the equator). The last equation explains why the Chern number is a winding number, because it counts how many times the function $\phi$ winds the equator.
however, this isn't our case.
When the Berry curvature is identically zero, the Berry connection can be locally a pure gauge, but not globally. This happens when there is a net flux flowing in the direction perpendicular to the surface of the path $\Gamma$, for example in the case of the Aharonov-Bohm effect. In this case the Berry phase does not depend on the shape of the integration path $\Gamma$ , but it can assume any value between $0$ and $2 \pi$, since the Berry phase is:
$$\phi_B = \frac{q \Phi}{\hbar c}$$
which can be controlled modulo $2 \pi$ by changing the flux. Thus , in this case the Berry phase becomes a topological invariant (independent of the shape of the trajectory), but not through its quantization, but through its periodicity (mod$-2\pi$). The Berry phase in this second case is called a topological phase.
Now, please observe the Berry connection in the case of graphene:
$$ \vec{A}_B \propto \vec{ \nabla}_{\vec{q}}\phi(\vec{q})$$
The Berry connection is locally a pure gauge. Thus this Berry phase belongs to the second type (a topological Berry phase). The additional phenomenon that takes place here is that we do not control the flux in the momentum space, it is determined by the solution of the (free) Dirac equation. This flux generates exactly a Berry phase of $\pm \pi$ (depending on the band).
The term winding number can be justified as we are integrating over a gradient of an angular variable as in the Chern number. The difference that in our case the integral divided by $2 \pi$ is half integer. 
Now, the origin of this Berry phase is quite simple. When the vector $\vec{q}$ is rotated a full revolution, then,  as very well known, the phase of the fermion  wave function rotates only by half a revolution and this is the reason of the half integral winding number. This phenomenon takes place in relativistic as well as non-relativistic systems. 
It was Aharonov and Susskind who first suggested that this phenomenon (rotation of $\pi$ of a spinor when the spatial rotation is $2 \pi$) is measurable even before the notion of the Berry phase was discovered. Their suggestion was based on a (non-relativistic) spin adiabatically following a slowly rotating magnetic field. In this case also, the wave function acquires a phase of $\pi$.
In the case of graphene, the Berry phase of $\pi$ was actually measured, as reported in the following work by: Liu, Bian, Miller and Chiang
