Consider a Hamiltonian that does not explicitly depend on time,
but does depend on a given parameter .
The Schrödinger equations then read:
The general full solution has the following form,
where we allow to evolve in time,
and we have abbreviated the traditional phase of the “wiggle factor” as :
The geometric phase is more interesting.
It is not included in ,
because it depends on the path
rather than only the present and .
Its dynamics can be found by inserting the above
into the time-dependent Schrödinger equation:
Here we recognize the Schrödinger equation, so those terms cancel.
We are then left with:
Front-multiplying by gives us
the equation of motion of the geometric phase :
Where we have defined the so-called Berry connection as follows:
Importantly, note that is real,
provided that is always normalized for all .
To prove this, we start from the fact that :
Consequently, is always real,
because is imaginary.
Suppose now that the parameter is changed adiabatically
(i.e. so slow that the system stays in the same eigenstate)
for , along a circuit with .
Integrating the phase over this contour then yields
the Berry phase :
But we have a problem: is not unique!
Due to the Schrödinger equation’s gauge invariance,
any function can be added to
without making an immediate physical difference to the state.
Consider the following general gauge transformation:
To find for a particular choice of ,
we need to evaluate the inner product
Unfortunately, does not vanish as we would have liked,
so depends on our choice of .
However, the curl of a gradient is always zero,
so although is not unique,
its curl is guaranteed to be.
Conveniently, we can introduce a curl in the definition of
by applying Stokes’ theorem, under the assumption
that has no singularities in the area enclosed by
(fortunately, can always be chosen to satisfy this):
Where we defined as the curl of .
Now is guaranteed to be unique.
Note that is analogous to a magnetic field,
and to a magnetic vector potential:
Unfortunately, is difficult to evaluate explicitly,
so we would like to rewrite such that it does not enter.
We do this as follows, inserting along the way:
The fact that is imaginary
means it is parallel to its complex conjugate,
and thus the cross product vanishes, so we exclude from the sum:
From the Hellmann-Feynman theorem,
we know that the inner products can be rewritten:
Where we have assumed that there is no degeneracy.
This leads to the following result:
Which only involves ,
and is therefore easier to evaluate than any .
- M.V. Berry,
Quantal phase factors accompanying adiabatic changes,
1984, Royal Society.
- G. Grosso, G.P. Parravicini,
Solid state physics,
2nd edition, Elsevier.