THESIS
2017

xvii, 116 pages : illustrations ; 30 cm

**Abstract**
Traditional phase transitions in condensed matter physics have been well understood with Ginzburg-Landau theory. In this theory, different phases of matter are characterized by different order parameters
which emerge as a result of spontaneous symmetry breaking. However, in recent years,
physicists have found new kinds of phases in which no order parameters can be defined based on
symmetry breaking. Instead, these phases are distinguished from each other by the topology of the
electronic ground states. Correspondingly, they are called topological phases of matter. Examples
include topological insulators and topological superconductors. Topological superconductors are
especially interesting because they are able to host Majorana fermions, a kind of non-Abelian quasi-particles
that...[

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Traditional phase transitions in condensed matter physics have been well understood with Ginzburg-Landau theory. In this theory, different phases of matter are characterized by different order parameters
which emerge as a result of spontaneous symmetry breaking. However, in recent years,
physicists have found new kinds of phases in which no order parameters can be defined based on
symmetry breaking. Instead, these phases are distinguished from each other by the topology of the
electronic ground states. Correspondingly, they are called topological phases of matter. Examples
include topological insulators and topological superconductors. Topological superconductors are
especially interesting because they are able to host Majorana fermions, a kind of non-Abelian quasi-particles
that can be used for fault-tolerant quantum computation. In the Chapter 1 of this thesis, I
first give an introduction to topological phases of matter including the definition of the topology of
the electronic state and the relation between the topology and the appearance of boundary states.
After that, we talk about topological superconductors and Majorana fermions, including their non-Abelian properties. The final component of Chapter 1 introduce realistic models of topological
superconductors and the experimental signatures in transport measurements. Since previous experimental
signatures of Majorana fermions only focus on charge transport properties, in Chapter 2,
we discuss the additional transport properties Majoranas induce — that is, spin transport. I show
that the Andreev reflection processes induce selective equal spin Andreev reflection (SESAR) when
a Majorana is present, which can be tested as an additional experimental signature of Majoranas
and can also be used for spintronics. In Chapter 3 We further study this SESAR property in systems
with Majorana flat bands, which can be realized in (110) quantum wells with superconductivity and
in which giant spin currents can be generated. In Chapter 4, we turn our interest into a model where
one or two Majorana fermions can exist. They are the end states in one dimension (1D) and become
chiral Majorana modes when the model is generalized to two dimension(2D). In the 1D case where
two Majoranas coexist, local Andreev reflection is suppressed at zero bias due to the interference
between the two Majoranas. As a result, crossed Andreev reflection probability can be enhanced up
to a value as large as unity (resonant). Similar result is obtained in 2D at low energy. In Chapter 5, I
further study the 2D system with Majorana chiral modes which are experimentally confirmed very
recently. In the phase with single Majorana chiral mode, the two terminal conductance σ

_{12} of the
QAHI/TSC/QAHI (QAHI stands for quantum anomalous Hall insulator and TSC the topological
superconductor) is quantized at 1/2e

^{2}/h as confirmed in the experiment. This is a result of the chiral Majorana mode if one can show the edge states of the QAHI is well defined. In the experiment, the
existence of the edge states is in question because the corresponding normal Hall conductance σ

_{xy}
is not quantized. By considering magnetic domains, we show that the non-quantization of the Hall
conductance is a result of the additional channels on the domain walls but the edge states are well
defined. Thus, we further prove that the σ12 = 1/2e

^{2}/h is a signature of the Majorana chiral mode.

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