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  Berry phase, Chern numberÂ
Take a 3-parameter Hamiltonian H(R1,R2,R3), with a nondegenerate spectrum  The Berry phase of the n-th energy eigenstate is a mapping...
a) from the set of closed curves in the parameter space to the interval [0,2Ï€[  b) from the set of points of the parameter space to the set of real numbers c) from the set of open curves in the parameter space to the interval [0,2Ï€[  d) from the set of open curves in the parameter space to the set of real numbersÂ
 Take a quantum system parametrized by the continuous R ,which obeys the Schrodinger equation H(R)|n(R)⟩=en(R)|n(R)⟩ What is characterized by an adiabatic phase?
a) The operator H(R) along a closed curve in the parameter space b) The energy eigenvalues en(R) at a given point R0 c) A state vector |n(R)⟩ along a continuous curve in the parameter space d) the linear combination of state vectors |n(R)⟩,|m(R)⟩ at two points R1 és R2 connected by a continuous curve Â
 Which of these is gauge invariant?
a) adiabatic phase   b) Berry connection   c) Berry curvature     Â
 Take the SSH model with hopping amplitudes v and w both real.When will the Berry phase of the lower band of the system be 0?
a) v>w b) v=w c) v<w d) not enough information to decide   Â
We adiabatically slowly compress a 1D potential well (continuous thick red) until its width is reduced by a factor of 1/2 (thin blue dashed). We then decompress it back.V(t+T,x)=V(t,x)V(T/2-t,x)=V(T/2+t,x) What is the Berry phase accumulated by the lowest energy bound state ? Â
x
V
a) There is no Berry phase defined to this process b) 2π c) 0 d) The answer depends on the precise shape of the potential.
We adiabatically slowly lift a 1D potential well (continuous thick red) until at every x it is positive (thin blue dashed). We then lower it back.V(t+T,x)=V(t,x)V(T/2-t,x)=V(T/2+t,x) What is the Berry phase accumulated by the lowest energy bound state ?  Â
x
V
Since raising the potential will increasethe energy of the bound state well aboveits original confinement, the particlewill not remain a bound state.   2π 0 Can not be decided. Â
a) Â Â Â Â Â b) Â c) Â d)
 Take a 2-dimensional, 2-band lattice model, with a bulk momentum-space Hamiltonian,Â
a) In the vicinity of k values where    b) In the vicinity of k values where   c) In the vicinity of k values where dz(k)=maximum d) The Chern number is a global quantity, we need the all of the d(k) to calculate it.
To find the Chern number of the lower band, you need to sample d(k)...
Let us denote states on the Bloch sphere by |θ, φ⟩ ! Which sequence of states has afinite Berry phase associated to it? Â
a) |0, 0⟩ ;|0, π/2⟩ ;|0, π⟩; |0, 3π/2⟩;|0,2 π⟩b) |0, 0⟩ ;|π/4, 0⟩ ;|π/2, 0⟩; |π/4, 0⟩;|0,0⟩  c) |π/2, 0⟩ ;|π/2, 2π/3⟩ ;|0, π⟩; |π/4, -π/7⟩;|π/2,0⟩  d) |π/2, 0⟩ ;|π/2, 2π/3⟩ ;|0, π⟩; |π/4, -π/7⟩;|π,0⟩
|0, 0⟩
|π, 0⟩
|π/2,π/2⟩
 Â
a) The Berry phase of the eigen states are the same.b) We can only define a Berry phase if we consider the cyclic adiabatic evolution of the ground state.c) We can not define a Berry phase from a sequence of states containing the states |π/2, 2π/3,+⟩ and |π/2, 2π/3,-⟩.d) We can define a Berry phase from a sequence of states containing the states |π/2, 2π/3,+⟩ and |π/2, 2π/3,-⟩.
Let us denote the ground(+) and excited(-) states of the operator σ·eθ,φ by |θ,φ,±⟩ which statement is true?
 Â
Consider the integral of the Berry curvature on a small sphericalcup on the Bloch sphere, located either (1) at the North Pole, or (2) near the Equator. In which case is the absolute value of theintegral bigger, in (1) or (2)?a) in (1), at the North Poleb) in (2), near the Equatorc) they are equald) depends on the gauge
(1)
(2)
Take the Hamiltonian H(R) defined in a three-dimensional parameter space, and consider its ground-state manifold ψ0(R).Assume that the ground-state manifold is non-degenerate,E0(R) < Ej(R) for any j>0.Then the Chern number of the ground-state manifold is afunction that mapsa) closed loops in the parameter space to integer numbersb) closed loops in the parameter space to [0,2π[ c) closed surfaces in the parameter space to integer numbersd) closed surfaces in the parameter space to real numbers
Take the Hamiltonian H(B) = - Bz σz. What is the Chern number associated to the ground-statemanifold on the sphere |B| = B0?a) 0b) 1c) 2d) undefined
Take the Hamiltonian H(R) defined in a three-dimensionalparameter space. Assume that the ground-state manifoldis non-degenerate on the sphere R = R0, and that the Chernnumber associated to this ground-state manifold on thissphere is 1.Which is the most precise statement about the ground-statedegeneracies in the interior of this sphere?a) There is no ground-state degeneracy.b) There is exactly one degeneracy point, and there the ground state is twofold degenerate.c) There is exactly one point where the ground state is degenerate.d) There is at least one point where the ground state is degenerate.
Take the Hamiltonian H(R) defined in a three-dimensionalparameter space. Assume that the ground-state manifoldis non-degenerate on the sphere |R| = R0,and that the Chern number associated to this ground-statemanifold on this sphere is 0.Which is the most precise statement about the ground-statedegeneracies in the interior of this sphere?a) There is no ground-state degeneracy.b) There is exactly one point where the ground state is degenerate.c) The number of degeneracy points is even.d) None of the above is true in general.