Microscopic Study of Statistics, Spin and Excitons in the Fractional Quantum Hall Effect

Open Access
- Author:
- Zhang, Yuhe
- Graduate Program:
- Physics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 12, 2017
- Committee Members:
- Jainendra Jain, Dissertation Advisor/Co-Advisor
Jainendra Jain, Committee Chair/Co-Chair
Chaoxing Liu, Committee Member
Nathan David Gemelke, Committee Member
Ismaila Dabo, Outside Member - Keywords:
- FQHE
composite fermion
braid statistics
spin
exciton
microscopic study
Monte Carlo - Abstract:
- Fractional quantum Hall effect (FQHE) is a marvelous physical phenomenon that arises in a two-dimensional electron gas subjected to a strong perpendicular magnetic field at low temperature. The microscopic origin of the FQHE for general filling factors is understood by the Composite Fermion theory, which explains FQHE of electrons as integer quantum Hall effect (IQHE) of bound objects of electrons and an even number of vortices, known as composite fermions (CFs). Despite tremendous advances over the last three decades in our understanding of the FQHE since its discovery, many puzzles remain unsolved and many hypotheses remain untested. The most interesting such hypothesis is probably the realization of certain exotic particles called ``anyons'', which interpolate between bosons and fermions. Anyons obey exotic braid statistics: fractional braid statistics refers to the property that braiding one particle around another produces a phase that is a nonintegral multiple of $2\pi$; non-Abelian braid statistics means that braiding non-Abelian anyons not only produces a phase in the wave function, but also transforms the system to a topologically distinct state. Anyon is of great interest both for a demonstration of new physics and because of its potential application in fault-tolerant topological quantum computers. Despite tremendous efforts in the last three decades, anyons have not been definitely detected in experiments. We proposed to create and manipulate anyons by introducing auxiliary particles into a FQHE system. These auxiliary particles bind quasiholes of the fractional quantum Hall (FQH) state, and thus inherit the exotic braiding properties of the quasiholes. We showed that the braiding property of a pair of auxiliary particles is manifested through the density distribution of one particle when holding another particle fixed. This density distribution can be directly measured with existing experimental techniques in cold atom system. The above demonstration of exotic braiding statistics, while still being at the level of a thought experiment, has the advantage that all the physics happens away from the edge of a FQHE system. Earlier proposals generally used interference of anyons moving along the edge of a FQHE system to reveal their non-Abelian braiding. The edge of a FQH state, however, may undergo a reconstruction due to edge exciton, and therefore lose its topological features. Motivated by certain discrepancies between theory and experimental measurements on the edge, we study in this thesis the edge physics of FQHE. We focus on the $\nu = 5/2$ state, which is believed to realize the Moore-Read Pfaffian state that supports non-Abelian excitations, and the $\nu = 7/3$ state, another state in the second Landau level that is studied in volumes of experiments. Using numerical simulations we find that the edge is unstable to a reconstruction for all experimental systems investigated so far, which must be taken into account when analyzing experimental results. Apart from the topological properties of the FQHE systems, we have also been interested in pursuing the quantitative accuracy of the theory of FQHE and explaining various experimental results in a quantitative way. Previous theories usually considered the ideal limit in which all the electrons in a FQH state are in the lowest Landau level (LLL). In reality, however, some electrons are excited to higher Landau levels in order to lower their correlation energy. This behavior is called Landau level (LL) mixing which can make significant quantitative corrections for measurable physical quantities, and sometimes may even play a qualitative role. Other physical factors that can lead to quantitative changes include the finite thickness of an two-dimensional electron layer. We study the transitions between differently spin-polarized states in the FQHE which provide a direct measure of the tiny energy differences between those states and thereby serve as an extremely sensitive test of the quantitative accuracy of the theory. Previous theoretical results for spin transitions were generally off by a factor of 2 or 3 compared to experimental data. We use a ``fixed-phase diffusion Monte Carlo'' algorithm to address the effect of LL mixing and also include finite-thickness effect based on realistic charge distribution. We find excellent agreement between our calculation and experiments. Our work demonstrates the most accurate comparison between theory and experiment in the FQHE field achieved so far, and also solves a long-standing puzzle about what the role LL mixing plays in breaking the particle-hole symmetry. In addition to spin physics, we notice that a quantitative understanding of the tunneling transport in a bilayer FQHE system is lacking. We identify the interlayer exciton responsible for the peak tunnel current and provide a quantitative account of the bias voltage at the peak current (denoted as $V_{max}$). The excitonic attraction is shown to be quantitatively significant, and its variation accounts for the increase of $V_{max}$ with the application of an in-plane magnetic field. We have included the effect of finite thickness but not the LL mixing in this calculation because LL mixing does not affect the excitation gaps substantially. The excellent agreements with experiment also validate our calculation.