Download Homework 3: Due in class on Monday, Oct 21st, 2013

Homework 3: Due in class on Monday, Oct 21st, 2013
October 10, 2013
Problem 1: Gauge-independence of Berry curvature.
While considering the two-level system, we had two forms for the |+i eigenstate:
cos(θ/2)
cos(θ/2)e−iφ
and
.
sin(θ/2)eiφ
sin(θ/2)
(1)
Check that the expression for the Berry connection, Ωθφ is the same in both cases.
Problem 2: Berry curvature in Cartesian coordinates
Find the eigenstates of HT LS = ~h~σ = hx σx + hy σy + hz σz in terms of hx , hy , hz , rather then in
the spherical angles in the space of ~h. Calculate the vectors of Berry curvature corresponding to the
|+i and |−i states, corresponding to the “spin” directed parallel and antiparallel to the field ~h. You
should be able to recover the field of the monopole located at the origin of the parameter space, with
particular values of the monopole strength (how are those related for |+i and |−i states?)
Problem 3: Berry phase for an arbitrary spin
~ where S
~ now is the operator of spin S ≥ 1/2,
Consider a spin-S particle in a magnetic field, H = ~hS,
whose specific form we will not need, but for S = 1/2 it is ~σ /2. Such a system has 2S + 1 degenerate
states for ~h = 0, which is 2 for S = 1/2. Initially, the system is prepared in a state having a definite
projection of the spin on the field, |N i, where N measures the component of the spin along the field.
(In the case of spin-1/2 we considered the |+i state, which corresponds to N = 1/2.) Generalize the
~ from page 95 of Lecture 12 to this case of an arbitrary spin. The only
method of calculation of Ω
new piece of information you will need is the expressions for the matrix elements for the ladder (aka
“raising and lowering”) operators for an arbitrary angular momentum, S± = Sx ± iSy , which can be
found in any quantum mechanics textbook.
What is the resulting strength of the Berry curvature monopole? What is the Berry phase accumulated over a closed path when the direction of the magnetic field is varied?
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