PhD Thesis: On Aspects of Anyons and Quantum Graphs
Tuesday 13 June 2017
to 16:00 at
Babak Majidzadeh Garjani (Stockholm University, Department of Physics)
This thesis consists of two distinct parts. The first part, based on the first two accompanied papers, is in the field of
topological phases of matter and the second part, based on the third accompanied paper, looks at a problem in the field of
quantum graphs, a rapidly growing field of mathematical physics.
First, we investigate the entanglement property of the Laughlin state by looking at the rank of the reduced density
operator when particles are divided into two groups. We show that the problem of determining this rank translates itself
into a question about symmetric polynomials, namely, one has to determine the lower bound for the degree in each variable
of the symmetric polynomials that vanish under a transformation that clusters the particles into groups of equal size and
then brings the particles in each group together. Although we were not able to prove this, but we were able to determine
the lower bound for the total degree of symmetric polynomials that vanish under the transformation described. Moreover,
we were able to characterize all symmetric polynomials that vanish under this transformation.
In the second paper, we introduce a one-dimensional model of interacting su(2)k anyons. The specific feature of this
model is that, through pairing terms present in the Hamiltonian, the number of anyons of the chain can fluctuate. We also
take into account the possibility that anyons hop to empty neighboring sites. We investigate the model in five different
points of the parameter space. At one of these points, the Hamiltonian of the model becomes a sum of projectors and we
determine the explicit form of all the zero-energy ground states for odd values of k. At the other four points, the system is
integrable and we determine the behavior of the model at these integrable points. In particular, we show that the system is
critical and determine the CFT describing the system at these points.
It is known that there are non-Hermitian Hamiltonians whose spectra are entirely real. This property can be understood
in terms of a certain symmetry of these Hamiltonians, known as PT-symmetry. It is also known that the spectrum of a
non-Hermitian PT-symmetric Hamiltonian has reflection symmetry with respect to the real axis. We then ask the reverse
question whether or not the reflection symmetry of a non-Hermitian Hamiltonian necessarily implies that the Hamiltonian
is PT-symmetric. In the context of quantum graphs, we introduce a model for which the answer to this question is positive.