Browsing by Subject "Polarization distribution"

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Open Access
A study of the modern theory of polarization on extensions of one dimensional topological insulators and disordered systems
(2021-12) Parlak, Selçuk
We work on identifying topological and quantum phase transition via the modern theory of polarization. We go through the problem of electrostatics definition of absolute polarization via a discrete chain of anions and cations. We move from this discrete approach to a continuous one and prove the emergence of the Berry phase under adiabatic evolution. Introducing the Resta's position operator, we show the correspondence to the definition of polarization and go through the polarization distribution of the system. Using these concepts we studied band structures, topological invariants, and symmetries in the Su-Schrie er-Heeger model. With these basics, we analyzed the emergence of torus knots and its identification of topological transition on the distance-dependent SSH model. We go through the formal definitions of knot theory and how to identify them to establish a new system using Klein bottle knots. Following the distance vector structure in the distance-dependent SSH model, we try to establish a real-life system using Klein bottle parametrizations; Pinched torus, figure 8, and the bottle shape. In all of the parametrizations, the interpretation of the real-life system had hoppings on empty sites. Klein bottle knots are tried to observe via parametrizations of the Klein bottle. In all cases, due to the four-dimensional nature of the Klein bottle, we encountered intersecting curves on 3-dimensional interpretations of the Klein bottle. To understand the system better, we go through the Berry phase and dispersion relation of the system. After not encountering anything significant, we try to acquire a knot structure on fundamental polygons. We obtain a Hopf link and unlink with a possible intersection point. The intersection becomes hard to judge due to computational approximation on determining the knot diagrams. Due to the complexity of the project, we go through the Anderson model to study the modern theory of polarization. A background on Anderson localization with a computational method: Transfer matrix method is given. We state Mott's relation of conductivity and discuss the concept of mobility edge. We show the localization theory of Resta by introducing the complex number z. With this, we prove that the jzj becomes 1 when the states are localized and 0 when we have extended states. We also go through the scaling theory of localization. To understand the scaling theory, we introduce the concept of the renormalization group and discuss how this idea is used in the gang of four paper. We go through the assumptions and results of the gang of four by observing that the scaling exponent of conductance exhibits a critical value only in the three dimensions. Using Resta's quantity, we try to establish the gang of four results by examining the size scaling of the variance of polarization of the system and introducing a renormalization flow with Resta's quantity. We observe the low-temperature limit by examining a single state in the ground state. With this, we recover the results of the gang of four. Based on the discussion of Mott's on mobility edge, we further examine the high-temperature limit, where the system is in the average of all states. We identify the transition point via two fixed points(repulsive and attractive) on flow diagrams and size scaling exponent in all dimensions. We recover the behavior of the gang of four in one and three dimensions by examination of the fixed points, and we state the ambiguity of two-dimensional solutions due to the convergence problem of the fixed points. We further solidify the analogy of conductance and Resta's number by observing the Binder cumulant and analogical mobility edge of the system. We found the same repulsive fixed point on Binder cumulant and analogical behavior on the mobility edge argument.
Open Access
Topological aspects of charge transport in quantum many-body systems
(2019-01) Yahyavi, Mohammad
Motivated by the recent proposals and developments of topological insulators and topological superconductors for their potential applications in electronic devices and quantum computing, we have theoretically studied topological properties of quantum many-body systems. First, we calculate the gauge-invariant cumulants (and moments) associated with the Zak phase. The first cumulant corresponds to the Berry phase itself, the others turn out to be the associated spread, skew, kurtosis, etc. The cumulants are shown to be gauge invariant. We reconstruct the underlying probability distribution of the polarization by maximizing the information entropy and applying the moments as constraints in the Rice-Mele model and in the interacting, spinless Su-Schrieffer-Heeger model. When the Wannier functions are localized within one-unit cell, the probability distribution so obtained corresponds to that of the Wannier function. We follow the probability distribution of the polarization in cycles around the topologically nontrivial point of these models. Secondly, we have constructed a topological one-dimensional analog of the Haldane and Kane-Mele models in two dimensions, with hexagonal lattices. Our Haldane one-dimensional analog model belongs to the C and CI symmetry classes, depending on the parameters, but, due to re ection, it exhibits topological insulation. The model consists of two superimposed Creutz models with onsite potentials. The topological invariants of each Creutz model sum to give the mirror winding number, with winding numbers which are nonzero individually but equal and opposite in the topological phase, and both zero in the trivial phase. We also construct a topological one-dimensional ladder model following the steps which lead to the Kane-Mele model in two dimensions. We couple two Haldane-type ladder models, one for each spin channel, in such a way that time-reversal invariance is restored. We also add a Rashba spin-orbit coupling term. The model falls in the CII symmetry class. We demonstrate the presence of edge states and quantized Hall response in the topological region. Our model exhibits two distinct topological regions, distinguished by the different types of re ection symmetries. Thirdly, we consider the edge at the interface of a simple tight-binding model and a band insulator. We find that crossings in the band structure (one dimensional Dirac points) appear when an interface is present in the system. We calculate the hopping energy resolved along lines of bonds parallel to the interface as a function of distance from the interface. Similarly, we introduce a transport coe cient (Drude weight) for charge currents running parallel to the interface. We find that charge mobility (both the kinetic energy and the Drude weight) is significantly enhanced in the surface of the tight-binding part of the model near the interface. Finally, we study a variant of the generalized Aubry-Andre-Harper model with the effect of introducing next nearest-neighbor p-wave superconducting pairing with incommensurate and commensurate cosine modulations. We extend generalized Aubry-Andr e-Harper model with p-wave superconducting to topologically equivalent and nontrivial "anancestor" two-dimensional p-wave superconducting model. It is found that in incommensurate (commensurate) modulation, by varying next nearest-neighbor p-wave pairing order parameter, the system can switch between extended states and localized states (fully gapped phase and a gapless phase).