I have a question on Berry phase for this very simple example:

Define D as the symmetric 4X4 diagonal 0-1 matrix

[1000]
[0000]
[0010]
[0000]

Define M as the symmetric 4X4 0-1 matrix

[0001]
[0011]
[0100]
[1100]

Define the parameterized Hamiltonian

H(x,y) = (x^2+y^2)*M + D (x,y real)

(I believe) The maximum eigenvalue of H(x,y) has only one
degeneracy at (x,y) = (0,0)

Let P(t) be the circular path (sin(t),cos(t)) [0=t=2*pi].

The maximum eigenvalue of H on P(t) is always 2 and has an
eigenvector = transpose of |1,1,1,1/2 independent of t.

Since P(t) encircles the single degeneracy, shouldn't
adiabatically transporting the max eigenvector around P(t)
accumulate a Berry phase of pi, i.e., change the polarity
of the max eigenvector?

However this seems not to occur since the eigenvector
(and the Hamiltonian) is fixed.

What is my mistake?

Thanks,
Lou Pagnucco

On Jul 5, 5:30Êam, "Lou Pagnucco" wrote:

[...]

What is my mistake?

The Hamiltonian is not a function of time.

Berry's phase only occurs when you change the state of the system very
slowly [adiabatic approximation and such]. A system that has no real
time evolution isn't going to give a change in phase. The phase factor
picked up by traveling around the loop will be constant.


Thanks,
Lou Pagnucco

On Jul 10, 7:51Êam, Eric Gisse wrote:
On Jul 5, 5:30æam, "Lou Pagnucco" wrote:

[...]

What is my mistake?

The Hamiltonian is not a function of time.

Berry'sphaseonly occurs when you change the state of the system very
slowly [adiabatic approximation and such]. A system that has no real
time evolution isn't going to give a change inphase. Thephasefactor
picked up by traveling around the loop will be constant.





Thanks,
Lou Pagnucco- Hide quoted text -

- Show quoted text -

Thanks for the reply, Eric.

Yes, I realize that the Hamiltonian is fixed
as a function of x^2+y^2, but are there any other
cases which violate the Longuet-Higgins criterion?,
i.e., if the path encirles an odd number of degeneracies,
it will result in a phase inversion?

Regards,
Lou Pagnucco