Quantum Geometry of Topological Phases of Matter

Ying-Kang Chen, Purdue University

Abstract

Quantum Hall states are prototypical topological states of matter whose Hall conductance is topologically quantized to an integer or rational fraction multiple of the fundamental conductance quantum. A significant consequence of this quantization is that the Hall conductance value can be made independent of variations from device to device, within acceptable limits. Such topologically quantized properties are thus highly desirable for metrology or industrial purposes. Formulating a microscopic picture of fractional quantum Hall states and the characterization of all topological responses of quantum Hall states are frontier areas of condensed matter research, with far reaching technological consequences such as realizing anyonic topological quantum computation. In this dissertation, I will present my research on these topics. We will begin with a brief review of integer and fractional quantum Hall effects, a recounting of topological reasons underlying the universal quantization of the Hall conductance in insulators and a presentation of basic quantum mechanical microscopic descriptions of these states. In Chapter 2, we introduce the framework of gauge-invariant variables to describe fractional quantum Hall states, and prove that the wave function can always be represented by a unique holomorphic multivariable complex function. As a special case, within the lowest Landau level, this function reproduces the well-known holomorphic representation of wave functions in the symmetric gauge. Using this framework, we derive an analytic guiding center Schrödinger’s equation governing FQH states, establishing a new avenue for deriving the properties of FQH states from first principles. In Chapter 3, again using the language of gauge-invariant variables to analyze the quantum mechanics of quantum Hall states, we provide tangible connections between the response of quantum Hall fluids to nonuniform electric fields and the characteristic geometry of electronic motion in the presence of magnetic and electric fields. The geometric picture we provide motivates the following ansatz: nonuniform electric fields mimic the presence of spatial curvature. Consequently, the gravitational coupling constant also appears in the charge response to nonuniform electric fields.

Degree

Ph.D.

Advisors

Biswas, Purdue University.

Subject Area

Physics|Energy|Astronomy|Atomic physics|Electromagnetics

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