## Description

**Strongly Correlated Systems Under High Magnetic Field is a well-researched Physical Sciences and Mathematics Thesis/Dissertation topic, it is to be used as a guide or framework for your Academic Research.**

## Abstract

Strong correlation among electrons under a high magnetic field gives rise to an entirely new arena of emergent physics, namely the fractional quantum Hall effect. Such systems have entirely different elementary degrees of freedom and generally, demand non-perturbative approaches to develop a better understanding.

In the literature, there are several analytical methodologies and numerical toolkits available to study such a system. Clustering of zeros, parent Hamiltonian, off-diagonal order parameter, Parton construction, matrix product states is to be named among a few of those popular methodologies in the existing literature.

Most of these methods work well in the lowest Landau level or holo-morphic wavefunction framework. It is, however, imperative to develop such methodology to study systems with Landau levels mixing to study more exotic as well as experimentally relevant states.

In this work, we have developed particular methodologies, which denounce the traditional importance of the analytic properties of first quantified model wavefunction thereby extend the existing parent Hamiltonian, topological order-parameter, matrix product states descriptions to mixed Landau level systems. Such ex-tension produces a deeper, compact, and holistic understanding of universal physics of exotic phases in strongly correlated systems from the microscopic viewpoint, as well as produces interesting new results.

Our second quantized/ non-analytic approach allows us to construct the “entangled Pauli principle”, a guidebook to extract universal/topological properties such as braid-ing statistics, fractional charge quantization, topological degeneracy of the ground States starting from a relatively simple many-body wavefunction, “root pattern” of fractional quantum Hall state.

Such an entangled Pauli principle can be derived from a microscopic parent Hamiltonian setting, thereby provide us a potential tool to probe the non-universal physics in quantum Hall fluids as well. Essentially, the entangled Pauli principle is the “DNA” of fractional quantum Hall states.

Using this guiding principle, we have shown ground states with non-abelian excitations, such as Majorana fermion or Fibonacci fermion can be stabilized for two-particle interaction. Fibonacci fermion supports universal quantum gates, thereby a potential candidate for the topologically protected universal quantum computer. The entangled Pauli principle, along with a recently developed topological order parameter for composite fermions, gives rise to Parent Hamiltonian description for composite fermions as well.

### Introduction

In 1879, Edwin Hall, a graduate student at Johns Hopkins University, first observed “Hall effect” phenomena in a thin gold leaf [6]. From the modern point of view, this phenomenon gives a rather mundane observation, once an out of the plane magnetic field B is applied to a constant current flow along the x-direction, a voltage is induced along the y-direction in the gold leaf.

One, however, must note that such a discovery precedes the discovery of the electron. Until that time, electrical measurements provided only the carrier density and mobility product, and the separation of these two important physical quantities had to rely on other difficult measurements.

The discovery of the Hall effect enabled a direct measure of the carrier density. The polarity of this transverse Hall voltage proved that it is in fact electrons (negative charge career) that are physically moving in an electric current. This earliest variant of the Hall effect can be easily explained using the classical linear response theory or Drude model. One can, in fact, calculate the resistivity matrix in two dimensions.

Longitudinal resistivity depends on the scattering time τ, the effective mass of the electron m, charge density n, and the square of effective carrier charge e. ρxx goes to zero as scattering processes become less important (τ → ∞), while off-diagonal resistivity (Hall resistivity) ρH goes linearly with the magnetic strength B. One should, however, realize that once the scattering process gets less important, Drude model starts to break down.

Once the magnetic field is very high, electrons get pinned down to small cyclotron orbits, thus no longer get governed by thermal physics or scattering. More than a hundred years after Edwin Hall’s discovery, in 1980, von Klitzing measured Hall conductivity under magnetic field 18 T in a sample prepared by Dorda and Pepper. One can easily notice that both longitudinal and Hall resistivity (voltage is shown in the graph) no longer follows the classical explanation based on the Drude model.

Hall resistivity stays on a plateau for a range of magnetic field, before jumping to the next plateau. On the plateau, the Hall resistivity has the following form, This phenomenon, however, has been already been explained by Ando [7] in terms of

Landau levels.

He has suggested that the electrons in impurity bands, arising from short-range scatterers, do not contribute to the Hall current; whereas the electrons in the Landau level give rise to the same Hall current as that obtained when all the electrons are in the level and can move freely. These Landau levels, as shown be Lev Landau, have the Figure 1.1: [Graph is taken from [1]] Recordings of the Hall voltage UH and the voltage drop between the potential probes, UPP, as a function of the gate voltage V at T = 1.5 K.

The constant magnetic field (B) is 18 T and the source-drain current, I, is 1 µA. The inset shows a top view of the device with a length of L =400 µm, a width of W =50 µm, and a distance between the potential probes of LPP= 130 µm. purely quantum origin, thus Eq. (1.2), describes the Quantum Hall effect. As ρH in eq. (1.2), is characterized by integer N, this phenomenon is named Integer quantum Hall (IQH)

effect. In the rest of the introduction, we will talk about IQH systems.

Understanding of such systems helps us also to create the foundation of strongly correlated systems under a high magnetic field.

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