ABSTRACT - DRUM - University of Maryland
ABSTRACT - DRUM - University of Maryland
ABSTRACT - DRUM - University of Maryland
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Chapter 6<br />
Non-adiabaticity in the Braiding <strong>of</strong> Majorana<br />
Fermions<br />
Mathematical definition <strong>of</strong> quantum statistics necessarily builds upon the concept<br />
<strong>of</strong> Berry phase <strong>of</strong> many-body wavefunctions. This implies that the adiabaticity<br />
<strong>of</strong> braiding is an essential ingredient for non-Abelian statistics, since the quantum<br />
state has to stay in the ground state manifold during the entire process <strong>of</strong> the braiding<br />
[131, 132]. In the real world, however, braidings are necessarily performed within<br />
a finite time interval, i.e., they are always non-Adiabatic. As known from the adiabatic<br />
perturbation theory, Berry phase is the leading-order term in the adiabatic<br />
perturbative expansion. [133, 134, 135] Given the fundamental role played by adiabatic<br />
braiding in TQC, it is therefore important to understand quantitatively the<br />
higher order corrections arising from non-adiabatic evolution,<br />
In this chapter, we present a systematic study <strong>of</strong> the non-adiabatic corrections<br />
to the braiding <strong>of</strong> non-Abelian anyons and develop formalism to describe their<br />
dynamical aspects. In our treatment, braidings are considered as dynamical evolutions<br />
<strong>of</strong> the many-body system, essentially using the time-dependent Schrödinger<br />
equation <strong>of</strong> the BCS condensate whose solutions are derived from time-dependent<br />
Bogoliubov-de Gennes (BdG) equation. Generally, adiabaticity may break down in<br />
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